fix small typos

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

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}
 

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

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
 $$

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}

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}

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) =

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 $A$,

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)