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 \input{preamble} % OK, start here. % \begin{document} \title{Obsolete} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent In this chapter we put some lemmas that have become obsolete'' (see \cite{Miller}). \section{Homological algebra} \label{section-homological-algebra} \begin{remark} \label{remark-weak-serre-subcategory} The following remarks are obsolete as they are subsumed in Homology, Lemmas \ref{homology-lemma-biregular-ss-converges} and \ref{homology-lemma-first-quadrant-ss}. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{C} \subset \mathcal{A}$ be a weak Serre subcategory (see Homology, Definition \ref{homology-definition-serre-subcategory}). Suppose that $K^{\bullet, \bullet}$ is a double complex to which Homology, Lemma \ref{homology-lemma-first-quadrant-ss} applies such that for some $r \geq 0$ all the objects ${}'E_r^{p, q}$ belong to $\mathcal{C}$. Then all the cohomology groups $H^n(sK^\bullet)$ belong to $\mathcal{C}$. Namely, the assumptions imply that the kernels and images of ${}'d_r^{p, q}$ are in $\mathcal{C}$. Whereupon we see that each ${}'E_{r + 1}^{p, q}$ is in $\mathcal{C}$. By induction we see that each ${}'E_\infty^{p, q}$ is in $\mathcal{C}$. Hence each $H^n(sK^\bullet)$ has a finite filtration whose subquotients are in $\mathcal{C}$. Using that $\mathcal{C}$ is closed under extensions we conclude that $H^n(sK^\bullet)$ is in $\mathcal{C}$ as claimed. The same result holds for the second spectral sequence associated to $K^{\bullet, \bullet}$. Similarly, if $(K^\bullet, F)$ is a filtered complex to which Homology, Lemma \ref{homology-lemma-biregular-ss-converges} applies and for some $r \geq 0$ all the objects $E_r^{p, q}$ belong to $\mathcal{C}$, then each $H^n(K^\bullet)$ is an object of $\mathcal{C}$. \end{remark} \section{Obsolete algebra lemmas} \label{section-algebra} \begin{lemma} \label{lemma-finite-presentation-module-independent} Let $M$ be an $R$-module of finite presentation. For any surjection $\alpha : R^{\oplus n} \to M$ the kernel of $\alpha$ is a finite $R$-module. \end{lemma} \begin{proof} This is a special case of Algebra, Lemma \ref{algebra-lemma-extension}. \end{proof} \begin{lemma} \label{lemma-p-ring-map} Let $\varphi : R \to S$ be a ring map. If \begin{enumerate} \item for any $x \in S$ there exists $n > 0$ such that $x^n$ is in the image of $\varphi$, and \item for any $x \in \Ker(\varphi)$ there exists $n > 0$ such that $x^n = 0$, \end{enumerate} then $\varphi$ induces a homeomorphism on spectra. Given a prime number $p$ such that \begin{enumerate} \item[(a)] $S$ is generated as an $R$-algebra by elements $x$ such that there exists an $n > 0$ with $x^{p^n} \in \varphi(R)$ and $p^nx \in \varphi(R)$, and \item[(b)] the kernel of $\varphi$ is generated by nilpotent elements, \end{enumerate} then (1) and (2) hold, and for any ring map $R \to R'$ the ring map $R' \to R' \otimes_R S$ also satisfies (a), (b), (1), and (2) and in particular induces a homeomorphism on spectra. \end{lemma} \begin{proof} This is a combination of Algebra, Lemmas \ref{algebra-lemma-powers} and \ref{algebra-lemma-p-ring-map}. \end{proof} \noindent The following technical lemma says that you can lift any sequence of relations from a fibre to the whole space of a ring map which is essentially of finite type, in a suitable sense. \begin{lemma} \label{lemma-lift-elements-ideal} Let $R \to S$ be a ring map. Let $\mathfrak p \subset R$ be a prime. Let $\mathfrak q \subset S$ be a prime lying over $\mathfrak p$. Assume $S_{\mathfrak q}$ is essentially of finite type over $R_\mathfrak p$. Assume given \begin{enumerate} \item an integer $n \geq 0$, \item a prime $\mathfrak a \subset \kappa(\mathfrak p)[x_1, \ldots, x_n]$, \item a surjective $\kappa(\mathfrak p)$-homomorphism $$\psi : (\kappa(\mathfrak p)[x_1, \ldots, x_n])_{\mathfrak a} \longrightarrow S_{\mathfrak q}/\mathfrak p S_{\mathfrak q},$$ and \item elements $\overline{f}_1, \ldots, \overline{f}_e$ in $\Ker(\psi)$. \end{enumerate} Then there exist \begin{enumerate} \item an integer $m \geq 0$, \item and element $g \in S$, $g \not\in \mathfrak q$, \item a map $$\Psi : R[x_1, \ldots, x_n, x_{n + 1}, \ldots, x_{n + m}] \longrightarrow S_g,$$ and \item elements $f_1, \ldots, f_e, f_{e + 1}, \ldots, f_{e + m}$ of $\Ker(\Psi)$ \end{enumerate} such that \begin{enumerate} \item the following diagram commutes $$\xymatrix{ R[x_1, \ldots, x_{n + m}] \ar[d]_\Psi \ar[rr]_-{x_{n + j} \mapsto 0} & & (\kappa(\mathfrak p)[x_1, \ldots, x_n])_{\mathfrak a} \ar[d]^\psi \\ S_g \ar[rr] & & S_{\mathfrak q}/\mathfrak p S_{\mathfrak q} },$$ \item the element $f_i$, $i \leq n$ maps to a unit times $\overline{f}_i$ in the local ring $$(\kappa(\mathfrak p)[x_1, \ldots, x_{n + m}])_{ (\mathfrak a, x_{n + 1}, \ldots, x_{n + m})},$$ \item the element $f_{e + j}$ maps to a unit times $x_{n + j}$ in the same local ring, and \item the induced map $R[x_1, \ldots, x_{n + m}]_{\mathfrak b} \to S_{\mathfrak q}$ is surjective, where $\mathfrak b = \Psi^{-1}(\mathfrak qS_g)$. \end{enumerate} \end{lemma} \begin{proof} We claim that it suffices to prove the lemma in case $R$ and $S$ are local with maximal ideals $\mathfrak p$ and $\mathfrak q$. Namely, suppose we have constructed $$\Psi' : R_{\mathfrak p}[x_1, \ldots, x_{n + m}] \longrightarrow S_{\mathfrak q}$$ and $f_1', \ldots, f_{e + m}' \in R_{\mathfrak p}[x_1, \ldots, x_{n + m}]$ with all the required properties. Then there exists an element $f \in R$, $f \not \in \mathfrak p$ such that each $ff_k'$ comes from an element $f_k \in R[x_1, \ldots, x_{n + m}]$. Moreover, for a suitable $g \in S$, $g \not \in \mathfrak q$ the elements $\Psi'(x_i)$ are the image of elements $y_i \in S_g$. Let $\Psi$ be the $R$-algebra map defined by the rule $\Psi(x_i) = y_i$. Since $\Psi(f_i)$ is zero in the localization $S_{\mathfrak q}$ we may after possibly replacing $g$ assume that $\Psi(f_i) = 0$. This proves the claim. \medskip\noindent Thus we may assume $R$ and $S$ are local with maximal ideals $\mathfrak p$ and $\mathfrak q$. Pick $y_1, \ldots, y_n \in S$ such that $y_i \bmod \mathfrak pS = \psi(x_i)$. Let $y_{n + 1}, \ldots, y_{n + m} \in S$ be elements which generate an $R$-subalgebra of which $S$ is the localization. These exist by the assumption that $S$ is essentially of finite type over $R$. Since $\psi$ is surjective we may write $y_{n + j} \bmod \mathfrak pS = \psi(h_j)$ for some $h_j \in \kappa(\mathfrak p)[x_1, \ldots, x_n]_{\mathfrak a}$. Write $h_j = g_j/d$, $g_j \in \kappa(\mathfrak p)[x_1, \ldots, x_n]$ for some common denominator $d \in \kappa(\mathfrak p)[x_1, \ldots, x_n]$, $d \not \in \mathfrak a$. Choose lifts $G_j, D \in R[x_1, \ldots, x_n]$ of $g_j$ and $d$. Set $y_{n + j}' = D(y_1, \ldots, y_n) y_{n + j} - G_j(y_1, \ldots, y_n)$. By construction $y_{n + j}' \in \mathfrak p S$. It is clear that $y_1, \ldots, y_n, y_n', \ldots, y_{n + m}'$ generate an $R$-subalgebra of $S$ whose localization is $S$. We define $$\Psi : R[x_1, \ldots, x_{n + m}] \to S$$ to be the map that sends $x_i$ to $y_i$ for $i = 1, \ldots, n$ and $x_{n + j}$ to $y'_{n + j}$ for $j = 1, \ldots, m$. Properties (1) and (4) are clear by construction. Moreover the ideal $\mathfrak b$ maps onto the ideal $(\mathfrak a, x_{n + 1}, \ldots, x_{n + m})$ in the polynomial ring $\kappa(\mathfrak p)[x_1, \ldots, x_{n + m}]$. \medskip\noindent Denote $J = \Ker(\Psi)$. We have a short exact sequence $$0 \to J_{\mathfrak b} \to R[x_1, \ldots, x_{n + m}]_{\mathfrak b} \to S_{\mathfrak q} \to 0.$$ The surjectivity comes from our choice of $y_1, \ldots, y_n, y_n', \ldots, y_{n + m}'$ above. This implies that $$J_{\mathfrak b}/ \mathfrak pJ_{\mathfrak b} \to \kappa(\mathfrak p)[x_1, \ldots, x_{n + m}]_{ (\mathfrak a, x_{n + 1}, \ldots, x_{n + m})} \to S_{\mathfrak q}/\mathfrak pS_{\mathfrak q} \to 0$$ is exact. By construction $x_i$ maps to $\psi(x_i)$ and $x_{n + j}$ maps to zero under the last map. Thus it is easy to choose $f_i$ as in (2) and (3) of the lemma. \end{proof} \begin{remark}[Projective resolutions] \label{remark-projective-resolution} Let $R$ be a ring. For any set $S$ we let $F(S)$ denote the free $R$-module on $S$. Then any left $R$-module has the following two step resolution $$F(M \times M) \oplus F(R \times M) \to F(M) \to M \to 0.$$ The first map is given by the rule $$[m_1, m_2] \oplus [r, m] \mapsto [m_1 + m_2] - [m_1] - [m_2] + [rm] - r[m].