coherent-lemma-coherent-proper-ample

Let $R$ be a Noetherian ring. Let $X \to \Spec(R)$ be a proper morphism. Let $\mathcal{L}$ be an ample invertible sheaf on $X$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. \begin{enumerate} \item The graded ring $A = \bigoplus_{d \geq 0} H^0(X, \mathcal{L}^{\otimes d})$ is a finitely generated $R$-algebra. \item There exists an $r \geq 0$ and $d_1, \ldots, d_r \in \mathbf{Z}$ and a surjection $$ \bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{L}^{\otimes d_j} \longrightarrow \mathcal{F}. $$ \item For any $p$ the cohomology group $H^p(X, \mathcal{F})$ is a finite $R$-module. \item If $p > 0$, then $H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) = 0$ for all $d$ large enough. \item For any $k \in \mathbf{Z}$ the graded $A$-module $$ \bigoplus\nolimits_{d \geq k} H^0(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) $$ is a finite $A$-module. \end{enumerate}


coherent-lemma-kill-by-twisting

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{L}$ be an invertible sheaf on $X$. Assume that \begin{enumerate} \item $S$ is Noetherian, \item $f$ is proper, \item $\mathcal{F}$ is coherent, and \item $\mathcal{L}$ is relatively ample on $X/S$. \end{enumerate} Then there exists an $n_0$ such that for all $n \geq n_0$ we have $$ R^pf_*\left(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}\right) = 0 $$ for all $p > 0$.


coherent-lemma-locally-projective-pushforward

Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a locally projective morphism. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $R^if_*\mathcal{F}$ is a coherent $\mathcal{O}_S$-module for all $i \geq 0$.


coherent-lemma-vanshing-gives-ample

\begin{reference} \cite[III Proposition 2.6.1]{EGA} \end{reference} Let $R$ be a Noetherian ring. Let $f : X \to \Spec(R)$ be a proper morphism. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{L}$ is ample on $X$ (this is equivalent to many other things, see Properties, Proposition \ref{properties-proposition-characterize-ample} and Morphisms, Lemma \ref{morphisms-lemma-finite-type-over-affine-ample-very-ample}), \item for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ there exists an $n_0 \geq 0$ such that $H^p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$ for all $n \geq n_0$ and $p > 0$, and \item for every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$, there exists an $n \geq 1$ such that $H^1(X, \mathcal{I} \otimes \mathcal{L}^{\otimes n}) = 0$. \end{enumerate}


coherent-lemma-surjective-finite-morphism-ample

Let $R$ be a Noetherian ring. Let $f : Y \to X$ be a morphism of schemes proper over $R$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume $f$ is finite and surjective. Then $\mathcal{L}$ is ample if and only if $f^*\mathcal{L}$ is ample.


coherent-lemma-invert-s-cohomology

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible sheaf on $X$. Let $s \in \Gamma(X, \mathcal{L})$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $X$ is quasi-compact and quasi-separated, the canonical map $$ H^p_*(X, \mathcal{L}, \mathcal{F})_{(s)} \longrightarrow H^p(X_s, \mathcal{F}) $$ which maps $\xi/s^n$ to $s^{-n}\xi$ is an isomorphism.


coherent-lemma-ample-on-reduction

Let $i : Z \to X$ be a closed immersion of Noetherian schemes inducing a homeomorphism of underlying topological spaces. Let $\mathcal{L}$ be an invertible sheaf on $X$. Then $i^*\mathcal{L}$ is ample on $Z$, if and only if $\mathcal{L}$ is ample on $X$.


coherent-lemma-affine-in-presence-ample

Let $X$ be a scheme. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module. Let $n_0$ be an integer. If $H^p(X, \mathcal{L}^{\otimes -n}) = 0$ for $n \geq n_0$ and $p > 0$, then $X$ is affine.


coherent-lemma-affine-if-quasi-affine

Let $X$ be a quasi-affine scheme. If $H^p(X, \mathcal{O}_X) = 0$ for $p > 0$, then $X$ is affine.


coherent-lemma-chow-Noetherian

\begin{reference} \cite[II Theorem 5.6.1(a)]{EGA} \end{reference} Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a separated morphism of finite type. Then there exists an $n \geq 0$ and a diagram $$ \xymatrix{ X \ar[rd] & X' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_S \ar[dl] \\ & S & } $$ where $X' \to \mathbf{P}^n_S$ is an immersion, and $\pi : X' \to X$ is proper and surjective. Moreover, we may arrange it such that there exists a dense open subscheme $U \subset X$ such that $\pi^{-1}(U) \to U$ is an isomorphism.


coherent-lemma-proper-over-affine-cohomology-finite

Let $S = \Spec(A)$ with $A$ a Noetherian ring. Let $f : X \to S$ be a proper morphism. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $H^i(X, \mathcal{F})$ is finite $A$-module for all $i \geq 0$.


coherent-lemma-graded-finiteness

Let $A$ be a Noetherian ring. Let $B$ be a finitely generated graded $A$-algebra. Let $f : X \to \Spec(A)$ be a proper morphism. Set $\mathcal{B} = f^*\widetilde B$. Let $\mathcal{F}$ be a quasi-coherent graded $\mathcal{B}$-module of finite type. \begin{enumerate} \item For every $p \geq 0$ the graded $B$-module $H^p(X, \mathcal{F})$ is a finite $B$-module. \item If $\mathcal{L}$ is an ample invertible $\mathcal{O}_X$-module, then there exists an integer $d_0$ such that $H^p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes d}) = 0$ for all $p > 0$ and $d \geq d_0$. \end{enumerate}


coherent-lemma-cohomology-powers-ideal-times-F

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Set $B = \bigoplus_{n \geq 0} I^n$. Let $f : X \to \Spec(A)$ be a proper morphism. Let $\mathcal{F}$ be a coherent sheaf on $X$. Then for every $p \geq 0$ the graded $B$-module $\bigoplus_{n \geq 0} H^p(X, I^n\mathcal{F})$ is a finite $B$-module.


coherent-lemma-cohomology-powers-ideal-application

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $f : X \to \Spec(A)$ be a proper morphism. Let $\mathcal{F}$ be a coherent sheaf on $X$. Then for every $p \geq 0$ there exists an integer $c \geq 0$ such that \begin{enumerate} \item the multiplication map $I^{n - c} \otimes H^p(X, I^c\mathcal{F}) \to H^p(X, I^n\mathcal{F})$ is surjective for all $n \geq c$, \item the image of $H^p(X, I^{n + m}\mathcal{F}) \to H^p(X, I^n\mathcal{F})$ is contained in the submodule $I^{m - e} H^p(X, I^n\mathcal{F})$ where $e = \max(0, c - n)$ for $n + m \geq c$, $n, m \geq 0$, \item we have $$ \Ker(H^p(X, I^n\mathcal{F}) \to H^p(X, \mathcal{F})) = \Ker(H^p(X, I^n\mathcal{F}) \to H^p(X, I^{n - c}\mathcal{F})) $$ for $n \geq c$, \item there are maps $I^nH^p(X, \mathcal{F}) \to H^p(X, I^{n - c}\mathcal{F})$ for $n \geq c$ such that the compositions $$ H^p(X, I^n\mathcal{F}) \to  I^{n - c}H^p(X, \mathcal{F}) \to H^p(X, I^{n - 2c}\mathcal{F}) $$ and $$ I^nH^p(X, \mathcal{F}) \to H^p(X, I^{n - c}\mathcal{F}) \to I^{n - 2c}H^p(X, \mathcal{F}) $$ for $n \geq 2c$ are the canonical ones, and \item the inverse systems $(H^p(X, I^n\mathcal{F}))$ and $(I^nH^p(X, \mathcal{F}))$ are pro-isomorphic. \end{enumerate}


coherent-lemma-ML-cohomology-powers-ideal

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $f : X \to \Spec(A)$ be a proper morphism. Let $\mathcal{F}$ be a coherent sheaf on $X$. Fix $p \geq 0$. There exists a $c \geq 0$ such that \begin{enumerate} \item for all $n \geq c$ we have $$ \Ker(H^p(X, \mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F})) \subset I^{n - c}H^p(X, \mathcal{F}). $$ \item the inverse system $$ \left(H^p(X, \mathcal{F}/I^n\mathcal{F})\right)_{n \in \mathbf{N}} $$ satisfies the Mittag-Leffler condition (see Homology, Definition \ref{homology-definition-Mittag-Leffler}), and \item we have $$ \Im(H^p(X, \mathcal{F}/I^k\mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F})) = \Im(H^p(X, \mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F})) $$ for all $k \geq n + c$. \end{enumerate}


coherent-lemma-spell-out-theorem-formal-functions

Let $A$ be a ring. Let $I \subset A$ be an ideal. Assume $A$ is Noetherian and complete with respect to $I$. Let $f : X \to \Spec(A)$ be a proper morphism. Let $\mathcal{F}$ be a coherent sheaf on $X$. Then $$ H^p(X, \mathcal{F}) = \lim_n H^p(X, \mathcal{F}/I^n\mathcal{F}) $$ for all $p \geq 0$.


coherent-lemma-formal-functions-stalk

Given a morphism of schemes $f : X \to Y$ and a quasi-coherent sheaf $\mathcal{F}$ on $X$. Assume \begin{enumerate} \item $Y$ locally Noetherian, \item $f$ proper, and \item $\mathcal{F}$ coherent. \end{enumerate} Let $y \in Y$ be a point. Consider the infinitesimal neighbourhoods $$ \xymatrix{ X_n = \Spec(\mathcal{O}_{Y, y}/\mathfrak m_y^n) \times_Y X \ar[r]_-{i_n} \ar[d]_{f_n} & X \ar[d]^f \\ \Spec(\mathcal{O}_{Y, y}/\mathfrak m_y^n) \ar[r]^-{c_n} & Y } $$ of the fibre $X_1 = X_y$ and set $\mathcal{F}_n = i_n^*\mathcal{F}$. Then we have $$ \left(R^pf_*\mathcal{F}\right)_y^\wedge \cong \lim_n H^p(X_n, \mathcal{F}_n) $$ as $\mathcal{O}_{Y, y}^\wedge$-modules.


coherent-lemma-higher-direct-images-zero-finite-fibre

Let $f : X \to Y$ be a morphism of schemes. Let $y \in Y$. Assume \begin{enumerate} \item $Y$ locally Noetherian, \item $f$ is proper, and \item $f^{-1}(\{y\})$ is finite. \end{enumerate} Then for any coherent sheaf $\mathcal{F}$ on $X$ we have $(R^pf_*\mathcal{F})_y = 0$ for all $p > 0$.


