From 09309903611659bb240146dc664c16dd5a9744cb Mon Sep 17 00:00:00 2001 From: mbertucci47 <61854785+mbertucci47@users.noreply.github.com> Date: Mon, 17 Jun 2024 18:26:55 -0400 Subject: [PATCH] fix small typos --- adequate.tex | 2 +- algebra.tex | 12 ++++++------ algebraization.tex | 16 ++++++++-------- artin.tex | 12 ++++++------ categories.tex | 2 +- chow.tex | 22 +++++++++++----------- cohomology.tex | 10 +++++----- cotangent.tex | 2 +- curves.tex | 6 +++--- decent-spaces.tex | 8 ++++---- defos.tex | 2 +- derham.tex | 8 ++++---- derived.tex | 8 ++++---- dga.tex | 10 +++++----- documentation/make-project | 2 +- documentation/rules | 4 ++-- documentation/todo-list | 2 +- duality.tex | 4 ++-- dualizing.tex | 6 +++--- equiv.tex | 4 ++-- etale-cohomology.tex | 18 +++++++++--------- etale.tex | 4 ++-- examples-defos.tex | 8 ++++---- examples-stacks.tex | 2 +- examples.tex | 4 ++-- exercises.tex | 6 +++--- fields.tex | 4 ++-- flat.tex | 14 +++++++------- formal-defos.tex | 4 ++-- formal-spaces.tex | 12 ++++++------ functors.tex | 2 +- groupoids.tex | 4 ++-- homology.tex | 2 +- limits.tex | 2 +- local-cohomology.tex | 2 +- models.tex | 2 +- modules.tex | 2 +- moduli-curves.tex | 8 ++++---- moduli.tex | 2 +- more-algebra.tex | 34 +++++++++++++++++----------------- more-etale.tex | 12 ++++++------ more-morphisms.tex | 30 +++++++++++++++--------------- obsolete.tex | 2 +- perfect.tex | 30 +++++++++++++++--------------- pic.tex | 2 +- pione.tex | 6 +++--- preamble.tex | 2 +- proetale.tex | 2 +- properties.tex | 2 +- quot.tex | 2 +- relative-cycles.tex | 22 +++++++++++----------- restricted.tex | 16 ++++++++-------- scripts/functions.py | 2 +- scripts/latex.sh | 2 +- scripts/second-homotopy.gp | 2 +- scripts/web_book.py | 2 +- sdga.tex | 10 +++++----- sheaves.tex | 2 +- simplicial.tex | 8 ++++---- sites-cohomology.tex | 12 ++++++------ sites-modules.tex | 2 +- sites.tex | 8 ++++---- spaces-descent.tex | 4 ++-- spaces-divisors.tex | 2 +- spaces-flat.tex | 6 +++--- spaces-groupoids.tex | 2 +- spaces-limits.tex | 2 +- spaces-more-cohomology.tex | 2 +- spaces-more-morphisms.tex | 8 ++++---- spaces-morphisms.tex | 4 ++-- spaces-perfect.tex | 10 +++++----- spaces-pushouts.tex | 4 ++-- spaces-simplicial.tex | 8 ++++---- spaces.tex | 2 +- stacks-cohomology.tex | 2 +- stacks-geometry.tex | 4 ++-- stacks-more-morphisms.tex | 4 ++-- stacks-morphisms.tex | 10 +++++----- stacks-perfect.tex | 2 +- stacks-properties.tex | 2 +- stacks-sheaves.tex | 6 +++--- stacks.tex | 2 +- tags/latex.sh | 2 +- tags/tags | 4 ++-- topology.tex | 2 +- trace.tex | 2 +- varieties.tex | 8 ++++---- weil.tex | 10 +++++----- 88 files changed, 285 insertions(+), 285 deletions(-) diff --git a/adequate.tex b/adequate.tex index 5db7e82e2..5d3d8971e 100644 --- a/adequate.tex +++ b/adequate.tex @@ -2620,7 +2620,7 @@ \section{Pure extensions} \medskip\noindent Proof of (2). Because $P_0$ is pure projective we can find a map $P_0 \to N_0$ lifting the map $P_0 \to M \to N$. -We obtain an induced map $P_1 \to F_0 \to N_0$ wich ends up in $K_0$. +We obtain an induced map $P_1 \to F_0 \to N_0$ which ends up in $K_0$. Since $P_1$ is pure projective we may lift this to a map $P_1 \to N_1$. This in turn induces a map $P_2 \to P_1 \to N_1$ which maps to zero into diff --git a/algebra.tex b/algebra.tex index 7aa31ccaf..ff0b6dea4 100644 --- a/algebra.tex +++ b/algebra.tex @@ -3922,7 +3922,7 @@ \section{Open and closed subsets of spectra} $R$ annihilated by a positive power of $e'$. Since $e'$ is idempotent this is the ideal of elements annihilated by $e'$ which is the ideal $I = (e)$ as $e + e' = 1$ is a pair -of orthognal idempotents. This proves (2). +of orthogonal idempotents. This proves (2). \end{proof} @@ -10297,7 +10297,7 @@ \section{Separable extensions, continued} is a transcendence base of $K/k$. Thus $L$ is the fraction field of the polynomial ring over $k$ in $x_1, \ldots, x_{i - 1}, x_{i + 1}, \ldots, x_{n + 1}$. -By Gaus' Lemma we conclude that +By Gauss' Lemma we conclude that $$ P(T) = F(x_1, \ldots, x_{i - 1}, T, x_{i + 1}, \ldots, x_{n + 1}) \in L[T] @@ -13135,7 +13135,7 @@ \section{K-groups} By Lemma \ref{lemma-K0-local} the module $P$ is finite free. So $P \cong R^{\oplus n}$ for some $n \geq 0$. Then $\text{rank}_R(P) = n$ and $\text{length}_R(R^{\oplus n}) = n \text{length}_R(R)$ -by additivity of lenghts (Lemma \ref{lemma-length-additive}). +by additivity of lengths (Lemma \ref{lemma-length-additive}). Thus the result holds. \end{proof} @@ -15580,7 +15580,7 @@ \section{Relative assassin} \item $B_{fin}$: for some finite $R$-module $M$ we have $\mathfrak q \in \text{Ass}_S(N \otimes_R M)$. \end{enumerate} -Let us determine some of the relations between theses sets. +Let us determine some of the relations between these sets. \begin{lemma} \label{lemma-compare-relative-assassins} @@ -33688,7 +33688,7 @@ \section{Differentials} Since $\varphi$ is surjective, the terms $s_j'[a_j', b_j']$ and $s_k'[f_k', g_k']$ are in the image of elements in the lower right corner. Thus, modifying -$\eta$ and $\theta$ by substracting the images of these +$\eta$ and $\theta$ by subtracting the images of these elements, we may assume $\theta = \sum s_l'[r_l']$. In other words, we see $\sum \varphi(s_i)[\varphi(a_i)]$ is of the form $\sum s'_l [\beta(r'_l)]$. @@ -43445,7 +43445,7 @@ \section{Serre's criterion for normality} Lemma \ref{lemma-characterize-zero-local}. This implies that $R_{\mathfrak p}$ is a subring of a product of localizations of $R$ at $\mathfrak p \supset \mathfrak q$ with -$t \not \in \mathfrak q$. Since theses primes have smaller height +$t \not \in \mathfrak q$. Since these primes have smaller height by induction on the height we conclude that $R$ is reduced. \end{proof} diff --git a/algebraization.tex b/algebraization.tex index f22cea91c..954fafc29 100644 --- a/algebraization.tex +++ b/algebraization.tex @@ -1716,7 +1716,7 @@ \section{The theorem on formal functions} see Remark \ref{remark-compare-with-completion}. Derived completion commutes with passing to open subschemes (Remark \ref{remark-localization-and-completion}). -Formation of $R\lim$ commutes with passsing to open subschemes. +Formation of $R\lim$ commutes with passing to open subschemes. It follows that to check our map is an isomorphism, we may work locally. Thus we may assume $X = U = \Spec(A)$. Say $I = (f_1, \ldots, f_r)$. Let @@ -3381,7 +3381,7 @@ \section{Algebraization of local cohomology, III} We want to show that $\xi$ maps to zero in $H^{s + 1}_T(I^mM)$. Namely, this will show that $\{H^s_T(M/I^nM)\}_{n \geq 0}$ is Mittag-Leffler exactly as in the proof of Lemma \ref{lemma-final-bootstrap}. -Picture to help vizualize the argument: +Picture to help visualize the argument: $$ \xymatrix{ & @@ -3660,7 +3660,7 @@ \section{Algebraization of formal sections, II} \label{section-algebraization-sections-coherent} \noindent -It is a bit difficult to succintly state all possible +It is a bit difficult to succinctly state all possible consequences of the results in Sections \ref{section-algebraization-sections-general} and \ref{section-bootstrap} @@ -3711,7 +3711,7 @@ \section{Algebraization of formal sections, II} corresponds to a module map $M' \to M$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-homs-over-open}. After replacing $M'$ by the image of $M' \to M$ -and seting $M'' = M/M'$ we see that our short exact sequence +and setting $M'' = M/M'$ we see that our short exact sequence corresponds to the restriction of the short exact sequence of coherent modules associated to the short exact sequence $0 \to M' \to M \to M'' \to 0$ of $A$-modules. @@ -3733,7 +3733,7 @@ \section{Algebraization of formal sections, II} Choose a presentation $A^{\oplus m} \to A^{\oplus n} \to J \to 0$. Denote $g_1, \ldots, g_n \in J$ the images of the basis vectors of $A^{\oplus n}$, so that $J = (g_1, \ldots, g_n)$. Let -$A^{\oplus m} \to A^{\oplus n}$ be given by the matric $(a_{ji})$ +$A^{\oplus m} \to A^{\oplus n}$ be given by the matrix $(a_{ji})$ so that $\sum a_{ji} g_i = 0$, $j = 1, \ldots, m$. Since $M \to M''$ is surjective, for each $i$ we can choose $m_i \in M$ mapping to $\varphi(g_i) \in M''$. Then the element $g_i \hat s - m_i$ @@ -3762,7 +3762,7 @@ \section{Algebraization of formal sections, II} \end{proof} \noindent -The following lemma will be superceded by +The following lemma will be superseded by Proposition \ref{proposition-application-H0}. \begin{lemma} @@ -4051,7 +4051,7 @@ \section{Algebraization of formal sections, II} Since $f$ is a nonzerodivisor on both $N$ and $M$ we conclude that $M_g \to N_g$ induces an isomorphism on $f$-adic completions which in turn implies $M_g \to N_g$ is an isomorphism in an open -neightbourhood of $V(f) \cap D(g)$. +neighbourhood of $V(f) \cap D(g)$. Since $g \in \mathfrak a$ was arbitrary, we conclude that $M$ and $N$ determine isomorphic coherent modules over an open $V$ as in the statement of the lemma. This finishes the proof. @@ -7447,7 +7447,7 @@ \section{Algebraization of coherent formal modules, V} \delta^Y_Z(y) - (d - 1) \end{aligned} \end{equation} -This inequality will allow us to check the remaning conditions. +This inequality will allow us to check the remaining conditions. \medskip\noindent Conditions (b) and (d) of diff --git a/artin.tex b/artin.tex index ec3b13b53..42bebf107 100644 --- a/artin.tex +++ b/artin.tex @@ -48,7 +48,7 @@ \section{Introduction} associated to $x_0$ over $k$. We introduce the Rim-Schlessinger condition (RS) for $\mathcal{X}$ and show it guarantees that $\mathcal{F}_{\mathcal{X}, k, x_0}$ is a deformation category, i.e., -$\mathcal{F}_{\mathcal{X}, k, x_0}$ satisies (RS) itself. +$\mathcal{F}_{\mathcal{X}, k, x_0}$ satisfies (RS) itself. We discuss how $\mathcal{F}_{\mathcal{X}, k, x_0}$ changes if one replaces $k$ by a finite extension and we discuss tangent spaces. @@ -3348,7 +3348,7 @@ \section{Infinitesimal deformations} \text{Mod}_A \longrightarrow \textit{Sets},\quad M \longrightarrow \text{Lift}(x, A[M]) $$ -of isomorphism classes of infintesimal deformations of $x$ to +of isomorphism classes of infinitesimal deformations of $x$ to $\Spec(A[M])$. We apply Formal Deformation Theory, Lemma \ref{formal-defos-lemma-linear-functor} to $\text{Inf}_x$ and $T_x$. This lemma is applicable, since @@ -5339,7 +5339,7 @@ \section{Artin's theorem on contractions} \ref{spaces-cohomology-lemma-proper-over-affine-cohomology-finite} and \ref{spaces-cohomology-lemma-flat-base-change-cohomology}). Denote $J'' = \Ker(A \to A')$\footnote{Contrary to what the reader -may expect, the ideals $J$ and $J''$ won't agreee in general.}. +may expect, the ideals $J$ and $J''$ won't agree in general.}. We have $J_i = J''A[1/f_i]$ as follows from base change to the spectrum of $A[1/f_i]$. Observe that we have a commutative diagram @@ -5803,7 +5803,7 @@ \section{Artin's theorem on contractions} Proof of surjectivity: algebra, more ring maps. Denote $Z_n \subset V$ the $n$th infinitesimal neighbourhood of $Z$ and denote $Z_{\mu, n} \subset V_\mu$ -the $n$th infinitesimal neighbourhoof of $Z_\mu$. +the $n$th infinitesimal neighbourhood of $Z_\mu$. By the theorem on formal functions (Cohomology of Spaces, Theorem \ref{spaces-cohomology-theorem-formal-functions}) @@ -6156,7 +6156,7 @@ \section{Artin's theorem on contractions} $W' \to W$ such that the image of $v$ in $W_{red}$ is in the image of $W'_{red} \to W_{red}$. Then $V_{/Z} \times_{g, W} W' \to V_{/Z}$ is an adic \'etale morphism of formal algebraic spaces over $S$ -and $V_{/Z} \times_{g, W} W'$ is an affine fromal algebraic space. +and $V_{/Z} \times_{g, W} W'$ is an affine formal algebraic space. By Algebraization of Formal Spaces, Lemma \ref{restricted-lemma-algebraize-rig-etale-affine} there exists an \'etale morphism $\varphi : V' \to V$ of affine schemes @@ -6190,7 +6190,7 @@ \section{Artin's theorem on contractions} $$ is smooth at $\mathfrak m$ for all $n$. By the discussion above we may and do assume that $B_1 \to A_1$ is a smooth ring map. -Denote $\mathfrak m_1 \subset A_1$ the maximal ideal corresponing +Denote $\mathfrak m_1 \subset A_1$ the maximal ideal corresponding to $\mathfrak m$. Since smoothness implies flatness, we see that: for all $n \geq 1$ the map $$ diff --git a/categories.tex b/categories.tex index c6e95c3ab..92220a695 100644 --- a/categories.tex +++ b/categories.tex @@ -3001,7 +3001,7 @@ \section{Essentially constant systems} \end{lemma} \begin{proof} -Omitted. This is a good excercise in the definitions. +Omitted. This is a good exercise in the definitions. \end{proof} \begin{remark} diff --git a/chow.tex b/chow.tex index 51a2bbccc..33fd7d464 100644 --- a/chow.tex +++ b/chow.tex @@ -273,7 +273,7 @@ \section{Periodic complexes and Herbrand quotients} 0 \to (M, \Im(\varphi), \varphi, 0) \to (M, N, \varphi, \psi) \to (0, N/\Im(\varphi), 0, 0) \to 0 $$ -The intial remark combined with the additivity of +The initial remark combined with the additivity of Lemma \ref{lemma-additivity-periodic-length} reduces us to the cases (a) $M = 0$ and (b) $\varphi$ is surjective. We leave those cases to the reader. @@ -457,7 +457,7 @@ \section{Calculation of some multiplicities} a_2 = a_1 + b_{n - 2} - b_1,\quad a_3 = a_2 + b_{n - 3} - b_2,\quad \ldots $$ -It is straighforward to see that this implies $a_i = a_{n - i}$ as desired. +It is straightforward to see that this implies $a_i = a_{n - i}$ as desired. \end{proof} \begin{lemma} @@ -4004,7 +4004,7 @@ \section{Chow groups and K-groups} $$ \text{length}_A(A/I) - \text{length}_A(A/fI) $$ -Using the distance fuction of +Using the distance function of Algebra, Definition \ref{algebra-definition-distance} we can rewrite this as $$ @@ -6190,7 +6190,7 @@ \section{Bivariant intersection theory} c \longmapsto res(c) $$ obtained by viewing a scheme $Y''$ locally of finite type over $Y'$ -as a scheme locally of finite type over $Y$ and settting +as a scheme locally of finite type over $Y$ and setting $res(c) \cap \alpha'' = c \cap \alpha''$ for any $\alpha'' \in \CH_k(Y'')$. This restriction operation is compatible with compositions in an obvious manner. @@ -6568,7 +6568,7 @@ \section{Lemmas on bivariant classes} (j'')^*(c \cap g^*\alpha') = c \cap (i'')^*g^*\alpha' $$ -by our assuptions on $c$; note that the modified version of (3) +by our assumptions on $c$; note that the modified version of (3) assumed in the statement of the lemma applies to $i''$ and its base change $j''$. We similarly know that $$ @@ -9510,7 +9510,7 @@ \section{Chern classes and the derived category} (Lemma \ref{lemma-splitting-principle}) we may assume each $\mathcal{E}^i$ has a filtration whose successive quotients $\mathcal{L}_{i, j}$ are invertible modules. -Settting $x_{i, j} = c_1(\mathcal{L}_{i, j})$ we see that +Setting $x_{i, j} = c_1(\mathcal{L}_{i, j})$ we see that $$ c(E) = \prod\nolimits_{i\text{ even}} (1 + x_{i, j}) @@ -9624,7 +9624,7 @@ \section{Chern classes and the derived category} \begin{proof} After choosing an envelope $f : Y \to X$ such that $Lf^*E$ and $Lf^*F$ can be represented by locally bounded complexes of finite locally -free $\mathcal{O}_X$-modules this follows by a compuation from the +free $\mathcal{O}_X$-modules this follows by a computation from the corresponding result for vector bundles in Lemmas \ref{lemma-chern-classes-tensor-product} and \ref{lemma-chern-character-multiplicative}. @@ -11222,7 +11222,7 @@ \section{Two technical lemmas} by Lemma \ref{lemma-base-change-loc-chern}. The fourth equality, in which $i'_\infty : W'_\infty \to W'$ is the inclusion morphism, follows from the fact that $c_p(W' \to W, Q)$ -is a bivariant class. For the fith equality, observe that +is a bivariant class. For the fifth equality, observe that $c_p(W' \to W, Q)$ and $c_p(Z' \to X, F)$ restrict to the same bivariant class in $A^p((b')^{-1} \to b^{-1}(\mathbf{A}^1_X))$ by @@ -15503,7 +15503,7 @@ \section{Change of base scheme} chow groups $\CH_k(X_i)$ as well as maps $\colim Z_k(X_i) \to Z_k(X')$ and $\colim \CH_i(X_i) \to \CH_k(X')$. We may replace $S$ by a quasi-compact open through which $X \to S$ -factors, hence we may and do assume all the schemes occuring in +factors, hence we may and do assume all the schemes occurring in this proof are Noetherian (and hence quasi-compact and quasi-separated). \medskip\noindent @@ -15512,7 +15512,7 @@ \section{Change of base scheme} be an integral closed subscheme of $\delta'$-dimension $k$. By Limits, Lemma \ref{limits-lemma-descend-finite-presentation} we can find an $i$ and a morphism $Z_i \to X_i$ of finite presentation -whose base change is $Z'$. Afer increasing $i$ we may assume $Z_i$ +whose base change is $Z'$. After increasing $i$ we may assume $Z_i$ is a closed subscheme of $X_i$, see Limits, Lemma \ref{limits-lemma-descend-closed-immersion-finite-presentation}. Then $Z' \to X_i$ factors through $Z_i$ and we may replace $Z_i$ @@ -15556,7 +15556,7 @@ \section{Change of base scheme} Arguing as above we can find an $i$ and integral closed subschemes $W_{i, l} \subset X_i$ of $\delta_i$-dimension $k + 1$ whose base change is $W'_l$. -After increasin $i$ we may assume we have rational functions +After increasing $i$ we may assume we have rational functions $f_{i, l}$ on $W_{i, l}$. Namely, we may think of $f'_l$ as a section of the structure sheaf over a nonempty open $U'_l \subset W'_l$, we can descend these opens by Limits, Lemma \ref{limits-lemma-descend-opens} diff --git a/cohomology.tex b/cohomology.tex index 91e0c5ad3..232108829 100644 --- a/cohomology.tex +++ b/cohomology.tex @@ -5073,7 +5073,7 @@ \section{{\v C}ech cohomology of complexes} } $$ where the left horizontal arrows are (\ref{equation-global-sections-to-cech}). -Since in this case the horizonal arrows are isomorphisms in the derived +Since in this case the horizontal arrows are isomorphisms in the derived category (see proof of Lemma \ref{lemma-cech-complex-complex}) it suffices to show that the left square commutes. This is true because the map $\gamma$ uses the sign $1$ on the summands @@ -7473,7 +7473,7 @@ \section{Cup product} \text{Tot}(\mathcal{A}^\bullet \otimes \mathcal{B}^\bullet)$. Finally, the commutativity of the remaining square is true on the level of complexes and may be viewed as the -definiton of the naive cup product (by the adjointness +definition of the naive cup product (by the adjointness of $f^*$ and $f_*$). The proof is finished because going around the diagram on the outside are the two maps given above. @@ -8831,7 +8831,7 @@ \section{Cohomology with support in a closed subset, II} (follows by checking what happens on stalks using that $i_*$ and $i'_*$ are exact and that $\mathcal{O}_{Z, z} = \mathcal{O}_{X, z}$ and similarly for $Z'$). Hence it suffices to construct a the top -horizonal arrow in the following diagram +horizontal arrow in the following diagram $$ \xymatrix{ Lf^* i_* R\mathcal{H}_Z(K) \ar[rr] \ar[rd] & & @@ -9634,7 +9634,7 @@ \section{Derived limits} H^{-n - 1}(E)[n + 1] \to K_{n + 1} \to K_n \to H^{-n - 1}(E)[n + 2] $$ -Looking at the asssociated long exact cohomology sequence the claim follows if +Looking at the associated long exact cohomology sequence the claim follows if $$ H^{m + n}(U, H^{-n - 1}(E)),\quad H^{m + n + 1}(U, H^{-n - 1}(E)),\quad @@ -10272,7 +10272,7 @@ \section{Hom complexes} It is a good idea to think of $\SheafHom^n$ as the sheaf of $\mathcal{O}_X$-modules of all $\mathcal{O}_X$-linear maps from $\mathcal{L}^\bullet$ to $\mathcal{M}^\bullet$ -(viewed as graded $\mathcal{O}_X$-modules) which are homogenous +(viewed as graded $\mathcal{O}_X$-modules) which are homogeneous of degree $n$. In this terminology, we define the differential by the rule $$ \text{d}(f) = diff --git a/cotangent.tex b/cotangent.tex index 53a4263a5..ec0c797d8 100644 --- a/cotangent.tex +++ b/cotangent.tex @@ -1114,7 +1114,7 @@ \section{The fundamental triangle} \medskip\noindent Consider the category $\mathcal{C}_{C/B/A}$ -wich is the {\bf opposite} of the category whose objects are +which is the {\bf opposite} of the category whose objects are $(P \to B, Q \to C)$ where \begin{enumerate} \item $P$ is a polynomial algebra over $A$, diff --git a/curves.tex b/curves.tex index 02add34bd..2d58e1896 100644 --- a/curves.tex +++ b/curves.tex @@ -994,7 +994,7 @@ \section{Riemann-Roch} capped with the cycle $[X]_1$ associated to $X$ is a natural zero cycle on $X$ with half-integer coefficients whose degree is $\chi(X, \mathcal{O}_X)$. -The occurence of fractions in the statement of Riemann-Roch cannot +The occurrence of fractions in the statement of Riemann-Roch cannot be avoided. \begin{lemma}[Riemann-Roch] @@ -5880,7 +5880,7 @@ \section{Contracting rational bridges} \begin{enumerate} \item $X' \not = \emptyset$ where $X' \subset X$ is the scheme theoretic closure of $X \setminus C$, -\item the scheme theoretic interesection $C \cap X'$ +\item the scheme theoretic intersection $C \cap X'$ has degree $2$ over $H^0(C, \mathcal{O}_C)$, and \item $C$ has genus zero. \end{enumerate} @@ -5906,7 +5906,7 @@ \section{Contracting rational bridges} } $$ with many good properties (all of which we will use below without -futher mention). Let $y \in Y$ be the image of $\Spec(k') \to Y$. +further mention). Let $y \in Y$ be the image of $\Spec(k') \to Y$. Then $$ \mathcal{O}_{Y, y}^\wedge \cong k'[[s, t]]/(st) diff --git a/decent-spaces.tex b/decent-spaces.tex index 40b58f3d8..f51575bc6 100644 --- a/decent-spaces.tex +++ b/decent-spaces.tex @@ -615,7 +615,7 @@ \section{Finiteness conditions and points} (1) -- (5) are equivalent. \medskip\noindent -It is clear that (6) implies (5). Conversly, assume $\Spec(k) \to X$ is +It is clear that (6) implies (5). Conversely, assume $\Spec(k) \to X$ is as in (4) and let $\Spec(k') \to X$ be another morphism with $k'$ a field in the equivalence class of $x$. By Properties of Spaces, Lemma @@ -2536,7 +2536,7 @@ \section{Residue fields and henselian local rings} and Categories, Lemma \ref{categories-lemma-cofinal} (initial is turned into cofinal because the colimit -definining henselian local rings is over the +defining henselian local rings is over the opposite of the category of elementary \'etale neighbourhoods). \end{proof} @@ -5122,7 +5122,7 @@ \section{Birational morphisms} \Spec(\mathcal{O}_{Y, f(\xi)}) \times_Y V \ar[r] & V } $$ -the vertical arrow on the left is an isomorphism. The horizonal arrows +the vertical arrow on the left is an isomorphism. The horizontal arrows have image contained in the schematic locus of $U$ and $V$ and identify local rings (some details omitted). Since the image of the horizontal arrows are the points of $|U|$, resp.\ $|V|$ @@ -5395,7 +5395,7 @@ \section{Jacobson spaces} Combining Lemma \ref{lemma-locally-Noetherian-decent-quasi-separated} with Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-quasi-separated-spectral} -we see that $|X|$ is a spectral toplogical space. +we see that $|X|$ is a spectral topological space. By Topology, Lemma \ref{topology-lemma-make-spectral-space} we see that $W \cup \{x\}$ is a spectral topological space. Now $W$ is a quasi-compact open of $W \cup \{x\}$ and hence $W$ is diff --git a/defos.tex b/defos.tex index 485227c41..fecd1d2b5 100644 --- a/defos.tex +++ b/defos.tex @@ -5694,7 +5694,7 @@ \section{Deformations of complexes on ringed topoi} \mathcal{G}_0^\bullet$ in $D(\mathcal{O}_0)$. In other words, we have produced a pair $(\mathcal{H}^\bullet, \alpha_0)$. We omit the verification that $o(\alpha_0) = \xi$; hint: $o(\alpha_0)$ -can be computed explitly in this case as we have maps +can be computed explicitly in this case as we have maps $\mathcal{H}^n \to \mathcal{F}^n$ (not compatible with differentials) lifting the components of $\alpha_0$. This finishes the proof. \end{proof} diff --git a/derham.tex b/derham.tex index 41424b35a..47ce56c12 100644 --- a/derham.tex +++ b/derham.tex @@ -227,7 +227,7 @@ \section{de Rham cohomology} formation commutes with arbitrary change of base by Lemma \ref{lemma-base-change-de-rham}. Hence $Rf_*\Omega^p_{X/S}$ is a perfect object of $D(\mathcal{O}_S)$ -whose formation commutes with abitrary base change, see +whose formation commutes with arbitrary base change, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-perfect-direct-image-general}. This proves the first assertion of the lemma. @@ -1657,7 +1657,7 @@ \section{The spectral sequence for a smooth morphism} F^1\Omega^p_{X/S} \supset F^2\Omega^p_{X/S} \supset \ldots \supset F^{p + 1}\Omega^p_{X/S} = 0 $$ -whose succesive quotients are +whose successive quotients are $$ \text{gr}^k\Omega^p_{X/S} = F^k\Omega^p_{X/S}/F^{k + 1}\Omega^p_{X/S} = @@ -2300,7 +2300,7 @@ \section{Log poles along a divisor} \end{lemma} \begin{proof} -This follows by direct calcuation from the local construction +This follows by direct calculation from the local construction of the complex in the proof of Lemma \ref{lemma-log-complex}. Details omitted. \end{proof} @@ -2722,7 +2722,7 @@ \section{Calculations} with zero section $0 : P \to L$, it suffices to show that on a line bundle there are no nonzero sections of a sheaf of differentials which vanish identically outside the zero section. -The reader sees this is true either (preferably) by a local caculation +The reader sees this is true either (preferably) by a local calculation or by using that $\Omega_{L/S, k} \subset \Omega_{L^\star/S, k}$ (see references above). \end{proof} diff --git a/derived.tex b/derived.tex index 8bf4a072c..7e35a44f1 100644 --- a/derived.tex +++ b/derived.tex @@ -9905,7 +9905,7 @@ \section{Bounded cohomological dimension} and the other values are unchanged. \medskip\noindent -To finish the proof we carefuly choose the order in which to do +To finish the proof we carefully choose the order in which to do the elementary replacements so that for every integer $m$ the complex $\sigma_{\geq m}K^\bullet$ is changed only a finite number of times. To do this set @@ -11794,7 +11794,7 @@ \section{Admissible subcategories} by Lemma \ref{lemma-pre-prepare-adjoint}. \medskip\noindent -Assume (2). We will contruct the adjoint $v$ explictly. +Assume (2). We will construct the adjoint $v$ explicitly. Let $X \in \Ob(\mathcal{D})$. Choose $A \to X \to B \to A[1]$ as in (2). Set $v(X) = A$. Let $f : X \to Y$ be a morphism in $\mathcal{D}$. Choose $A' \to Y \to B' \to A'[1]$ as in (2). Since @@ -11903,7 +11903,7 @@ \section{Admissible subcategories} \end{proposition} \begin{proof} -The equivalence between (1), (2), and (3) follows in a straighforward manner +The equivalence between (1), (2), and (3) follows in a straightforward manner from Lemmas \ref{lemma-right-adjoint} and \ref{lemma-left-adjoint} (small detail omitted). Denote $v : \mathcal{D} \to \mathcal{A}$ the right adjoint of the inclusion functor $i : \mathcal{A} \to \mathcal{D}$. @@ -12383,7 +12383,7 @@ \section{Essentially constant systems} $$ A \to B \to C \xrightarrow{\delta} A[1] $$ -Next, choose a morphism of distingished triangles +Next, choose a morphism of distinguished triangles $$ (A_1 \to B_1 \to C_1 \to A_1[1]) \to (A \to B \to C \to A[1]) diff --git a/dga.tex b/dga.tex index 4d06fc692..4083d3499 100644 --- a/dga.tex +++ b/dga.tex @@ -1597,7 +1597,7 @@ \section{Hom complexes and differential graded modules} \tau_{p, q}(a) : M^{-q} \longrightarrow M^p,\quad x \longmapsto (-1)^{\deg(a)\deg(x)}x a = (-1)^{-nq}xa $$ -This is not compatible with the product on $A$ as the readed should +This is not compatible with the product on $A$ as the reader should expect from the discussion in Section \ref{section-left-modules}. Namely, we have $$ @@ -4000,7 +4000,7 @@ \section{Obtaining triangulated categories} \end{align*} where we have used equation (6) of Lemma \ref{lemma-get-triangle} for the first equality and -the preceeding remark for the second. Similarly, we obtain +the preceding remark for the second. Similarly, we obtain $d(\tilde{p}) = i\delta[-1]$. Hence \begin{align*} d(i\tilde{\alpha} + \tilde{p}\beta) @@ -4394,7 +4394,7 @@ \section{Obtaining triangulated categories} } $$ In these diagrams, the maps $\delta_i$ are defined as -$\delta_i = \pi_i d(s_i)$ analagous to the maps defined in +$\delta_i = \pi_i d(s_i)$ analogous to the maps defined in Lemma \ref{lemma-get-triangle}. They fit in the following solid commutative diagram $$ @@ -4914,7 +4914,7 @@ \section{Bimodules and internal hom} Let $f : N_1 \to N_2$ be a $B$-module homomorphism which is homogeneous of degree $n$, in other words, $f \in H_{12}^n$. The functor in the lemma sends $f$ to $c_f : H_1 \to H_2$, $g \mapsto c(f, g)$. -Simlarly for $\text{d}(f)$. On the other hand, the differential on +Similarly for $\text{d}(f)$. On the other hand, the differential on $$ \Hom_{\text{Mod}^{dg}_{(A, \text{d})}}(H_1, H_2) $$ @@ -7189,7 +7189,7 @@ \section{Resolutions of differential graded algebras} From the claim it is clear that $P = \bigcup P'_n$ is a countable rising union of $P'_n$ as above. By construction each $P'_n$ is a differential graded module with -property (P) such that the filtration is finite and the succesive +property (P) such that the filtration is finite and the successive quotients are finite direct sums of shifts of $A$. Hence $P'_n$ defines a compact object of $D(A, \text{d})$, see for example Proposition \ref{proposition-compact}. Since diff --git a/documentation/make-project b/documentation/make-project index 1ca051a51..28a739ac3 100644 --- a/documentation/make-project +++ b/documentation/make-project @@ -32,7 +32,7 @@ The command make web creates files book.tex, tags, and my.bib in a target directory WEBDIR; these -files are ment to be parsed by plastex. The command will fail if WEBDIR does +files are meant to be parsed by plastex. The command will fail if WEBDIR does not exist. Please email if you run into any problems. diff --git a/documentation/rules b/documentation/rules index 4f5208333..077e4f937 100644 --- a/documentation/rules +++ b/documentation/rules @@ -65,7 +65,7 @@ occurs in the chapter varieties.tex hence its long label is varieties-lemma-locally-finite-type-Jacobson -There is a one to one correspondence between long labels and occurences +There is a one to one correspondence between long labels and occurrences of the latex command '\label{...}' @@ -154,7 +154,7 @@ Latex style ----------- Obviously you can argue about this forever. Some of these rules have to do with -readability of the latex files and some have to do with the overal setup of the +readability of the latex files and some have to do with the overall setup of the project as well as the tags system (which is discussed elsewhere). * Every line has at most 80 characters diff --git a/documentation/todo-list b/documentation/todo-list index f0fb2fd7b..b7ee78571 100644 --- a/documentation/todo-list +++ b/documentation/todo-list @@ -43,7 +43,7 @@ locally of finite type is locally of finite presentation. Also Define quasi-finite morphisms of algebraic stacks. -Limits of Schemes: Absloute Noetherian approximation. Add a second proof +Limits of Schemes: Absolute Noetherian approximation. Add a second proof following Temkin's proof in [Relative RZ-spaces, section 1.1]. Look also at David Rydh's paper [Noetherian approximation of algebraic spaces and stacks]. In fact, using this method one gets a short proof of a more diff --git a/duality.tex b/duality.tex index 821df0d1a..43152cf0d 100644 --- a/duality.tex +++ b/duality.tex @@ -2359,7 +2359,7 @@ \section{Right adjoint of pushforward for closed immersions and base change} Since $i'_*$ is exact and faithful we see that it suffices to show the map (\ref{equation-base-change-exact-support}) is an isomorphism after applying $Ri'_*$. Since -$Ri'_* \circ Lg^* = Lf^* \circ Ri_*$ by the assumed tor indepence and +$Ri'_* \circ Lg^* = Lf^* \circ Ri_*$ by the assumed tor independence and Derived Categories of Schemes, Lemma \ref{perfect-lemma-compare-base-change-closed-immersion} we obtain a map @@ -3391,7 +3391,7 @@ \section{Right adjoint of pushforward in examples} $a'(\mathcal{O}_{\Spec(k)}) = \mathcal{O}_E(-2)[1]$ which is an invertible sheaf of degree $-2$ placed in cohomological degree $-1$, so different. In this example -the hypothesis of Tor indepence in Lemma \ref{lemma-more-base-change} +the hypothesis of Tor independence in Lemma \ref{lemma-more-base-change} is violated. \end{example} diff --git a/dualizing.tex b/dualizing.tex index b937d2f0a..148e66cfd 100644 --- a/dualizing.tex +++ b/dualizing.tex @@ -2669,7 +2669,7 @@ \section{Dualizing complexes} Let $G$ be a quasi-inverse to our functor $F$. Arguing as above we find an integer $b$ such that $G(\kappa) = \kappa[-b]$. On composing we find $a + b = 0$. Let $E$ be a finite $A$-module -wich is annihilated by a power of $\mathfrak m$. Arguing by +which is annihilated by a power of $\mathfrak m$. Arguing by induction on the length of $E$ we find that $G(E) = E'[-b]$ for some finite $A$-module $E'$ annihilated by a power of $\mathfrak m$. Then $E[-a] = F(E')$. @@ -4220,7 +4220,7 @@ \section{Formal fibres} complex for the fraction field $K$ of $A$ (Lemma \ref{lemma-dualizing-localize}). Hence $\omega_A^\bullet \otimes_A K$ -is isomorphic ot $K[n]$ for some $n \in \mathbf{Z}$. +is isomorphic to $K[n]$ for some $n \in \mathbf{Z}$. Similarly, we conclude a dualizing complex for the formal fibre $A^\wedge \otimes_A K$ is $$ @@ -4302,7 +4302,7 @@ \section{Upper shriek algebraically} relative dualizing complex: it lies in $D^b_{\textit{Coh}}(A)$ and restricts to a dualizing complex on the fibres provided $R \to A$ is flat. \end{enumerate} -These statemens are Lemmas \ref{lemma-shriek-dualizing-algebraic} and +These