$$ \end{remark} \begin{lemma} \label{lemma-spec-localization-first} Let $S$ be a multiplicative set of $A$. Then the map $$f: \Spec(S^{-1}A)\longrightarrow \Spec(A)$$ induced by the canonical ring map $A \to S^{-1}A$ is a homeomorphism onto its image and $\Im(f) = \{ \mathfrak p \in \Spec(A) : \mathfrak p\cap S = \emptyset \}$. \end{lemma} \begin{proof} This is a duplicate of Algebra, Lemma \ref{algebra-lemma-spec-localization}. \end{proof} \begin{lemma} \label{lemma-finite-type-flat-over-integral-algebra} Let $A \to B$ be a finite type, flat ring map with $A$ an integral domain. Then $B$ is a finitely presented $A$-algebra. \end{lemma} \begin{proof} Special case of More on Flatness, Proposition \ref{flat-proposition-flat-finite-type-finite-presentation-domain}. \end{proof} \begin{lemma} \label{lemma-helper-finite-type-flat-finite-presentation} Let $R$ be a domain with fraction field $K$. Let $S = R[x_1, \ldots, x_n]$ be a polynomial ring over $R$. Let $M$ be a finite $S$-module. Assume that $M$ is flat over $R$. If for every subring $R \subset R' \subset K$, $R \not = R'$ the module $M \otimes_R R'$ is finitely presented over $S \otimes_R R'$, then $M$ is finitely presented over $S$. \end{lemma} \begin{proof} This lemma is true because $M$ is finitely presented even without the assumption that $M \otimes_R R'$ is finitely presented for every $R'$ as in the statement of the lemma. This follows from More on Flatness, Proposition \ref{flat-proposition-flat-finite-type-finite-presentation-domain}. Originally this lemma had an erroneous proof (thanks to Ofer Gabber for finding the gap) and was used in an alternative proof of the proposition cited. To reinstate this lemma, we need a correct argument in case $R$ is a local normal domain using only results from the chapters on commutative algebra; please email \href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com} if you have an argument. \end{proof} \begin{lemma} \label{lemma-relative-effective-cartier-algebra} Let $A \to B$ be a ring map. Let $f \in B$. Assume that \begin{enumerate} \item $A \to B$ is flat, \item $f$ is a nonzerodivisor, and \item $A \to B/fB$ is flat. \end{enumerate} Then for every ideal $I \subset A$ the map $f : B/IB \to B/IB$ is injective. \end{lemma} \begin{proof} Note that $IB = I \otimes_A B$ and $I(B/fB) = I \otimes_A B/fB$ by the flatness of $B$ and $B/fB$ over $A$. In particular $IB/fIB \cong I \otimes_A B/fB$ maps injectively into $B/fB$. Hence the result follows from the snake lemma applied to the diagram $$\xymatrix{ 0 \ar[r] & I \otimes_A B \ar[r] \ar[d]^f & B \ar[r] \ar[d]^f & B/IB \ar[r] \ar[d]^f & 0 \\ 0 \ar[r] & I \otimes_A B \ar[r] & B \ar[r] & B/IB \ar[r] & 0 }$$ with exact rows. \end{proof} \begin{lemma} \label{lemma-faithfully-flat-injective} If $R \to S$ is a faithfully flat ring map then for every $R$-module $M$ the map $M \to S \otimes_R M$, $x \mapsto 1 \otimes x$ is injective. \end{lemma} \begin{proof} This lemma is a duplicate of Algebra, Lemma \ref{algebra-lemma-faithfully-flat-universally-injective}. \end{proof} \begin{remark} \label{remark-section-colimits} This reference/tag used to refer to a Section in the chapter Smoothing Ring Maps, but the material has since been subsumed in Algebra, Section \ref{algebra-section-colimits-flat}. \end{remark} \begin{lemma} \label{lemma-bound-primes} Let $A$ be a Noetherian local normal domain of dimension $2$. For $f \in \mathfrak m$ nonzero denote $\text{div}(f) = \sum n_i (\mathfrak p_i)$ the divisor associated to $f$ on the punctured spectrum of $A$. We set $|f| = \sum n_i$. There exist integers $N$ and $M$ such that $|f + g| \leq M$ for all $g \in \mathfrak m^N$. \end{lemma} \begin{proof} Pick $h \in \mathfrak m$ such that $f, h$ is a regular sequence in $A$ (this follows from Algebra, Lemmas \ref{algebra-lemma-criterion-normal} and \ref{algebra-lemma-depth-drops-by-one}). We will prove the lemma with $M = \text{length}_A(A/(f, h))$ and with $N$ any integer such that $\mathfrak m^N \subset (f, h)$. Such an integer $N$ exists because $\sqrt{(f, h)} = \mathfrak m$. Note that $M = \text{length}_A(A/(f + g, h))$ for all $g \in \mathfrak m^N$ because $(f, h) = (f + g, h)$. This moreover implies that $f + g, h$ is a regular sequence in $A$ too, see Algebra, Lemma \ref{algebra-lemma-reformulate-CM}. Now suppose that $\text{div}(f + g ) = \sum m_j (\mathfrak q_j)$. Then consider the map $$c : A/(f + g) \longrightarrow \prod A/\mathfrak q_j^{(m_j)}$$ where $\mathfrak q_j^{(m_j)}$ is the symbolic power, see Algebra, Section \ref{algebra-section-symbolic-power}. Since $A$ is normal, we see that $A_{\mathfrak q_i}$ is a discrete valuation ring and hence $$A_{\mathfrak q_i}/(f + g) = A_{\mathfrak q_i}/\mathfrak q_i^{m_i} A_{\mathfrak q_i} = (A/\mathfrak q_i^{(m_i)})_{\mathfrak q_i}$$ Since $V(f + g, h) = \{\mathfrak m\}$ this implies that $c$ becomes an isomorphism on inverting $h$ (small detail omitted). Since $h$ is a nonzerodivisor on $A/(f + g)$ we see that the length of $A/(f + g, h)$ equals the Herbrand quotient $e_A(A/(f + g), 0, h)$ as defined in Chow Homology, Section \ref{chow-section-periodic-complexes}). Similarly the length of $A/(h, \mathfrak q_j^{(m_j)})$ equals $e_A(A/\mathfrak q_j^{(m_j)}, 0, h)$. Then we have \begin{align*} M & = \text{length}_A(A/(f + g, h) \\ & = e_A(A/(f + g), 0, h) \\ & = \sum\nolimits_i e_A(A/\mathfrak q_j^{(m_j)}, 0, h) \\ & = \sum\nolimits_i \sum\nolimits_{m = 0, \ldots, m_j - 1} e_A(\mathfrak q_j^{(m)}/\mathfrak q_j^{(m + 1)}, 0, h) \end{align*} The equalities follow from Chow Homology, Lemma \ref{chow-lemma-periodic-length} using in particular that the cokernel of $c$ has finite length as discussed above. It is straightforward to prove that $e_A(\mathfrak q^{(m)}/\mathfrak q^{(m + 1)}, 0, h)$ is at least $1$ by Nakayama's lemma. This finishes the proof of the lemma. \end{proof} \begin{lemma} \label{lemma-flat-over-gorenstein-gorenstein-fibre} Let $A \to B$ be a flat local homomorphism of Noetherian local rings. If $A$ and $B/\mathfrak m_A B$ are Gorenstein, then $B$ is Gorenstein. \end{lemma} \begin{proof} Follows immediately from Dualizing Complexes, Lemma \ref{dualizing-lemma-flat-under-gorenstein}. \end{proof} \section{Lemmas related to ZMT} \label{section-ZMT} \noindent The lemmas in this section were originally used in the proof of the (algebraic version of) Zariski's Main Theorem, Algebra, Theorem \ref{algebra-theorem-main-theorem}. \begin{lemma} \label{lemma-make-integral-less-trivial} Let $\varphi : R \to S$ be a ring map. Suppose $t \in S$ satisfies the relation $\varphi(a_0) + \varphi(a_1)t + \ldots + \varphi(a_n) t^n = 0$. Set $u_n = \varphi(a_n)$, $u_{n-1} = u_n t + \varphi(a_{n-1})$, and so on till $u_1 = u_2 t + \varphi(a_1)$. Then all of $u_n, u_{n-1}, \ldots, u_1$ and $u_nt, u_{n-1}t, \ldots, u_1t$ are integral over $R$, and the ideals $(\varphi(a_0), \ldots, \varphi(a_n))$ and $(u_n, \ldots, u_1)$ of $S$ are equal. \end{lemma} \begin{proof} We prove this by induction on $n$. As $u_n = \varphi(a_n)$ we conclude from Algebra, Lemma \ref{algebra-lemma-make-integral-trivial} that $u_nt$ is integral over $R$. Of course $u_n = \varphi(a_n)$ is integral over $R$. Then $u_{n - 1} = u_n t + \varphi(a_{n - 1})$ is integral over $R$ (see Algebra, Lemma \ref{algebra-lemma-integral-closure-is-ring}) and we have $$\varphi(a_0) + \varphi(a_1)t + \ldots + \varphi(a_{n - 1})t^{n - 1} + u_{n - 1}t^{n - 1} = 0.$$ Hence by the induction hypothesis applied to the map $S' \to S$ where $S'$ is the integral closure of $R$ in $S$ and the displayed equation we see that $u_{n-1}, \ldots, u_1$ and $u_{n-1}t, \ldots, u_1t$ are all in $S'$ too. The statement on the ideals is immediate from the shape of the elements and the fact that $u_1t + \varphi(a_0) = 0$. \end{proof} \begin{lemma} \label{lemma-make-integral-not-in-ideal} Let $\varphi : R \to S$ be a ring map. Suppose $t \in S$ satisfies the relation $\varphi(a_0) + \varphi(a_1)t + \ldots + \varphi(a_n) t^n = 0$. Let $J \subset S$ be an ideal such that for at least one $i$ we have $\varphi(a_i) \not \in J$. Then there exists a $u \in S$, $u \not\in J$ such that both $u$ and $ut$ are integral over $R$. \end{lemma} \begin{proof} This is immediate from Lemma \ref{lemma-make-integral-less-trivial} since one of the elements $u_i$ will not be in $J$. \end{proof} \noindent The following two lemmas are a way of describing closed subschemes of $\mathbf{P}^1_R$ cut out by one (nondegenerate) equation. \begin{lemma} \label{lemma-P1} Let $R$ be a ring. Let $F(X, Y) \in R[X, Y]$ be homogeneous of degree $d$. Assume that for every prime $\mathfrak p$ of $R$ at least one coefficient of $F$ is not in $\mathfrak p$. Let $S = R[X, Y]/(F)$ as a graded ring. Then for all $n \geq d$ the $R$-module $S_n$ is finite locally free of rank $d$. \end{lemma} \begin{proof} The $R$-module $S_n$ has a presentation $$R[X, Y]_{n-d} \to R[X, Y]_n \to S_n \to 0.