coherent-lemma-higher-direct-images-zero-above-dimension-fibre

Let $f : X \to Y$ be a morphism of schemes. Let $y \in Y$. Assume \begin{enumerate} \item $Y$ locally Noetherian, \item $f$ is proper, and \item $\dim(X_y) = d$. \end{enumerate} Then for any coherent sheaf $\mathcal{F}$ on $X$ we have $(R^pf_*\mathcal{F})_y = 0$ for all $p > d$.


coherent-lemma-characterize-finite

(For a more general version see More on Morphisms, Lemma \ref{more-morphisms-lemma-characterize-finite}.) Let $f : X \to S$ be a morphism of schemes. Assume $S$ is locally Noetherian. The following are equivalent \begin{enumerate} \item $f$ is finite, and \item $f$ is proper with finite fibres. \end{enumerate}


coherent-lemma-proper-finite-fibre-finite-in-neighbourhood

\begin{slogan} A proper morphism is finite in a neighbourhood of a finite fiber. \end{slogan} (For a more general version see More on Morphisms, Lemma \ref{more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood}.) Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Assume \begin{enumerate} \item $S$ is locally Noetherian, \item $f$ is proper, and \item $f^{-1}(\{s\})$ is a finite set. \end{enumerate} Then there exists an open neighbourhood $V \subset S$ of $s$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is finite.


coherent-lemma-ample-on-fibre

Let $f : X \to Y$ be a proper morphism of schemes with $Y$ Noetherian. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Let $y \in Y$ be a point such that $\mathcal{L}_y$ is ample on $X_y$. Then there exists a $d_0$ such that for all $d \geq d_0$ we have $$ R^pf_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})_y = 0 \text{ for }p > 0 $$ and the map $$ f_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})_y \longrightarrow H^0(X_y, \mathcal{F}_y \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d}) $$ is surjective.


coherent-lemma-ample-in-neighbourhood

(For a more general version see More on Morphisms, Lemma \ref{more-morphisms-lemma-ample-in-neighbourhood}.) Let $f : X \to Y$ be a proper morphism of schemes with $Y$ Noetherian. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $y \in Y$ be a point such that $\mathcal{L}_y$ is ample on $X_y$. Then there is an open neighbourhood $V \subset Y$ of $y$ such that $\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$.


coherent-lemma-perfect-direct-image

Let $A$ be a Noetherian ring and set $S = \Spec(A)$. Let $f : X \to S$ be a proper morphism of schemes. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module flat over $S$. Then \begin{enumerate} \item $R\Gamma(X, \mathcal{F})$ is a perfect object of $D(A)$, and \item for any ring map $A \to A'$ the base change map $$ R\Gamma(X, \mathcal{F}) \otimes_A^{\mathbf{L}} A' \longrightarrow R\Gamma(X_{A'}, \mathcal{F}_{A'}) $$ is an isomorphism. \end{enumerate}


coherent-lemma-inverse-systems-affine

If $X = \Spec(A)$ is the spectrum of a Noetherian ring and $\mathcal{I}$ is the quasi-coherent sheaf of ideals associated to the ideal $I \subset A$, then $\textit{Coh}(X, \mathcal{I})$ is equivalent to the category of finite $A^\wedge$-modules where $A^\wedge$ is the completion of $A$ with respect to $I$.


coherent-lemma-inverse-systems-abelian

Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. \begin{enumerate} \item The category $\textit{Coh}(X, \mathcal{I})$ is abelian. \item For $U \subset X$ open the restriction functor $\textit{Coh}(X, \mathcal{I}) \to \textit{Coh}(U, \mathcal{I}|_U)$ is exact. \item Exactness in $\textit{Coh}(X, \mathcal{I})$ may be checked by restricting to the members of an open covering of $X$. \end{enumerate}


coherent-lemma-inverse-systems-surjective

Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. A map $(\mathcal{F}_n) \to (\mathcal{G}_n)$ is surjective in $\textit{Coh}(X, \mathcal{I})$ if and only if $\mathcal{F}_1 \to \mathcal{G}_1$ is surjective.


coherent-lemma-exact

The functor (\ref{equation-completion-functor}) is exact.


coherent-lemma-completion-internal-hom

Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. Set $\mathcal{H} = \SheafHom_{\mathcal{O}_X}(\mathcal{G}, \mathcal{F})$. Then $$ \lim H^0(X, \mathcal{H}/\mathcal{I}^n\mathcal{H}) = \Mor_{\textit{Coh}(X, \mathcal{I})} (\mathcal{G}^\wedge, \mathcal{F}^\wedge). $$


coherent-lemma-existence-easy

Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{G}$ be a coherent $\mathcal{O}_X$-module. Let $(\mathcal{F}_n)$ an object of $\textit{Coh}(X, \mathcal{I})$. \begin{enumerate} \item If $\alpha : (\mathcal{F}_n) \to \mathcal{G}^\wedge$ is a map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$, then there exists a unique (up to unique isomorphism) triple $(\mathcal{F}, a, \beta)$ where \begin{enumerate} \item $\mathcal{F}$ is a coherent $\mathcal{O}_X$-module, \item $a : \mathcal{F} \to \mathcal{G}$ is an $\mathcal{O}_X$-module map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$, \item $\beta : (\mathcal{F}_n) \to \mathcal{F}^\wedge$ is an isomorphism, and \item $\alpha = a^\wedge \circ \beta$. \end{enumerate} \item If $\alpha : \mathcal{G}^\wedge \to (\mathcal{F}_n)$ is a map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$, then there exists a unique (up to unique isomorphism) triple $(\mathcal{F}, a, \beta)$ where \begin{enumerate} \item $\mathcal{F}$ is a coherent $\mathcal{O}_X$-module, \item $a : \mathcal{G} \to \mathcal{F}$ is an $\mathcal{O}_X$-module map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$, \item $\beta : \mathcal{F}^\wedge \to (\mathcal{F}_n)$ is an isomorphism, and \item $\alpha = \beta \circ a^\wedge$. \end{enumerate} \end{enumerate}


coherent-lemma-finite-over-rees-algebra

Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. If $(\mathcal{F}_n)$ is an object of $\textit{Coh}(X, \mathcal{I})$ then $\bigoplus \Ker(\mathcal{F}_{n + 1} \to \mathcal{F}_n)$ is a finite type, graded, quasi-coherent $\bigoplus \mathcal{I}^n/\mathcal{I}^{n + 1}$-module.


coherent-lemma-inverse-systems-pullback

Let $f : X \to Y$ be a morphism of Noetherian schemes. Let $\mathcal{J} \subset \mathcal{O}_Y$ be a quasi-coherent sheaf of ideals and set $\mathcal{I} = f^{-1}\mathcal{J} \mathcal{O}_X$. Then there is a right exact functor $$ f^* : \textit{Coh}(Y, \mathcal{J}) \longrightarrow \textit{Coh}(X, \mathcal{I}) $$ which sends $(\mathcal{G}_n)$ to $(f^*\mathcal{G}_n)$. If $f$ is flat, then $f^*$ is an exact functor.


coherent-lemma-inverse-systems-pullback-equivalence

Let $f : X' \to X$ be a morphism of Noetherian schemes. Let $Z \subset X$ be a closed subscheme and denote $Z' = f^{-1}Z$ the scheme theoretic inverse image. Let $\mathcal{I} \subset \mathcal{O}_X$, $\mathcal{I}' \subset \mathcal{O}_{X'}$ be the corresponding quasi-coherent sheaves of ideals. If $f$ is flat and the induced morphism $Z' \to Z$ is an isomorphism, then the pullback functor $f^* : \textit{Coh}(X, \mathcal{I}) \to \textit{Coh}(X', \mathcal{I}')$ (Lemma \ref{lemma-inverse-systems-pullback}) is an equivalence.


coherent-lemma-inverse-systems-ideals-equivalence

Let $X$ be a Noetherian scheme. Let $\mathcal{I}, \mathcal{J} \subset \mathcal{O}_X$ be quasi-coherent sheaves of ideals. If $V(\mathcal{I}) = V(\mathcal{J})$ is the same closed subset of $X$, then $\textit{Coh}(X, \mathcal{I})$ and $\textit{Coh}(X, \mathcal{J})$ are equivalent.


coherent-lemma-fully-faithful

Let $A$ be Noetherian ring complete with respect to an ideal $I$. Let $f : X \to \Spec(A)$ be a proper morphism. Let $\mathcal{I} = I\mathcal{O}_X$. Then the functor (\ref{equation-completion-functor}) is fully faithful.


coherent-lemma-vanishing-projective

Let $A$ be Noetherian ring and $I \subset A$ and ideal. Let $f : X \to \Spec(A)$ be a proper morphism and let $\mathcal{L}$ be an $f$-ample invertible sheaf. Let $\mathcal{I} = I\mathcal{O}_X$. Let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(X, \mathcal{I})$. Then there exists an integer $d_0$ such that $$ H^1(X, \Ker(\mathcal{F}_{n + 1} \to \mathcal{F}_n) \otimes \mathcal{L}^{\otimes d} ) = 0 $$ for all $n \geq 0$ and all $d \geq d_0$.


coherent-lemma-existence-projective

Let $A$ be Noetherian ring complete with respect to an ideal $I$. Let $f : X \to \Spec(A)$ be a projective morphism. Let $\mathcal{I} = I\mathcal{O}_X$. Then the functor (\ref{equation-completion-functor}) is an equivalence.


coherent-lemma-existence-tricky

Let $X$ be a Noetherian scheme. Let $\mathcal{I}, \mathcal{K} \subset \mathcal{O}_X$ be quasi-coherent sheaves of ideals. Let $X_e \subset X$ be the closed subscheme cut out by $\mathcal{K}^e$. Let $\mathcal{I}_e = \mathcal{I}\mathcal{O}_{X_e}$. Let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(X, \mathcal{I})$. Assume \begin{enumerate} \item the functor $\textit{Coh}(\mathcal{O}_{X_e}) \to \textit{Coh}(X_e, \mathcal{I}_e)$ is an equivalence for all $e \geq 1$, and \item there exists a coherent sheaf $\mathcal{H}$ on $X$ and a map $\alpha : (\mathcal{F}_n) \to \mathcal{H}^\wedge$ whose kernel and cokernel are annihilated by a power of $\mathcal{K}$. \end{enumerate} Then $(\mathcal{F}_n)$ is in the essential image of (\ref{equation-completion-functor}).