statements are Lemmas \ref{lemma-shriek-dualizing-algebraic} and \ref{lemma-relative-dualizing-algebraic}. \medskip\noindent diff --git a/equiv.tex b/equiv.tex index dbcebe091..d690f6ec8 100644 --- a/equiv.tex +++ b/equiv.tex @@ -1642,7 +1642,7 @@ \section{Resolutions and bounds} and we get the vanishing of $\Hom(K, L)$ from the vanishing of $\Hom(H^i(K)[-i], L)$ and $\Hom(\tau_{\geq i + 1}K, L)$ by Derived Categories, Lemma \ref{derived-lemma-representable-homological}. -Simlarly if $L$ has more than one nonzero cohomology sheaf. +Similarly if $L$ has more than one nonzero cohomology sheaf. \end{proof} \begin{lemma} @@ -2225,7 +2225,7 @@ \section{Sibling functors} let us say $X$ has {\it width} $w = w(X)$ if $w \geq 0$ is minimal such that there exists an integer $a \in \mathbf{Z}$ with $H^i(X) = 0$ for $i \not \in [a, a + w - 1]$. Since $F$ is a sibling of the identity -and since $F \circ [n] = [n] \circ F$ we are aready given isomorphisms +and since $F \circ [n] = [n] \circ F$ we are already given isomorphisms $$ c_X : X \to F(X) $$ diff --git a/etale-cohomology.tex b/etale-cohomology.tex index 655f726b2..0a1ae3c0a 100644 --- a/etale-cohomology.tex +++ b/etale-cohomology.tex @@ -2183,7 +2183,7 @@ \section{Cohomology of quasi-coherent sheaves} \end{lemma} \begin{proof} -The defintion of a standard $\tau$-covering is given in +The definition of a standard $\tau$-covering is given in Topologies, Definition \ref{topologies-definition-standard-Zariski}, \ref{topologies-definition-standard-etale}, @@ -5302,7 +5302,7 @@ \section{Recovering morphisms} \end{lemma} \begin{proof} -Let us argue this carefuly since it is a bit confusing. +Let us argue this carefully since it is a bit confusing. Let $t : a_{small}^{-1} \to b_{small}^{-1}$ be the $2$-isomorphism. Consider any open $V \subset Y$. Note that $h_V$ is a subsheaf of the final sheaf $*$. Thus both $a_{small}^{-1}h_V = h_{a^{-1}(V)}$ @@ -5569,7 +5569,7 @@ \section{Recovering morphisms} \end{theorem} \begin{proof} -You can prove this theorem by carefuly adjusting the arguments of +You can prove this theorem by carefully adjusting the arguments of the proof of Lemma \ref{lemma-morphism-ringed-etale-topoi-affines} to the global setting. However, we want to indicate how we @@ -10541,7 +10541,7 @@ \section{Galois cohomology} Fix an $\ell$-power torsion sheaf $\mathcal{G}$ on $S$ with $H^n_\etale(S, \mathcal{G}) \not = 0$. The system given by $\mathcal{G}_i = g_i^{-1}\mathcal{G}$ -satisify the conditions of Theorem \ref{theorem-colimit} +satisfy the conditions of Theorem \ref{theorem-colimit} with colimit sheaf given by $g^{-1}\mathcal{G}$. This tells us that: $$ @@ -13125,7 +13125,7 @@ \section{Specializations and \'etale sheaves} is a transformation of functors. \medskip\noindent -For sheaves in the \'etale topology we can mimick this construction, see +For sheaves in the \'etale topology we can mimic this construction, see \cite[Exposee VIII, 7.7, page 397]{SGA4}. To do this suppose we have a scheme $S$, a geometric point $\overline{s}$ of $S$, and a geometric point $\overline{t}$ of $\Spec(\mathcal{O}^{sh}_{S, \overline{s}})$. @@ -13260,7 +13260,7 @@ \section{Specializations and \'etale sheaves} K_{\overline{t}} $$ The equality signs are valid as taking global sections over -the striclty henselian schemes +the strictly henselian schemes $\Spec(\mathcal{O}^{sh}_{S, \overline{s}})$ and $\Spec(\mathcal{O}^{sh}_{S, \overline{t}})$ is exact (and the same as taking stalks at $\overline{s}$ and $\overline{t}$) @@ -17720,7 +17720,7 @@ \section{Smooth base change} Then $T \to S$ is a disjoint union of dominant morphisms of integral normal schemes, see Morphisms, Lemma \ref{morphisms-lemma-normal-normalization}. -Clearly we may argue one connnected component +Clearly we may argue one connected component at a time, hence we may assume $T \to S$ is a dominant morphism of integral normal schemes. @@ -18838,7 +18838,7 @@ \section{Applications of proper base change} $$ whose fibres have dimension $< n$. Hence by induction we see that $R^if_*b^{-1}\mathcal{F} = 0$ for $i > 2(n - 1)$. -We conclude by the Leray spectral seqence +We conclude by the Leray spectral sequence $$ H^i(\mathbf{P}^1_k, R^jf_*b^{-1}\mathcal{F}) \Rightarrow @@ -22991,7 +22991,7 @@ \section{Descending \'etale sheaves} is an isomorphism; details omitted. In order to prove this it suffices to check after pulling back by any collection of morphisms $\Spec(k) \to Y$ where $k$ is an algebraically closed field. -Namely, the corresponing base changes $X_k \to X$ are jointly +Namely, the corresponding base changes $X_k \to X$ are jointly surjective and we can check whether a map of sheaves on $X_\etale$ is an isomorphism by looking at stalks on geometric points, see Theorem \ref{theorem-exactness-stalks}. diff --git a/etale.tex b/etale.tex index e14399a52..09ec6ecd8 100644 --- a/etale.tex +++ b/etale.tex @@ -1520,7 +1520,7 @@ \section{Topological invariance of the \'etale topology} \end{proof} \noindent -From the proof of preceeding theorem, we also obtain one direction of the +From the proof of preceding theorem, we also obtain one direction of the promised functorial characterization of \'etale morphisms. The following theorem will be strengthened in \'Etale Cohomology, @@ -2005,7 +2005,7 @@ \section{Permanence properties} \end{proof} \noindent -The preceeding propositions give some indication as to why we'd like to think +The preceding propositions give some indication as to why we'd like to think of \'etale maps as ``local isomorphisms''. Another property that gives an excellent indication that we have the ``right'' definition is the fact that for $\mathbf{C}$-schemes of finite type, a morphism is \'etale if and only if diff --git a/examples-defos.tex b/examples-defos.tex index ecada3a5f..db3f319a8 100644 --- a/examples-defos.tex +++ b/examples-defos.tex @@ -72,7 +72,7 @@ \section{General outline} \medskip\noindent Step II. In each section we define a category $\mathcal{F}$ -cofibred in groupoids over $\mathcal{C}_\Lambda$. Occassionally +cofibred in groupoids over $\mathcal{C}_\Lambda$. Occasionally we will consider instead a functor $F : \mathcal{C}_\Lambda \to \textit{Sets}$. @@ -1595,7 +1595,7 @@ \section{Morphisms of Schemes} By Deformation Theory, Lemma \ref{defos-lemma-huge-diagram-ringed-spaces} for an arbitrary pair $(\alpha, \beta)$ the difference between the morphism $f' : X' \to Y'$ and the morphism -$\beta^{-1} \circ f' \circ \alpha : X' \to Y'$ defines an elment +$\beta^{-1} \circ f' \circ \alpha : X' \to Y'$ defines an element in $$ \text{Der}_k(\mathcal{O}_Y, f_*\mathcal{O}_X) = @@ -2006,7 +2006,7 @@ \section{Deformations of completions} } $$ with cartesian squares, with flat horizontal arrows, and with -surjective vertial arrows whose kernels are nilpotent. +surjective vertical arrows whose kernels are nilpotent. Let $J' \subset P'$ be an ideal such that $P'/J' = Q'/J'Q'$. Let $M$ be an $A$-flat $P$-module. Assume for all $g \in J'$ there exists an $A'$-flat $(P')_g$-module @@ -2997,7 +2997,7 @@ \section{Smoothings} \begin{lemma} \label{lemma-smoothing-affine-lci} -Let $k$ be a field. Let $A$ be a global complete interesection +Let $k$ be a field. Let $A$ be a global complete intersection over $k$. There exists a flat finite type ring map $k[[t]] \to B$ with $B/tB \cong A$ such that $B[1/t]$ is smooth over $k((t))$. diff --git a/examples-stacks.tex b/examples-stacks.tex index e6826d1b1..c343d0d58 100644 --- a/examples-stacks.tex +++ b/examples-stacks.tex @@ -786,7 +786,7 @@ \section{Classifying torsors} \label{section-torsors} \noindent -We want to carefuly explain a number of variants of what it could +We want to carefully explain a number of variants of what it could mean to study the stack of torsors for a group algebraic space $G$ or a sheaf of groups $\mathcal{G}$. diff --git a/examples.tex b/examples.tex index 17efabdff..74c393074 100644 --- a/examples.tex +++ b/examples.tex @@ -1383,7 +1383,7 @@ \section{A non catenary Noetherian local ring} \label{section-non-catenary-Noetherian-local} \noindent -Even though there is a succesful dimension theory of Noetherian local rings +Even though there is a successful dimension theory of Noetherian local rings there are non-catenary Noetherian local rings. An example may be found in \cite[Appendix, Example 2]{Nagata}. In fact, we will present this example in the simplest case. Namely, we will construct a local Noetherian domain $A$ @@ -6731,7 +6731,7 @@ \section{The spectrum of the integers is not quasi-compact} $$ I' \longrightarrow M $$ -of $R$-modules wich recovers the map $I_A \to M_A$ for all $A \in U$. +of $R$-modules which recovers the map $I_A \to M_A$ for all $A \in U$. This is true by all the compatibilities mentioned above and the claim proved previously that $M$ is the equalizer of the two maps from $\prod_{A \in U} M_A$ to $\prod_{A, B \in U} M_{A \cup B}$. diff --git a/exercises.tex b/exercises.tex index dce7ff040..7d01cd72c 100644 --- a/exercises.tex +++ b/exercises.tex @@ -270,7 +270,7 @@ \section{Additive and abelian categories} Homology, Definition \ref{homology-definition-filtered} for the definition of a filtered object of any category. \begin{enumerate} -\item Show that this is an additive category (explain carefuly what the +\item Show that this is an additive category (explain carefully what the direct sum of two objects is). \item Let $f : (V, F) \to (W, F)$ be a morphism of $\mathcal{C}$. Show that $f$ has a kernel and cokernel (explain precisely @@ -1410,7 +1410,7 @@ \section{Catenary rings} there exists a $z \in X$ with $x \leadsto z \leadsto y$ and $\delta(x) > \delta(z) > \delta(y)$. \end{enumerate} -Describe your space clearly and succintly explain why there +Describe your space clearly and succinctly explain why there cannot be a dimension function. \end{exercise} @@ -4554,7 +4554,7 @@ \section{Curves} $X^r \times X^s$ is in $Z$ if and only if $x_1 + \ldots + x_r - y_1 - \ldots - y_s$ is linearly equivalent to an effective divisor. -Hint: Choose an auxilliary invertible module $\mathcal{L}$ +Hint: Choose an auxiliary invertible module $\mathcal{L}$ of very high degree so that $\mathcal{L}(-D)$ has a nonvanshing section for any effective divisor $D$ of degree $r$. Then use the result of Exercise \ref{exercise-effective} twice. diff --git a/fields.tex b/fields.tex index 1f917d128..1792fc71d 100644 --- a/fields.tex +++ b/fields.tex @@ -1444,7 +1444,7 @@ \section{Linear independence of characters} & = \sum\nolimits_{i = 1, \ldots, n - 1} \lambda_i\chi_i(hg) \\ & = \sum\nolimits_{i = 1, \ldots, n - 1} \lambda_i\chi_i(h) \chi_i(g) \end{align*} -Multiplying the previous relation by $\chi_n(h)$ and substracting we obtain +Multiplying the previous relation by $\chi_n(h)$ and subtracting we obtain $$ 0 = \sum\nolimits_{i = 1, \ldots, n - 1} \lambda_i (\chi_n(h) - \chi_i(h)) \chi_i(g) @@ -3070,7 +3070,7 @@ \section{Infinite Galois theory} \medskip\noindent Assume $H \subset G$ is open. Arguing as above we find that -$H$ containes $\text{Gal}(L/E)$ for some large enough finite +$H$ contains $\text{Gal}(L/E)$ for some large enough finite Galois subextension $E$ and we find that $L^H$ is contained in $E$ whence finite over $K$. Conversely, if $M$ is a finite subextension, then $M$ is generated by a finite subset $S$ diff --git a/flat.tex b/flat.tex index 7072d3e0c..1c5860ecd 100644 --- a/flat.tex +++ b/flat.tex @@ -3742,7 +3742,7 @@ \section{Flat finite type modules, Part II} \end{proof} \noindent -The following lemma has a sligthly stronger variant +The following lemma has a slightly stronger variant Lemma \ref{lemma-weak-bourbaki} below. @@ -9867,7 +9867,7 @@ \section{Nagata compactification} $V_{12} = V_1 \times_U V_2$. By Lemma \ref{lemma-blowup-iso-along} there exists a $V_1$-admissible blowing up $X_1' \to X_1$ -such that the strict tranform $p'_1 : X'_{12} \to X'_1$ +such that the strict transform $p'_1 : X'_{12} \to X'_1$ of $p_1$ is an isomorphism over an open neighbourhood of the closure of $Z_{1, 2}$ in $X'_1$. After replacing $X_1$ by $X'_1$ and $X_{12}$ by $X'_{12}$ @@ -11719,7 +11719,7 @@ \section{Almost blow up squares and the h topology} \begin{proof} This lemma is a formal consequence of Lemma \ref{lemma-refine-check-h} -and our defnition of stacks in groupoids. +and our definition of stacks in groupoids. For example, assume (1), (2), (3). To show that $\mathcal{S}$ is a stack, we have to prove descent for morphisms and objects, see @@ -12147,7 +12147,7 @@ \section{Absolute weak normalization and h coverings} \medskip\noindent Suppose $x, y \in A$ with $\ell^\ell x = y^\ell$ for some prime -number $\ell$. If $\ell \not = p$, then $a = y/\ell$ satsifies +number $\ell$. If $\ell \not = p$, then $a = y/\ell$ satisfies $a^\ell = x$ and $\ell a = y$. If $\ell = p$, then $y = 0$ and $x = a^p$ for some $a$. \end{proof} @@ -12782,7 +12782,7 @@ \section{Descent vector bundles in positive characteristic} \medskip\noindent Formulated in this way, the condition descends to an absolute -Noetherian approximation. We stronly urge the reader to skip +Noetherian approximation. We strongly urge the reader to skip this paragraph. First write $Y = \lim_{i \in I} Y_i$ as a cofiltered limit of schemes of finite type over $\mathbf{F}_p$ with affine transition morphisms (Limits, Lemma \ref{limits-lemma-relative-approximation}). @@ -13503,7 +13503,7 @@ \section{Blowing up complexes, II} \end{lemma} \begin{proof} -The first statement follows from the folowing observations: +The first statement follows from the following observations: $g$ is a nonzerodivisor in $B$ which defines $E \cap V \subset V$ and $M^\bullet \otimes_A B$ represents $M^\bullet \otimes_A^\mathbf{L} B$ and hence represents the pullback of $M$ to $V$ by @@ -13670,7 +13670,7 @@ \section{Blowing up complexes, II} $$ b_{U_2}|_{b_{U_2}^{-1}(U_1)} : b_{U_2}^{-1}(U_1) \longrightarrow U_1 $$ -by the universal propery. In other words, we get an isomorphism +by the universal property. In other words, we get an isomorphism $U'_1 \to b_{U_2}^{-1}(U_1)$ over $U_1$. These isomorphisms satisfy the cocycle condition (again by the universal property) and hence by Constructions, Lemma \ref{constructions-lemma-relative-glueing} diff --git a/formal-defos.tex b/formal-defos.tex index 50b0c1e52..2edf8df13 100644 --- a/formal-defos.tex +++ b/formal-defos.tex @@ -5955,7 +5955,7 @@ \section{Applications} Part (2). Let $w_0 \in \Ob(\mathcal{W}(k))$ and let $x_0, y_0, z_0$ be the image of $w_0$ in $\mathcal{F}, \mathcal{H}, \mathcal{G}$. Then $\text{Inf}(\mathcal{W}) = \text{Inf}_{w_0}(\mathcal{W})$ -and simlarly for $\mathcal{H}$, $\mathcal{G}$, and $\mathcal{F}$, see +and similarly for $\mathcal{H}$, $\mathcal{G}$, and $\mathcal{F}$, see Remark \ref{remark-trivial-aut-point}. We apply Lemmas \ref{lemma-k-linear-differential} and \ref{lemma-k-linear-infaut} to get all the linear maps @@ -7481,7 +7481,7 @@ \section{Change of residue field} \item an $S \in \Ob(\widehat{\mathcal{C}}_{\Lambda, l})$ and a local $\Lambda$-algebra homomorphism $R \to S$ which is formally smooth in the $\mathfrak m_S$-adic topology and induces -the given field extension $l/k$ on residue fieds, and +the given field extension $l/k$ on residue fields, and \item a versal formal object of $\mathcal{F}_{l/k}$ lying over $S$. \end{enumerate} \end{lemma} diff --git a/formal-spaces.tex b/formal-spaces.tex index 7c5cc27e8..dbc29fe71 100644 --- a/formal-spaces.tex +++ b/formal-spaces.tex @@ -1611,7 +1611,7 @@ \section{Descending properties} of $I'B$. Then $A/I' \to B/J'$ is faithfully flat, hence injective. Thus in order to show that $f^n \in I'$ it suffices to show that $\varphi(f)^n \in J'$. This holds for $n \gg 0$ since -$\varphi(f) \in J$, the ideal $J$ is a weak ideal of defintion of $B$, +$\varphi(f) \in J$, the ideal $J$ is a weak ideal of definition of $B$, and $J'$ is open in $B$. \end{proof} @@ -1629,7 +1629,7 @@ \section{Descending properties} of $I'B$. Then $A/I' \to B/J'$ is faithfully flat, hence injective. Thus in order to show that $I^n \subset I'$ it suffices to show that $\varphi(I)^n \subset J'$. This holds for $n \gg 0$ since -$\varphi(I) \subset J$, the ideal $J$ is an ideal of defintion of $B$, +$\varphi(I) \subset J$, the ideal $J$ is an ideal of definition of $B$, and $J'$ is open in $B$. \end{proof} @@ -2487,7 +2487,7 @@ \section{Formal algebraic spaces} \medskip\noindent In fact, since we've shown above that the morphisms of schemes -$U \times_X V \to U \times_S V$ are aways monomorphisms and +$U \times_X V \to U \times_S V$ are always monomorphisms and locally quasi-finite we conclude that $\Delta : X \to X \times_S X$ is a monomorphism and locally quasi-finite, see Spaces, Lemma \ref{spaces-lemma-transformation-diagonal-properties}. @@ -2715,7 +2715,7 @@ \section{The reduction} is a smooth morphism of algebraic spaces. Hence $X \times_Y Y_{red}$ is a reduced algebraic space by Descent on Spaces, Lemma \ref{spaces-descent-lemma-reduced-local-smooth}. -Then the univeral property of reduction shows that the canonical morphism +Then the universal property of reduction shows that the canonical morphism $X_{red} \to X \times_Y Y_{red}$ is an isomorphism. \end{proof} @@ -7119,7 +7119,7 @@ \section{Maps out of affine formal schemes} is an algebraic space endowed with a closed immersion to $\text{Spf}(R)$. (Equality holds because $X_{/T} \to X$ is a monomorphism.) Thus this fibre product is equal to $\Spec(R/J)$ for some -ideal $J \subset R$ wich contains $\mathfrak m_R^{n_0}$ for some +ideal $J \subset R$ which contains $\mathfrak m_R^{n_0}$ for some $n_0 \geq 1$. This implies that $\Spec(R) \times_X Z$ is a closed subscheme of $\Spec(R)$, say $\Spec(R) \times_X Z = \Spec(R/I)$, whose intersection with @@ -7505,7 +7505,7 @@ \section{The structure sheaf} \prod\nolimits_i \mathcal{O}_X(U_{i, red}) \to \prod\nolimits_{i, j} \mathcal{O}_X((U_i \times_U U_j)_{red}) $$ -is exact and hence the sheaf propery holds as desired. +is exact and hence the sheaf property holds as desired. \end{proof} \begin{remark} diff --git a/functors.tex b/functors.tex index c0add3820..b9442fded 100644 --- a/functors.tex +++ b/functors.tex @@ -1377,7 +1377,7 @@ \section{Functors between categories of quasi-coherent modules} \Gamma(U \times_R Y \times_R W, \text{pr}_{12}^*\mathcal{K} \otimes \text{pr}_{23}^*\mathcal{L}) \end{align*} -as desired. We omit the verication that these isomorphisms are +as desired. We omit the verification that these isomorphisms are compatible with restriction mappings. \end{proof} diff --git a/groupoids.tex b/groupoids.tex index f4f4f83ff..29a96997d 100644 --- a/groupoids.tex +++ b/groupoids.tex @@ -1000,7 +1000,7 @@ \section{Properties of group schemes over a field} \medskip\noindent By the claim $W \subset G$ is an open and closed subset of $G$. Now $W \cap Z' = \emptyset$ since otherwise by the argument given in -the precending paragraph we would get $Z' = Zg$ for some $g \in W(k)$. +the preceding paragraph we would get $Z' = Zg$ for some $g \in W(k)$. Then as $Z$ is a subgroup we could even pick $g \in U(k)$ which would contradict $Z' \cap U = \emptyset$. Hence $W \subset G$ is a proper open and closed subset which contradicts the assumption that $G$ is connected. @@ -3050,7 +3050,7 @@ \section{Colimits of quasi-coherent modules} \mathcal{F}_i \ar[r] \ar[u] & s_*t^*\mathcal{H}_i \ar[u] } $$ -where the right vertical arrow is the result of appplying $s_*$ to the map +where the right vertical arrow is the result of applying $s_*$ to the map $$ t^*\mathcal{H}_i \to t^*\mathcal{F} \xrightarrow{\beta} s^*\mathcal{F} $$ diff --git a/homology.tex b/homology.tex index 72e0d6b63..c89b9428d 100644 --- a/homology.tex +++ b/homology.tex @@ -5092,7 +5092,7 @@ \section{Spectral sequences} \label{remark-allow-translation-functors} It is often the case that the terms of a spectral sequence have additional structure, for example a grading or a bigrading. -To accomodate this (and to get around certain technical issues) +To accommodate this (and to get around certain technical issues) we introduce the following notion. Let $\mathcal{A}$ be an abelian category. Let $(T_r)_{r \geq 1}$ be a sequence of {\it translation} or {\it shift} functors, i.e., diff --git a/limits.tex b/limits.tex index d50fa91a3..3aaeb0c86 100644 --- a/limits.tex +++ b/limits.tex @@ -5648,7 +5648,7 @@ \section{Descending finite type schemes} $X^j = \lim_{i \geq i_j} X^j_i$ be the corresponding limit descriptions as in Lemma \ref{lemma-morphism-good-diagram}. Let $(a_i)_{i \geq i_5}$, $(b_i)_{i \geq i_6}$, $(p_i)_{i \geq i_7}$, and -$(q_i)_{i \geq i_8}$ be the corresponding morphisms of systems contructed +$(q_i)_{i \geq i_8}$ be the corresponding morphisms of systems constructed in Lemma \ref{lemma-morphism-good-diagram}. Then there exists an $i_9 \geq \max(i_5, i_6, i_7, i_8)$ such that for $i \geq i_9$ we have $a_i \circ p_i = b_i \circ q_i$ and such that diff --git a/local-cohomology.tex b/local-cohomology.tex index 5d9b213eb..787dbf6fe 100644 --- a/local-cohomology.tex +++ b/local-cohomology.tex @@ -3418,7 +3418,7 @@ \section{Hartshorne-Lichtenbaum vanishing} of $H^{-i}(\omega_A^\bullet)$ has dimension $\leq i$, see Lemma \ref{lemma-sitting-in-degrees}. Set $\omega_A = H^{-d}(\omega_A^\bullet)$. -By prime avoidence (Algebra, Lemma \ref{algebra-lemma-silly}) +By prime avoidance (Algebra, Lemma \ref{algebra-lemma-silly}) we can find $f \in A$, $f \not \in \mathfrak p_i$ which annihilates $H^{-i}(\omega_A^\bullet)$ for $i < d$. Consider the distinguished triangle diff --git a/models.tex b/models.tex index 7c623fcd2..a5e00cc4f 100644 --- a/models.tex +++ b/models.tex @@ -148,7 +148,7 @@ \section{Linear algebra} Thus equality holds everywhere. In particular, we see that $a_{rk} = 0$ if $r \in I$, $k \not \in I$ and equality holds in (\ref{equation-ineq}) for $r \in I$. Then we see that we -can substract a suitable multiple of $x_I$ from $x$ to decrease +can subtract a suitable multiple of $x_I$ from $x$ to decrease the number of nonzero coordinates. \end{proof} diff --git a/modules.tex b/modules.tex index e5d853a2b..917ad3064 100644 --- a/modules.tex +++ b/modules.tex @@ -2186,7 +2186,7 @@ \section{Bilinear maps} \mathcal{F}(U) \times \mathcal{G}(U) \to \mathcal{H}(U) $$ is an $\mathcal{O}_X(U)$-bilinear map of modules. Equivalently you can ask -certain diagrams of maps of sheaves of sets commute, immitating the usual +certain diagrams of maps of sheaves of sets commute, imitating the usual axioms for bilinear maps of modules. For example, the axiom $f(x + y, z) = f(x, z) + f(y, z)$ is represented by the commutativity of the diagram diff --git a/moduli-curves.tex b/moduli-curves.tex index 089b5abd9..096e33869 100644 --- a/moduli-curves.tex +++ b/moduli-curves.tex @@ -461,7 +461,7 @@ \section{Open substacks of the stack of curves} \noindent Below we will often characterize an open substack of $\Curvesstack$ -by a propery $P$ of morphisms of algebraic spaces. To see that $P$ +by a property $P$ of morphisms of algebraic spaces. To see that $P$ defines an open substack it suffices to check \begin{enumerate} \item[(o)] given a family of curves $f : X \to S$ there exists @@ -1873,7 +1873,7 @@ \section{The relative dualizing sheaf} \ref{duality-remark-relative-dualizing-complex}. Thus to prove a property of $\omega_{X/S}^\bullet$ which is \'etale local, we may assume $X \to S$ is a morphism of schemes -and use the theory developped in the chapter on duality for schemes. +and use the theory developed in the chapter on duality for schemes. More generally, for any base change of $X$ which is a scheme, the relative dualizing complex agrees with the relative dualizing complex of Duality for Schemes, Remark @@ -2494,7 +2494,7 @@ \section{Stable curves} Algebraic Curves, Lemmas \ref{curves-lemma-contracting-rational-tails} and \ref{curves-lemma-contracting-rational-bridges} -or because ampleness is insenstive to base field extensions by +or because ampleness is insensitive to base field extensions by Descent, Lemma \ref{descent-lemma-descending-property-ample}. \end{proof} @@ -2984,7 +2984,7 @@ \section{Contraction morphisms} \overline{\mathcal{M}}_g $$ which sends a prestable family of curves $X \to S$ of genus $g$ -to the stable family $Y \to S$ asssociated to it in +to the stable family $Y \to S$ associated to it in Lemma \ref{lemma-contract-prestable-to-stable}. \end{lemma} diff --git a/moduli.tex b/moduli.tex index bdba0fced..904f99a0a 100644 --- a/moduli.tex +++ b/moduli.tex @@ -587,7 +587,7 @@ \section{Properties of Quot} \ref{more-algebra-lemma-valuation-ring-torsion-free-flat}). Since $M$ is finite as an $A$-module so is $L$ and we conclude that $L$ is of finite presentation as an $A$-module -(by the reference above). Clearly $M/L$ is the unqiue such +(by the reference above). Clearly $M/L$ is the unique such quotient with $(M/L)_K = N_K$. \medskip\noindent diff --git a/more-algebra.tex b/more-algebra.tex index c07eea1eb..96602b114 100644 --- a/more-algebra.tex +++ b/more-algebra.tex @@ -1961,7 +1961,7 @@ \section{Lifting} where the image of $\overline{g}$, resp.\ $\overline{h}$ in $A_j/I_j$ has degree $j$, resp.\ $d - j$ with invertible leading coefficient. Clearly, it suffices to prove the result for each factor $A_j$ -separatedly. Hence the lemma follows from +separately. Hence the lemma follows from Lemma \ref{lemma-lift-factorization-easy}. \end{proof} @@ -2934,7 +2934,7 @@ \section{Henselian pairs} be a factorization with $g_0, h_0 \in A/I[T]$ monic generating the unit ideal in $A/I[T]$. Write $1 = g_0 g'_0 + h_0 h'_0$ for some $g'_0, h'_0 \in A/I[T]$. Since $A = \colim A_j$ and $A/I = \colim A_j/I_j$ -are filterd colimits we can find a $j \in J$ and $f_j \in A_j$ and +are filtered colimits we can find a $j \in J$ and $f_j \in A_j$ and a factorization $\overline{f}_j = g_{j, 0} h_{j, 0}$ with $g_{j, 0}, h_{j, 0} \in A_j/I_j[T]$ monic and $1 = g_{j, 0} g'_{j, 0} + h_{j, 0} h'_{j, 0}$ @@ -5920,7 +5920,7 @@ \section{Blowing up and flatness} \begin{lemma} \label{lemma-blowup-fitting-ideal-locally-free} Let $R$ be a ring. Let $M$ be a finite $R$-module. -Let $k \geq 0$ and $I = \text{Fit}_k(M)$. Asssume that +Let $k \geq 0$ and $I = \text{Fit}_k(M)$. Assume that $M_\mathfrak p$ is free of rank $k$ for every $\mathfrak p \not \in V(I)$. Then for every $a \in I$ with $R' = R[\frac{I}{a}]$ the strict transform @@ -7320,7 +7320,7 @@ \section{Koszul regular sequences} $(\overline{a}_{n + 1}, \ldots, \overline{a}_{n + m})$ is the image of some element $\overline{\alpha}$ of $\wedge^2(A/I^{\oplus m})$. We can lift $\overline{\alpha}$ to an element -$\alpha \in \wedge^2(A^{\oplus n + m})$ and substract the image of it +$\alpha \in \wedge^2(A^{\oplus n + m})$ and subtract the image of it in $A^{\oplus n + m}$ from our element $(a_1, \ldots, a_{n + m})$. Thus we may assume that $a_{n + 1}, \ldots, a_{n + m} \in I$. Since $I = (f_1, \ldots, f_n)$ we can modify our element @@ -15873,7 +15873,7 @@ \section{Pseudo-coherent modules, II} $$ in $D(R)$. Then $H^i(L) = 0$ for $i \geq t$. Moreover, the long exact sequence of $\Ext$ associated to this distinguished triangle -shows that $L$ inherts the assumption we made on $K$ by a small +shows that $L$ inherits the assumption we made on $K$ by a small argument we omit. By induction on $t$ we conclude that $L$ is $m$-pseudo-coherent. Hence $K$ is $m$-pseudo-coherent by Lemma \ref{lemma-cone-pseudo-coherent}. @@ -17076,7 +17076,7 @@ \section{Hom complexes} $$ It is a good idea to think of $\Hom^n$ as the $R$-module of all $R$-linear maps from $L^\bullet$ to $M^\bullet$ (viewed as graded modules) -which are homogenous of degree $n$. In this terminology, we define the +which are homogeneous of degree $n$. In this terminology, we define the differential by the rule $$ \text{d}(f) = \text{d}_M \circ f - (-1)^n f \circ \text{d}_L @@ -21841,7 +21841,7 @@ \section{Two term complexes} \begin{proof} Set $M = \Ker(H^{-1}(\alpha))$. Then $\alpha$ fits into a distinguished -trangle +triangle $$ M[1] \to K \to K' \to M[2] $$ @@ -23836,7 +23836,7 @@ \section{Formal glueing of module categories} in general). The category $\text{Glue}(R \to S, f_1, \ldots, f_t)$ has direct sums and cokernels. The functor $\text{Can}$ commutes with direct sums and cokernels. From these observations we find the displayed map is bijective -by writing $N$ as a cokernel of a map betweeen free $R$-modules. +by writing $N$ as a cokernel of a map between free $R$-modules. We omit the details. \end{proof} @@ -24069,7 +24069,7 @@ \section{Formal glueing of module categories} \noindent Next, we specialize Proposition \ref{proposition-equivalence} to get something -more useable. Namely, if $I = (f)$ is a principal ideal then the objects +more usable. Namely, if $I = (f)$ is a principal ideal then the objects of $\text{Glue}(R \to S, f)$ are simply triples $(M', M_1, \alpha_1)$ and there is {\it no} cocycle condition to check! @@ -29044,7 +29044,7 @@ \section{Miscellany} \sum\nolimits_\eta M_1(E_\eta, F_\eta)^\bullet $$ where the sum is over $\xi$ and $\eta$ as in the previous two paragraphs -and the sum is taken inside $M^\bullet$. Similiarly we set +and the sum is taken inside $M^\bullet$. Similarly we set $$ N_2^\bullet = \sum\nolimits_\xi N_1(E_\xi, F_\xi)^\bullet + @@ -29533,7 +29533,7 @@ \section{Weakly \'etale ring maps} Then $A$ is the product of its local rings, e.g., by Algebra, Proposition \ref{algebra-proposition-dimension-zero-ring}. Moreover, since $A$ is reduced, all local rings are equal -to their residue fields wich are finite separable over $K$. +to their residue fields which are finite separable over $K$. This means that $A$ is \'etale over $K$ by Algebra, Lemma \ref{algebra-lemma-etale-over-field} and finishes the proof. @@ -34743,7 +34743,7 @@ \section{Eliminating ramification, II} $$ Namely, $\varphi'$ is the unique $A_1$-algebra isomorphism sending the class of $x$ to the class of $x$. -Simiarly, $\psi'$ is the unique $B_1$-algebra isomorphism sending +Similarly, $\psi'$ is the unique $B_1$-algebra isomorphism sending the class of $x$ to the class of $x$. By our choice of $n$ we obtain, via Lemma \ref{lemma-construct-extension} and Remark \ref{remark-functoriality-construct-extension} @@ -35301,7 +35301,7 @@ \section{Determinants} \xymatrix{ \det(M') \otimes \det(M'') \ar[r] \ar[d]_{\epsilon \cdot (\text{switch tensors})} & -\det(M' \oplus M'') \ar[d]^{\det(\text{swith summands})} \\ +\det(M' \oplus M'') \ar[d]^{\det(\text{switch summands})} \\ \det(M'') \otimes \det(M') \ar[r] & \det(M'' \oplus M') } @@ -35543,7 +35543,7 @@ \section{Determinants of endomorphisms of finite length modules} \det\nolimits_\kappa(\varphi),\quad \text{Trace}_\kappa(\varphi) $$ as elements of $\kappa$ -using linear algebra. Simlarly for +using linear algebra. Similarly for the characteristic polynomial of $\varphi$ in this case. \medskip\noindent @@ -35611,7 +35611,7 @@ \section{Determinants of endomorphisms of finite length modules} $\text{length}_R(M') = \text{length}_{R'}(M') [\kappa' : \kappa]$. If $M' = \kappa'$, then the equality holds by definition of the norm as the determinant of the linear operator given by multiplication -by $u$. In general one reduces to this case by chosing a suitable +by $u$. In general one reduces to this case by choosing a suitable filtration and using the multiplicativity of Lemma \ref{lemma-ses}. Some details omitted. \end{proof} @@ -36176,7 +36176,7 @@ \section{Determinants of complexes} $b^\bullet$ and consider varying good diagrams for $a^\bullet$. Then the result of the previous paragraphs is that no matter what choices, the composition always equals the identity -map on $\det(L^\bullet)$. This clearly proves indepence of those choices. +map on $\det(L^\bullet)$. This clearly proves independence of those choices. \medskip\noindent The statement on canonical elements follows immediately from @@ -37597,7 +37597,7 @@ \section{Invertible objects in the derived category} $R = \prod_{a \leq n \leq b} R_n$ as in (d) such that $U_n$ corresponds to $\Spec(R_n)$, see Algebra, Lemma \ref{algebra-lemma-disjoint-implies-product}. -Since $D(R) = \prod_{a \leq n \leq b} D(R_n)$ and similary +Since $D(R) = \prod_{a \leq n \leq b} D(R_n)$ and similarly for the category of modules parts (b), (c), and (d) follow immediately. \end{proof} diff --git a/more-etale.tex b/more-etale.tex index f70633e01..78eaca31d 100644 --- a/more-etale.tex +++ b/more-etale.tex @@ -208,7 +208,7 @@ \section{Sections with compact support} \noindent An alternative would be to move this lemma later and prove this -using the descrition of the stalks of both sheaves. +using the description of the stalks of both sheaves. \begin{proof} The construction of $j_!\mathcal{F} \to j_*\mathcal{F}$ in the proof of @@ -522,7 +522,7 @@ \section{Sections with compact support} $j : X \to \overline{X}$ over $Y$. Set $\mathcal{G} = j_!\mathcal{F}$ so that $\mathcal{F} = \mathcal{G}|_X$ by \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-jshriek-open}. -By the disussion above we get a commutative diagram +By the discussion above we get a commutative diagram $$ \xymatrix{ (f_!