$$ Thus by Algebra, Lemma \ref{algebra-lemma-cokernel-flat} it is enough to show that multiplication by $F$ induces an injective map $\kappa(\mathfrak p)[X, Y] \to \kappa(\mathfrak p)[X, Y]$ for all primes $\mathfrak p$. This is clear from the assumption that $F$ does not map to the zero polynomial mod $\mathfrak p$. The assertion on ranks is clear from this as well. \end{proof} \begin{lemma} \label{lemma-rel-prime-pols} Let $k$ be a field. Let $F, G \in k[X, Y]$ be homogeneous of degrees $d, e$. Assume $F, G$ relatively prime. Then multiplication by $G$ is injective on $S = k[X, Y]/(F)$. \end{lemma} \begin{proof} This is one way to define relatively prime''. If you have another definition, then you can show it is equivalent to this one. \end{proof} \begin{lemma} \label{lemma-P1-localize} Let $R$ be a ring. Let $F(X, Y) \in R[X, Y]$ be homogeneous of degree $d$. Let $S = R[X, Y]/(F)$ as a graded ring. Let $\mathfrak p \subset R$ be a prime such that some coefficient of $F$ is not in $\mathfrak p$. There exists an $f \in R$ $f \not\in \mathfrak p$, an integer $e$, and a $G \in R[X, Y]_e$ such that multiplication by $G$ induces isomorphisms $(S_n)_f \to (S_{n + e})_f$ for all $n \geq d$. \end{lemma} \begin{proof} During the course of the proof we may replace $R$ by $R_f$ for $f\in R$, $f\not\in \mathfrak p$ (finitely often). As a first step we do such a replacement such that some coefficient of $F$ is invertible in $R$. In particular the modules $S_n$ are now locally free of rank $d$ for $n \geq d$ by Lemma \ref{lemma-P1}. Pick any $G \in R[X, Y]_e$ such that the image of $G$ in $\kappa(\mathfrak p)[X, Y]$ is relatively prime to the image of $F(X, Y)$ (this is possible for some $e$). Apply Algebra, Lemma \ref{algebra-lemma-cokernel-flat} to the map induced by multiplication by $G$ from $S_d \to S_{d + e}$. By our choice of $G$ and Lemma \ref{lemma-rel-prime-pols} we see $S_d \otimes \kappa(\mathfrak p) \to S_{d + e} \otimes \kappa(\mathfrak p)$ is bijective. Thus, after replacing $R$ by $R_f$ for a suitable $f$ we may assume that $G : S_d \to S_{d + e}$ is bijective. This in turn implies that the image of $G$ in $\kappa(\mathfrak p')[X, Y]$ is relatively prime to the image of $F$ for all primes $\mathfrak p'$ of $R$. And then by Algebra, Lemma \ref{algebra-lemma-cokernel-flat} again we see that all the maps $G : S_d \to S_{d + e}$, $n \geq d$ are isomorphisms. \end{proof} \begin{remark} \label{remark-algebra} Let $R$ be a ring. Suppose that we have $F \in R[X, Y]_d$ and $G \in R[X, Y]_e$ such that, setting $S = R[X, Y]/(F)$ we have (1) $S_n$ is finite locally free of rank $d$ for all $n \geq d$, and (2) multiplication by $G$ defines isomorphisms $S_n \to S_{n + e}$ for all $n \geq d$. In this case we may define a finite, locally free $R$-algebra $A$ as follows: \begin{enumerate} \item as an $R$-module $A = S_{ed}$, and \item multiplication $A \times A \to A$ is given by the rule that $H_1 H_2 = H_3$ if and only if $G^d H_3 = H_1 H_2$ in $S_{2ed}$. \end{enumerate} This makes sense because multiplication by $G^d$ induces a bijective map $S_{de} \to S_{2de}$. It is easy to see that this defines a ring structure. Note the confusing fact that the element $G^d$ defines the unit element of the ring $A$. \end{remark} \begin{lemma} \label{lemma-finite-after-localization} Let $R$ be a ring, let $f \in R$. Suppose we have $S$, $S'$ and the solid arrows forming the following commutative diagram of rings $$\xymatrix{ & S'' \ar@{-->}[rd] \ar@{-->}[dd] & \\ R \ar[rr] \ar@{-->}[ru] \ar[d] & & S \ar[d] \\ R_f \ar[r] & S' \ar[r] & S_f }$$ Assume that $R_f \to S'$ is finite. Then we can find a finite ring map $R \to S''$ and dotted arrows as in the diagram such that $S' = (S'')_f$. \end{lemma} \begin{proof} Namely, suppose that $S'$ is generated by $x_i$ over $R_f$, $i = 1, \ldots, w$. Let $P_i(t) \in R_f[t]$ be a monic polynomial such that $P_i(x_i) = 0$. Say $P_i$ has degree $d_i > 0$. Write $P_i(t) = t^{d_i} + \sum_{j < d_i} (a_{ij}/f^n) t^j$ for some uniform $n$. Also write the image of $x_i$ in $S_f$ as $g_i / f^n$ for suitable $g_i \in S$. Then we know that the element $\xi_i = f^{nd_i} g_i^{d_i} + \sum_{j < d_i} f^{n(d_i - j)} a_{ij} g_i^j$ of $S$ is killed by a power of $f$. Hence upon increasing $n$ to $n'$, which replaces $g_i$ by $f^{n' - n}g_i$ we may assume $\xi_i = 0$. Then $S'$ is generated by the elements $f^n x_i$, each of which is a zero of the monic polynomial $Q_i(t) = t^{d_i} + \sum_{j < d_i} f^{n(d_i - j)} a_{ij} t^j$ with coefficients in $R$. Also, by construction $Q_i(f^ng_i) = 0$ in $S$. Thus we get a finite $R$-algebra $S'' = R[z_1, \ldots, z_w]/(Q_1(z_1), \ldots, Q_w(z_w))$ which fits into a commutative diagram as above. The map $\alpha : S'' \to S$ maps $z_i$ to $f^ng_i$ and the map $\beta : S'' \to S'$ maps $z_i$ to $f^nx_i$. It may not yet be the case that $\beta$ induces an isomorphism $(S'')_f \cong S'$. For the moment we only know that this map is surjective. The problem is that there could be 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 $J = \{h \in S'' \mid \alpha(h) = 0 \text{ and } \beta(h) = 0\}$ of $S''$. OK, and it is easy to see that $S''/J$ does the job. \end{proof} \section{Formally smooth ring maps} \label{section-formally-smooth} \begin{lemma} \label{lemma-formally-smooth-smooth} Let $R$ be a ring. Let $S$ be a $R$-algebra. If $S$ is of finite presentation and formally smooth over $R$ then $S$ is smooth over $R$. \end{lemma} \begin{proof} See Algebra, Proposition \ref{algebra-proposition-smooth-formally-smooth}. \end{proof} \section{Cohomology} \label{section-cohomology} \noindent The following lemma computes the cohomology sheaves of the derived limit in a special case. \begin{lemma} \label{lemma-Rlim-of-system} Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K_n)$ be an inverse system of objects of $D(\mathcal{O})$. Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset. Let $d \in \mathbf{N}$. Assume \begin{enumerate} \item $K_n$ is an object of $D^+(\mathcal{O})$ for all $n$, \item for $q \in \mathbf{Z}$ there exists $n(q)$ such that $H^q(K_{n + 1}) \to H^q(K_n)$ is an isomorphism for $n \geq n(q)$, \item every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$, \item for every $U \in \mathcal{B}$ we have $H^p(U, H^q(K_n)) = 0$ for $p > d$ and all $q$. \end{enumerate} Then we have $H^m(R\lim K_n) = \lim H^m(K_n)$ for all $m \in \mathbf{Z}$. \end{lemma} \begin{proof} Set $K = R\lim K_n$. Let $U \in \mathcal{B}$. For each $n$ there is a spectral sequence $$H^p(U, H^q(K_n)) \Rightarrow H^{p + q}(U, K_n)$$ which converges as $K_n$ is bounded below, see Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}. If we fix $m \in \mathbf{Z}$, then we see from our assumption (4) that only $H^p(U, H^q(K_n))$ contribute to $H^m(U, K_n)$ for $0 \leq p \leq d$ and $m - d \leq q \leq m$. By assumption (2) this implies that $H^m(U, K_{n + 1}) \to H^m(U, K_n)$ is an isomorphism as soon as $n \geq \max{n(m), \ldots, n(m - d)}$. The functor $R\Gamma(U, -)$ commutes with derived limits by Injectives, Lemma \ref{injectives-lemma-RF-commutes-with-Rlim}. Thus we have $$H^m(U, K) = H^m(R\lim R\Gamma(U, K_n))$$ On the other hand we have just seen that the complexes $R\Gamma(U, K_n)$ have eventually constant cohomology groups. Thus by More on Algebra, Remark \ref{more-algebra-remark-compare-derived-limit} we find that $H^m(U, K)$ is equal to $H^m(U, K_n)$ for all $n \gg 0$ for some bound independent of $U \in \mathcal{B}$. Pick such an $n$. Finally, recall that $H^m(K)$ is the sheafification of the presheaf $U \mapsto H^m(U, K)$ and $H^m(K_n)$ is the sheafification of the presheaf $U \mapsto H^m(U, K_n)$. On the elements of $\mathcal{B}$ these presheaves have the same values. Therefore assumption (3) guarantees that the sheafifications are the same too. The lemma follows. \end{proof} \section{Simplicial methods} \label{section-simplicial} \begin{lemma} \label{lemma-equiv} Assumptions and notation as in Simplicial, Lemma \ref{simplicial-lemma-section}. There exists a section $g : U \to V$ to the morphism $f$ and the composition $g \circ f$ is homotopy equivalent to the identity on $V$. In particular, the morphism $f$ is a homotopy equivalence. \end{lemma} \begin{proof} Immediate from Simplicial, Lemmas \ref{simplicial-lemma-section} and \ref{simplicial-lemma-trivial-kan-homotopy}. \end{proof} \begin{lemma} \label{lemma-cosk-hom-deltak} Let $\mathcal{C}$ be a category with finite coproducts and finite limits. Let $X$ be an object of $\mathcal{C}$. Let $k \geq 0$. The canonical map $$\Hom(\Delta[k], X) \longrightarrow \text{cosk}_1 \text{sk}_1 \Hom(\Delta[k], X)$$ is an isomorphism. \end{lemma} \begin{proof} For any simplicial object $V$ we have \begin{eqnarray*} \Mor(V, \text{cosk}_1 \text{sk}_1 \Hom(\Delta[k], X)) & = & \Mor(\text{sk}_1 V, \text{sk}_1 \Hom(\Delta[k], X)) \\ & = & \Mor(i_{1!} \text{sk}_1 V, \Hom(\Delta[k], X)) \\ & = & \Mor(i_{1!} \text{sk}_1 V \times \Delta[k], X) \end{eqnarray*} The first equality by the adjointness of $\text{sk}$ and $\text{cosk}$, the second equality by the adjointness of $i_{1!}$ and $\text{sk}_1$, and the first equality by Simplicial, Definition \ref{simplicial-definition-hom-from-simplicial-set} where the last $X$ denotes the constant simplicial object with value $X$. By Simplicial, Lemma \ref{simplicial-lemma-augmentation-howto} an element in this set depends only on the terms of degree $0$ and $1$ of $i_{1!