coherent-lemma-inverse-systems-push-pull

Let $Y$ be a Noetherian scheme. Let $\mathcal{J}, \mathcal{K} \subset \mathcal{O}_Y$ be quasi-coherent sheaves of ideals. Let $f : X \to Y$ be a proper morphism which is an isomorphism over $V = Y \setminus V(\mathcal{K})$. Set $\mathcal{I} = f^{-1}\mathcal{J} \mathcal{O}_X$. Let $(\mathcal{G}_n)$ be an object of $\textit{Coh}(Y, \mathcal{J})$, let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module, and let $\beta : (f^*\mathcal{G}_n)  \to \mathcal{F}^\wedge$ be an isomorphism in $\textit{Coh}(X, \mathcal{I})$. Then there exists a map $$ \alpha : (\mathcal{G}_n) \longrightarrow (f_*\mathcal{F})^\wedge $$ in $\textit{Coh}(Y, \mathcal{J})$ whose kernel and cokernel are annihilated by a power of $\mathcal{K}$.


coherent-lemma-closed-proper-over-base

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $Z \subset X$ be a closed subset. The following are equivalent \begin{enumerate} \item the morphism $Z \to S$ is proper if $Z$ is endowed with the reduced induced closed subscheme structure (Schemes, Definition \ref{schemes-definition-reduced-induced-scheme}), \item for some closed subscheme structure on $Z$ the morphism $Z \to S$ is proper, \item for any closed subscheme structure on $Z$ the morphism $Z \to S$ is proper. \end{enumerate}


coherent-lemma-closed-closed-proper-over-base

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $Y \subset Z \subset X$ be closed subsets. If $Z$ is proper over $S$, then the same is true for $Y$.


coherent-lemma-base-change-closed-proper-over-base

Consider a cartesian diagram of schemes $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ with $f$ locally of finite type. If $Z$ is a closed subset of $X$ proper over $S$, then $(g')^{-1}(Z)$ is a closed subset of $X'$ proper over $S'$.


coherent-lemma-functoriality-closed-proper-over-base

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes which are locally of finite type over $S$. \begin{enumerate} \item If $Y$ is separated over $S$ and $Z \subset X$ is a closed subset proper over $S$, then $f(Z)$ is a closed subset of $Y$ proper over $S$. \item If $f$ is universally closed and $Z \subset X$ is a closed subset proper over $S$, then $f(Z)$ is a closed subset of $Y$ proper over $S$. \item If $f$ is proper and $Z \subset Y$ is a closed subset proper over $S$, then $f^{-1}(Z)$ is a closed subset of $X$ proper over $S$. \end{enumerate}


coherent-lemma-union-closed-proper-over-base

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $Z_i \subset X$, $i = 1, \ldots, n$ be closed subsets. If $Z_i$, $i = 1, \ldots, n$ are proper over $S$, then the same is true for $Z_1 \cup \ldots \cup Z_n$.


coherent-lemma-module-support-proper-over-base

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item the support of $\mathcal{F}$ is proper over $S$, \item the scheme theoretic support of $\mathcal{F}$ (Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-support}) is proper over $S$, and \item there exists a closed subscheme $Z \subset X$ and a finite type, quasi-coherent $\mathcal{O}_Z$-module $\mathcal{G}$ such that (a) $Z \to S$ is proper, and (b) $(Z \to X)_*\mathcal{G} = \mathcal{F}$. \end{enumerate}


coherent-lemma-support-proper-over-base-pushforward

Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module with support proper over $S$. Then $R^pf_*\mathcal{F}$ is a coherent $\mathcal{O}_S$-module for all $p \geq 0$.


coherent-lemma-systems-with-proper-support

Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a finite type morphism. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. The following are Serre subcategories of $\textit{Coh}(X, \mathcal{I})$ \begin{enumerate} \item the full subcategory of $\textit{Coh}(X, \mathcal{I})$ consisting of those objects $(\mathcal{F}_n)$ such that the support of $\mathcal{F}_1$ is proper over $S$, \item the full subcategory of $\textit{Coh}(X, \mathcal{I})$ consisting of those objects $(\mathcal{F}_n)$ such that there exists a closed subscheme $Z \subset X$ proper over $S$ with $\mathcal{I}_Z \mathcal{F}_n = 0$ for all $n \geq 1$. \end{enumerate}


coherent-lemma-algebraize-formal-closed-subscheme

Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \Spec(A)$ and $S_n = \Spec(A/I^n)$. Let $X \to S$ be a separated morphism of finite type. For $n \geq 1$ we set $X_n = X \times_S S_n$. Suppose given a commutative diagram $$ \xymatrix{ Z_1 \ar[r] \ar[d] & Z_2 \ar[r] \ar[d] & Z_3 \ar[r] \ar[d] & \ldots \\ X_1 \ar[r]^{i_1} & X_2 \ar[r]^{i_2} & X_3 \ar[r] & \ldots } $$ of schemes with cartesian squares. Assume that \begin{enumerate} \item $Z_1 \to X_1$ is a closed immersion, and \item $Z_1 \to S_1$ is proper. \end{enumerate} Then there exists a closed immersion of schemes $Z \to X$ such that $Z_n = Z \times_S S_n$. Moreover, $Z$ is proper over $S$.


coherent-lemma-algebraize-formal-scheme-finite-over-proper

Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \Spec(A)$ and $S_n = \Spec(A/I^n)$. Let $X \to S$ be a separated morphism of finite type. For $n \geq 1$ we set $X_n = X \times_S S_n$. Suppose given a commutative diagram $$ \xymatrix{ Y_1 \ar[r] \ar[d] & Y_2 \ar[r] \ar[d] & Y_3 \ar[r] \ar[d] & \ldots \\ X_1 \ar[r]^{i_1} & X_2 \ar[r]^{i_2} & X_3 \ar[r] & \ldots } $$ of schemes with cartesian squares. Assume that \begin{enumerate} \item $Y_n \to X_n$ is a finite morphism, and \item $Y_1 \to S_1$ is proper. \end{enumerate} Then there exists a finite morphism of schemes $Y \to X$ such that $Y_n = Y \times_S S_n$. Moreover, $Y$ is proper over $S$.


coherent-lemma-algebraize-morphism

Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \Spec(A)$ and $S_n = \Spec(A/I^n)$. Let $X$, $Y$ be schemes over $S$. For $n \geq 1$ we set $X_n = X \times_S S_n$ and $Y_n = Y \times_S S_n$. Suppose given a compatible system of commutative diagrams $$ \xymatrix{ & & X_{n + 1} \ar[rd] \ar[rr]_{g_{n + 1}} & & Y_{n + 1} \ar[ld] \\ X_n \ar[rru] \ar[rd] \ar[rr]_{g_n} & & Y_n \ar[rru] \ar[ld] & S_{n + 1} \\ & S_n \ar[rru] } $$ Assume that \begin{enumerate} \item $X \to S$ is proper, and \item $Y \to S$ is separated of finite type. \end{enumerate} Then there exists a unique morphism of schemes $g : X \to Y$ over $S$ such that $g_n$ is the base change of $g$ to $S_n$.


coherent-proposition-coherent-modules-on-proj

Let $A$ be a graded ring such that $A_0$ is Noetherian and $A$ is generated by finitely many elements of $A_1$ over $A_0$. Set $X = \text{Proj}(A)$. The functor $M \mapsto \widetilde M$ induces an equivalence $$ \text{Mod}^{fg}_A/\text{Mod}^{fg}_{A, torsion} \longrightarrow \textit{Coh}(\mathcal{O}_X) $$ whose quasi-inverse is given by $\mathcal{F} \longmapsto \bigoplus_{n \geq 0} \Gamma(X, \mathcal{F}(n))$.


coherent-proposition-coherent-modules-on-proj-general

Let $A$ be a Noetherian graded ring. Set $X = \text{Proj}(A)$. The functor $M \mapsto \widetilde M$ induces an equivalence $$ \text{Mod}^{fg}_A/\text{Mod}^{fg}_{A, irrelevant} \longrightarrow \textit{Coh}(\mathcal{O}_X) $$ whose quasi-inverse is given by $\mathcal{F} \longmapsto \bigoplus_{n \geq 0} \Gamma(X, \mathcal{F}(n))$.


coherent-proposition-proper-pushforward-coherent

\begin{reference} \cite[III Theorem 3.2.1]{EGA} \end{reference} Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a proper morphism. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $R^if_*\mathcal{F}$ is a coherent $\mathcal{O}_S$-module for all $i \geq 0$.


coherent-proposition-existence-proper

Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Let $f : X \to \Spec(A)$ be a proper morphism of schemes. Set $\mathcal{I} = I\mathcal{O}_X$. Then the functor (\ref{equation-completion-functor}) is an equivalence.


formal-defos-theorem-miniversal-object-existence

Let $\mathcal{F}$ be a predeformation category. Consider the following conditions \begin{enumerate} \item $\mathcal{F}$ has a minimal versal formal object satisfying (\ref{equation-bijective}), \item $\mathcal{F}$ has a minimal versal formal object satisfying (\ref{equation-bijective-orbits}), \item the following conditions hold: \begin{enumerate} \item $\mathcal{F}$ satisfies (S1). \item $\mathcal{F}$ satisfies (S2). \item $\dim_k T\mathcal{F}$ is finite. \end{enumerate} \end{enumerate} We always have $$ (1) \Rightarrow (3) \Rightarrow (2). $$ If $k' \subset k$ is separable, then all three are equivalent.


formal-defos-theorem-Schlessinger-prorepresentability

Let $F: \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor. Then $F$ is prorepresentable if and only if (a) $F$ is a deformation functor, (b) $\dim_k TF$ is finite, and (c) $\gamma : \text{Der}_\Lambda(k, k) \to TF$ is injective.


formal-defos-theorem-presentation-deformation-groupoid

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. Then $\mathcal{F}$ admits a presentation by a smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda$ if and only if the following conditions hold: \begin{enumerate} \item $\mathcal{F}$ is a deformation category. \item $\dim_k T\mathcal{F}$ is finite. \item $\dim_k \text{Inf}(\mathcal{F})$ is finite. \end{enumerate}


formal-defos-lemma-factor-small-extension

Let $f: B \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$. Then $f$ can be factored as a composition of small extensions.


formal-defos-lemma-length

Let $A$ be a local $\Lambda$-algebra with residue field $k$. Let $M$ be an $A$-module. Then $[k : k'] \text{length}_A(M) = \text{length}_\Lambda(M)$. In the classical case we have $\text{length}_A(M) = \text{length}_\Lambda(M)$.