\mathcal{F})_{\overline{y}} \ar[r] \ar[d] & @@ -701,7 +701,7 @@ \section{Sections with compact support} and the fact that the isomorphisms in Lemmas \ref{lemma-base-change-f-shriek-separated} and \ref{lemma-f-shriek-composition} -are constructed using the corresponding maps fo pushforwards. +are constructed using the corresponding maps of pushforwards. \end{remark} \begin{lemma} @@ -1121,7 +1121,7 @@ \section{Sections with finite support} the bottom horizontal arrow is constructed in the proof of Lemma \ref{lemma-finite-support-stalk}, the right vertical arrow is the obvious equality, and -the left veritical arrow is the map defined in the previous +the left vertical arrow is the map defined in the previous paragraph on stalks. The claim follows in a straightforward manner from the explicit description of all of the arrows involved here and in the references given. @@ -1948,7 +1948,7 @@ \section{Weightings and trace maps for locally quasi-finite morphisms} $$ The sum is locally finite on $V$ and hence makes sense; details omitted (in the whole discussion the reader may first choose affine opens and -make sure all the schemes occuring in the argument are quasi-compact so +make sure all the schemes occurring in the argument are quasi-compact so the sum is finite). We omit the verification that this construction is compatible with the maps in the colimit and with the restriction mappings defining $f_{p!}\underline{\mathbf{Z}}$. @@ -4307,7 +4307,7 @@ \section{A constructibility result} Assume $S$ is Noetherian. We may write $S = U \cup Z$ where $U$ is the open subscheme defined by the nonvanishing of $\ell$ and $Z = V(\ell) \subset S$. Since the formation of -$R^qf_*\underline{\mathbf{Z}/\ell\mathbf{Z}}$ commutes with arbtrary +$R^qf_*\underline{\mathbf{Z}/\ell\mathbf{Z}}$ commutes with arbitrary base change (\'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-proper-base-change}), it suffices to prove the result over $U$ and over $Z$. diff --git a/more-morphisms.tex b/more-morphisms.tex index cb5ebc88f..30b795309 100644 --- a/more-morphisms.tex +++ b/more-morphisms.tex @@ -4072,7 +4072,7 @@ \section{Pushouts in the category of schemes, I} m'_*\mathcal{O}_T \times_{m_*\mathcal{O}_T} n_*\mathcal{O}_S $$ given by $(m')^\sharp$ and $n^\sharp$. -Thus $(n', (n')^\sharp)$ is the unque morphism of +Thus $(n', (n')^\sharp)$ is the unique morphism of ringed spaces $T' \to X$ compatible with $m'$ and $n$. To finish the proof it suffices to show that $n'$ is a morphism of schemes, i.e., a morphism of locally ringed spaces. @@ -4863,7 +4863,7 @@ \section{Closed immersions between smooth schemes} \quad\text{with}\quad A = R[x_1, \ldots, x_m, y_1, \ldots, y_n] $$ -Then $X'$ is the Proj of the Rees algebra of $A$ with respect ot +Then $X'$ is the Proj of the Rees algebra of $A$ with respect to the ideal $(y_1, \ldots, y_n)$. This Rees algebra is equal to $B = A[T_1, \ldots, T_n]/(y_iT_j - y_jT_i)$; details omitted. Hence $X' = \text{Proj}(B)$ is smooth over $S$ as it is @@ -8406,7 +8406,7 @@ \section{Dimension of fibres} $y$ is in the image of $U_n \to Y$. By Morphisms, Lemma \ref{morphisms-lemma-fppf-open} we see that $f(U_n)$ is open in $Y$. -Hence there is an open neighbourhoof of $y$ where +Hence there is an open neighbourhood of $y$ where $n_{X/Y}$ is $\geq n$. \end{proof} @@ -8863,7 +8863,7 @@ \section{Bertini theorems} $\xi = (a_2 \xi_1 - a_1 \xi_2)/(a_2 - a_1)$, and $\delta = (\xi_1 - \xi_2)/(a_1 - a_2)$. Since $K_2/K_0$ is purely transcendental of degree $2$ we conclude -that $\xi_1$ and $\xi_2$ are algebraically indepedent over $K_0$. +that $\xi_1$ and $\xi_2$ are algebraically independent over $K_0$. Since $\theta_1$ is algebraic over $K_0(\xi_1)$ we conclude that $\xi_2$ is transcendental over $K_0(\xi_1, \theta_1)$. @@ -9169,7 +9169,7 @@ \section{Theorem of the cube} (small detail omitted). \medskip\noindent -In this sitation if $g : T \to S$ be a morphism and there exists an +In this situation if $g : T \to S$ be a morphism and there exists an $\mathcal{N}$ as in the lemma, then the map (cup product in degree $0$) $$ p_*(q^*\mathcal{E}) @@ -13206,7 +13206,7 @@ \section{Application to morphisms with connected fibres} Denote $x' \in (S' \times_S X)_{s'} = X_{k'}$ the image of $\overline{x}$. Thus after replacing $(S, s)$ by $(S', s')$ and $(X, x)$ by $(S' \times_S X, x')$ we reduce to the case handled in the next -paragrah. +paragraph. \medskip\noindent Assume there is a quasi-compact open $V \subset X_s$ @@ -14096,7 +14096,7 @@ \section{Projective schemes} is ample on $X_{i, y_i}$ by Limits, Lemma \ref{limits-lemma-limit-ample}. By Cohomology of Schemes, Lemma \ref{coherent-lemma-ample-in-neighbourhood} -we find an open neigbourhood +we find an open neighbourhood $V_i \subset Y_i$ of $y_i$ such that $\mathcal{L}_i$ restricted to $f_i^{-1}(V_i)$ is ample relative to $V_i$. @@ -14207,7 +14207,7 @@ \section{Proj and Spec} Taking $m' = 0$ we see that $\pi_*\mathcal{O}_L$ is the direct sum of $\mathcal{O}_P(m)$ for $m \geq 0$\footnote{It similarly follows that $\pi_*\mathcal{O}_L(i) = \bigoplus_{m \geq -i} \mathcal{O}_P(m)$.}. -We conclude $L$ is idendified with the relative spectrum: +We conclude $L$ is identified with the relative spectrum: $$ L = \underline{\Spec}_P \left( @@ -15602,7 +15602,7 @@ \section{Generic flatness stratification} A_g/A = \colim (1/g^m)A/A \cong \colim A/g^mA $$ and the module $A/g^mA$ has a filtration of length $m$ whose -succesive quotients are isomorphic to $A/gA$. Again by generic +successive quotients are isomorphic to $A/gA$. Again by generic flatness we may assume $A/gA$ is $R$-flat and hence each $A/g^mA$ is $R$-flat, and hence so is $A_g/A$. \end{proof} @@ -17050,7 +17050,7 @@ \section{Relative pseudo-coherence} \medskip\noindent Since $j$ is a closed immersion, the functor $j_*$ is exact and -$Rj_*$ is computed by applying $j_*$ to any representating complex +$Rj_*$ is computed by applying $j_*$ to any representing complex of sheaves. Thus we have to show that $j_*\text{Tot}(p^*\mathcal{E}^\bullet \otimes_{\mathcal{O}_{\mathbf{A}_Y^n}} \mathcal{F}^\bullet)$ is $m$-pseudo-coherent @@ -20238,7 +20238,7 @@ \section{Characterizing pseudo-coherent complexes, III} Derived Categories of Schemes, Lemma \ref{perfect-lemma-compare-base-change}. This also uses that $R(b', c')_*\mathcal{O}_{Z'} = (b', c')_*\mathcal{O}_{Z'}$ -because $(b', c')$ is a closed immersion and simiarly for +because $(b', c')$ is a closed immersion and similarly for $(b, c)_*\mathcal{O}_Z$. Since $E'|_{U' \times_{A'} V'} = (b', c')_*\mathcal{O}_{W'}$ we obtain @@ -21383,7 +21383,7 @@ \section{Affine stratifications} to prove the assertion that $X$ has a finite affine stratification. If $X$ is empty, then it has an empty affine stratification. If $X$ is nonempty affine then it has an affine stratification -with one stratum. Next, asssume $X = U \cup V$ where $U$ is quasi-compact open, +with one stratum. Next, assume $X = U \cup V$ where $U$ is quasi-compact open, $V$ is affine open, and we have a finite affine stratifications $U = \bigcup_{i \in I} U_i$ and $U \cap V = \coprod_{j \in J} W_j$. Denote $Z = X \setminus V$ and $Z' = X \setminus U$. @@ -22318,7 +22318,7 @@ \section{More on weightings} \label{section-more-weightings} \noindent -We prove a few more basic properties of weightings. Allthough +We prove a few more basic properties of weightings. Although at first it appears that weightings can be very wild, it actually turns out the condition imposed in Definition \ref{definition-weighting} is rather strong. @@ -22585,7 +22585,7 @@ \section{More on weightings} $$ where $I$ is the ideal generated by the coefficients in $A[z_1, \ldots, z_n, y_1, \ldots, y_m]$ -of the basis elements $b'_1, \ldots, b'_N$ of the expresssion +of the basis elements $b'_1, \ldots, b'_N$ of the expression $$ (\sum z_j b'_j)^2 - \sum z_j b'_j + \sum y_k f_k $$ @@ -22738,7 +22738,7 @@ \section{Weightings and affine stratification numbers} \label{lemma-affineness-of-large-open} Let $f : X \to Y$ be a morphism of affine schemes which is quasi-finite and of finite presentation. -Let $w : X \to \mathbf{Z}_{> 0}$ be a postive weighting of $f$. +Let $w : X \to \mathbf{Z}_{> 0}$ be a positive weighting of $f$. Let $d < \infty$ be the maximum value of $\int_f w$. The open $$ Y_d = \{y \in Y \mid (\textstyle{\int}_f w)(y) = d \} diff --git a/obsolete.tex b/obsolete.tex index 5b95d90bb..021c93f6f 100644 --- a/obsolete.tex +++ b/obsolete.tex @@ -795,7 +795,7 @@ \section{Lemmas related to ZMT} elements $h/f^n \in (S'')_f$ which map to zero in $S'$ but are not zero. In this case $\beta(h)$ is an element of $S$ such that $f^N \beta(h) = 0$ -for some $N$. Thus $f^N h$ is an element ot the ideal +for some $N$. Thus $f^N h$ is an element of the ideal $J = \{h \in S'' \mid \alpha(h) = 0 \text{ and } \beta(h) = 0\}$ of $S''$. OK, and it is easy to see that $S''/J$ does the job. diff --git a/perfect.tex b/perfect.tex index 90b5390b6..48f07cbb8 100644 --- a/perfect.tex +++ b/perfect.tex @@ -1194,7 +1194,7 @@ \section{Cohomology with support in a closed subset} $$ where $\lambda(i_0 \ldots i_p, j_0 \ldots j_{c - p - 2})$ is the coefficient of $e_1 \wedge \ldots \wedge e_c$ in -the formal expresssion +the formal expression $$ e_{i_0} \wedge \ldots \wedge e_{i_p} \wedge (a_{j_01} e_1 + \ldots + a_{j_0c}e_c) \wedge \ldots \wedge @@ -1268,7 +1268,7 @@ \section{Cohomology with support in a closed subset} (i')^*\mathcal{F}' \ar[u]^-{c_{f'_1, \ldots, f'_c}} } $$ -is commutative where the top horizonal arrow is +is commutative where the top horizontal arrow is the map of Cohomology, Remark \ref{cohomology-remark-support-functorial} on cohomology sheaves in degree $c$. Namely, denote $\mathcal{F}^\bullet$, resp.\ $(\mathcal{F}')^\bullet$ @@ -1295,7 +1295,7 @@ \section{Cohomology with support in a closed subset} as is the explanation for the equal sign. The arrows pointing up are from Remark \ref{remark-extended-alternating-map-to-support}. -The lower horizonal arrow is the map $Lg^*\mathcal{F}^\bullet +The lower horizontal arrow is the map $Lg^*\mathcal{F}^\bullet \to g^*\mathcal{F}^\bullet = (\mathcal{F}')^\bullet$ and the arrow pointing down is induced by $Lg^*\mathcal{F} \to g^*\mathcal{F} = \mathcal{F}'$. @@ -1708,7 +1708,7 @@ \section{The coherator} of the left exact functor $\Gamma(U, -)$. Since this is a universal $\delta$-functor, and since the functors $H^i(U, -)$ (defined for all abelian sheaves on $X$) restricted to $\QCoh(\mathcal{O}_X)$ form -a $\delta$-functor, we obtain canonical tranformations +a $\delta$-functor, we obtain canonical transformations $$ t^i : R^i(\QCoh)\Gamma(U, -) \to H^i(U, -). $$ @@ -3638,7 +3638,7 @@ \section{Approximation by perfect complexes} quasi-compact and quasi-separated schemes. It seems that the second condition is necessary for our method of proof. It is possible that the first condition may be weakened to ``$E$ is $m$-pseudo-coherent'' -by carefuly analyzing the arguments below. +by carefully analyzing the arguments below. \begin{lemma} \label{lemma-open} @@ -3960,7 +3960,7 @@ \section{Generating derived categories} supported on $T$ (Definition \ref{definition-supported-on}). We similarly denote $D_{\QCoh, T}(\mathcal{O}_X)$ the strictly full, saturated, triangulated subcategory of $D(\mathcal{O}_X)$ consisting of those -complexes whose cohomology sheaves are quasi-coherent and are suppported +complexes whose cohomology sheaves are quasi-coherent and are supported on $T$. \begin{lemma} @@ -4974,7 +4974,7 @@ \section{The coherator revisited} See Lemmas \ref{lemma-affine-compare-bounded} and \ref{lemma-affine-coherator}. \medskip\noindent -Asssume $U, V$ are quasi-compact open in $X$ +Assume $U, V$ are quasi-compact open in $X$ and the lemma holds for $U$, $V$, and $U \cap V$. Say with integers $N(U)$, $N(V)$, and $N(U \cap V)$. Now suppose $K$ is in $D(\mathcal{O}_X)$ with @@ -4982,7 +4982,7 @@ \section{The coherator revisited} Then $K|_U$, $K|_V$, $K|_{U \cap V}$ have the same property. Hence we see that $RQ_U(K|_U)$ and $RQ_V(K|_V)$ and $RQ_{U \cap V}(K|_{U \cap V})$ have vanishing cohomology -sheaves outside the inverval $[a, b + \max(N(U), N(V), N(U \cap V))$. +sheaves outside the interval $[a, b + \max(N(U), N(V), N(U \cap V))$. Since the functors $Rj_{U, *}$, $Rj_{V, *}$, $Rj_{U \cap V, *}$ have finite cohomological dimension on $D_\QCoh$ by Lemma \ref{lemma-quasi-coherence-direct-image} @@ -8067,7 +8067,7 @@ \section{Applications} (Lemma \ref{lemma-flat-proper-perfect-direct-image-general}) the complex $E = Rf_*\mathcal{O}_X$ is perfect and its formation commutes with arbitrary base change. -This first implies that $E$ has tor aplitude in $[0, \infty)$. +This first implies that $E$ has tor amplitude in $[0, \infty)$. Second, it implies that for $s \in S$ we have $H^0(E \otimes^\mathbf{L} \kappa(s)) = H^0(X_s, \mathcal{O}_{X_s}) = \kappa(s)$. @@ -8201,7 +8201,7 @@ \section{Applications} (Lemma \ref{lemma-flat-proper-perfect-direct-image-general}) the complex $E = Rf_*\mathcal{O}_X$ is perfect and its formation commutes with arbitrary base change. -This first implies that $E$ has tor aplitude in $[0, \infty)$. +This first implies that $E$ has tor amplitude in $[0, \infty)$. \medskip\noindent We claim that after replacing $S$ by an open neighbourhood of @@ -8330,7 +8330,7 @@ \section{Other applications} \medskip\noindent To finish the proof it suffices to show: if $U = V \cup W$ and the result holds for $V$, $W$, and $V \cap W$, then -the result holds for $U$. This is an immediate consquence +the result holds for $U$. This is an immediate consequence of the Mayer-Vietoris sequence, see Cohomology, Lemma \ref{cohomology-lemma-unbounded-mayer-vietoris}. \end{proof} @@ -9237,7 +9237,7 @@ \section{The resolution property} \end{lemma} \begin{proof} -Immediate consquence of Properties, Proposition +Immediate consequence of Properties, Proposition \ref{properties-proposition-characterize-ample}. \end{proof} @@ -10545,7 +10545,7 @@ \section{Detecting Boundedness} $R\Hom_X(P, E)$ is in $D^-(\mathbf{Z})$. Clearly, $T$ is inherited by direct sums, satisfies the 2-out-of-three property for distinguished -triangles, is inherited by direct summands, and is perserved by shifts. +triangles, is inherited by direct summands, and is preserved by shifts. Hence by Derived Categories, Remark \ref{derived-remark-check-on-generator} we see that (4) implies $T$ holds on all of $D_{perf}(\mathcal{O}_X)$. The same argument works for all other properties, except that for property @@ -10663,7 +10663,7 @@ \section{Detecting Boundedness} $R\Hom_X(P, E)$ is in $D^+(\mathbf{Z})$. Clearly, $T$ is inherited by direct sums, satisfies the 2-out-of-three property for distinguished -triangles, is inherited by direct summands, and is perserved by shifts. +triangles, is inherited by direct summands, and is preserved by shifts. Hence by Derived Categories, Remark \ref{derived-remark-check-on-generator} we see that (4) implies $T$ holds on all of $D_{perf}(\mathcal{O}_X)$. The same argument works for all other properties, except that for property @@ -10782,7 +10782,7 @@ \section{Quasi-coherent objects in the derived category} \begin{lemma} \label{lemma-DQCoh-alternative-small} -In the sitation above there are canonical exact equivalences between +In the situation above there are canonical exact equivalences between the following triangulated categories \begin{enumerate} \item $D_\QCoh(\mathcal{O}_X)$, diff --git a/pic.tex b/pic.tex index 1ed432ce8..81a7bd570 100644 --- a/pic.tex +++ b/pic.tex @@ -246,7 +246,7 @@ \section{Hilbert scheme of points} Representable, i.e., condition (2)(a) of the lemma. Let $W \subset A$ have cardinality $d$. We claim that $F_W$ is representable by an affine scheme over $R$. We will construct this affine scheme here, but we encourage -the reader to think it trough for themselves. Choose a numbering +the reader to think it through for themselves. Choose a numbering $f_1, \ldots, f_d$ of the elements of $W$. We will construct a universal element $Z_{univ} = \Spec(B_{univ})$ of $F_W$ over $T_{univ} = \Spec(R_{univ})$ which will be the spectrum of diff --git a/pione.tex b/pione.tex index 08d81004c..38a6b9dfa 100644 --- a/pione.tex +++ b/pione.tex @@ -1347,7 +1347,7 @@ \section{Galois covers of connected schemes} Since $F_{\overline{x}} : \textit{F\'Et}_X \to \textit{Sets}$ is a Galois category (Lemma \ref{lemma-finite-etale-connected-galois-category}) the material in Section \ref{section-galois} applies. -In this section we explicity transfer some of the terminology +In this section we explicitly transfer some of the terminology and results to the setting of schemes and finite \'etale morphisms. \medskip\noindent @@ -4046,7 +4046,7 @@ \section{Restriction to a closed subscheme} recovers $U_1 \to Y_1$. Consider the sheaves $\mathcal{F}_n = \pi_{n, *}\mathcal{O}_{U_n}$. We may and do view $\mathcal{F}_n$ as an $\mathcal{O}_X$-module on $X$ -wich is locally isomorphic to +which is locally isomorphic to $(\mathcal{O}_X/f^{n + 1}\mathcal{O}_X)^{\oplus r}$. This $(\mathcal{F}_n)$ is a finite locally free object of $\textit{Coh}(X, \mathcal{I})$. @@ -4218,7 +4218,7 @@ \section{Restriction to a closed subscheme} recovers $U_1 \to Y_1$. Consider