} \text{sk}_1 V \times \Delta[k]$. These agree with the degree $0$ and $1$ terms of $V \times \Delta[k]$, see Simplicial, Lemma \ref{simplicial-lemma-recovering-U-for-real}. Thus the set above is equal to $\Mor(V \times \Delta[k], X) = \Mor(V, \Hom(\Delta[k], X))$. \end{proof} \begin{lemma} \label{lemma-cosk0-hom-deltak} Let $\mathcal{C}$ be a category. Let $X$ be an object of $\mathcal{C}$ such that the self products $X \times \ldots \times X$ exist. Let $k \geq 0$ and let $C[k]$ be as in Simplicial, Example \ref{simplicial-example-simplex-cosimplicial-set}. With notation as in Simplicial, Lemma \ref{simplicial-lemma-morphism-into-product} the canonical map $$\Hom(C[k], X)_1 \longrightarrow (\text{cosk}_0 \text{sk}_0 \Hom(C[k], X))_1$$ is identified with the map $$\prod\nolimits_{\alpha : [k] \to [1]} X \longrightarrow X \times X$$ which is the projection onto the factors where $\alpha$ is a constant map. \end{lemma} \begin{proof} This is shown in the proof of Hypercoverings, Lemma \ref{hypercovering-lemma-covering}. \end{proof} \section{Obsolete lemmas on schemes} \label{section-devissage} \noindent Lemmas that seem superfluous. \begin{lemma} \label{lemma-stein-projective} Let $(R, \mathfrak m, \kappa)$ be a local ring. Let $X \subset \mathbf{P}^n_R$ be a closed subscheme. Assume that $R = \Gamma(X, \mathcal{O}_X)$. Then the special fibre $X_k$ is geometrically connected. \end{lemma} \begin{proof} This is a special case of More on Morphisms, Theorem \ref{more-morphisms-theorem-stein-factorization-general}. \end{proof} \begin{lemma} \label{lemma-property-irreducible-higher-rank} Let $X$ be a Noetherian scheme. Let $Z_0 \subset X$ be an irreducible closed subset with generic point $\xi$. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that \begin{enumerate} \item For any short exact sequence of coherent sheaves if two out of three of them have property $\mathcal{P}$ then so does the third. \item If $\mathcal{P}$ holds for a direct sum of coherent sheaves then it holds for both. \item For every integral closed subscheme $Z \subset Z_0 \subset X$, $Z \not = Z_0$ and every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_Z$ we have $\mathcal{P}$ for $(Z \to X)_*\mathcal{I}$. \item There exists some coherent sheaf $\mathcal{G}$ on $X$ such that \begin{enumerate} \item $\text{Supp}(\mathcal{G}) = Z_0$, \item $\mathcal{G}_\xi$ is annihilated by $\mathfrak m_\xi$, and \item property $\mathcal{P}$ holds for $\mathcal{G}$. \end{enumerate} \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z_0$. \end{lemma} \begin{proof} The proof is a variant on the proof of Cohomology of Schemes, Lemma \ref{coherent-lemma-property-irreducible}. In exactly the same manner as in that proof we see that any coherent sheaf whose support is strictly contained in $Z_0$ has property $\mathcal{P}$. \medskip\noindent Consider a coherent sheaf $\mathcal{G}$ as in (3). By Cohomology of Schemes, Lemma \ref{coherent-lemma-prepare-filter-irreducible} there exists a sheaf of ideals $\mathcal{I}$ on $Z_0$ and a short exact sequence $$0 \to \left((Z_0 \to X)_*\mathcal{I}\right)^{\oplus r} \to \mathcal{G} \to \mathcal{Q} \to 0$$ where the support of $\mathcal{Q}$ is strictly contained in $Z_0$. In particular $r > 0$ and $\mathcal{I}$ is nonzero because the support of $\mathcal{G}$ is equal to $Z$. Since $\mathcal{Q}$ has property $\mathcal{P}$ we conclude that also $\left((Z_0 \to X)_*\mathcal{I}\right)^{\oplus r}$ has property $\mathcal{P}$. By (2) we deduce property $\mathcal{P}$ for $(Z_0 \to X)_*\mathcal{I}$. Slotting this into the proof of Cohomology of Schemes, Lemma \ref{coherent-lemma-property-irreducible} at the appropriate point gives the lemma. Some details omitted. \end{proof} \begin{lemma} \label{lemma-property-higher-rank} Let $X$ be a Noetherian scheme. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that \begin{enumerate} \item For any short exact sequence of coherent sheaves if two out of three of them have property $\mathcal{P}$ then so does the third. \item If $\mathcal{P}$ holds for a direct sum of coherent sheaves then it holds for both. \item For every integral closed subscheme $Z \subset X$ with generic point $\xi$ there exists some coherent sheaf $\mathcal{G}$ such that \begin{enumerate} \item $\text{Supp}(\mathcal{G}) = Z$, \item $\mathcal{G}_\xi$ is annihilated by $\mathfrak m_\xi$, and \item property $\mathcal{P}$ holds for $\mathcal{G}$. \end{enumerate} \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf on $X$. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-property-irreducible-higher-rank} in exactly the same way that Cohomology of Schemes, Lemma \ref{coherent-lemma-property} follows from Cohomology of Schemes, Lemma \ref{coherent-lemma-property-irreducible}. \end{proof} \begin{lemma} \label{lemma-section-maps-back-into} Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$ be a section. Let $\mathcal{F}' \subset \mathcal{F}$ be quasi-coherent $\mathcal{O}_X$-modules. Assume that \begin{enumerate} \item $X$ is quasi-compact, \item $\mathcal{F}$ is of finite type, and \item $\mathcal{F}'|_{X_s} = \mathcal{F}|_{X_s}$. \end{enumerate} Then there exists an $n \geq 0$ such that multiplication by $s^n$ on $\mathcal{F}$ factors through $\mathcal{F}'$. \end{lemma} \begin{proof} In other words we claim that $s^n\mathcal{F} \subset \mathcal{F}' \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}$ for some $n \geq 0$. In other words, we claim that the quotient map $\mathcal{F} \to \mathcal{F}/\mathcal{F}'$ becomes zero after multiplying by a power of $s$. This follows from Properties, Lemma \ref{properties-lemma-section-maps-backwards}. \end{proof} \section{Functor of quotients} \label{section-quotients} \begin{lemma} \label{lemma-factors-through-quotient} Let $S = \Spec(R)$ be an affine scheme. Let $X$ be an algebraic space over $S$. Let $q_i : \mathcal{F} \to \mathcal{Q}_i$, $i = 1, 2$ be surjective maps of quasi-coherent $\mathcal{O}_X$-modules. Assume $\mathcal{Q}_1$ flat over $S$. Let $T \to S$ be a quasi-compact morphism of schemes such that there exists a factorization $$\xymatrix{ & \mathcal{F}_T \ar[rd]^{q_{2, T}} \ar[ld]_{q_{1, T}} \\ \mathcal{Q}_{1, T} & & \mathcal{Q}_{2, T} \ar@{..>}[ll] }$$ Then exists a closed subscheme $Z \subset S$ such that (a) $T \to S$ factors through $Z$ and (b) $q_{1, Z}$ factors through $q_{2, Z}$. If $\Ker(q_2)$ is a finite type $\mathcal{O}_X$-module and $X$ quasi-compact, then we can take $Z \to S$ of finite presentation. \end{lemma} \begin{proof} Apply Flatness on Spaces, Lemma \ref{spaces-flat-lemma-F-zero-somewhat-closed} to the map $\Ker(q_2) \to \mathcal{Q}_1$. \end{proof} \section{Spaces and fpqc coverings} \label{section-fpqc} \noindent The material here was made obsolete by Gabber's argument showing that algebraic spaces satisfy the sheaf condition with respect to fpqc coverings. Please visit Properties of Spaces, Section \ref{spaces-properties-section-fpqc}. \begin{lemma} \label{lemma-separated-fpqc} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\{f_i : T_i \to T\}_{i \in I}$ be a fpqc covering of schemes over $S$. Then the map $$\Mor_S(T, X) \longrightarrow \prod\nolimits_{i \in I} \Mor_S(T_i, X)$$ is injective. \end{lemma} \begin{proof} Immediate consequence of Properties of Spaces, Proposition \ref{spaces-properties-proposition-sheaf-fpqc}. \end{proof} \begin{lemma} \label{lemma-sheaf-fpqc-open-covering} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $X = \bigcup_{j \in J} X_j$ be a Zariski covering, see Spaces, Definition \ref{spaces-definition-Zariski-open-covering}. If each $X_j$ satisfies the sheaf property for the fpqc topology then $X$ satisfies the sheaf property for the fpqc topology. \end{lemma} \begin{proof} This is true because all algebraic spaces satisfy the sheaf property for the fpqc topology, see Properties of Spaces, Proposition \ref{spaces-properties-proposition-sheaf-fpqc}. \end{proof} \begin{lemma} \label{lemma-sheaf-fpqc-quasi-separated} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is Zariski locally quasi-separated over $S$, then $X$ satisfies the sheaf condition for the fpqc topology. \end{lemma} \begin{proof} Immediate consequence of the general Properties of Spaces, Proposition \ref{spaces-properties-proposition-sheaf-fpqc}. \end{proof} \begin{remark} \label{remark-proof-works-when} This remark used to discuss to what extend the original proof of Lemma \ref{lemma-sheaf-fpqc-quasi-separated} (of December 18, 2009) generalizes. \end{remark} \section{Very reasonable algebraic spaces} \label{section-very-reasonable} \noindent Material that is somewhat obsolete. \begin{lemma} \label{lemma-reasonable-kolmogorov} Let $S$ be a scheme. Let $X$ be a reasonable algebraic space over $S$. Then $|X|$ is Kolmogorov (see Topology, Definition \ref{topology-definition-generic-point}). \end{lemma} \begin{proof} Follows from the definitions and Decent Spaces, Lemma \ref{decent-spaces-lemma-kolmogorov}. \end{proof} \noindent In the rest of this section we make some remarks about very reasonable algebraic spaces. If there exists a scheme $U$ and a surjective, \'etale, quasi-compact morphism $U \to X$, then $X$ is very reasonable, see Decent Spaces, Lemma \ref{decent-spaces-lemma-characterize-very-reasonable}. \begin{lemma} \label{lemma-scheme-very-reasonable} A scheme is very reasonable. \end{lemma} \begin{proof} This is true because the identity map is a quasi-compact, surjective \'etale morphism. \end{proof} \begin{lemma} \label{lemma-very-reasonable-Zariski-local} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If there exists a Zariski open covering $X = \bigcup X_i$ such that each $X_i$ is very reasonable, then $X$ is very