formal-defos-lemma-surjective

Let $A \to B$ be a ring map in $\mathcal{C}_\Lambda$. The following are equivalent \begin{enumerate} \item $f$ is surjective, \item $\mathfrak m_A/\mathfrak m_A^2 \to \mathfrak m_B/\mathfrak m_B^2$ is surjective, and \item $\mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2) \to \mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)$ is surjective. \end{enumerate}


formal-defos-lemma-fiber-product-CLambda

Let $f_1 : A_1 \to A$ and $f_2 : A_2 \to A$ be ring maps in $\mathcal{C}_\Lambda$. Then: \begin{enumerate} \item If $f_1$ or $f_2$ is surjective, then $A_1 \times_A A_2$ is in $\mathcal{C}_\Lambda$. \item If $f_2$ is a small extension, then so is $A_1 \times_A A_2 \to A_1$. \item If the field extension $k' \subset k$ is separable, then $A_1 \times_A A_2$ is in $\mathcal{C}_\Lambda$. \end{enumerate}


formal-defos-lemma-essential-surjection-mod-squares

Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$. The following are equivalent \begin{enumerate} \item $f$ is an essential surjection, \item the map $B/\mathfrak m_B^2 \to A/\mathfrak m_A^2$ is an essential surjection, and \item the map $B/(\mathfrak m_\Lambda B + \mathfrak m_B^2) \to A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$ is an essential surjection. \end{enumerate}


formal-defos-lemma-H1-separable-case

There is a canonical map $$ \mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \longrightarrow H_1(L_{k/\Lambda}). $$ If $k' \subset k$ is separable (for example if the characteristic of $k$ is zero), then this map induces an isomorphism $\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k = H_1(L_{k/\Lambda})$. If $k = k'$ (for example in the classical case), then $\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 = H_1(L_{k/\Lambda})$. The composition $$ \mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \longrightarrow H_1(L_{k/\Lambda}) \longrightarrow \mathfrak m_A/\mathfrak m_A^2 $$ comes from the canonical map $\mathfrak m_\Lambda \to \mathfrak m_A$.


formal-defos-lemma-essential-surjection

Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$. Notation as in (\ref{equation-sequence-functorial}). \begin{enumerate} \item The equivalent conditions of Lemma \ref{lemma-essential-surjection-mod-squares} characterizing when $f$ is surjective are also equivalent to \begin{enumerate} \item $\Im(\text{d}_B) \to \Im(\text{d}_A)$ is surjective, and \item the map $\Omega_{B/\Lambda} \otimes_B k \to \Omega_{A/\Lambda} \otimes_A k$ is surjective. \end{enumerate} \item The following are equivalent \begin{enumerate} \item $f$ is an essential surjection, \item the map $\Im(\text{d}_B) \to \Im(\text{d}_A)$ is an isomorphism, and \item the map $\Omega_{B/\Lambda} \otimes_B k \to \Omega_{A/\Lambda} \otimes_A k$ is an isomorphism. \end{enumerate} \item If $k/k'$ is separable, then $f$ is an essential surjection if and only if the map $\mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2) \to \mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)$ is an isomorphism. \item If $f$ is a small extension, then $f$ is not essential if and only if $f$ has a section $s: A \to B$ in $\mathcal{C}_\Lambda$ with $f \circ s = \text{id}_A$. \end{enumerate}


formal-defos-lemma-surjective-cotangent-space

Let $f: R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda$. The following are equivalent \begin{enumerate} \item $f$ is surjective, \item the map $\mathfrak m_R/\mathfrak m_R^2 \to \mathfrak m_S/\mathfrak m_S^2$ is surjective, and \item the map $\mathfrak m_R/(\mathfrak m_\Lambda R + \mathfrak m_R^2) \to \mathfrak m_S/(\mathfrak m_\Lambda S + \mathfrak m_S^2)$ is surjective. \end{enumerate}


formal-defos-lemma-CLambdahat-pushouts

The category $\widehat{\mathcal{C}}_\Lambda$ admits pushouts.


formal-defos-lemma-CLambdahat-coproducts

The category $\widehat{\mathcal{C}}_\Lambda$ admits coproducts of pairs of objects.


formal-defos-lemma-derivations-finite

Let $S$ be an object of $\widehat{\mathcal{C}}_\Lambda$. Then $\dim_k \text{Der}_\Lambda(S, k) < \infty$.


formal-defos-lemma-derivations-surjective

Let $f : R \to S$ be a morphism of $\widehat{\mathcal{C}}_\Lambda$. If $\text{Der}_\Lambda(S, k) \to \text{Der}_\Lambda(R, k)$ is injective, then $f$ is surjective.


formal-defos-lemma-m-adic-topology

Let $R$ be an object of $\widehat{\mathcal{C}}_\Lambda$. Let $(J_n)$ be a decreasing sequence of ideals such that $\mathfrak m_R^n \subset J_n$. Set $J = \bigcap J_n$. Then the sequence $(J_n/J)$ defines the $\mathfrak m_{R/J}$-adic topology on $R/J$.


formal-defos-lemma-limit-artinian

Let $\ldots \to A_3 \to A_2 \to A_1$ be a sequence of surjective ring maps in $\mathcal{C}_\Lambda$. If $\dim_k (\mathfrak m_{A_n}/\mathfrak m_{A_n}^2)$ is bounded, then $S = \lim A_n$ is an object in $\widehat{\mathcal{C}}_\Lambda$ and the ideals $I_n = \Ker(S \to A_n)$ define the $\mathfrak m_S$-adic topology on $S$.


formal-defos-lemma-power-series

Let $R', R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$. Suppose that $R = R' \oplus I$ for some ideal $I$ of $R$. Let $x_1, \ldots, x_r \in I$ map to a basis of $I/\mathfrak m_R I$. Set $S = R'[[X_1, \ldots, X_r]]$ and consider the $R'$-algebra map $S \to R$ mapping $X_i$ to $x_i$. Assume that for every $n \gg 0$ the map $S/\mathfrak m_S^n \to R/\mathfrak m_R^n$ has a left inverse in $\mathcal{C}_\Lambda$. Then $S \to R$ is an isomorphism.


formal-defos-lemma-completion-cofibred

Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in groupoids. Then $\widehat{p} : \widehat{\mathcal{F}} \to \widehat{\mathcal{C}}_\Lambda$ is a category cofibered in groupoids.


formal-defos-lemma-formal-objects-different-filtration

In the situation above, $\widehat{\mathcal{F}}_\mathcal{I}(R)$ is equivalent to the category $\widehat{\mathcal{F}}(R)$.


formal-defos-lemma-smoothness-small-extensions

Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$.  Then $\varphi$ is smooth if the condition in Definition \ref{definition-smooth-morphism} is assumed to hold only for small extensions $B \to A$.


formal-defos-lemma-smooth-morphism-power-series

Let $R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda$. Then the induced morphism $\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}$ is smooth if and only if $S$ is a power series ring over $R$.


formal-defos-lemma-smooth-properties

Let $\varphi : \mathcal{F} \to \mathcal{G}$ and $\psi : \mathcal{G} \to \mathcal{H}$ be morphisms of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. \begin{enumerate} \item If $\varphi$ and $\psi$ are smooth, then $\psi \circ \varphi$ is smooth. \item If $\varphi$ is essentially surjective and $\psi \circ \varphi$ is smooth, then $\psi$ is smooth. \item If $\mathcal{G}' \to \mathcal{G}$ is a morphism of categories cofibered in groupoids and $\varphi$ is smooth, then $\mathcal{F} \times_\mathcal{G} \mathcal{G}' \to \mathcal{G}'$ is smooth. \end{enumerate}


formal-defos-lemma-smooth-morphism-essentially-surjective

Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a smooth morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$.  Assume $\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$ is essentially surjective. Then $\varphi : \mathcal{F} \to \mathcal{G}$ and $\widehat{\varphi} : \widehat{\mathcal{F}} \to \widehat{\mathcal{G}}$ are essentially surjective.


formal-defos-lemma-versal-object-quasi-initial

Let $\mathcal{F}$ be a predeformation category. Let $\xi$ be a versal formal object of $\mathcal{F}$. For any formal object $\eta$ of $\widehat{\mathcal{F}}$, there exists a morphism $\xi \to \eta$.


formal-defos-lemma-smooth

Let $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$. The following are equivalent \begin{enumerate} \item $\underline{R}|_{\mathcal{C}_\Lambda}$ is smooth, \item $\Lambda \to R$ is formally smooth in the $\mathfrak m_R$-adic topology, \item $\Lambda \to R$ is flat and $R \otimes_\Lambda k'$ is geometrically regular over $k'$, and \item $\Lambda \to R$ is flat and $k' \to R \otimes_\Lambda k'$ is formally smooth in the $\mathfrak m_R$-adic topology. \end{enumerate} In the classical case, these are also equivalent to \begin{enumerate} \item[(5)] $R$ is isomorphic to $\Lambda[[x_1, \ldots, x_n]]$ for some $n$. \end{enumerate}


formal-defos-lemma-smooth-power-series-classical

Let $\mathcal{F}$ be a predeformation category. Let $\xi$ be a versal formal object of $\mathcal{F}$ lying over $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is unobstructed, and \item $\Lambda \to R$ is formally smooth in the $\mathfrak m_R$-adic topology. \end{enumerate} In the classical case these are also equivalent to \begin{enumerate} \item[(3)] $R \cong \Lambda[[x_1, \ldots, x_n]]$ for some $n$.\ \end{enumerate}


formal-defos-lemma-exists-smooth

There exists an $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$ such that the equivalent conditions of Lemma \ref{lemma-smooth} hold and moreover $H_1(L_{k/\Lambda}) = \mathfrak m_R/\mathfrak m_R^2$ and $\Omega_{R/\Lambda} \otimes_R k = \Omega_{k/\Lambda}$.


formal-defos-lemma-S2-extensions

Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda$. If $\mathcal{F}$ satisfies (S2), then the condition of (S2) also holds when $k[\epsilon]$ is replaced by $k[V]$ for any finite dimensional $k$-vector space $V$.


formal-defos-lemma-S1-S2-associated-functor

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. \begin{enumerate} \item If $\mathcal{F}$ satisfies (S1), then so does $\overline{\mathcal{F}}$. \item If $\mathcal{F}$ satisfies (S2), then so does $\overline{\mathcal{F}}$ provided at least one of the following conditions is satisfied \begin{enumerate} \item $\mathcal{F}$ is a predeformation category, \item the category $\mathcal{F}(k)$ is a set or a setoid, or \item for any morphism $x_\epsilon \to x_0$ of $\mathcal{F}$ lying over $k[\epsilon] \to k$ the pushforward map $\text{Aut}_{k[\epsilon]}(x_\epsilon) \to \text{Aut}_k(x_0)$ is surjective. \end{enumerate} \end{enumerate}


formal-defos-lemma-S1-S2-localize

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. Let $x_0 \in \Ob(\mathcal{F}(k))$. Let $\mathcal{F}_{x_0}$ be the category cofibred in groupoids over $\mathcal{C}_\Lambda$ constructed in Remark \ref{remark-localize-cofibered-groupoid}. \begin{enumerate} \item If $\mathcal{F}$ satisfies (S1), then so does $\mathcal{F}_{x_0}$. \item If $\mathcal{F}$ satisfies (S2), then so does $\mathcal{F}_{x_0}$. \end{enumerate}


formal-defos-lemma-lifting-section

Let $p: \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in groupoids. Consider a diagram of $\mathcal{F}$ $$ \vcenter{ \xymatrix{ y \ar[r] \ar[d]_a & x_\epsilon \ar[d]_e \\ x \ar[r]^d        & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[\epsilon] \ar[r] \ar[d] & k[\epsilon] \ar[d] \\ A \ar[r] & k. } } $$ in $\mathcal{C}_\Lambda$. Assume $\mathcal{F}$ satisfies (S2). Then there exists a morphism $s : x \to y$ with $a \circ s = \text{id}_x$ if and only if there exists a morphism $s_\epsilon : x \to x_\epsilon$ with $e \circ s_\epsilon = d$.