the sheaves $\mathcal{F}_n = \pi_{n, *}\mathcal{O}_{U_n}$. We may and do view $\mathcal{F}_n$ as an $\mathcal{O}_X$-module on $X$ -wich is locally isomorphic to +which is locally isomorphic to $(\mathcal{O}_X/f^{n + 1}\mathcal{O}_X)^{\oplus r}$. This $(\mathcal{F}_n)$ is a finite locally free object of $\textit{Coh}(X, \mathcal{I})$. diff --git a/preamble.tex b/preamble.tex index cf8dc13de..02d79de01 100644 --- a/preamble.tex +++ b/preamble.tex @@ -21,7 +21,7 @@ % We use multicol for the list of chapters between chapters \usepackage{multicol} -% This is generall recommended for better output +% This is generally recommended for better output \usepackage{lmodern} \usepackage[T1]{fontenc} diff --git a/proetale.tex b/proetale.tex index b6dda9eed..8db7098b6 100644 --- a/proetale.tex +++ b/proetale.tex @@ -3298,7 +3298,7 @@ \section{Comparing topologies} u_p\mathcal{F} : U \mapsto \colim_{U \to V} \mathcal{F}(V) $$ on $X_\etale$, see Sites, Sections \ref{sites-section-morphism-sites} and -\ref{sites-section-continuous-functors}; we've surpressed from the notation +\ref{sites-section-continuous-functors}; we've suppressed from the notation that the colimit is over the opposite of the category $\{U \to V, V \in Y_\proetale\}$. By Lemma \ref{lemma-presheaf-value-weakly-contractible} diff --git a/properties.tex b/properties.tex index ef2bc5693..c0c9eaf1e 100644 --- a/properties.tex +++ b/properties.tex @@ -2860,7 +2860,7 @@ \section{Extending quasi-coherent sheaves} \begin{proof} The beginning of the proof is a repeat of the beginning of the -proof of Lemma \ref{lemma-extend}. We write it out carefuly anyway. +proof of Lemma \ref{lemma-extend}. We write it out carefully anyway. \medskip\noindent Let $n$ be the minimal number of affine opens $U_i \subset X$, diff --git a/quot.tex b/quot.tex index 63d24b761..5f4e4f16e 100644 --- a/quot.tex +++ b/quot.tex @@ -81,7 +81,7 @@ \section{Introduction} parametrizing flat families of proper algebraic spaces satisfies all of Artin's axioms (including openness of versality) except for formal effectiveness. We've chosen the very awkward notation for -this stack intentionally, because the reader should be carefull +this stack intentionally, because the reader should be careful in using its properties. \medskip\noindent diff --git a/relative-cycles.tex b/relative-cycles.tex index ad989fe2e..3557a5c4e 100644 --- a/relative-cycles.tex +++ b/relative-cycles.tex @@ -212,7 +212,7 @@ \section{Specialization of cycles} \ref{more-morphisms-lemma-relative-dimension-support-flat}. Let $x$ be a generic point of an integral closed subscheme $Z \subset X_\kappa$ of dimension $r$. To finish the proof -we wil show that the coefficient of $[Z]$ +we will show that the coefficient of $[Z]$ in the left (L) and right hand side (R) of equality are the same. \medskip\noindent @@ -671,7 +671,7 @@ \section{Families of cycles on fibres} $$ It follows from the associativity of base change that we have $\text{pr}_1^* \circ g^* = \text{pr}_2^* \circ g^*$ (small detail -omitted). To be explicity, in terms of the maps of sets this +omitted). To be explicitly, in terms of the maps of sets this equality just means that for $x'' \in (X''/S'')^{(r)}$ we have $$ m(x'', \text{pr}_1) m(\text{pr}_1(x''), g) = @@ -920,7 +920,7 @@ \section{Relative cycles} $U \subset X$ induces an isomorphism on the groups of $r$-cycles on fibres after any base change, compatible with specialization maps and with formation of the relative cycle associated to $\mathcal{F}$. -Thus it suffices to show compability with +Thus it suffices to show compatibility with specializations for $[\mathcal{F}|_U / U /S]_r$. Since $\mathcal{F}|_U$ is flat over $S$, this follows from Lemma \ref{lemma-specialization-module} and the definitions. @@ -1157,7 +1157,7 @@ \section{Relative cycles} $E' \subset E'' = \{s \in S: \gamma''_s = 0\}$ and $s \not \in E''$ because the coefficient of $Z''$ in $\gamma''_s$ is nonzero, where $Z'' \subset Y_s$ is the closure of $y$. By -Lemma \ref{lemma-relative-cycle-smooth} and openess of $Y \to S$ +Lemma \ref{lemma-relative-cycle-smooth} and openness of $Y \to S$ we see that an open neighbourhood of $s$ is disjoint from $E''$ and the proof is complete. \end{proof} @@ -1297,7 +1297,7 @@ \section{Relative cycles} $h : \Spec(R) \to S$ factors as $g \circ h'$ for some unique morphism $h' : \Spec(R) \to S'$. Namely, the morphism $S' \times_S \Spec(R) \to \Spec(R)$ is -a univeral homomorphism inducing bijections on residue fields, +a universal homomorphism inducing bijections on residue fields, and hence has a section (for example because $R$ is a seminormal ring, see Morphisms, Section \ref{morphisms-section-seminormalization}). Thus the condition that $\alpha$ is compatible with @@ -1349,7 +1349,7 @@ \section{Equidimensional relative cycles} the idea is that $a/b$ and $(a + b^2)/b = a/b + b$ limit to the same point in $\mathbf{P}^1$ over the residue field of any valuation $v$ on $\kappa(s)$ with $v(b) > 0$. On the other hand, the closure of the support of $\alpha$ -containes the whole fibre over $(0, 0)$. +contains the whole fibre over $(0, 0)$. \end{example} \begin{lemma} @@ -1760,7 +1760,7 @@ \section{Effective relative cycles} the characteristic of $k$ is $2$. Looking at the generic point $\eta$ of $V$, the cycle $\beta_\eta$ would be a zero cycle of degree $2$ on $\mathbf{A}^2_{k(u_1^2, v_1^2)}$ -whose pullback to $\mathbf{A}^2_{k(u_1, u_2)}$ whould be +whose pullback to $\mathbf{A}^2_{k(u_1, u_2)}$ would be $2[\text{the point with coordinates} (u_1, v_2)]$. This is clearly impossible. @@ -1963,7 +1963,7 @@ \section{Action on cycles} \end{lemma} \begin{proof} -Immediate from the explict description of $\alpha \cap \beta$ +Immediate from the explicit description of $\alpha \cap \beta$ given above. \end{proof} @@ -2059,7 +2059,7 @@ \section{Action on cycles} \medskip\noindent The statement in the lemma on flat pullback means the following. Let $(S, \delta)$, $X \to Y$, $\alpha$, and $\beta$ be as in the -constuction of $\alpha \cap \beta$ above. Let $Y' \to Y$ be a flat +construction of $\alpha \cap \beta$ above. Let $Y' \to Y$ be a flat morphism, locally of finite type, and of relative dimension $c$. Then we can let $\alpha'$ be the base change of $\alpha$ by $Y' \to Y$ and $\beta'$ the flat pullback of $\beta$. Compatibility with flat @@ -2120,7 +2120,7 @@ \section{Action on cycles} \text{length}_{\mathcal{O}_{Y, y_i}}(\mathcal{G}_{y_i}) $$ By the first paragraph of the proof the left hand side is equal to -the lenth of the $B = \mathcal{O}_{X, \xi_{ij}}$-module +the length of the $B = \mathcal{O}_{X, \xi_{ij}}$-module $$ \mathcal{G}_{y_i} \otimes_{\mathcal{O}_{Y, y_i}} @@ -2788,7 +2788,7 @@ \section{Comparison with Suslin and Voevodsky} \item For $X \to Y \to S$ the composition law $z(X/Y, r) \otimes_\mathbf{Z} z(Y/S, e) \longrightarrow z(X/S, r + e)$ defined in Section \ref{section-compose} -agrees with the opration $Cor_{X/Y}(-, -)$ in +agrees with the operation $Cor_{X/Y}(-, -)$ in \cite[Corollary 3.7.5]{SV}. \end{enumerate} diff --git a/restricted.tex b/restricted.tex index 97412ef01..bf4e0d856 100644 --- a/restricted.tex +++ b/restricted.tex @@ -502,7 +502,7 @@ \section{A naive cotangent complex} The induced arrow $J_1/J_1^2 \otimes_{B_1} B_2 \to J_2/J_2^2$ is surjective because $J_2$ is generated by the image of $J_1$. This determines the arrow displayed in the lemma. We omit the proof -that this arrow is well defined up to homotopy (i.e., indepedent +that this arrow is well defined up to homotopy (i.e., independent of the choice of the presentations up to homotopy). The statement about the induced map on cohomology modules follows easily from the discussion (details omitted). @@ -1648,7 +1648,7 @@ \section{Rig-\'etale algebras} \end{enumerate} \item \label{item-condition-artin} -choosing generaters $f_1, \ldots, f_t$ for $J$ we have +choosing generators $f_1, \ldots, f_t$ for $J$ we have \begin{enumerate} \item the Jacobian ideal of $B$ over $A$, namely the ideal in $B$ generated by the $r \times r$ minors of the matrx @@ -3104,7 +3104,7 @@ \section{Rig-closed points} \noindent We will need a few lemmas which essentially tell us there are -plenty of rig-closed primes even in a relative settting. +plenty of rig-closed primes even in a relative setting. \begin{lemma} \label{lemma-rig-closed-point-relative-residue-field} @@ -5003,7 +5003,7 @@ \section{Rig-\'etale morphisms} Proof of (3) $\Rightarrow$ (2). Assume (3). By Lemma \ref{lemma-rig-etale-morphisms} it suffices to show that the ring maps -$B \to A$ occuring in (3) are rig-\'etale in the +$B \to A$ occurring in (3) are rig-\'etale in the sense of Definition \ref{definition-rig-etale-continuous-homomorphism}. Let $I$ be as in (3). The naive cotangent complex $\NL_{A/B}^\wedge$ of $A$ over $(B, I)$ is the complex of $A$-modules @@ -6385,7 +6385,7 @@ \section{Completions and morphisms, I} Spaces, Lemma \ref{spaces-lemma-glueing-algebraic-spaces}). The two projections $U' \times_U U' \to U'$ extend to $R$ and we obtain two \'etale morphisms $s, t : R \to X'$. -Checking on each piece separatedly we find that $R$ +Checking on each piece separately we find that $R$ is an \'etale equivalence relation on $X'$. Set $X'' = X'/R$ which is an algebraic space by Bootstrap, Theorem \ref{bootstrap-theorem-final-bootstrap}. @@ -6456,7 +6456,7 @@ \section{Rig glueing of morphisms} The assumptions on $T$ and $f$ are preserved by base change by any \'etale morphism $X_1 \to X$ of algebraic spaces. Since formal algebraic spaces are sheaves for the -\'etale topology and since we aready have the uniqueness, +\'etale topology and since we already have the uniqueness, it suffices to prove existence after replacing $X$ by the members of an \'etale covering. Thus we may assume $X$ is an affine Noetherian scheme. @@ -6611,7 +6611,7 @@ \section{Algebraization of rig-\'etale morphisms} \mathcal{C}_{X_{1, /T_1}} } $$ -is commutative where the horizonal arrows are given by +is commutative where the horizontal arrows are given by (\ref{equation-completion-functor}) and the vertical arrows by base change along $X_1 \to X$ and along $X_{1, /T_1} \to X_{/T}$. @@ -7192,7 +7192,7 @@ \section{Artin's theorem on dilatations} \end{proof} \noindent -Here is the characterization of formal modifcations +Here is the characterization of formal modifications as promised in Section \ref{section-formal-modifications}. \begin{lemma} diff --git a/scripts/functions.py b/scripts/functions.py index 4c4353c09..c43986fa8 100644 --- a/scripts/functions.py +++ b/scripts/functions.py @@ -33,7 +33,7 @@ def find_defined_terms(def_text): n = def_text.find("{\\it ", m) return def_terms -# Spits out the git version of the git repositor in the path +# Spits out the git version of the git repository in the path def git_version(path): from subprocess import Popen, PIPE, STDOUT cmd = 'git --git-dir=' + path + '.git log --pretty=format:%h -n1' diff --git a/scripts/latex.sh b/scripts/latex.sh index d4fe4d2d4..6d2404395 100755 --- a/scripts/latex.sh +++ b/scripts/latex.sh @@ -38,7 +38,7 @@ cp $STEM.toc $TMPD cd $TMPD $LATEX $STEM.tex -# Move newly created files back to stacks project directoy +# Move newly created files back to stacks project directory [ -f $STEM.pdf ] && mv $STEM.pdf $SPD [ -f $STEM.dvi ] && mv $STEM.dvi $SPD [ -f $STEM.aux ] && mv $STEM.aux $SPD diff --git a/scripts/second-homotopy.gp b/scripts/second-homotopy.gp index 0060ad356..6648b0336 100644 --- a/scripts/second-homotopy.gp +++ b/scripts/second-homotopy.gp @@ -197,7 +197,7 @@ while(i, if(test(m) <> [], error("Did not work for m = ", m)); j = 1; while((j <= d - 1) && (bits[j] == 1), j = j + 1); - if(j == d, print("Succes!"); return); + if(j == d, print("Success!"); return); bits[j] = 1; j = j - 1; while(j >= 1, bits[j] = 0; j = j - 1); diff --git a/scripts/web_book.py b/scripts/web_book.py index 3c6cfc1d4..8041fee06 100644 --- a/scripts/web_book.py +++ b/scripts/web_book.py @@ -5,7 +5,7 @@ # documentclass book # load amsmath package for plastex # Ignore reference, slogan, history environments -# Do not bother with multicol and xr-hyper pacakges +# Do not bother with multicol and xr-hyper packages def print_preamble(path): preamble = open(path + "preamble.tex", 'r') next(preamble) diff --git a/sdga.tex b/sdga.tex index 2309f7e16..df5f23be3 100644 --- a/sdga.tex +++ b/sdga.tex @@ -766,7 +766,7 @@ \section{Pull and push for sheaves of graded modules} \textit{Mod}^{gr}(\mathcal{A}) $$ which do the same thing on underlying objects and are defined -by functoriality of the constructions on homogenous morphisms +by functoriality of the constructions on homogeneous morphisms of degree $n$. \begin{lemma} @@ -1993,7 +1993,7 @@ \section{Pull and push for sheaves of differential graded modules} \textit{Mod}^{dg}(\mathcal{A}, \text{d}) $$ which do the same thing on underlying objects and are defined -by functoriality of the constructions on homogenous morphisms +by functoriality of the constructions on homogeneous morphisms of degree $n$. \begin{lemma} @@ -2493,7 +2493,7 @@ \section{Cones and triangles} $$ associated to this (see discussion above) is equal to $f[1]$. For the section $s : \mathcal{K}[1] \to C(f)$ we use in degree -$n$ the embeddding $\mathcal{K}^{n + 1} \to C(f)^n$. Then +$n$ the embedding $\mathcal{K}^{n + 1} \to C(f)^n$. Then in degree $n$ the map $\pi$ is given by the projections $C(f)^n \to \mathcal{L}^n$. Then finally we have to compute $$ @@ -5471,7 +5471,7 @@ \section{Resolutions of differential graded algebras} $$ The weight of any section of $\mathcal{S} \times \ldots \times \mathcal{S}$ is defined locally. The reader checks easily that we obtain a disjoint union -decompostion +decomposition $$ \mathcal{S} \times \ldots \times \mathcal{S} = \coprod\nolimits_{w \in W} \left( @@ -6126,7 +6126,7 @@ \section{Differential graded modules on a category, bis} $$ \mathcal{E} \to \mathcal{M} $$ -wich over $U$ in $\mathcal{C}$ is given by the map +which over $U$ in $\mathcal{C}$ is given by the map $$ \mathcal{E}(U) \longrightarrow diff --git a/sheaves.tex b/sheaves.tex index 31e957054..33380be2e 100644 --- a/sheaves.tex +++ b/sheaves.tex @@ -3510,7 +3510,7 @@ \section{Limits and colimits of sheaves} $i_{jj'} \geq i_j$, $i_{jj'} \geq i_{j'}$ such that $\varphi_{i_ji_{jj'}}(s_j)$ and $\varphi_{i_{j'}i_{jj'}}(s_{j'})$ agree over $U_j \cap U_{j'}$. Choose an index $i \in I$ -wich is bigger or equal than all the $i_{jj'}$. Then we see that +which is bigger or equal than all the $i_{jj'}$. Then we see that the sections $\varphi_{i_ji}(s_j)$ of $\mathcal{F}_i$ glue to a section of $\mathcal{F}_i$ over $U$. This section is mapped to the element $s$ as desired. diff --git a/simplicial.tex b/simplicial.tex index 25183afe6..7286d7a7d 100644 --- a/simplicial.tex +++ b/simplicial.tex @@ -1277,7 +1277,7 @@ \section{Hom from cosimplicial sets into simplicial objects} \longrightarrow \Mor_\mathcal{C}(V_k, X). $$ -wich maps $\gamma$ to the morphism $\gamma_k$ composed with +which maps $\gamma$ to the morphism $\gamma_k$ composed with the projection onto the factor corresponding to $\text{id}_{[k]}$. \item Similarly, if $W$ is an $k$-truncated simplicial object of $\mathcal{C}$, then we have @@ -4922,7 +4922,7 @@ \section{Homotopies and cosimplicial objects} constant map with value $0$. Similarly, the condition that $b = e_1 \circ h$ means that $b_n = h_{n, 1 : [n] \to [1]}$ where $1 : [n] \to [1]$ is the constant map with value $1$. -Conversly, given a family of morphisms $\{h_{n, \alpha}\}$ +Conversely, given a family of morphisms $\{h_{n, \alpha}\}$ such that (\ref{equation-property-homotopy-cosimplicial}) holds for all morphisms $f$ of $\Delta$ and such that $a_n = h_{n, 0 : [n] \to [1]}$ and $b_n = h_{n, 1 : [n] \to [1]}$ @@ -6691,7 +6691,7 @@ \section{Standard resolutions} \longrightarrow U \circ V \circ U \circ V = Y \circ Y $$ -This places us in the sitation of Example \ref{example-godement}. +This places us in the situation of Example \ref{example-godement}. It is immediate from the formulas that the $X, d^n_i, s^n_i$ constructed above and the $X, s^n_i, s^n_i$ constructed from $Y, d, s$ @@ -6740,7 +6740,7 @@ \section{Standard resolutions} \begin{proof} As in the proof of Lemma \ref{lemma-standard-simplicial} we set $Y = U \circ V$ so that we are -in the sitation of Example \ref{example-godement}. +in the situation of Example \ref{example-godement}. \medskip\noindent Proof of the first homotopy equivalence. By Lemma \ref{lemma-godement-section} diff --git a/sites-cohomology.tex b/sites-cohomology.tex index b12f39bfb..15e843918 100644 --- a/sites-cohomology.tex +++ b/sites-cohomology.tex @@ -2737,7 +2737,7 @@ \section{Cohomology and colimits} $K' \to K^n$ with $K' \in S$. Then $(K' \to K^n) \to (K^n \to K^n)$ is a