reasonable. \end{lemma} \begin{proof} This is case $(\epsilon)$ of Decent Spaces, Lemma \ref{decent-spaces-lemma-properties-local}. \end{proof} \begin{lemma} \label{lemma-quasi-separated-very-reasonable} An algebraic space which is Zariski locally quasi-separated is very reasonable. In particular any quasi-separated algebraic space is very reasonable. \end{lemma} \begin{proof} This is one of the implications of Decent Spaces, Lemma \ref{decent-spaces-lemma-bounded-fibres}. \end{proof} \begin{lemma} \label{lemma-representable-very-reasonable} Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $Y \to X$ be a representable morphism. If $X$ is very reasonable, so is $Y$. \end{lemma} \begin{proof} This is case $(\epsilon)$ of Decent Spaces, Lemma \ref{decent-spaces-lemma-representable-properties}. \end{proof} \begin{remark} \label{remark-very-reasonable-Zariski-locally-quasi-separated} Very reasonable algebraic spaces form a strictly larger collection than Zariski locally quasi-separated algebraic spaces. Consider an algebraic space of the form $X = [U/G]$ (see Spaces, Definition \ref{spaces-definition-quotient}) where $G$ is a finite group acting without fixed points on a non-quasi-separated scheme $U$. Namely, in this case $U \times_X U = U \times G$ and clearly both projections to $U$ are quasi-compact, hence $X$ is very reasonable. On the other hand, the diagonal $U \times_X U \to U \times U$ is not quasi-compact, hence this algebraic space is not quasi-separated. Now, take $U$ the infinite affine space over a field $k$ of characteristic $\not = 2$ with zero doubled, see Schemes, Example \ref{schemes-example-not-quasi-separated}. Let $0_1, 0_2$ be the two zeros of $U$. Let $G = \{+1, -1\}$, and let $-1$ act by $-1$ on all coordinates, and by switching $0_1$ and $0_2$. Then $[U/G]$ is very reasonable but not Zariski locally quasi-separated (details omitted). \end{remark} \noindent Warning: The following lemma should be used with caution, as the schemes $U_i$ in it are not necessarily separated or even quasi-separated. \begin{lemma} \label{lemma-very-reasonable-quasi-compact-pieces} Let $S$ be a scheme. Let $X$ be a very reasonable algebraic space over $S$. There exists a set of schemes $U_i$ and morphisms $U_i \to X$ such that \begin{enumerate} \item each $U_i$ is a quasi-compact scheme, \item each $U_i \to X$ is \'etale, \item both projections $U_i \times_X U_i \to U_i$ are quasi-compact, and \item the morphism $\coprod U_i \to X$ is surjective (and \'etale). \end{enumerate} \end{lemma} \begin{proof} Decent Spaces, Definition \ref{decent-spaces-definition-very-reasonable} says that there exist $U_i \to X$ such that (2), (3) and (4) hold. Fix $i$, and set $R_i = U_i \times_X U_i$, and denote $s, t : R_i \to U_i$ the projections. For any affine open $W \subset U_i$ the open $W' = t(s^{-1}(W)) \subset U_i$ is a quasi-compact $R_i$-invariant open (see Groupoids, Lemma \ref{groupoids-lemma-constructing-invariant-opens}). Hence $W'$ is a quasi-compact scheme, $W' \to X$ is \'etale, and $W' \times_X W' = s^{-1}(W') = t^{-1}(W')$ so both projections $W' \times_X W' \to W'$ are quasi-compact. This means the family of $W' \to X$, where $W \subset U_i$ runs through the members of affine open coverings of the $U_i$ gives what we want. \end{proof} \section{Obsolete lemma on algebraic spaces} \label{section-obsolete-on-spaces} \noindent Lemmas that seem superfluous. \begin{lemma} \label{lemma-vanishing-surjective} In Cohomology of Spaces, Situation \ref{spaces-cohomology-situation-vanishing} the morphism $p : X \to \Spec(A)$ is surjective. \end{lemma} \begin{proof} This lemma was originally used in the proof of Cohomology of Spaces, Proposition \ref{spaces-cohomology-proposition-vanishing-affine} but now is a consequence of it. \end{proof} \begin{lemma} \label{lemma-vanishing-universally-closed} In Cohomology of Spaces, Situation \ref{spaces-cohomology-situation-vanishing} the morphism $p : X \to \Spec(A)$ is universally closed. \end{lemma} \begin{proof} This lemma was originally used in the proof of Cohomology of Spaces, Proposition \ref{spaces-cohomology-proposition-vanishing-affine} but now is a consequence of it. \end{proof} \section{Variants of cotangent complexes for schemes} \label{section-cotangent-schemes-variant} \noindent This section gives an alternative construction of the cotangent complex of a morphism of schemes. This section is currently in the obsolete chapter as we can get by with the easier version discussed in Cotangent, Section \ref{cotangent-section-cotangent-schemes-variant} for applications. \medskip\noindent Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{C}_{X/Y}$ be the category whose objects are commutative diagrams \begin{equation} \label{equation-object} \vcenter{ \xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_i & A \ar[ld] \\ Y & V \ar[l] } } \end{equation} of schemes where \begin{enumerate} \item $U$ is an open subscheme of $X$, \item $V$ is an open subscheme of $Y$, and \item there exists an isomorphism $A = V \times \Spec(P)$ over $V$ where $P$ is a polynomial algebra over $\mathbf{Z}$ (on some set of variables). \end{enumerate} In other words, $A$ is an (infinite dimensional) affine space over $V$. Morphisms are given by commutative diagrams. \medskip\noindent {\bf Notation.} An object of $\mathcal{C}_{X/Y}$, i.e., a diagram (\ref{equation-object}), is often denoted $U \to A$ where it is understood that (a) $U$ is an open subscheme of $X$, (b) $U \to A$ is a morphism over $Y$, (c) the image of the structure morphism $A \to Y$ is an open $V \subset Y$, and (d) $A \to V$ is an affine space. We'll write $U \to A/V$ to indicate $V \subset Y$ is the image of $A \to Y$. Recall that $X_{Zar}$ denotes the small Zariski site $X$. There are forgetful functors $$\mathcal{C}_{X/Y} \to X_{Zar},\ (U \to A) \mapsto U \quad\text{and}\quad \mathcal{C}_{X/Y} \mapsto Y_{Zar},\ (U \to A/V) \mapsto V.$$ \begin{lemma} \label{lemma-category-fibred} Let $X \to Y$ be a morphism of schemes. \begin{enumerate} \item The category $\mathcal{C}_{X/Y}$ is fibred over $X_{Zar}$. \item The category $\mathcal{C}_{X/Y}$ is fibred over $Y_{Zar}$. \item The category $\mathcal{C}_{X/Y}$ is fibred over the category of pairs $(U, V)$ where $U \subset X$, $V \subset Y$ are open and $f(U) \subset V$. \end{enumerate} \end{lemma} \begin{proof} Ad (1). Given an object $U \to A$ of $\mathcal{C}_{X/Y}$ and a morphism $U' \to U$ of $X_{Zar}$ consider the object $i' : U' \to A$ of $\mathcal{C}_{X/Y}$ where $i'$ is the composition of $i$ and $U' \to U$. The morphism $(U' \to A) \to (U \to A)$ of $\mathcal{C}_{X/Y}$ is strongly cartesian over $X_{Zar}$. \medskip\noindent Ad (2). Given an object $U \to A/V$ and $V' \to V$ we can set $U' = U \cap f^{-1}(V')$ and $A' = V' \times_V A$ to obtain a strongly cartesian morphism $(U' \to A') \to (U \to A)$ over $V' \to V$. \medskip\noindent Ad (3). Denote $(X/Y)_{Zar}$ the category in (3). Given $U \to A/V$ and a morphism $(U', V') \to (U, V)$ in $(X/Y)_{Zar}$ we can consider $A' = V' \times_V A$. Then the morphism $(U' \to A'/V') \to (U \to A/V)$ is strongly cartesian in $\mathcal{C}_{X/Y}$ over $(X/Y)_{Zar}$. \end{proof} \noindent We obtain a topology $\tau_X$ on $\mathcal{C}_{X/Y}$ by using the topology inherited from $X_{Zar}$ (see Stacks, Section \ref{stacks-section-topology}). If not otherwise stated this is the topology on $\mathcal{C}_{X/Y}$ we will consider. To be precise, a family of morphisms $\{(U_i \to A_i) \to (U \to A)\}$ is a covering of $\mathcal{C}_{X/Y}$ if and only if \begin{enumerate} \item $U = \bigcup U_i$, and \item $A_i = A$ for all $i$. \end{enumerate} We obtain the same collection of sheaves if we allow $A_i \cong A$ in (2). The functor $u$ defines a morphism of topoi $\pi : \Sh(\mathcal{C}_{X/Y}) \to \Sh(X_{Zar})$. \medskip\noindent The site $\mathcal{C}_{X/Y}$ comes with several sheaves of rings. \begin{enumerate} \item The sheaf $\mathcal{O}$ given by the rule $(U \to A) \mapsto \mathcal{O}(A)$. \item The sheaf $\underline{\mathcal{O}}_X = \pi^{-1}\mathcal{O}_X$ given by the rule $(U \to A) \mapsto \mathcal{O}(U)$. \item The sheaf $\underline{\mathcal{O}}_Y$ given by the rule $(U \to A/V) \mapsto \mathcal{O}(V)$. \end{enumerate} We obtain morphisms of ringed topoi \begin{equation} \label{equation-pi-schemes} \vcenter{ \xymatrix{ (\Sh(\mathcal{C}_{X/Y}), \underline{\mathcal{O}}_X) \ar[r]_i \ar[d]_\pi & (\Sh(\mathcal{C}_{X/Y}), \mathcal{O}) \\ (\Sh(X_{Zar}), \mathcal{O}_X) } } \end{equation} The morphism $i$ is the identity on underlying topoi and $i^\sharp : \mathcal{O} \to \underline{\mathcal{O}}_X$ is the obvious map. The map $\pi$ is a special case of Cohomology on Sites, Situation \ref{sites-cohomology-situation-fibred-category}. An important role will be played in the following by the derived functors $Li^* : D(\mathcal{O}) \longrightarrow D(\underline{\mathcal{O}}_X)$ left adjoint to $Ri_* = i_* : D(\underline{\mathcal{O}}_X) \to D(\mathcal{O})$ and $L\pi_! : D(\underline{\mathcal{O}}_X) \longrightarrow D(\mathcal{O}_X)$ left adjoint to $\pi^* = \pi^{-1} : D(\mathcal{O}_X) \to