formal-defos-lemma-lifting-along-small-extension

Consider a commutative diagram in a predeformation category $\mathcal{F}$ $$ \vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_2 \ar[d]^{a_2} \\ x_1 \ar[r]^{a_1}        & x } } \quad\text{lying over} \vcenter{ \xymatrix{ A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d]^{f_2} \\ A_1 \ar[r]^{f_1} & A } } $$ in $\mathcal{C}_\Lambda$ where $f_2 : A_2 \to A$ is a small extension. Assume there is a map $h : A_1 \to A_2$ such that $f_2 = f_1 \circ h$. Let $I = \Ker(f_2)$. Consider the ring map $$ g : A_1 \times_A A_2 \longrightarrow k[I] = k \oplus I, \quad (u, v) \longmapsto \overline{u} \oplus (v - h(u)) $$ Choose a pushforward $y \to g_*y$. Assume $\mathcal{F}$ satisfies (S2). If there exists a morphism $x_1 \to g_*y$, then there exists a morphism $b: x_1 \to x_2$ such that $a_1 =  a_2 \circ b$.


formal-defos-lemma-linear-functor

Let $L: \text{Mod}^{fg}_R \to \textit{Sets}$, resp.\ $L: \text{Mod}_R \to \textit{Sets}$ be a functor.  Suppose $L(0)$ is a one element set and $L$ preserves finite products.  Then there exists a unique $R$-linear functor $\widetilde{L} : \text{Mod}^{fg}_R \to \text{Mod}_R$, resp.\  $\widetilde{L} : \text{Mod}^{fg}_R \to \text{Mod}_R$, such that $$ \vcenter{ \xymatrix{ & \text{Mod}_R \ar[dr]^{\text{forget}} &   \\ \text{Mod}^{fg}_R  \ar[ur]^{\widetilde{L}} \ar[rr]^{L} &  & \textit{Sets} } } \quad\text{resp.}\quad \vcenter{ \xymatrix{ & \text{Mod}_R \ar[dr]^{\text{forget}} &   \\ \text{Mod}_R  \ar[ur]^{\widetilde{L}} \ar[rr]^{L} &  & \textit{Sets} } } $$ commutes.


formal-defos-lemma-morphism-linear-functors

Let $L_1, L_2: \text{Mod}^{fg}_R \to \textit{Sets}$ be functors that take $0$ to a one element set and preserve finite products. Let $t : L_1 \to L_2$ be a morphism of functors. Then $t$ induces a morphism $\widetilde{t} : \widetilde{L}_1 \to \widetilde{L}_2$ between the functors guaranteed by Lemma \ref{lemma-linear-functor}, which is given simply by $\widetilde{t}_M = t_M: \widetilde{L}_1(M) \to \widetilde{L}_2(M)$ for each $M \in \Ob(\text{Mod}^{fg}_R)$. In other words, $t_M: \widetilde{L}_1(M) \to \widetilde{L}_2(M)$ is a map of $R$-modules.


formal-defos-lemma-linear-functor-over-field

Let $K$ be a field. Let $L: \text{Mod}^{fg}_K \to \text{Mod}_K$ be a $K$-linear functor.  Then $L$ is isomorphic to the functor $L(K) \otimes_K - : \text{Mod}^{fg}_K \to \text{Mod}_K$.


formal-defos-lemma-preserves-products

Let $R$ be an $S$-algebra. Then the functor $\text{Mod}_R \to S\text{-Alg}/R$ described above preserves finite products.


formal-defos-lemma-tangent-space-functor

Let $R$ be an $S$-algebra, and let $\mathcal{C}$ be a strictly full subcategory of $S\text{-Alg}/R$ containing $R[M]$ for all $M \in \Ob(\text{Mod}^{fg}_R)$. Let $F: \mathcal{C} \to \textit{Sets}$ be a functor. Suppose that $F(R)$ is a one element set and that for any $M, N \in \Ob(\text{Mod}^{fg}_R)$, the induced map $$ F(R[M] \times_R R[N]) \to F(R[M]) \times F(R[N]) $$ is a bijection. Then $F(R[M])$ has a natural $R$-module structure for any $M \in \Ob(\text{Mod}^{fg}_R)$.


formal-defos-lemma-morphism-tangent-spaces

Let $F, G: \mathcal{C} \to \textit{Sets}$ be functors satisfying the hypotheses of Lemma \ref{lemma-tangent-space-functor}. Let $t : F \to G$ be a morphism of functors. For any $M \in \Ob(\text{Mod}^{fg}_R)$, the map $t_{R[M]}: F(R[M]) \to G(R[M])$ is a map of $R$-modules, where $F(R[M])$ and $G(R[M])$ are given the $R$-module structure from Lemma \ref{lemma-tangent-space-functor}. In particular, $t_{R[\epsilon]} : TF \to TG$ is a map of $R$-modules.


formal-defos-lemma-tangent-space-tensor

Let $F: \mathcal{C} \to \textit{Sets}$ be a functor satisfying the hypotheses of Lemma \ref{lemma-tangent-space-functor}. Assume $R = K$ is a field. Then $F(K[V]) \cong TF \otimes_K V$ for any finite dimensional $K$-vector space $V$.


formal-defos-lemma-tangent-space-vector-space

Let $\mathcal{F}$ be a predeformation category such that $\overline{\mathcal{F}}$ satisfies (S2)\footnote{For example if $\mathcal{F}$ satisfies (S2), see Lemma \ref{lemma-S1-S2-associated-functor}.}. Then $T \mathcal{F}$ has a natural $k$-vector space structure. For any finite dimensional vector space $V$ we have $\overline{\mathcal{F}}(k[V]) = T\mathcal{F} \otimes_k V$ functorially in $V$.


formal-defos-lemma-k-linear-differential

Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of predeformation categories. Assume $\overline{\mathcal{F}}$ and $\overline{\mathcal{G}}$ both satisfy (S2). Then $d \varphi : T \mathcal{F} \to T \mathcal{G}$ is $k$-linear.


formal-defos-lemma-action-linear

Let $\mathcal{F}$ be a predeformation category over $\mathcal{C}_\Lambda$. If $\overline{\mathcal{F}}$ has (S2) then the maps $\gamma_V$ are $k$-linear and we have $a_V(D, x) = x + \gamma_V(D)$.


formal-defos-lemma-versal-object-S1

Let $\mathcal{F}$ be a predeformation category. Assume $\mathcal{F}$ has a versal formal object. Then $\mathcal{F}$ satisfies (S1).


formal-defos-lemma-versal-criterion

Let $\mathcal{F}$ be a predeformation category satisfying (S1) and (S2). Let $\xi$ be a formal object of $\mathcal{F}$ corresponding to $\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$, see Remark \ref{remark-formal-objects-yoneda}. Then $\xi$ is versal if and only if the following two conditions hold: \begin{enumerate} \item the map $d\underline{\xi} : T\underline{R}|_{\mathcal{C}_\Lambda} \to T\mathcal{F}$ on tangent spaces is surjective, and \item given a diagram in $\widehat{\mathcal{F}}$ $$ \vcenter{ \xymatrix{             &  y \ar[d] \\ \xi \ar[r]  &  x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{          &   B  \ar[d]^{f} \\ R \ar[r] &   A } } $$ in $\widehat{\mathcal{C}}_\Lambda$ with $B \to A$ a small extension of Artinian rings, then there exists a ring map $R \to B$ such that $$ \xymatrix{          &   B  \ar[d]^{f} \\ R \ar[ur] \ar[r] &   A } $$ commutes. \end{enumerate}


formal-defos-lemma-largest-closed-where-lift

Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda$ which has (S1). Let $B \to A$ be a surjection in $\mathcal{C}_\Lambda$ with kernel $I$ annihilated by $\mathfrak m_B$. Let $x \in \mathcal{F}(A)$. The set of ideals $$ \mathcal{J} = \{ J \subset I \mid \text{there exists an }y \to x\text{ lying over }B/J \to A\} $$ has a smallest element.


formal-defos-lemma-versal-object-existence

Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda$. Assume the following conditions hold: \begin{enumerate} \item $\mathcal{F}$ is a predeformation category. \item $\mathcal{F}$ satisfies (S1). \item $\mathcal{F}$ satisfies (S2). \item $\dim_k T\mathcal{F}$ is finite. \end{enumerate} Then $\mathcal{F}$ has a versal formal object.


formal-defos-lemma-smallest-where-descends

Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda$ which has (S1). \begin{enumerate} \item For $y \to x$ in $\mathcal{F}$ a minimal object in $\mathcal{S}_y$ maps to a minimal object of $\mathcal{S}_x$. \item For $y \to x$ in $\mathcal{F}$ lying over a surjection $f : B \to A$ in $\mathcal{C}_\Lambda$ every minimal object of $\mathcal{S}_x$ is the image of a minimal object of $\mathcal{S}_y$. \end{enumerate}


formal-defos-lemma-smallest-where-descends-versal

Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda$ which has (S1). Let $\xi$ be a versal formal object of $\mathcal{F}$ lying over $R$. There exists a morphism $\xi' \to \xi$ lying over $R' \subset R$ with the following minimality properties \begin{enumerate} \item for every $f : R \to A$ with $A \in \Ob(\mathcal{C}_\Lambda)$ the pushforwards $$ \vcenter{ \xymatrix{ \xi' \ar[d] \ar[r] & x' \ar[d] \\ \xi \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ R' \ar[d] \ar[r] & f(R') \ar[d] \\ R \ar[r] & A } } $$ produce a minimal object $x' \to x$ of $\mathcal{S}_x$, and \item for any morphism of formal objects $\xi'' \to \xi'$ the corresponding morphism $R'' \to R'$ is surjective. \end{enumerate}


formal-defos-lemma-descends-versal

Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda$ which has (S1). Let $\xi$ be a versal formal object of $\mathcal{F}$ lying over $R$. Let $\xi' \to \xi$ be a morphism of formal objects lying over $R' \subset R$ as constructed in Lemma \ref{lemma-smallest-where-descends-versal}. Then $$ R \cong R'[[x_1, \ldots, x_r]] $$ is a power series ring over $R'$. Moreover, $\xi'$ is a versal formal object too.