surjection in $\Sh(\mathcal{C})/K^n$ and $K^n \to K^n$ is the final object of $\Sh(\mathcal{C})/K^n$. Hence (1) holds for $S_n$, -\item Property (2) for $S_n$ is an immediate consquence +\item Property (2) for $S_n$ is an immediate consequence of (2) for $S$. \item Let $a : K_1 \to K^n$ and $b : K_2 \to K^n$ be in $S_n$. Then $(K_1 \to K^n) \times (K_2 \to K^n)$ is the object @@ -5374,7 +5374,7 @@ \section{Derived and homotopy limits} H^{-n - 1}(E)[n + 1] \to K_{n + 1} \to K_n \to H^{-n - 1}(E)[n + 2] $$ -Looking at the asssociated long exact cohomology sequence the claim follows if +Looking at the associated long exact cohomology sequence the claim follows if $$ H^{m + n}(V_i, H^{-n - 1}(E)),\quad H^{m + n + 1}(V_i, H^{-n - 1}(E)),\quad @@ -5844,7 +5844,7 @@ \section{Bounded cohomological dimension} \noindent It turns out that we sometimes need a variant of the lemma above -where the assumptions are sligthly different. +where the assumptions are slightly different. \begin{situation} \label{situation-olsson-laszlo-prime} @@ -8409,7 +8409,7 @@ \section{Cup product} \text{Tot}(\mathcal{A}^\bullet \otimes \mathcal{B}^\bullet)$. Finally, the commutativity of the remaining square is true on the level of complexes and may be viewed as the -definiton of the naive cup product (by the adjointness +definition of the naive cup product (by the adjointness of $f^*$ and $f_*$). The proof is finished because going around the diagram on the outside are the two maps given above. @@ -8546,7 +8546,7 @@ \section{Hom complexes} It is a good idea to think of $\SheafHom^n$ as the sheaf of $\mathcal{O}$-modules of all $\mathcal{O}$-linear maps from $\mathcal{L}^\bullet$ to $\mathcal{M}^\bullet$ -(viewed as graded $\mathcal{O}$-modules) which are homogenous +(viewed as graded $\mathcal{O}$-modules) which are homogeneous of degree $n$. In this terminology, we define the differential by the rule $$ \text{d}(f) = @@ -13160,7 +13160,7 @@ \section{Duals} \medskip\noindent Let $U$ be an object of $\mathcal{C}$. It suffices to find a covering -$\{U_i \to U\}_{i \in I}$ fo $\mathcal{C}$ such that $M|_{U_i}$ is +$\{U_i \to U\}_{i \in I}$ of $\mathcal{C}$ such that $M|_{U_i}$ is a perfect object of $D(\mathcal{O}_{U_i})$. Hence we may replace $\mathcal{C}, \mathcal{O}, M, N, \eta, \epsilon$ by $\mathcal{C}/U, \mathcal{O}_U, M|_U, N|_U, \eta|_U, \epsilon|_U$ diff --git a/sites-modules.tex b/sites-modules.tex index 9a9c00da6..cd5bec585 100644 --- a/sites-modules.tex +++ b/sites-modules.tex @@ -7096,7 +7096,7 @@ \section{Skyscraper sheaves} ( r f ) (\sigma) = \sigma(r) f(\sigma) $$ for $r \in R$ and $f \in \text{Map}(G, M)$. This is true because it is the -unique $G$-invariant $R$-module strucure compatible with evaluation at $1$. +unique $G$-invariant $R$-module structure compatible with evaluation at $1$. The reader observes that in general the image of $M \to \text{Map}(G, M)$ is not an $R$-submodule (for example take $M = R$ and assume the $G$-action is nontrivial), which concludes the proof. diff --git a/sites.tex b/sites.tex index dfdddb81c..58a19afbd 100644 --- a/sites.tex +++ b/sites.tex @@ -3490,7 +3490,7 @@ \section{Quasi-compact objects and colimits} $i_{jj'} \geq i_j$, $i_{jj'} \geq i_{j'}$ such that $\varphi_{i_ji_{jj'}}(s_j)$ and $\varphi_{i_{j'}i_{jj'}}(s_{j'})$ agree over $U_j \times_U U_{j'}$. Choose an index $i \in I$ -wich is bigger or equal than all the $i_{jj'}$. Then we see that +which is bigger or equal than all the $i_{jj'}$. Then we see that the sections $\varphi_{i_ji}(s_j)$ of $\mathcal{F}_i$ glue to a section of $\mathcal{F}_i$ over $U$. This section is mapped to the element $s$ as desired. @@ -3546,7 +3546,7 @@ \section{Quasi-compact objects and colimits} Let $a_i, b_i : * \to \mathcal{F}_i$ and for $i' \geq i$ denote $a_{i'}, b_{i'} : * \to \mathcal{F}_{i'}$ the composition with the transition maps of the system. -Set $a = \colim_{i' \geq i} a_{i'}$ and similary for $b$. +Set $a = \colim_{i' \geq i} a_{i'}$ and similarly for $b$. For $i' \geq i$ denote $$ E_{i'} = \text{Equalizer}(a_{i'}, b_{i'}) \subset * @@ -3617,7 +3617,7 @@ \section{Quasi-compact objects and colimits} $a_{i'}, b_{i'} : \mathcal{K} \times \mathcal{K} \to \mathcal{F}_{i'}$ the composition of $a_i, b_i$ with the transition maps of the system. Set $a = \colim_{i' \geq i} a_{i'} : -\mathcal{K} \times \mathcal{K} \to \mathcal{F}$ and similary for $b$. +\mathcal{K} \times \mathcal{K} \to \mathcal{F}$ and similarly for $b$. We have $a = b$ by the commutativity of the diagram above. For $i' \geq i$ denote $$ @@ -11683,7 +11683,7 @@ \section{Topologies and sheaves} Suppose $S$ is a covering sieve. It clearly suffices to find a morphism $h_U \to S^\#$ such that the composition $h_U \to h_U^\#$ is the canonical map. To find such a map -it suffices to find a section $s \in S^\#(U)$ wich restricts +it suffices to find a section $s \in S^\#(U)$ which restricts to $\text{id}_U$. But since $S$ is a covering sieve, the element $\text{id}_S \in \Mor_{\textit{PSh}(\mathcal{C})}(S, S)$ diff --git a/spaces-descent.tex b/spaces-descent.tex index 21f5e8091..d22ff821e 100644 --- a/spaces-descent.tex +++ b/spaces-descent.tex @@ -2141,7 +2141,7 @@ \section{Application of descent of properties of morphisms} The implication (1) $\Rightarrow$ (2) follows from Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-ample-base-change}. Assume (2). To check $\mathcal{L}$ is ample on $X/Y$ we may -work \'etale localy on $Y$, see +work \'etale locally on $Y$, see Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-relatively-ample-local}. Thus we may assume that $Y$ is a scheme and then we may in turn assume each $Y_i$ is a scheme too, see @@ -2282,7 +2282,7 @@ \section{Application of descent of properties of morphisms} is ample on $X_{i, y_i}$ by Limits, Lemma \ref{limits-lemma-limit-ample}. By Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-ample-in-neighbourhood} -we find an open neigbourhood +we find an open neighbourhood $V_i \subset Y_i$ of $y_i$ such that $\mathcal{L}_i$ restricted to $f_i^{-1}(V_i)$ is ample relative to $V_i$. diff --git a/spaces-divisors.tex b/spaces-divisors.tex index 0ecafb295..cbfe402de 100644 --- a/spaces-divisors.tex +++ b/spaces-divisors.tex @@ -1958,7 +1958,7 @@ \section{Meromorphic functions and sections} \ref{algebra-lemma-characterize-henselian} (13), and \ref{algebra-lemma-mop-up} we see that $A_\mathfrak p \otimes_A B$ is a finite product of local rings finite \'etale over $A_\mathfrak p$. -This cleary implies that $A_\mathfrak p \otimes_A B = +This clearly implies that $A_\mathfrak p \otimes_A B = \prod_{\mathfrak q\text{ lies over }\mathfrak p} B_\mathfrak q$ as desired. diff --git a/spaces-flat.tex b/spaces-flat.tex index 8336c6479..948a4a8ed 100644 --- a/spaces-flat.tex +++ b/spaces-flat.tex @@ -729,7 +729,7 @@ \section{Flat finitely presented modules} \end{proposition} \begin{proof} -As formulated this proposition immmediately reduces +As formulated this proposition immediately reduces to the case of schemes, which is More on Flatness, Proposition \ref{flat-proposition-finite-presentation-flat-at-point}. @@ -761,7 +761,7 @@ \section{Flat finitely presented modules} \end{lemma} \begin{proof} -As formulated this lemma immmediately reduces +As formulated this lemma immediately reduces to the case of schemes, which is More on Flatness, Lemma \ref{flat-lemma-finite-presentation-flat-along-fibre}. @@ -904,7 +904,7 @@ \section{A criterion for purity} \begin{proof} Consider the morphism $\Spec(\mathcal{O}_{Y, \overline{y}}) \to Y$. -This is a flat morphism from the spectrum of a stricly henselian +This is a flat morphism from the spectrum of a strictly henselian local ring which maps the closed point to $y$. By Lemma \ref{lemma-flat-descend-pure} we reduce to the case described in the next paragraph. diff --git a/spaces-groupoids.tex b/spaces-groupoids.tex index 412e48fe2..e65fadf15 100644 --- a/spaces-groupoids.tex +++ b/spaces-groupoids.tex @@ -1434,7 +1434,7 @@ \section{Colimits of quasi-coherent modules} \mathcal{F}_i \ar[r] \ar[u] & s_*t^*\mathcal{H}_i \ar[u] } $$ -where the right vertical arrow is the result of appplying $s_*$ to the map +where the right vertical arrow is the result of applying $s_*$ to the map $$ t^*\mathcal{H}_i \to t^*\mathcal{F} \xrightarrow{\beta} s^*\mathcal{F} $$ diff --git a/spaces-limits.tex b/spaces-limits.tex index 4aa359b46..ef2413635 100644 --- a/spaces-limits.tex +++ b/spaces-limits.tex @@ -5296,7 +5296,7 @@ \section{Descending finite type spaces} as in Lemma \ref{lemma-morphism-good-diagram}. Let $(a_i)_{i \geq i_5}$, $(b_i)_{i \geq i_6}$, $(p_i)_{i \geq i_7}$, and $(q_i)_{i \geq i_8}$ be the corresponding morphisms of inverse systems -contructed in Lemma \ref{lemma-morphism-good-diagram}. Then there exists an +constructed in Lemma \ref{lemma-morphism-good-diagram}. Then there exists an $i_9 \geq \max(i_5, i_6, i_7, i_8)$ such that for $i \geq i_9$ we have $a_i \circ p_i = b_i \circ q_i$ and such that $$ diff --git a/spaces-more-cohomology.tex b/spaces-more-cohomology.tex index 160426a30..6b763b416 100644 --- a/spaces-more-cohomology.tex +++ b/spaces-more-cohomology.tex @@ -1454,7 +1454,7 @@ \section{Comparing fppf and \'etale topologies: modules} \medskip\noindent Let $\mathcal{U} = \{g_i : U_i \to U\}_{i \in I}$ be an fppf covering -with $f : U \to X$ as above. The sheaf propery holds for +with $f : U \to X$ as above. The sheaf property holds for $\mathcal{H}_\etale$ and the covering $\mathcal{U}$ by (1) of Lemma \ref{lemma-review-quasi-coherent} applied to $\mathcal{H}_\etale|_{U_\etale}$ and the above. diff --git a/spaces-more-morphisms.tex b/spaces-more-morphisms.tex index 52d1b21fe..d982ec14f 100644 --- a/spaces-more-morphisms.tex +++ b/spaces-more-morphisms.tex @@ -5984,7 +5984,7 @@ \section{Crit\`ere de platitude par fibres} we see that $x$ is in our set if and only if $(\mathcal{F}_{\overline{y}})_{\overline{x}}$ is a flat $\mathcal{O}_{X_{\overline{y}}, \overline{x}}$-module. -Simiarly, $x'$ is in the analogue of our set for the pullback +Similarly, $x'$ is in the analogue of our set for the pullback $\mathcal{F}'$ of $\mathcal{F}$ to $X'$ if and only if $(\mathcal{F}'_{\overline{y}'})_{\overline{x}'}$ is a flat $\mathcal{O}_{X'_{\overline{y}'}, \overline{x}'}$-module @@ -9305,7 +9305,7 @@ \section{Chow's lemma} $U_i \subset X_i$ and the morphism $X_i' \to \mathbf{P}^n_{Y_i}$ is an immersion. Then the strict transform $X' \to X$ of $X$ relative to $X_i' \to X_i$ is a $U$-admissible blowing up where $U \subset X$ -is the inverse image of $U_i$ in $X$. Because of our carefuly chosen +is the inverse image of $U_i$ in $X$. Because of our carefully chosen index $i$ it follows that $\eta_1, \ldots, \eta_n \in |U|$ and $U \subset X$ is dense. Moreover, $X' \to \mathbf{P}^n_Y$ is an immersion as $X'$ is closed in @@ -12044,7 +12044,7 @@ \section{Characterizing pseudo-coherent complexes, II} Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-compare-base-change}. This also uses that $R(b', c')_*\mathcal{O}_{Z'} = (b', c')_*\mathcal{O}_{Z'}$ -because $(b', c')$ is a closed immersion and simiarly for +because $(b', c')$ is a closed immersion and similarly for $(b, c)_*\mathcal{O}_Z$. Since $E'|_{U' \times_{A'} V'} = (b', c')_*\mathcal{O}_{W'}$ we obtain @@ -12902,7 +12902,7 @@ \section{Theorem of the cube} whose formation commutes with arbitrary change of base. \medskip\noindent -In this sitation if $g : Y' \to Y$ is a morphism and there exists an +In this situation if $g : Y' \to Y$ is a morphism and there exists an $\mathcal{N}$ as in the lemma, then the map (cup product in degree $0$) $$ f'_*((g')^*\mathcal{E}) diff --git a/spaces-morphisms.tex b/spaces-morphisms.tex index d3f768827..8ed1e99ef 100644 --- a/spaces-morphisms.tex +++ b/spaces-morphisms.tex @@ -4623,7 +4623,7 @@ \section{Points and geometric points} \lambda(T) = \sup\{\aleph_0, |\kappa(t)| ; t \in T\}. $$ In words $\lambda(T)$ is the smallest infinite cardinal bounding -all the cardinalities of residue fields ot $T$. Note that if $R$ +all the cardinalities of residue fields of $T$. Note that if $R$ is a ring then the cardinality of any residue field $\kappa(\mathfrak p)$ of $R$ is bounded by the cardinality of $R$ (details omitted). This implies that $\lambda(T) \leq \text{size}(T)$ where @@ -10275,7 +10275,7 @@ \section{Normalization} $\nu_X : X^\nu \to X$ and $g^\nu : U^\nu \to X^\nu$ be as above. Let $Z$ be a normal scheme and let $h : Z \to U$ and $a : Z \to X^\nu$ be morphisms over $S$ such that $g \circ h = \nu_X \circ a$ and such that -every irreducible compoent of $Z$ dominates an irreducible component of $U$ +every irreducible component of $Z$ dominates an irreducible component of $U$ (via $h$). By Morphisms, Lemma \ref{morphisms-lemma-normalization-normal} part (4) we obtain a unique morphism $h^\nu : Z \to U^\nu$ such diff --git a/spaces-perfect.tex b/spaces-perfect.tex index 033d51c50..b6eccc86f 100644 --- a/spaces-perfect.tex +++ b/spaces-perfect.tex @@ -4594,7 +4594,7 @@ \section{The coherator revisited} Then $K|_U$, $K|_V$, $K|_{U \times_W V}$ have the same property. Hence we see that $RQ_U(K|_U)$ and $RQ_V(K|_V)$ and $RQ_{U \cap V}(K|_{U \times_W V})$ have vanishing cohomology -sheaves outside the inverval $[a, b + \max(N(U), N(V), N(U \times_W V))$. +sheaves outside the interval $[a, b + \max(N(U), N(V), N(U \times_W V))$. Since the functors $Rj_{U, *}$, $Rj_{V, *}$, $Rj_{U \times_W V, *}$ have finite cohomological dimension on $D_\QCoh$ by Lemma \ref{lemma-quasi-coherence-direct-image} @@ -6737,7 +6737,7 @@ \section{Other applications} To finish the proof it suffices to show: if $(U \subset W, V \to W)$ is an elementary distinguished triangle and the result holds for $U$, $V$, and $U \times_W V$, then -the result holds for $W$. This is an immediate consquence +the result holds for $W$. This is an immediate consequence of the Mayer-Vietoris sequence, see Lemma \ref{lemma-unbounded-mayer-vietoris}. \end{proof} @@ -7123,7 +7123,7 @@ \section{Detecting Boundedness} $R\Hom_X(P, E)$ is in $D^-(\mathbf{Z})$. Clearly, $T$ is inherited by direct sums, satisfies the 2-out-of-three property for distinguished -triangles, is inherited by direct summands, and is perserved by shifts. +triangles, is inherited by direct summands, and is preserved by shifts. Hence by Derived Categories, Remark \ref{derived-remark-check-on-generator} we see that (4) implies $T$ holds on all of $D_{perf}(\mathcal{O}_X)$. The same argument works for all other properties, except that for property @@ -7246,7 +7246,7 @@ \section{Detecting Boundedness} $R\Hom_X(P, E)$ is in $D^+(\mathbf{Z})$. Clearly, $T$ is inherited by direct sums, satisfies the 2-out-of-three property for distinguished -triangles, is inherited by direct summands, and is perserved by shifts. +triangles, is inherited by direct summands, and is preserved by shifts. Hence by Derived Categories, Remark \ref{derived-remark-check-on-generator} we see that (4) implies $T$ holds on all of $D_{perf}(\mathcal{O}_X)$. The same argument works for all other properties, except that for property @@ -7372,7 +7372,7 @@ \section{Quasi-coherent objects in the derived category} \begin{lemma} \label{lemma-DQCoh-alternative-small} -In the sitation above there are canonical exact equivalences between +In the situation above there are canonical exact equivalences between the following triangulated categories \begin{enumerate} \item $D_\QCoh(\mathcal{O}_X)$, diff --git a/spaces-pushouts.tex b/spaces-pushouts.tex index 990d784a2..9b43f1b44 100644 --- a/spaces-pushouts.tex +++ b/spaces-pushouts.tex @@ -2247,7 +2247,7 @@ \section{Glueing and the Beauville-Laszlo theorem} $$ of algebraic spaces with both squares cartesian and whose morphism are defined in the obvious manner. An object of this category will -be denoted $(V, V', Y')$ with arrows surpressed from the notation. +be denoted $(V, V', Y')$ with arrows suppressed from the notation. There is a functor \begin{equation} \label{equation-beauville-laszlo-glueing-spaces} @@ -3365,7 +3365,7 @@ \section{Compactifications} is an isomorphism over an open neighbourhood of $Z_2$ and $V_{12} = V_1 \times_U V_2$. By Lemma \ref{lemma-blowup-iso-along} there exists a $V_1$-admissible blowing up $X_1' \to X_1$ -such that the strict tranform $p'_1 : X'_{12} \to X'_1$ +such that the strict transform $p'_1 : X'_{12} \to X'_1$ of $p_1$ is an isomorphism over an open neighbourhood of the closure of $|Z_{1, 2}|$ in $|X'_1|$. After replacing $X_1$ by $X'_1$ and $X_{12}$ by $X'_{12}$ diff --git a/spaces-simplicial.tex b/spaces-simplicial.tex index 2edfa7acb..7740a48ba 100644 --- a/spaces-simplicial.tex +++ b/spaces-simplicial.tex @@ -463,7 +463,7 @@ \section{Simplicial topological spaces} This is the analogue of Cohomology, Lemma \ref{cohomology-lemma-describe-higher-direct-images} or of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images} -and we strongly encourge the reader to skip the proof. +and we strongly encourage the reader to skip the proof. Choosing an injective resolution of $\mathcal{F}$ on $X_{Zar}$ and using the definitions we see that it suffices to show: (1) the restriction of an injective abelian @@ -3777,7 +3777,7 @@ \section{The site associate to a simplicial semi-representable object} $a^{-1} : \textit{Ab}(\mathcal{C}) \to \textit{Ab}((\mathcal{C}/K)_{total})$, \item the functor $a^{-1}$ associates to $\mathcal{F}$ in $\Sh(\mathcal{C})$ the sheaf on $(\mathcal{C}/K)_{total}$ -wich in degree $n$ is equal to $a_n^{-1}\mathcal{F}$, +which in degree $n$ is equal to $a_n^{-1}\mathcal{F}$, \item the functor $a_*$ associates to $\mathcal{G}$ in $\textit{Ab}((\mathcal{C}/K)_{total})$ the equalizer of the two maps $j_{0, *}\mathcal{G}_0 \to j_{1, *}\mathcal{G}_1$, @@ -4050,7 +4050,7 @@ \section{Cohomological descent for hypercoverings} \mathcal{G}_0 \times_{\mathcal{K}_0, \mathcal{K}(\delta^1_1)} \mathcal{K}_1 \to \mathcal{G}_0 \times_{\mathcal{K}_0, \mathcal{K}(\delta^1_0)} \mathcal{K}_1 $$ -over $\mathcal{K}_1$ satisfying a cocycle condtion in +over $\mathcal{K}_1$ satisfying a cocycle condition in $\Sh(\mathcal{C})/\mathcal{K}_2$, there exists $\mathcal{F}$ in $\Sh(\mathcal{C})$ and an isomorphism $\mathcal{F} \times \mathcal{K}_0 \to \mathcal{G}_0$ over $\mathcal{K}_0$ @@ -7532,7 +7532,7 @@ \section{Fppf hypercoverings of algebraic spaces: modules} $$ by Lemma \ref{lemma-sanity-check-modules}. Here on the right hand side we have the cohomology of the quasi-coherent -sheaf $\mathcal{F}_n$ on $U_n$ over the affine obect $V$ +sheaf $\mathcal{F}_n$ on $U_n$ over the affine object $V$ of $U_{n, \etale}$. This vanishes for $p > 0$ by the discussion in Cohomology of Spaces, Section \ref{spaces-cohomology-section-higher-direct-image} and diff --git a/spaces.tex b/spaces.tex index 19468497e..ee1f9ce2b 100644 --- a/spaces.tex +++ b/spaces.tex @@ -2869,7 +2869,7 @@ \section{Change of base scheme} \begin{proof} Let $F$ be an algebraic space over $S$. The functor from left to right -assigns the pair $(j_!F, j_!F \to S)$ ot $F$ +assigns the pair $(j_!F, j_!F \to S)$ to $F$ which is an object of the right hand side by Lemma \ref{lemma-change-base-scheme}. Since this defines an equivalence of categories of sheaves by diff --git a/stacks-cohomology.tex b/stacks-cohomology.tex index 113782127..69fcf4507 100644 --- a/stacks-cohomology.tex +++ b/stacks-cohomology.tex @@ -1720,7 +1720,7 @@ \section{Quasi-coherent modules} Let $\mathcal{X}$ be an algebraic stack. Let $\textit{Parasitic}(\mathcal{O}_\mathcal{X}) \subset \textit{Mod}(\mathcal{O}_\mathcal{X})$ denote the -full subcategory consiting of parasitic modules. The results of +full subcategory consisting of parasitic modules. The results of Lemmas \ref{lemma-adjoint} and \ref{lemma-adjoint-kernel-parasitic} imply that $$ diff --git a/stacks-geometry.tex b/stacks-geometry.tex index aba0c28bd..d88270636 100644 --- a/stacks-geometry.tex +++ b/stacks-geometry.tex @@ -112,7 +112,7 @@ \section{Versal rings} Formal Deformation Theory, Lemmas \ref{formal-defos-lemma-RS-implies-S1-S2} and Definition \ref{formal-defos-definition-deformation-category}). -By the uniquness result of +By the uniqueness result of Formal Deformation Theory, Lemma \ref{formal-defos-lemma-minimal-versal} there exists a ``minimal'' versal ring $A$ of $\mathcal{X}$ at $x_0$ such that any other versal ring of $\mathcal{X}$ at $x_0$ is @@ -2097,7 +2097,7 @@ \section{The dimension of the local ring} \medskip\noindent Assume $(U', u') \to (U, u)$ is a smooth morphism of smooth -neightbourhoods of $x$ with $u$ and $u'$ closed points. +neighbourhoods of $x$ with $u$ and $u'$ closed points. Goal: prove the invariant defined for $(U, u)$ is the same as the invariant defined for $(U', u')$. To see this observe that $\mathcal{O}_{U, u} \to \mathcal{O}_{U', u'}$ diff --git a/stacks-more-morphisms.tex b/stacks-more-morphisms.tex index 36b04d878..34962fbed 100644 --- a/stacks-more-morphisms.tex +++ b/stacks-more-morphisms.tex @@ -4515,7 +4515,7 @@ \section{Tensor functors} \noindent The following very simple version of K\"unneth should become obsoleted when we write a section on K\"unneth theorems for -cohomology of quasi-coherent modues on algebraic stacks. +cohomology of quasi-coherent modules on algebraic stacks. \begin{lemma} \label{lemma-easy-kunneth} @@ -4700,7 +4700,7 @@ \section{Tensor functors} The arrow $\Spec(C) \to U = \Spec(A)$ comes from the ring map $A \to C$ in the statement of Lemma \ref{lemma-affine}. The arrow $\Spec(C \otimes_B C) \to U \times_\mathcal{X} U$ -simlarly comes from the ring map $R \to C \otimes_B C$. +similarly comes from the ring map $R \to C \otimes_B C$. To verify the top square commutes use Lemma \ref{lemma-fully-faithful}; details omitted. We conclude we get the dotted arrow $\Spec(B) \to \mathcal{X}$ diff --git a/stacks-morphisms.tex b/stacks-morphisms.tex index bab15e045..c395f06de 100644 --- a/stacks-morphisms.tex +++ b/stacks-morphisms.tex @@ -2784,7 +2784,7 @@ \section{Universally injective morphisms} we find that the integral closures of the two images of $k'$ in $\Gamma(U, \mathcal{O}_U)$ are equal. Looking in $\kappa(\xi)$ means that any element of the form -$\lambda \otimes 1$ is algebraically dependend on +$\lambda \otimes 1$ is algebraically dependent on the subfield $$ 1 \otimes k' \subset @@ -7033,7 +7033,7 @@ \section{\'Etale local structure} \ref{algebraic-proposition-algebraic-stack-no-automorphisms}) and of course it is equal to the algebraic space $U/R$ (of Bootstrap, Theorem \ref{bootstrap-theorem-final-bootstrap}). -Since $U \to X$ is surjective, flat, and locally of finite presenation +Since $U \to X$ is surjective, flat, and locally of finite presentation (Bootstrap, Lemma \ref{bootstrap-lemma-covering-quotient}) we conclude that $X \to Y$ is flat and locally of finite presentation by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-flat-permanence} @@ -7049,7 +7049,7 @@ \section{\'Etale local structure} $$ Since the right vertical arrow is surjective, flat, and locally of finite presentation (small detail omitted), we -find that $X \to X \times_Y X$ is an open immersion as the top horizonal arrow +find that $X \to X \times_Y X$ is an open immersion as the top horizontal arrow has this property by assumption (use Properties of Stacks, Lemma \ref{stacks-properties-lemma-check-property-covering}). @@ -8608,7 +8608,7 @@ \section{Scheme theoretic image} t^{-1}\mathcal{I} \cdot \mathcal{O}_R $$ Hence $\mathcal{I}$ defines an $R$-invariant closed subscheme -$Z \subset U$ which is contained in every $Z_\alpha$ and containes +$Z \subset U$ which is contained in every $Z_\alpha$ and contains $Z_1$ as desired. \end{proof} @@ -10530,7 +10530,7 @@ \section{Residual gerbes} \begin{proof} The morphism $\mathcal{Z}_x \to \mathcal{X}$ is a monomorphism as the base change of the monomorphism $\mathcal{Z}_y \to \mathcal{Y}$. -The morphism $\pi$ is a univeral homeomorphism by +The morphism $\pi$ is a universal homeomorphism by Lemma \ref{lemma-gerbe-bijection-points} and hence $|\mathcal{Z}_x| = \{x\}$. Finally, the morphism $\mathcal{Z}_x \to \mathcal{Z}_y$ is smooth as a base change of the smooth morphism $\pi$, see diff --git a/stacks-perfect.tex b/stacks-perfect.tex index bfbf851e3..935bee401 100644 --- a/stacks-perfect.tex +++ b/stacks-perfect.tex @@ -375,7 +375,7 @@ \section{Cohomology and the lisse-\'etale and flat-fppf sites} By Lemma \ref{lemma-lisse-etale-functorial-derived} we find $f^{-1}Lg_!\mathbf{Z} = L(g')_!\mathbf{Z}$ where $g' : \Sh(\mathcal{U}_{flat, fppf}) \to \Sh(\mathcal{U}_{fppf})$ -is the corresponding comparision morphism for $\mathcal{U}$. +is the corresponding comparison morphism for $\mathcal{U}$. This reduces us to the case studied in the next paragraph. \medskip\noindent diff --git a/stacks-properties.tex b/stacks-properties.tex index 3ff204daf..7d5c47371 100644 --- a/stacks-properties.tex +++ b/stacks-properties.tex @@ -630,7 +630,7 @@ \section{Properties of morphisms representable by algebraic spaces} We omit the proof of both claims. Hints: part (1) follows from the construction of $f = f_{can}$ and $f' = f'_{can}$ in Algebraic Stacks, Lemma \ref{algebraic-lemma-map-space-into-stack}. -Part (2) follows by carefuly working through the definitions. +Part (2) follows by carefully working through the definitions. \end{proof} \begin{remark} diff --git a/stacks-sheaves.tex b/stacks-sheaves.tex index f46485a2e..b71e1a79d 100644 --- a/stacks-sheaves.tex +++ b/stacks-sheaves.tex @@ -4741,7 +4741,7 @@ \section{Quasi-coherent objects in the derived category} \begin{lemma} \label{lemma-QC-compare} Let $S$ be a scheme. Let $\mathcal{X} \to (\Sch/S)_{fppf}$ be a category -fibred in groupoids. The comparision morphism +fibred in groupoids. The comparison morphism $\epsilon : \mathcal{X}_{affine, \etale} \to \mathcal{X}_{affine}$ satisfies the assumptions and conclusions of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cartesion-plus-topology}. @@ -4809,7 +4809,7 @@ \section{Quasi-coherent objects in the derived category} prove that $G$ is an equivalence. \medskip\noindent -Consider the flat comparision morphisms +Consider the flat comparison morphisms $\epsilon_\mathcal{X} : \mathcal{X}_{affine, \etale} \to \mathcal{X}_{affine}$ and $\epsilon_X : X_{affine, \etale} \to X_{affine}$ of ringed sites. Lemma \ref{lemma-QC-compare} and @@ -4960,7 +4960,7 @@ \section{Quasi-coherent objects in the derived category} \begin{lemma} \label{lemma-QC-compare-fppf} Let $S$ be a scheme. Let $\mathcal{X} \to (\Sch/S)_{fppf}$ be a category -fibred in groupoids. The comparision morphism +fibred in groupoids. The comparison morphism $\epsilon : \mathcal{X}_{affine, fppf} \to \mathcal{X}_{affine}$ satisfies the assumptions and conclusions of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cartesion-plus-topology}. diff --git a/stacks.tex b/stacks.tex index 2a8598d52..eba1e2277 100644 --- a/stacks.tex +++ b/stacks.tex @@ -2192,7 +2192,7 @@ \section{Inherited topologies} We have to find a morphism $\varphi : y_1 \to y_2$ restricting to $\varphi_i$. Note that $q(\varphi_i) = \psi|_{U_i}$ for some morphism $\psi : x_1 \to x_2$ over $U$ because the second presheaf is a sheaf -(by assumption). Let $y_{12} \to y_2$ be the stronly $\mathcal{X}$-cartesian +(by assumption). Let $y_{12} \to y_2$ be the strongly $\mathcal{X}$-cartesian morphism of $\mathcal{Y}$ lying over $\psi$. Then $\varphi_i$ corresponds to a morphism $\varphi'_i : y_1|_{U_i} \to y_{12}|_{U_i}$ over $x_1|_{U_i}$. In other words, $\varphi'_i$ now define local sections of the presheaf diff --git a/tags/latex.sh b/tags/latex.sh index cbc6e2d46..93318960b 100755 --- a/tags/latex.sh +++ b/tags/latex.sh @@ -41,7 +41,7 @@ if [ ! $? == 0 ]; then exit 1; fi -# Move newly created files back to stacks project directoy +# Move newly created files back to stacks project directory mv $STEM.pdf $STEM.dvi $STEM.aux $STEM.toc $STEM.out $STEM.log $OLD # Remove temporary directory diff --git a/tags/tags b/tags/tags index 8d4fb7623..d87189423 100644 --- a/tags/tags +++ b/tags/tags @@ -2604,7 +2604,7 @@ 01ZR,limits-lemma-descend-modules-finite-presentation 01ZS,limits-section-affine 01ZT,limits-lemma-affine -# This section got moved to properies.tex +# This section got moved to properties.tex # This affects tags 01ZU, 01ZV, 01ZW, 01ZX, 01ZY 01ZU,properties-section-finding-affine-opens 01ZV,properties-lemma-maximal-points-affine @@ -9149,7 +9149,7 @@ 0760,stacks-cohomology-section-flat-comparison 0761,stacks-cohomology-equation-comparison-modules 0762,stacks-cohomology-definition-flat-base-change -# equation dissappeared due to reorganization material +# equation disappeared due to reorganization material #0763,stacks-cohomology-equation-fppf-comparison 0764,stacks-cohomology-lemma-check-flat-comparison-on-etale-covering 0765,stacks-cohomology-lemma-flat-comparison diff --git a/topology.tex b/topology.tex index e24af1184..3dcc47d3e 100644 --- a/topology.tex +++ b/topology.tex @@ -2290,7 +2290,7 @@ \section{Locally quasi-compact spaces} disjoint opens $z \in V_{z, z'}$ and $z' \in W_{z, z'}$ of $X$. Since $Z \setminus U_i$ is quasi-compact (Lemma \ref{lemma-closed-in-quasi-compact}), -we can choose a finite nunber $z'_1, \ldots, z'_r$ such that +we can choose a finite number $z'_1, \ldots, z'_r$ such that $Z \setminus U_i \subset W_{z, z'_1} \cup \ldots \cup W_{z, z'_r}$. Then we see that $V_z = V_{z, z'_1} \cap \ldots \cap V_{z, z'_r} \cap U_i$ diff --git a/trace.tex b/trace.tex index f4d6b9cc3..821838cf2 100644 --- a/trace.tex +++ b/trace.tex @@ -3843,7 +3843,7 @@ \section{Precise form of Chebotarev} (\# C)(2g - 2) \sqrt{q^n} $$ (Warning: Please check the coefficient $\# C$ on the right hand side -carefuly before using.) +carefully before using.) \begin{proof}[Sketch] Write diff --git a/varieties.tex b/varieties.tex index 50a8ebec1..199b056d7 100644 --- a/varieties.tex +++ b/varieties.tex @@ -536,7 +536,7 @@ \section{Geometrically reduced schemes} To prove (1) combine Lemma \ref{lemma-geometrically-reduced-at-point} with Algebra, Lemma \ref{algebra-lemma-geometrically-reduced-any-reduced-base-change}. -To prove (2) comine Lemma \ref{lemma-geometrically-reduced} +To prove (2) combine Lemma \ref{lemma-geometrically-reduced} with Algebra, Lemma \ref{algebra-lemma-geometrically-reduced-any-reduced-base-change}. To prove (3) note that @@ -6664,7 +6664,7 @@ \subsection{Regularity} shows we can lift $g_i$ to $f_i \in H^0(\mathbf{P}^n_k, \mathcal{F}(m))$. We can also lift $\overline{s}_i$ to $s_i \in H^0(\mathbf{P}^n_k, \mathcal{O}(1))$ (see proof of -Lemma \ref{lemma-hyperplane} for example). After substracting the +Lemma \ref{lemma-hyperplane} for example). After subtracting the image of $\sum f_i \otimes s_i$ from $t$ we see that we may assume $\overline{t} = 0$. But this exactly means that $t$ is the image of $f \otimes s$ for some $f \in H^0(\mathbf{P}^n_k, \mathcal{F}(m))$ @@ -8220,7 +8220,7 @@ \section{The delta invariant} Then $A_\mathfrak p$ is a field for $\mathfrak p \subset A$ minimal, see Algebra, Lemma \ref{algebra-lemma-minimal-prime-reduced-ring}. -Hence (3) is equivalent ot (6). +Hence (3) is equivalent to (6). \end{proof} \begin{lemma} @@ -10772,7 +10772,7 @@ \section{Bertini theorems} Choose $n_1$ such that for $n \geq n_1$ the sheaf $\mathcal{I} \otimes \mathcal{L}^{\otimes n}$ is globally generated, see Properties, Proposition \ref{properties-proposition-characterize-ample}. -It follows that $V_n$ gererates $\mathcal{L}^{\otimes n}|_{X \setminus Z}$ +It follows that $V_n$ generates $\mathcal{L}^{\otimes n}|_{X \setminus Z}$ for $n \geq n_1$. \medskip\noindent diff --git a/weil.tex b/weil.tex index 94933275f..f5bdaacc2 100644 --- a/weil.tex +++ b/weil.tex @@ -356,9 +356,9 @@ \section{Correspondences} c' = \text{pr}_{13, *}(\text{pr}_{12}^*[\Delta] \cdot \text{pr}_{23}^*c') $$ where $\text{pr}_{12} : X \times X \times Y \to X \times X$ is the -projection and simlarly for $\text{pr}_{13}$ and $\text{pr}_{23}$. +projection and similarly for $\text{pr}_{13}$ and $\text{pr}_{23}$. We may write $c' = \sum a_i [Z_i]$ for some integral closed subschemes -$Z_i \subset X \times Y$ and rational numers $a_i$. Thus it clearly +$Z_i \subset X \times Y$ and rational numbers $a_i$. Thus it clearly suffices to show that $$ [Z] = \text{pr}_{13, *}(\text{pr}_{12}^*[\Delta] \cdot \text{pr}_{23}^*[Z]) @@ -2356,7 +2356,7 @@ \section{Weil cohomology theories, I} axioms correspond to functors $G$ of symmetric monoidal categories from the category of motives to the category of graded vector spaces such that $G(\mathbf{1}(1))$ sits in degree $-2$. In Section \ref{section-old} -we will define a Weil cohomology theory by adding a single suplementary +we will define a Weil cohomology theory by adding a single supplementary condition. \medskip\noindent @@ -3200,7 +3200,7 @@ \section{Further properties} $(i^*, j^*) : H^*(X \amalg Y) \to H^*(X) \times H^*(Y)$ is an isomorphism. The statement of the lemma means that under the isomorphism $(i^*, j^*) : H^{2d}(X \amalg Y)(d) \to H^{2d}(X)(d) \oplus H^{2d}(Y)(d)$ -the map $\int_X + \int_Y$ is tranformed into $\int_{X \amalg Y}$. +the map $\int_X + \int_Y$ is transformed into $\int_{X \amalg Y}$. This is true because $$ \int_{X \amalg Y} a = @@ -4755,7 +4755,7 @@ \section{Weil cohomology theories, III} \eta_{X \times Y, i} \in H^i(X \times Y) \otimes H^{2d + 2e - i}(X \times Y)(d + e) $$ -and simiarly $\gamma([\Delta_X]) = \sum \eta_{X, i}$ and +and similarly $\gamma([\Delta_X]) = \sum \eta_{X, i}$ and $\gamma([\Delta_Y]) = \sum \eta_{Y, i}$. The observation above implies we have $$