D(\underline{\mathcal{O}}_X)$. \begin{remark} \label{remark-different-topologies} We obtain a second topology $\tau_Y$ on $\mathcal{C}_{X/Y}$ by taking the topology inherited from $Y_{Zar}$. There is a third topology $\tau_{X \to Y}$ where a family of morphisms $\{(U_i \to A_i) \to (U \to A)\}$ is a covering if and only if $U = \bigcup U_i$, $V = \bigcup V_i$ and $A_i \cong V_i \times_V A$. This is the topology inherited from the topology on the site $(X/Y)_{Zar}$ whose underlying category is the category of pairs $(U, V)$ as in Lemma \ref{lemma-category-fibred} part (3). The coverings of $(X/Y)_{Zar}$ are families $\{(U_i, V_i) \to (U, V)\}$ such that $U = \bigcup U_i$ and $V = \bigcup V_i$. There are morphisms of topoi $$\xymatrix{ \Sh(\mathcal{C}_{X/Y}) = \Sh(\mathcal{C}_{X/Y}, \tau_X) & \Sh(\mathcal{C}_{X/Y}, \tau_{X \to Y}) \ar[l] \ar[r] & \Sh(\mathcal{C}_{X/Y}, \tau_Y) }$$ (recall that $\tau_X$ is our default'' topology). The pullback functors for these arrows are sheafification and pushforward is the identity on underlying presheaves. The diagram of topoi $$\xymatrix{ \Sh(X_{Zar}) \ar[d]^f & \Sh(\mathcal{C}_{X/Y}) \ar[l]^\pi & \Sh(\mathcal{C}_{X/Y}, \tau_{X \to Y}) \ar[l] \ar[d] \\ \Sh(Y_{Zar}) & & \Sh(\mathcal{C}_{X/Y}, \tau_Y) \ar[ll] }$$ is {\bf not} commutative. Namely, the pullback of a nonzero abelian sheaf on $Y$ is a nonzero abelian sheaf on $(\mathcal{C}_{X/Y}, \tau_{X \to Y})$, but we can certainly find examples where such a sheaf pulls back to zero on $X$. Note that any presheaf $\mathcal{F}$ on $Y_{Zar}$ gives a sheaf $\underline{\mathcal{F}}$ on $\mathcal{C}_{Y/X}$ by the rule which assigns to $(U \to A/V)$ the set $\mathcal{F}(V)$. Even if $\mathcal{F}$ happens to be a sheaf it isn't true in general that $\underline{\mathcal{F}} = \pi^{-1}f^{-1}\mathcal{F}$. This is related to the noncommutativity of the diagram above, as we can describe $\underline{\mathcal{F}}$ as the pushforward of the pullback of $\mathcal{F}$ to $\Sh(\mathcal{C}_{X/Y}, \tau_{X \to Y})$ via the lower horizontal and right vertical arrows. An example is the sheaf $\underline{\mathcal{O}}_Y$. But what is true is that there is a map $\underline{\mathcal{F}} \to \pi^{-1}f^{-1}\mathcal{F}$ which is transformed (as we shall see later) into an isomorphism after applying $\pi_!$. \end{remark} \section{Deformations and obstructions of flat modules} \label{section-modules} \noindent In this section we sketch a construction of a deformation theory for the stack of coherent sheaves for any algebraic space $X$ over a ring $\Lambda$. This material is obsolete due to the improved discussion in Quot, Section \ref{quot-section-not-flat}. \medskip\noindent Our setup will be the following. We assume given \begin{enumerate} \item a ring $\Lambda$, \item an algebraic space $X$ over $\Lambda$, \item a $\Lambda$-algebra $A$, set $X_A = X \times_{\Spec(\Lambda)} \Spec(A)$, and \item a finitely presented $\mathcal{O}_{X_A}$-module $\mathcal{F}$ flat over $A$. \end{enumerate} In this situation we will consider all possible surjections $$0 \to I \to A' \to A \to 0$$ where $A'$ is a $\Lambda$-algebra whose kernel $I$ is an ideal of square zero in $A'$. Given $A'$ we obtain a first order thickening $X_A \to X_{A'}$ of algebraic spaces over $\Spec(\Lambda)$. For each of these we consider the problem of lifting $\mathcal{F}$ to a finitely presented module $\mathcal{F}'$ on $X_{A'}$ flat over $A'$. We would like to replicate the results of Deformation Theory, Lemma \ref{defos-lemma-flat-ringed-topoi} in this setting. \medskip\noindent To be more precise let $\textit{Lift}(\mathcal{F}, A')$ denote the category of pairs $(\mathcal{F}', \alpha)$ where $\mathcal{F}'$ is a finitely presented module on $X_{A'}$ flat over $A'$ and $\alpha : \mathcal{F}'|_{X_A} \to \mathcal{F}$ is an isomorphism. Morphisms $(\mathcal{F}'_1, \alpha_1) \to (\mathcal{F}'_2, \alpha_2)$ are isomorphisms $\mathcal{F}'_1 \to \mathcal{F}'_2$ which are compatible with $\alpha_1$ and $\alpha_2$. The set of isomorphism classes of $\textit{Lift}(\mathcal{F}, A')$ is denoted $\text{Lift}(\mathcal{F}, A')$. \medskip\noindent Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_X \otimes_\Lambda A$-modules on $X_\etale$ flat over $A$. We introduce the category $\textit{Lift}(\mathcal{G}, A')$ of pairs $(\mathcal{G}', \beta)$ where $\mathcal{G}'$ is a sheaf of $\mathcal{O}_X \otimes_\Lambda A'$-modules flat over $A'$ and $\beta$ is an isomorphism $\mathcal{G}' \otimes_{A'} A \to \mathcal{G}$. \begin{lemma} \label{lemma-equivalence} Notation and assumptions as above. Let $p : X_A \to X$ denote the projection. Given $A'$ denote $p' : X_{A'} \to X$ the projection. The functor $p'_*$ induces an equivalence of categories between \begin{enumerate} \item the category $\textit{Lift}(\mathcal{F}, A')$, and \item the category $\textit{Lift}(p_*\mathcal{F}, A')$. \end{enumerate} \end{lemma} \begin{proof} FIXME. \end{proof} \noindent Let $\mathcal{H}$ be a sheaf of $\mathcal{O} \otimes_\Lambda A$-modules on $\mathcal{C}_{X/\Lambda}$ flat over $A$. We introduce the category $\textit{Lift}_\mathcal{O}(\mathcal{H}, A')$ whose objects are pairs $(\mathcal{H}', \gamma)$ where $\mathcal{H}'$ is a sheaf of $\mathcal{O} \otimes_\Lambda A'$-modules flat over $A'$ and $\gamma : \mathcal{H}' \otimes_A A' \to \mathcal{H}$ is an isomorphism of $\mathcal{O} \otimes_\Lambda A$-modules. \medskip\noindent Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_X \otimes_\Lambda A$-modules on $X_\etale$ flat over $A$. Consider the morphisms $i$ and $\pi$ of Cotangent, Equation (\ref{cotangent-equation-pi-spaces}). Denote $\underline{\mathcal{G}} = \pi^{-1}(\mathcal{G})$. It is simply given by the rule $(U \to \mathbf{A}) \mapsto \mathcal{G}(U)$ hence it is a sheaf of $\underline{\mathcal{O}}_X \otimes_\Lambda A$-modules. Denote $i_*\underline{\mathcal{G}}$ the same sheaf but viewed as a sheaf of $\mathcal{O} \otimes_\Lambda A$-modules. \begin{lemma} \label{lemma-second-equivalence} Notation and assumptions as above. The functor $\pi_!$ induces an equivalence of categories between \begin{enumerate} \item the category $\textit{Lift}_\mathcal{O}(i_*\underline{\mathcal{G}}, A')$, and \item the category $\textit{Lift}(\mathcal{G}, A')$. \end{enumerate} \end{lemma} \begin{proof} FIXME. \end{proof} \begin{lemma} \label{lemma-second-equivalence-obs} Notation and assumptions as in Lemma \ref{lemma-second-equivalence}. Consider the object $$L = L(\Lambda, X, A, \mathcal{G}) = L\pi_!(Li^*(i_*(\underline{\mathcal{G}})))$$ of $D(\mathcal{O}_X \otimes_\Lambda A)$. Given a surjection $A' \to A$ of $\Lambda$-algebras with square zero kernel $I$ we have \begin{enumerate} \item The category $\textit{Lift}(\mathcal{G}, A')$ is nonempty if and only if a certain class $\xi \in \text{Ext}^2_{\mathcal{O}_X \otimes A}(L, \mathcal{G} \otimes_A I)$ is zero. \item If $\textit{Lift}(\mathcal{G}, A')$ is nonempty, then $\text{Lift}(\mathcal{G}, A')$ is principal homogeneous under $\text{Ext}^1_{\mathcal{O}_X \otimes A}(L, \mathcal{G} \otimes_A I)$. \item Given a lift $\mathcal{G}'$, the set of automorphisms of $\mathcal{G}'$ which pull back to $\text{id}_\mathcal{G}$ is canonically isomorphic to $\text{Ext}^0_{\mathcal{O}_X \otimes A}(L, \mathcal{G} \otimes_A I)$. \end{enumerate} \end{lemma} \begin{proof} FIXME. \end{proof} \noindent Finally, we put everything together as follows. \begin{proposition} \label{proposition-conclusion} With $\Lambda$, $X$, $A$, $\mathcal{F}$ as above. There exists a canonical object $L = L(\Lambda, X, A, \mathcal{F})$ of $D(X_A)$ such that given a surjection $A' \to A$ of $\Lambda$-algebras with square zero kernel $I$ we have \begin{enumerate} \item The category $\textit{Lift}(\mathcal{F}, A')$ is nonempty if and only if a certain class $\xi \in \text{Ext}^2_{X_A}(L, \mathcal{F} \otimes_A I)$ is zero. \item If $\textit{Lift}(\mathcal{F}, A')$ is nonempty, then $\text{Lift}(\mathcal{F}, A')$ is principal homogeneous under $\text{Ext}^1_{X_A}(L, \mathcal{F} \otimes_A I)$. \item Given a lift $\mathcal{F}'$, the set of automorphisms of $\mathcal{F}'$ which pull back to $\text{id}_\mathcal{F}$ is canonically isomorphic to $\text{Ext}^0_{X_A}(L, \mathcal{F} \otimes_A I)$. \end{enumerate} \end{proposition} \begin{proof} FIXME. \end{proof} \begin{lemma} \label{lemma-pseudo-coherent} In the situation of Proposition \ref{proposition-conclusion}, if $X \to \Spec(\Lambda)$ is locally of finite type and $\Lambda$ is Noetherian, then $L$ is pseudo-coherent. \end{lemma} \begin{proof} FIXME. \end{proof} \section{The stack of coherent sheaves in the non-flat case} \label{section-not-flat} \noindent In Quot, Theorem \ref{quot-theorem-coherent-algebraic} the assumption that $f : X \to B$ is flat is not necessary. In this section we modify the method of proof based on ideas from derived algebraic geometry to get around the flatness hypothesis. An entirely different method is used in Quot, Section \ref{quot-section-not-flat} to get exactly the same result; this is why the method from this section is obsolete. \medskip\noindent The only step in the proof of Quot, Theorem \ref{quot-theorem-coherent-algebraic} which