formal-defos-lemma-minimal-versal

Let $\mathcal{F}$ be a predeformation category which has a versal formal object. Then \begin{enumerate} \item $\mathcal{F}$ has a minimal versal formal object, \item minimal versal objects are unique up to isomorphism, and \item any versal object is the pushforward of a minimal versal object along a power series ring extension. \end{enumerate}


formal-defos-lemma-miniversal-object-existence-1

Let $\mathcal{F}$ be a predeformation category. Let $\xi$ be a versal formal object of $\mathcal{F}$ such that (\ref{equation-bijective}) holds. Then \begin{enumerate} \item $\mathcal{F}$ satisfies (S1). \item $\mathcal{F}$ satisfies (S2). \item $\dim_k T\mathcal{F}$ is finite. \end{enumerate}


formal-defos-lemma-construct-bijective-orbits

Let $\mathcal{F}$ be a predeformation category satisfying (S2) which has a versal formal object. Then its minimal versal formal object satisfies (\ref{equation-bijective-orbits}).


formal-defos-lemma-RS-fiber-square

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$ satisfying (RS). Given a commutative diagram in $\mathcal{F}$ $$ \vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_2 \ar[d]   \\ x_1 \ar[r]      & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1 \ar[r]      & A. } } $$ with $A_2 \to A$ surjective, then it is a fiber square.


formal-defos-lemma-RS-small-extension

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda$. Then $\mathcal{F}$ satisfies (RS) if the condition in Definition \ref{definition-RS} is assumed to hold only when $A_2 \to A$ is a small extension.


formal-defos-lemma-RS-2-categorical

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ satisfies (RS), \item the functor $\mathcal{F}(A_1 \times_A A_2) \to \mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)$ see (\ref{equation-compare}) is an equivalence of categories whenever $A_2 \to A$ is surjective, and \item same as in (2) whenever $A_2 \to A$ is a small extension. \end{enumerate}


formal-defos-lemma-RS-implies-S1-S2

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. The condition (RS) for $\mathcal{F}$ implies both (S1) and (S2) for $\mathcal{F}$.


formal-defos-lemma-RS-associated-functor

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$ satisfying (RS). The following conditions are equivalent: \begin{enumerate} \item $\overline{\mathcal{F}}$ satisfies (RS). \item Let $f_1: A_1 \to A$ and $f_2: A_2 \to A$ be ring maps in $\mathcal{C}_\Lambda$ with $f_2$ surjective. The induced map of sets of isomorphism classes $$ \overline{\mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)} \to \overline{\mathcal{F}}(A_1) \times_{\overline{\mathcal{F}}(A)} \overline{\mathcal{F}}(A_2) $$ is injective. \item For every morphism $x' \to x$ in $\mathcal{F}$ lying over a surjective ring map $A' \to A$, the map $\text{Aut}_{A'}(x') \to \text{Aut}_A(x)$ is surjective. \item For every morphism $x' \to x$ in $\mathcal{F}$ lying over a small extension $A' \to A$, the map $\text{Aut}_{A'}(x') \to \text{Aut}_A(x)$ is surjective. \end{enumerate}


formal-defos-lemma-localize-RS

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. Let $x_0 \in \Ob(\mathcal{F}(k))$. Let $\mathcal{F}_{x_0}$ be the category cofibred in groupoids over $\mathcal{C}_\Lambda$ constructed in Remark \ref{remark-localize-cofibered-groupoid}. If $\mathcal{F}$ satisfies (RS), then so does $\mathcal{F}_{x_0}$. In particular, $\mathcal{F}_{x_0}$ is a deformation category.


formal-defos-lemma-RS-fiber-product-morphisms

Let $$ \xymatrix{ \mathcal{H} \times_\mathcal{F} \mathcal{G} \ar[r] \ar[d] & \mathcal{G} \ar[d]^g \\ \mathcal{H} \ar[r]^f & \mathcal{F} } $$ be $2$-fibre product of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. If $\mathcal{F}, \mathcal{G}, \mathcal{H}$ all satisfy (RS), then $\mathcal{H} \times_\mathcal{F} \mathcal{G}$ satisfies (RS).


formal-defos-lemma-free-transitive-action

Let $\mathcal{F}$ be a deformation category. Let $A' \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$ whose kernel $I$ is annihilated by $\mathfrak m_{A'}$. Let $x \in \Ob(\mathcal{F}(A))$. If $\text{Lift}(x, A')$ is nonempty, then there is a free and transitive action of $T\mathcal{F} \otimes_k I$ on $\text{Lift}(x, A')$.


formal-defos-lemma-minimal-smooth-morphism-functors

Let $F, G: \mathcal{C}_\Lambda \to \textit{Sets}$ be deformation functors. Let $\varphi : F \to G$ be a smooth morphism which induces an isomorphism $d\varphi : TF \to TG$ of tangent spaces. Then $\varphi$ is an isomorphism.


formal-defos-lemma-Aut-functor-RS

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$ satisfying (RS). Let $x \in \Ob(\mathcal{F}(A))$. Then $\mathit{Aut}(x): \mathcal{C}_A \to \textit{Sets}$ satisfies (RS).


formal-defos-lemma-Aut-functor-tangent-space

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$ satisfying (RS). Let $x \in \Ob(\mathcal{F}(A))$. Let $x_0$ be a pushforward of $x$ to $\mathcal{F}(k)$. \begin{enumerate} \item $T_{\text{id}_{x_0}} \mathit{Aut}(x)$ has a natural $k$-vector space structure such that addition agrees with composition in $T_{\text{id}_{x_0}} \mathit{Aut}(x)$. In particular, composition in $T_{\text{id}_{x_0}} \mathit{Aut}(x)$ is commutative. \item There is a canonical isomorphism $T_{\text{id}_{x_0}} \mathit{Aut}(x) \to T_{\text{id}_{x_0}} \mathit{Aut}(x_0)$ of $k$-vector spaces. \end{enumerate}


formal-defos-lemma-k-linear-infaut

Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibred in groupoids over $\mathcal{C}_\Lambda$ satisfying (RS). Let $x_0 \in \Ob(\mathcal{F}(k))$. Then $\varphi$ induces a $k$-linear map $\text{Inf}_{x_0}(\mathcal{F}) \to \text{Inf}_{\varphi(x_0)}(\mathcal{G})$.


formal-defos-lemma-lifted-automorphisms-torsor

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$ satisfying (RS). Let $x' \to x$ be a morphism lying over a surjective ring map $A' \to A$ with kernel $I$ annihilated by $\mathfrak m_{A'}$. Let $x_0$ be a pushforward of $x$ to $\mathcal{F}(k)$. Then $\text{Inf}(x'/x)$ has a free and transitive action by $T_{\text{id}_{x_0}} \mathit{Aut}(x') \otimes_k I = \text{Inf}_{x_0}(\mathcal{F}) \otimes_k I$.


formal-defos-lemma-infaut-trivial

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$ satisfying (RS). Let $x' \to x$ be a morphism in $\mathcal{F}$ lying over a surjective ring map. Let $x_0$ be a pushforward of $x$ to $\mathcal{F}(k)$. If $\text{Inf}_{x_0}(\mathcal{F}) = 0$ then $\text{Inf}(x'/x) = 0$.


formal-defos-lemma-infdef-trivial

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$ satisfying (RS). Let $x_0 \in \Ob(\mathcal{F}(k))$. Then $\text{Inf}_{x_0}(\mathcal{F}) = 0$ if and only if the natural morphism $\mathcal{F}_{x_0} \to \overline{\mathcal{F}_{x_0}}$ of categories cofibered in groupoids is an equivalence.


formal-defos-lemma-deformation-categories-fiber-product-morphisms

Let $f : \mathcal{H} \to \mathcal{F}$ and $g : \mathcal{G} \to \mathcal{F}$ be $1$-morphisms of deformation categories. Then \begin{enumerate} \item $\mathcal{W} = \mathcal{H} \times_\mathcal{F} \mathcal{G}$ is a deformation category, and \item we have a $6$-term exact sequence of vector spaces $$ 0 \to \text{Inf}(\mathcal{W}) \to \text{Inf}(\mathcal{H}) \oplus \text{Inf}(\mathcal{G}) \to \text{Inf}(\mathcal{F}) \to T\mathcal{W} \to T\mathcal{H} \oplus T\mathcal{G} \to T\mathcal{F} $$ \end{enumerate}


formal-defos-lemma-map-fibre-products-smooth

Let $\mathcal{H}_1 \to \mathcal{G}$, $\mathcal{H}_2 \to \mathcal{G}$, and $\mathcal{G} \to \mathcal{F}$ be maps of categories cofibred in groupoids over $\mathcal{C}_\Lambda$. Assume \begin{enumerate} \item $\mathcal{F}$ and $\mathcal{G}$ are deformation categories, \item $T\mathcal{G} \to T\mathcal{F}$ is injective, and \item $\text{Inf}(\mathcal{G}) \to \text{Inf}(\mathcal{F})$ is surjective. \end{enumerate} Then $\mathcal{H}_1 \times_\mathcal{G} \mathcal{H}_2 \to \mathcal{H}_1 \times_\mathcal{F} \mathcal{H}_2$ is smooth.


formal-defos-lemma-easy-check-smooth

Let $f : \mathcal{F} \to \mathcal{G}$ be a map of deformation categories. Let $x_0 \in \Ob(\mathcal{F}(k))$ with image $y_0 \in \Ob(\mathcal{G}(k))$. If \begin{enumerate} \item the map $T\mathcal{F} \to T\mathcal{G}$ is surjective, and \item for every small extension $A' \to A$ in $\mathcal{C}_\Lambda$ and $x \in \mathcal{F}(A)$ with image $y \in \mathcal{G}(A)$ if there is a lift of $y$ to $A'$, then there is a lift of $x$ to $A'$, \end{enumerate} then $\mathcal{F} \to \mathcal{G}$ is smooth (and vice versa).


formal-defos-lemma-groupoid-in-functors-prorep-equivalences

Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda$. \begin{enumerate} \item $(U, R, s, t, c)$ is prorepresentable if and only if its completion is representable as a groupoid in functors on $\widehat{\mathcal{C}}_\Lambda$. \item $(U, R, s, t, c)$ is prorepresentable if and only if $U$ and $R$ are prorepresentable. \end{enumerate}


formal-defos-lemma-smooth-quotient-morphism

Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda$. The following are equivalent: \begin{enumerate} \item The groupoid in functors $(U, R, s, t, c)$ is smooth. \item The morphism $s : R \to U$ is smooth. \item The morphism $t : R \to U$ is smooth. \item The quotient morphism $U \to [U/R]$ is smooth. \end{enumerate}


formal-defos-lemma-smooth-RS-groupoid-in-functors-quotient

Let $(U, R, s, t, c)$ be a smooth groupoid in functors on $\mathcal{C}_\Lambda$. Assume $U$ and $R$ satisfy (RS). Then $[U/R]$ satisfies (RS).


formal-defos-lemma-deformation-groupoid-quotient

Let $(U, R, s, t, c)$ be a smooth groupoid in functors on $\mathcal{C}_\Lambda$. Assume $U$ and $R$ are deformation functors. Then: \begin{enumerate} \item The quotient $[U/R]$ is a deformation category. \item The tangent space of $[U/R]$ is $$ T[U/R] = \Coker(ds-dt: TR \to TU). $$ \item The space of infinitesimal automorphisms of $[U/R]$ is $$ \text{Inf}([U/R]) = \Ker(ds \oplus dt : TR \to TU \oplus TU). $$ \end{enumerate}


formal-defos-lemma-presentation-construction

Let $\mathcal{F}$ be category cofibered in groupoids over a category $\mathcal{C}$. Let $U : \mathcal{C} \to \textit{Sets}$ be a functor. Let $f : U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}$. Define $R, s, t, c$ as follows: \begin{enumerate} \item $R : \mathcal{C} \to \textit{Sets}$ is the functor $U \times_{f, \mathcal{F}, f} U$. \item $t, s : R \to U$ are the first and second projections, respectively. \item $c : R \times_{s, U, t} R \to R$ is the morphism given by projection onto the first and last factors of $U \times_{f, \mathcal{F}, f} U \times_{f, \mathcal{F}, f} U$ under the canonical isomorphism $R \times_{s, U, t} R \to U \times_{f, \mathcal{F}, f} U \times_{f, \mathcal{F}, f} U$. \end{enumerate} Then $(U, R, s, t, c)$ is a groupoid in functors on $\mathcal{C}$.


formal-defos-lemma-presentation-morphism

Let $\mathcal{F}$ be category cofibered in groupoids over a category $\mathcal{C}$. Let $U : \mathcal{C} \to \textit{Sets}$ be a functor. Let $f : U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}$. Let $(U, R, s, t, c)$ be the groupoid in functors on $\mathcal{C}$ constructed from $f : U \to \mathcal{F}$ in Lemma \ref{lemma-presentation-construction}. Then there is a natural morphism $[f] : [U/R] \to \mathcal{F}$ such that: \begin{enumerate} \item $[f]: [U/R] \to \mathcal{F}$ is fully faithful. \item $[f]: [U/R] \to \mathcal{F}$ is an equivalence if and only if $f : U \to \mathcal{F}$ is essentially surjective. \end{enumerate}


formal-defos-lemma-smooth-groupoid-in-functors-construction

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor. Let $f : U \to \mathcal{F}$ be a smooth morphism of categories cofibered in groupoids. Then: \begin{enumerate} \item If $(U, R, s, t, c)$ is the groupoid in functors on $\mathcal{C}_\Lambda$ constructed from $f : U \to \mathcal{F}$ in Lemma \ref{lemma-presentation-construction}, then $(U, R, s, t, c)$ is smooth. \item If $f : U(k) \to \mathcal{F}(k)$ is essentially surjective, then the morphism $[f] : [U/R] \to \mathcal{F}$ of Lemma \ref{lemma-presentation-morphism} is an equivalence. \end{enumerate}


formal-defos-lemma-deformation-functor-diagonal

Let $\mathcal{F}$ be a deformation category. Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$ be a deformation functor. Let $f: U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids. Then $U \times_{f, \mathcal{F}, f} U$ is a deformation functor with tangent space fitting into an exact sequence of $k$-vector spaces $$ 0 \to \text{Inf}(\mathcal{F}) \to T(U \times_{f, \mathcal{F}, f} U) \to TU \oplus TU $$


formal-defos-lemma-prorepresentable-groupoid-in-functors-construction

Let $\mathcal{F}$ be a deformation category. Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$ be a prorepresentable functor. Let $f : U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids. Let $(U, R, s, t, c)$ be the groupoid in functors on $\mathcal{C}_\Lambda$ constructed from $f : U \to \mathcal{F}$ in Lemma \ref{lemma-presentation-construction}. If $\dim_k \text{Inf}(\mathcal{F}) < \infty$, then $(U, R, s, t, c)$ is prorepresentable.


formal-defos-lemma-characterize-minimal-groupoid-in-functors

Let $(U, R, s, t, c)$ be a smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda$. \begin{enumerate} \item $(U, R, s, t, c)$ is normalized if and only if the morphism $U \to [U/R]$ induces an isomorphism on tangent spaces, and \item $(U, R, s, t, c)$ is minimal if and only if the kernel of $TU \to T[U/R]$ is contained in the image of $\text{Der}_\Lambda(k, k) \to TU$. \end{enumerate}


formal-defos-lemma-surjective-morphism-prorepresentable-functor

Let $U: \mathcal{C}_\Lambda \to \textit{Sets}$ be a prorepresentable functor. Let $\varphi : U \to U$ be a morphism such that $d\varphi : TU \to TU$ is an isomorphism.  Then $\varphi$ is an isomorphism.


formal-defos-lemma-minimal-prorepresentable-groupoid-autoequivalence

Let $(U, R, s, t, c)$ be a minimal smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda$. If $\varphi : [U/R] \to [U/R]$ is an equivalence of categories cofibered in groupoids, then $\varphi$ is an isomorphism.


formal-defos-lemma-minimal-prorepresentable-groupoid-equivalence

Let $(U, R, s, t, c)$ and $(U', R', s', t', c')$ be minimal smooth prorepresentable groupoids in functors on $\mathcal{C}_\Lambda$. If $\varphi : [U/R] \to [U'/R']$ is an equivalence of categories cofibered in groupoids, then $\varphi$ is an isomorphism.


formal-defos-lemma-minimal-groupoid-in-functors-construction

Let $\mathcal{F}$ be a deformation category such that $\dim_k T\mathcal{F} <\infty$ and $\dim_k \text{Inf}(\mathcal{F}) < \infty$. Then there exists a minimal versal formal object $\xi$ of $\mathcal{F}$. Say $\xi$ lies over $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$. Let $U = \underline{R}|_{\mathcal{C}_\Lambda}$. Let $f = \underline{\xi} : U \to \mathcal{F}$ be the associated morphism. Let $(U, R, s, t, c)$ be the groupoid in functors on $\mathcal{C}_\Lambda$ constructed from $f : U \to \mathcal{F}$ in Lemma \ref{lemma-presentation-construction}. Then $(U, R, s, t, c)$ is a minimal smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda$ and there is an equivalence $[U/R] \to \mathcal{F}$.


formal-defos-lemma-minimal-presentations-equivalent

Let $\mathcal{F}$ be category cofibered in groupoids over $\mathcal{C}_\Lambda$. Assume there exist presentations of $\mathcal{F}$ by minimal smooth prorepresentable groupoids in functors $(U, R, s, t, c)$ and $(U', R', s', t', c')$. Then $(U, R, s, t, c)$ and $(U', R', s', t', c')$ are isomorphic.


formal-defos-lemma-composition-homotopic

In the category $\widehat{\mathcal{C}}_\Lambda$, if $f_1, f_2 : R \to S$ are formally homotopic and $g : S \to S'$ is a morphism, then $g \circ f_1$ and $g \circ f_2$ are formally homotopic.


formal-defos-lemma-versal-unique-up-to-homotopy

Let $\mathcal{F}$ be a deformation category over $\mathcal{C}_\Lambda$ with $\dim_k T\mathcal{F} < \infty$ and $\dim_k \text{Inf}(\mathcal{F}) < \infty$. Let $\xi$ be a versal formal object lying over $R$. Let $\eta$ be a formal object lying over $S$. Then any two maps $$ f, g : R \to S $$ such that $f_*\xi \cong \eta \cong g_*\xi$ are formally homotopic.


formal-defos-lemma-homotopic-minimal-prime

In the category $\widehat{\mathcal{C}}_\Lambda$, if $f_1, f_2 : R \to S$ are formally homotopic and $\mathfrak p \subset R$ is a minimal prime ideal, then $f_1(\mathfrak p)S = f_2(\mathfrak p)S$ as ideals.


formal-defos-lemma-elementary-properties-change-of-field

With notation and assumptions as in Situation \ref{situation-change-of-fields}. \begin{enumerate} \item We have $\overline{\mathcal{F}_{l/k}} = (\overline{\mathcal{F}})_{l/k}$. \item If $\mathcal{F}$ is a predeformation category, then $\mathcal{F}_{l/k}$ is a predeformation category. \item If $\mathcal{F}$ satisfies (S1), then $\mathcal{F}_{l/k}$ satisfies (S1). \item If $\mathcal{F}$ satisfies (S2), then $\mathcal{F}_{l/k}$ satisfies (S2). \item If $\mathcal{F}$ satisfies (RS), then $\mathcal{F}_{l/k}$ satisfies (RS). \end{enumerate}


formal-defos-lemma-tangent-space-change-of-field

With notation and assumptions as in Situation \ref{situation-change-of-fields}. Assume $\mathcal{F}$ is a predeformation category and $\overline{\mathcal{F}}$ satisfies (S2). Then there is a canonical $l$-vector space isomorphism $$ T\mathcal{F} \otimes_k l \longrightarrow T\mathcal{F}_{l/k} $$ of tangent spaces.


formal-defos-lemma-inf-aut-change-of-field

With notation and assumptions as in Situation \ref{situation-change-of-fields}. Assume $\mathcal{F}$ is a deformation category. Then there is a canonical $l$-vector space isomorphism $$ \text{Inf}(\mathcal{F}) \otimes_k l \longrightarrow \text{Inf}(\mathcal{F}_{l/k}) $$ of infinitesimal automorphism spaces.


formal-defos-lemma-change-of-fields-smooth

With notation and assumptions as in Situation \ref{situation-change-of-fields}. If $\mathcal{F} \to \mathcal{G}$ is a smooth morphism of categories cofibred in groupoids over $\mathcal{C}_{\Lambda, k}$, then $\mathcal{F}_{l/k} \to \mathcal{G}_{l/k}$ is a smooth morphism of categories cofibred in groupoids over $\mathcal{C}_{\Lambda, l}$.