uses flatness is in the application of Quot, Lemma \ref{quot-lemma-coherent-defo-thy}. The lemma is used to construct an obstruction theory as in Artin's Axioms, Section \ref{artin-section-dual}. The proof of the lemma relies on Deformation Theory, Lemmas \ref{defos-lemma-flat-ringed-topoi} and \ref{defos-lemma-verify-iv-ringed-topoi} from Deformation Theory, Section \ref{defos-section-flat-ringed-topoi}. This is how the assumption that $f$ is flat comes about. Before we go on, note that results (2) and (3) of Deformation Theory, Lemmas \ref{defos-lemma-flat-ringed-topoi} do hold without the assumption that $f$ is flat as they rely on Deformation Theory, Lemmas \ref{defos-lemma-inf-ext-rel-ringed-topoi} and \ref{defos-lemma-inf-map-rel-ringed-topoi} which do not have any flatness assumptions. \medskip\noindent Before we give the details we give some motivation for the construction from derived algebraic geometry, since we think it will clarify what follows. Let $A$ be a finite type algebra over the locally Noetherian base $S$. Denote $X \otimes^\mathbf{L} A$ a derived base change'' of $X$ to $A$ and denote $i : X_A \to X \otimes^\mathbf{L} A$ the canonical inclusion morphism. The object $X \otimes^\mathbf{L} A$ does not (yet) have a definition in the Stacks project; we may think of it as the algebraic space $X_A$ endowed with a simplicial sheaf of rings $\mathcal{O}_{X \otimes^\mathbf{L} A}$ whose homology sheaves are $$H_i(\mathcal{O}_{X \otimes^\mathbf{L} A}) = \text{Tor}^{\mathcal{O}_S}_i(\mathcal{O}_X, A).$$ The morphism $X \otimes^\mathbf{L} A \to \Spec(A)$ is flat (the terms of the simplicial sheaf of rings being $A$-flat), so the usual material for deformations of flat modules applies to it. Thus we see that we get an obstruction theory using the groups $$\text{Ext}^i_{X \otimes^\mathbf{L} A}(i_*\mathcal{F}, i_*\mathcal{F} \otimes_A M)$$ where $i = 0, 1, 2$ for inf auts, inf defs, obstructions. Note that a flat deformation of $i_*\mathcal{F}$ to $X \otimes^\mathbf{L} A'$ is automatically of the form $i'_*\mathcal{F}'$ where $\mathcal{F}'$ is a flat deformation of $\mathcal{F}$. By adjunction of the functors $Li^*$ and $i_* = Ri_*$ these ext groups are equal to $$\text{Ext}^i_{X_A}(Li^*(i_*\mathcal{F}), \mathcal{F} \otimes_A M)$$ Thus we obtain obstruction groups of exactly the same form as in the proof of Quot, Lemma \ref{quot-lemma-coherent-defo-thy} with the only change being that one replaces the first occurrence of $\mathcal{F}$ by the complex $Li^*(i_*\mathcal{F})$. \medskip\noindent Below we prove the non-flat version of the lemma by a direct'' construction of $E(\mathcal{F}) = Li^*(i_*\mathcal{F})$ and direct proof of its relationship to the deformation theory of $\mathcal{F}$. In fact, it suffices to construct $\tau_{\geq -2}E(\mathcal{F})$, as we are only interested in the ext groups $\text{Ext}^i_{X_A}(Li^*(i_*\mathcal{F}), \mathcal{F} \otimes_A M)$ for $i = 0, 1, 2$. We can even identify the cohomology sheaves $$H^i(E(\mathcal{F})) = \left\{ \begin{matrix} 0 & \text{if }i > 0 \\ \mathcal{F} & \text{if } i = 0 \\ 0 & \text{if } i = -1 \\ \text{Tor}_1^{\mathcal{O}_S}(\mathcal{O}_X, A) \otimes_{\mathcal{O}_X} \mathcal{F} & \text{if } i = -2 \end{matrix} \right.$$ This observation will guide our construction of $E(\mathcal{F})$ in the remarks below. \begin{remark}[Direct construction] \label{remark-construction-E} Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $U$ be another algebraic space over $B$. Denote $q : X \times_B U \to U$ the second projection. Consider the distinguished triangle $$Lq^*L_{U/B} \to L_{X \times_B U/B} \to E \to Lq^*L_{U/B}[1]$$ of Cotangent, Section \ref{cotangent-section-fibre-product}. For any sheaf $\mathcal{F}$ of $\mathcal{O}_{X \times_B U}$-modules we have the Atiyah class $$\mathcal{F} \to L_{X \times_B U/B} \otimes_{\mathcal{O}_{X \times_B U}}^\mathbf{L} \mathcal{F}[1]$$ see Cotangent, Section \ref{cotangent-section-atiyah-general}. We can compose this with the map to $E$ and choose a distinguished triangle $$E(\mathcal{F}) \to \mathcal{F} \to \mathcal{F} \otimes_{\mathcal{O}_{X \times_B U}}^\mathbf{L} E[1] \to E(\mathcal{F})[1]$$ in $D(\mathcal{O}_{X \times_B U})$. By construction the Atiyah class lifts to a map $$e_\mathcal{F} : E(\mathcal{F}) \longrightarrow Lq^*L_{U/B} \otimes_{\mathcal{O}_{X \times_B U}}^\mathbf{L} \mathcal{F}[1]$$ fitting into a morphism of distinguished triangles $$\xymatrix{ \mathcal{F} \otimes^\mathbf{L} Lq^*L_{U/B}[1] \ar[r] & \mathcal{F} \otimes^\mathbf{L} L_{X \times_B U/B}[1] \ar[r] & \mathcal{F} \otimes^\mathbf{L} E[1] \\ E(\mathcal{F}) \ar[r] \ar[u]^{e_\mathcal{F}} & \mathcal{F} \ar[r] \ar[u]^{Atiyah} & \mathcal{F} \otimes^\mathbf{L} E[1] \ar[u]^{=} }$$ Given $S, B, X, f, U, \mathcal{F}$ we fix a choice of $E(\mathcal{F})$ and $e_\mathcal{F}$. \end{remark} \begin{remark}[Construction of obstruction class] \label{remark-construction-ob} With notation as in Remark \ref{remark-construction-E} let $i : U \to U'$ be a first order thickening of $U$ over $B$. Let $\mathcal{I} \subset \mathcal{O}_{U'}$ be the quasi-coherent sheaf of ideals cutting out $B$ in $B'$. The fundamental triangle $$Li^*L_{U'/B} \to L_{U/B} \to L_{U/U'} \to Li^*L_{U'/B}[1]$$ together with the map $L_{U/U'} \to \mathcal{I}[1]$ determine a map $e_{U'} : L_{U/B} \to \mathcal{I}[1]$. Combined with the map $e_\mathcal{F}$ of the previous remark we obtain $$(\text{id}_\mathcal{F} \otimes Lq^*e_{U'}) \cup e_\mathcal{F} : E(\mathcal{F}) \longrightarrow \mathcal{F} \otimes_{\mathcal{O}_{X \times_B U}} q^*\mathcal{I}[2]$$ (we have also composed with the map from the derived tensor product to the usual tensor product). In other words, we obtain an element $$\xi_{U'} \in \text{Ext}^2_{\mathcal{O}_{X \times_B U}}( E(\mathcal{F}), \mathcal{F} \otimes_{\mathcal{O}_{X \times_B U}} q^*\mathcal{I})$$ \end{remark} \begin{lemma} \label{lemma-ob-is-obstruction} In the situation of Remark \ref{remark-construction-ob} assume that $\mathcal{F}$ is flat over $U$. Then the vanishing of the class $\xi_{U'}$ is a necessary and sufficient condition for the existence of a $\mathcal{O}_{X \times_B U'}$-module $\mathcal{F}'$ flat over $U'$ with $i^*\mathcal{F}' \cong \mathcal{F}$. \end{lemma} \begin{proof}[Proof (sketch)] We will use the criterion of Deformation Theory, Lemma \ref{defos-lemma-inf-obs-ext-rel-ringed-topoi}. We will abbreviate $\mathcal{O} = \mathcal{O}_{X \times_B U}$ and $\mathcal{O}' = \mathcal{O}_{X \times_B U'}$. Consider the short exact sequence $$0 \to \mathcal{I} \to \mathcal{O}_{U'} \to \mathcal{O}_U \to 0.$$ Let $\mathcal{J} \subset \mathcal{O}'$ be the quasi-coherent sheaf of ideals cutting out $X \times_B U$. By the above we obtain an exact sequence $$\text{Tor}_1^{\mathcal{O}_B}(\mathcal{O}_X, \mathcal{O}_U) \to q^*\mathcal{I} \to \mathcal{J} \to 0$$ where the $\text{Tor}_1^{\mathcal{O}_B}(\mathcal{O}_X, \mathcal{O}_U)$ is an abbreviation for $$\text{Tor}_1^{h^{-1}\mathcal{O}_B}(p^{-1}\mathcal{O}_X, q^{-1}\mathcal{O}_U) \otimes_{(p^{-1}\mathcal{O}_X\otimes_{h^{-1}\mathcal{O}_B}q^{-1}\mathcal{O}_U)} \mathcal{O}.$$ Tensoring with $\mathcal{F}$ we obtain the exact sequence $$\mathcal{F} \otimes_\mathcal{O} \text{Tor}_1^{\mathcal{O}_B}(\mathcal{O}_X, \mathcal{O}_U) \to \mathcal{F} \otimes_\mathcal{O} q^*\mathcal{I} \to \mathcal{F} \otimes_\mathcal{O} \mathcal{J} \to 0$$ (Note that the roles of the letters $\mathcal{I}$ and $\mathcal{J}$ are reversed relative to the notation in Deformation Theory, Lemma \ref{defos-lemma-inf-obs-ext-rel-ringed-topoi}.) Condition (1) of the lemma is that the last map above is an isomorphism, i.e., that the first map is zero. The vanishing of this map may be checked on stalks at geometric points $\overline{z} = (\overline{x}, \overline{u}) : \Spec(k) \to X \times_B U$. Set $R = \mathcal{O}_{B, \overline{b}}$, $A = \mathcal{O}_{X, \overline{x}}$, $B = \mathcal{O}_{U, \overline{u}}$, and $C = \mathcal{O}_{\overline{z}}$. By Cotangent, Lemma \ref{cotangent-lemma-fibre-product} and the defining triangle for $E(\mathcal{F})$ we see that $$H^{-2}(E(\mathcal{F}))_{\overline{z}} = \mathcal{F}_{\overline{z}} \otimes \text{Tor}_1^R(A, B)$$ The map $\xi_{U'}$ therefore induces a map $$\mathcal{F}_{\overline{z}} \otimes \text{Tor}_1^R(A, B) \longrightarrow \mathcal{F}_{\overline{z}} \otimes_B \mathcal{I}_{\overline{u}}$$ We claim this map is the same as the stalk of the map described above (proof omitted; this is a purely ring theoretic statement). Thus we see that condition (1) of Deformation Theory, Lemma \ref{defos-lemma-inf-obs-ext-rel-ringed-topoi} is equivalent to the vanishing $H^{-2}(\xi_{U'}) : H^{-2}(E(\mathcal{F})) \to \mathcal{F} \otimes \mathcal{I}$. \medskip\noindent To finish the proof we show that, assuming that condition (1) is satisfied, condition (2) is equivalent to the vanishing of $\xi_{U'}$. In the rest of the proof we write $\mathcal{F} \otimes \mathcal{I}$ to denote $\mathcal{F} \otimes_\mathcal{O} q^*\mathcal{I} = \mathcal{F} \otimes_\mathcal{O} \mathcal{J}$. A consideration of the spectral sequence $$\text{Ext}^i(H^{-j}(E(\mathcal{F})), \mathcal{F} \otimes \mathcal{I}) \Rightarrow \text{Ext}^{i + j}(E(\mathcal{F}), \mathcal{F} \otimes \mathcal{I})$$ using that $H^0(E(\mathcal{F})) = \mathcal{F}$ and $H^{-1}(E(\mathcal{F})) = 0$ shows that there is an exact sequence $$0 \to \text{Ext}^2(\mathcal{F}, \mathcal{F} \otimes \mathcal{I}) \to \text{Ext}^2(E(\mathcal{F}), \mathcal{F} \otimes \mathcal{I}) \to \Hom(H^{-2}(E(\mathcal{F})), \mathcal{F} \otimes \mathcal{I})$$ Thus our element $\xi_{U'}$ is an element of $\text{Ext}^2(\mathcal{F}, \mathcal{F} \otimes \mathcal{I})$. The proof is finished by showing this element agrees with the element of Deformation Theory, Lemma \ref{defos-lemma-inf-obs-ext-rel-ringed-topoi} a verification we omit. \end{proof} \begin{lemma} \label{lemma-coherent-defo-thy-general} In Quot, Situation \ref{quot-situation-coherent} assume that $S$ is a locally Noetherian scheme and $S = B$. Let $\mathcal{X} = \textit{Coh}_{X/B}$. Then we have openness of versality for $\mathcal{X}$ (see Artin's Axioms, Definition \ref{artin-definition-openness-versality}). \end{lemma} \begin{proof}[Proof (sketch)] Let $U \to S$ be of finite type morphism of schemes, $x$ an object of $\mathcal{X}$ over $U$ and $u_0 \in U$ a finite type point such that $x$ is versal at $u_0$. After shrinking $U$ we may assume that $u_0$ is a closed point (Morphisms, Lemma \ref{morphisms-lemma-point-finite-type}) and $U = \Spec(A)$ with $U \to S$ mapping into an affine open $\Spec(\Lambda)$ of $S$. We will use Artin's Axioms, Lemma \ref{artin-lemma-dual-openness} to prove the lemma. Let $\mathcal{F}$ be the coherent module on $X_A = \Spec(A) \times_S X$ flat over $A$ corresponding to the given object $x$. \medskip\noindent Choose $E(\mathcal{F})$ and $e_\mathcal{F}$ as in Remark \ref{remark-construction-E}. The description of the cohomology sheaves of $E(\mathcal{F})$ shows that $$\text{Ext}^1(E(\mathcal{F}), \mathcal{F} \otimes_A M) = \text{Ext}^1(\mathcal{F}, \mathcal{F} \otimes_A M)$$ for any $A$-module $M$. Using this and using Deformation Theory, Lemma \ref{defos-lemma-inf-ext-rel-ringed-topoi} we have an isomorphism of functors $$T_x(M) = \text{Ext}^1_{X_A}(E(\mathcal{F}), \mathcal{F} \otimes_A M)$$ By Lemma \ref{lemma-ob-is-obstruction} given any surjection $A' \to A$ of $\Lambda$-algebras with square zero kernel $I$ we have an obstruction class $$\xi_{A'} \in \text{Ext}^2_{X_A}(E(\mathcal{F}), \mathcal{F} \otimes_A I)$$ Apply Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-compute-ext} to the computation of the Ext groups $\text{Ext}^i_{X_A}(E(\mathcal{F}), \mathcal{F} \otimes_A M)$ for $i \leq m$ with $m = 2$. We omit the verification that $E(\mathcal{F})$ is in $D^-_{\textit{Coh}}$; hint: use Cotangent, Lemma \ref{cotangent-lemma-cotangent-finite}. We find a perfect object $K \in D(A)$ and functorial isomorphisms $$H^i(K \otimes_A^\mathbf{L} M) \longrightarrow \text{Ext}^i_{X_A}(E(\mathcal{F}), \mathcal{F} \otimes_A M)$$ for $i \leq m$ compatible with boundary maps. This object $K$, together with the displayed identifications above gives us a datum as in Artin's Axioms, Situation \ref{artin-situation-dual}. Finally, condition (iv) of Artin's Axioms, Lemma \ref{artin-lemma-dual-obstruction} holds by a variant of Deformation Theory, Lemma \ref{defos-lemma-verify-iv-ringed-topoi} whose formulation and proof we omit. Thus Artin's Axioms, Lemma \ref{artin-lemma-dual-openness} applies and the lemma is proved. \end{proof} \begin{theorem} \label{theorem-coherent-algebraic-general} Let $S$ be a scheme. Let $f : X \to B$ be morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation and separated. Then $\textit{Coh}_{X/B}$ is an algebraic stack over $S$. \end{theorem} \begin{proof} This theorem is a copy of Quot, Theorem \ref{quot-theorem-coherent-algebraic-general}. The reason we have this copy here is that with the material in this section we get a second proof (as discussed at the beginning of this section). Namely, we argue exactly as in the proof of Quot, Theorem \ref{quot-theorem-coherent-algebraic} except that we substitute Lemma \ref{lemma-coherent-defo-thy-general} for Quot, Lemma \ref{quot-lemma-coherent-defo-thy}. \end{proof} \section{Modifications} \label{section-modifications} \noindent Here is a obsolete result on the category of Restricted Power Series, Equation (\ref{restricted-equation-modification}). Please visit Restricted Power Series, Section \ref{restricted-section-modifications} for the current material. \begin{lemma} \label{lemma-henselian} Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring. The category of Restricted Power Series, Equation (\ref{restricted-equation-modification}) for $A$ is equivalent to the category Restricted Power Series, Equation (\ref{restricted-equation-modification}) for the henselization $A^h$ of $A$. \end{lemma} \begin{proof} This is a special case of Restricted Power Series, Lemma \ref{restricted-lemma-equivalence-to-completion}. \end{proof} \noindent The following lemma on rational singularities is no longer needed in the chapter on resolving surface singularities. \begin{lemma} \label{lemma-double-dual-rational} In Resolution of Surfaces, Situation \ref{resolve-situation-rational}. Let $M$ be a finite reflexive $A$-module. Let $M \otimes_A \mathcal{O}_X$ denote the pullback of the associated $\mathcal{O}_S$-module. Then $M \otimes_A \mathcal{O}_X$ maps onto its double dual. \end{lemma} \begin{proof} Let $\mathcal{F} = (M \otimes_A \mathcal{O}_X)^{**}$ be the double dual and let $\mathcal{F}' \subset \mathcal{F}$ be the image of the evaluation map $M \otimes_A \mathcal{O}_X \to \mathcal{F}$. Then we have a short exact sequence $$0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{Q} \to 0$$ Since $X$ is normal, the local rings $\mathcal{O}_{X, x}$ are discrete valuation rings for points of codimension $1$ (see Properties, Lemma \ref{properties-lemma-criterion-normal}). Hence $\mathcal{Q}_x = 0$ for such points by More on Algebra, Lemma \ref{more-algebra-lemma-cokernel-map-double-dual-dvr}. Thus $\mathcal{Q}$ is supported in finitely many closed points and is globally generated by Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-support-dimension-0}. We obtain the exact sequence $$0 \to H^0(X, \mathcal{F}') \to H^0(X, \mathcal{F}) \to H^0(X, \mathcal{Q}) \to 0$$ because $\mathcal{F}'$ is generated by global sections (Resolution of Surfaces, Lemma \ref{resolve-lemma-globally-generated}). Since $X \to \Spec(A)$ is an isomorphism over the complement of the closed point, and since $M$ is reflexive, we see that the maps $$M \to H^0(X, \mathcal{F}') \to H^0(X, \mathcal{F})$$ induce isomorphisms after localization at any nonmaximal prime of $A$. Hence these maps are isomorphisms by More on Algebra, Lemma \ref{more-algebra-lemma-check-isomorphism-via-depth-and-ass} and the fact that reflexive modules over normal rings have property $(S_2)$ (More on Algebra, Lemma \ref{more-algebra-lemma-reflexive-over-normal}). Thus we conclude that $\mathcal{Q} = 0$ as desired. \end{proof} \section{Intersection theory} \label{section-intersection-theory} \begin{lemma} \label{lemma-gysin-factors-principal} Let $(S, \delta)$ be as in Chow Homology, Situation \ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Let $X$ be integral and $n = \dim_\delta(X)$. Let $a \in \Gamma(X, \mathcal{O}_X)$ be a nonzero function. Let $i : D = Z(a) \to X$ be the closed immersion of the zero scheme of $a$. Let $f \in R(X)^*$. In this case $i^*\text{div}_X(f) = 0$ in $A_{n - 2}(D)$. \end{lemma} \begin{proof} Special case of Chow Homology, Lemma \ref{chow-lemma-gysin-factors-general}. \end{proof} \section{Duplicate references} \label{section-duplicates} \noindent This section is a place where we collect duplicates. \begin{lemma} \label{lemma-points-monomorphism} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The map $\{\Spec(k) \to X \text{ monomorphism}\} \to |X|$ is injective. \end{lemma} \begin{proof} This is a duplicate of Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-monomorphism}. \end{proof} \begin{theorem} \label{theorem-equivalence-sheaves-point} Let $S = \Spec(K)$ with $K$ a field. Let $\overline{s}$ be a geometric point of $S$. Let $G = \text{Gal}_{\kappa(s)}$ denote the absolute Galois group. Then there is an equivalence of categories $\Sh(S_\etale) \to G\textit{-Sets}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$. \end{theorem} \begin{proof} This is a duplicate of \'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-equivalence-sheaves-point}. \end{proof} \begin{remark} \label{remark-tangent-spaces} You got here because of a duplicate tag. Please see Formal Deformation Theory, Section \ref{formal-defos-section-tangent-spaces} for the actual content. \end{remark} \begin{lemma} \label{lemma-locally-ringed-space-direct-summand-free} Let $X$ be a locally ringed space. A direct summand of a finite free $\mathcal{O}_X$-module is finite locally free. \end{lemma} \begin{proof} This is a duplicate of Modules, Lemma \ref{modules-lemma-direct-summand-of-locally-free-is-locally-free}. \end{proof} \input{chapters} \bibliography{my} \bibliographystyle{amsalpha} \end{document}