formal-defos-lemma-change-of-field-versal-ring

With notation and assumptions as in Situation \ref{situation-change-of-fields}. Let $\xi$ be a versal formal object for $\mathcal{F}$ lying over $R \in \Ob(\widehat{\mathcal{C}}_{\Lambda, k})$. Then there exist \begin{enumerate} \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 \item a versal formal object of $\mathcal{F}_{l/k}$ lying over $S$. \end{enumerate}


adequate-lemma-adequate-finite-presentation

Let $A$ be a ring. Let $F$ be an adequate functor on $\textit{Alg}_A$. If $B = \colim B_i$ is a filtered colimit of $A$-algebras, then $F(B) = \colim F(B_i)$.


adequate-lemma-adequate-flat

Let $A$ be a ring. Let $F$ be an adequate functor on $\textit{Alg}_A$. If $B \to B'$ is flat, then $F(B) \otimes_B B' \to F(B')$ is an isomorphism.


adequate-lemma-adequate-surjection-from-linear

Let $A$ be a ring. Let $F$ be an adequate functor on $\textit{Alg}_A$. Then there exists a surjection $L \to F$ with $L$ a direct sum of linearly adequate functors.


adequate-lemma-flat-functor-split

Let $A$ be a ring. Let $F$ be a module-valued functor on $\textit{Alg}_A$. Assume that for $B \to B'$ flat the map $F(B) \otimes_B B' \to F(B')$ is an isomorphism. Let $B$ be a graded $A$-algebra. Then \begin{enumerate} \item $F(B) = \bigoplus_{k \in \mathbf{Z}} F(B)^{(k)}$, and \item the map $B \to B_0 \to B$ induces map $F(B) \to F(B)$ whose image is contained in $F(B)^{(0)}$. \end{enumerate}


adequate-lemma-lift-map

Let $A$ be a ring. Given a solid diagram $$ \xymatrix{ 0 \ar[r] & L \ar[d]_\varphi \ar[r] & \underline{A^{\oplus n}} \ar[r] \ar@{..>}[ld] & \underline{A^{\oplus m}} \\ & \underline{M} } $$ of module-valued functors on $\textit{Alg}_A$ with exact row there exists a dotted arrow making the diagram commute.


adequate-lemma-cokernel-into-module

Let $A$ be a ring. Let $\varphi : F \to \underline{M}$ be a map of module-valued functors on $\textit{Alg}_A$ with $F$ adequate. Then $\Coker(\varphi)$ is adequate.


adequate-lemma-cokernel-adequate

\begin{slogan} The cokernel of a map of adequate functors on the category of algebras over a ring is adequate. \end{slogan} Let $A$ be a ring. Let $\varphi : F \to G$ be a map of adequate functors on $\textit{Alg}_A$. Then $\Coker(\varphi)$ is adequate.


adequate-lemma-kernel-adequate

Let $A$ be a ring. Let $\varphi : F \to G$ be a map of adequate functors on $\textit{Alg}_A$. Then $\Ker(\varphi)$ is adequate.


adequate-lemma-colimit-adequate

Let $A$ be a ring. An arbitrary direct sum of adequate functors on $\textit{Alg}_A$ is adequate. A colimit of adequate functors is adequate.


adequate-lemma-flat-linear-functor

Let $A$ be a ring. Let $F, G$ be module-valued functors on $\textit{Alg}_A$. Let $\varphi : F \to G$ be a transformation of functors. Assume \begin{enumerate} \item $\varphi$ is additive, \item for every $A$-algebra $B$ and $\xi \in F(B)$ and unit $u \in B^*$ we have $\varphi(u\xi) = u\varphi(\xi)$ in $G(B)$, and \item for any flat ring map $B \to B'$ we have $G(B) \otimes_B B' = G(B')$. \end{enumerate} Then $\varphi$ is a morphism of module-valued functors.


adequate-lemma-extension-adequate-key

Let $A$ be a ring. Let $0 \to \underline{M} \to G \to L \to 0$ be a short exact sequence of module-valued functors on $\textit{Alg}_A$ with $L$ linearly adequate. Then $G$ is adequate.


adequate-lemma-extension-adequate

Let $A$ be a ring. Let $0 \to F \to G \to H \to 0$ be a short exact sequence of module-valued functors on $\textit{Alg}_A$. If $F$ and $H$ are adequate, so is $G$.


adequate-lemma-base-change-adequate

Let $A \to A'$ be a ring map. If $F$ is an adequate functor on $\textit{Alg}_A$, then its restriction $F'$ to $\textit{Alg}_{A'}$ is adequate too.


adequate-lemma-pushforward-adequate

Let $A \to A'$ be a ring map. If $F'$ is an adequate functor on $\textit{Alg}_{A'}$, then the module-valued functor $F : B \mapsto F'(A' \otimes_A B)$ on $\textit{Alg}_A$ is adequate too.


adequate-lemma-adequate-product

Let $A = A_1 \times \ldots \times A_n$ be a product of rings. An adequate functor over $A$ is the same thing as a sequence $F_1, \ldots, F_n$ of adequate functors $F_i$ over $A_i$.


adequate-lemma-adequate-descent

Let $A \to A'$ be a ring map and let $F$ be a module-valued functor on $\textit{Alg}_A$ such that \begin{enumerate} \item the restriction $F'$ of $F$ to the category of $A'$-algebras is adequate, and \item for any $A$-algebra $B$ the sequence $$ 0 \to F(B) \to F(B \otimes_A A') \to F(B \otimes_A A' \otimes_A A') $$ is exact. \end{enumerate} Then $F$ is adequate.


adequate-lemma-adjoint

Let $A$ be a ring. For every module-valued functor $F$ on $\textit{Alg}_A$ there exists a morphism $Q(F) \to F$ of module-valued functors on $\textit{Alg}_A$ such that (1) $Q(F)$ is adequate and (2) for every adequate functor $G$ the map $\Hom(G, Q(F)) \to \Hom(G, F)$ is a bijection.


adequate-lemma-enough-injectives

Let $A$ be a ring. Denote $\mathcal{P}$ the category of module-valued functors on $\textit{Alg}_A$ and $\mathcal{A}$ the category of adequate functors on $\textit{Alg}_A$. Denote $i : \mathcal{A} \to \mathcal{P}$ the inclusion functor. Denote $Q : \mathcal{P} \to \mathcal{A}$ the construction of Lemma \ref{lemma-adjoint}. Then \begin{enumerate} \item $i$ is fully faithful, exact, and its image is a weak Serre subcategory, \item $\mathcal{P}$ has enough injectives, \item the functor $Q$ is a right adjoint to $i$ hence left exact, \item $Q$ transforms injectives into injectives, \item $\mathcal{A}$ has enough injectives. \end{enumerate}


adequate-lemma-tangent-functor

Let $A$ be a ring. Let $F$ be a module valued functor. For every $B \in \Ob(\textit{Alg}_A)$ and $B$-module $N$ there is a canonical decomposition $$ F(B[N]) = F(B) \oplus TF(B, N) $$ characterized by the following properties \begin{enumerate} \item $TF(B, N) = \Ker(F(B[N]) \to F(B))$, \item there is a $B$-module structure $TF(B, N)$ compatible with $B[N]$-module structure on $F(B[N])$, \item $TF$ is a functor from the category of pairs $(B, N)$, \item \label{item-mult-map} there are canonical maps $N \otimes_B F(B) \to TF(B, N)$ inducing a transformation between functors defined on the category of pairs $(B, N)$, \item $TF(B, 0) = 0$ and the map $TF(B, N) \to TF(B, N')$ is zero when $N \to N'$ is the zero map. \end{enumerate}


adequate-lemma-tangent-injective

Let $A$ be a ring. Let $I$ be an injective object of the category of module-valued functors. Then for any $B \in \Ob(\textit{Alg}_A)$ and short exact sequence $0 \to N_1 \to N \to N_2 \to 0$ of $B$-modules the sequence $$ TI(B, N_1) \to TI(B, N) \to TI(B, N_2) \to 0 $$ is exact.


adequate-lemma-exactness-implies

Let $A$ be a ring. Let $F$ be a module-valued functor such that for any $B \in \Ob(\textit{Alg}_A)$ the functor $TF(B, -)$ on $B$-modules transforms a short exact sequence of $B$-modules into a right exact sequence. Then \begin{enumerate} \item $TF(B, N_1 \oplus N_2) = TF(B, N_1) \oplus TF(B, N_2)$, \item there is a second functorial $B$-module structure on $TF(B, N)$ defined by setting $x \cdot b = TF(B, b\cdot 1_N)(x)$ for $x \in TF(B, N)$ and $b \in B$, \item \label{item-mult-map-linear} the canonical map $N \otimes_B F(B) \to TF(B, N)$ of Lemma \ref{lemma-tangent-functor} is $B$-linear also with respect to the second $B$-module structure, \item \label{item-tangent-right-exact} given a finitely presented $B$-module $N$ there is a canonical isomorphism $TF(B, B) \otimes_B N \to TF(B, N)$ where the tensor product uses the second $B$-module structure on $TF(B, B)$. \end{enumerate}


adequate-lemma-exactness-permanence

Let $A$ be a ring. For $F$ a module-valued functor on $\textit{Alg}_A$ say $(*)$ holds if for all $B \in \Ob(\textit{Alg}_A)$ the functor $TF(B, -)$ on $B$-modules transforms a short exact sequence of $B$-modules into a right exact sequence. Let $0 \to F \to G \to H \to 0$ be a short exact sequence of module-valued functors on $\textit{Alg}_A$. \begin{enumerate} \item If $(*)$ holds for $F, G$ then $(*)$ holds for $H$. \item If $(*)$ holds for $F, H$ then $(*)$ holds for $G$. \item If $H' \to H$ is morphism of module-valued functors on $\textit{Alg}_A$ and $(*)$ holds for $F$, $G$, $H$, and $H'$, then $(*)$ holds for $G \times_H H'$. \end{enumerate}


adequate-lemma-ext-group-zero-key

Let $A$ be a ring. Let $M$, $P$ be $A$-modules with $P$ of finite presentation. Then $\Ext^i_\mathcal{P}(\underline{P}, \underline{M}) = 0$ for $i > 0$ where $\mathcal{P}$ is the category of module-valued functors on $\textit{Alg}_A$.


adequate-lemma-ext-group-zero

Let $A$ be a ring. Let $M$ be an $A$-module. Let $L$ be a linearly adequate functor on $\textit{Alg}_A$. Then $\Ext^i_\mathcal{P}(L, \underline{M}) = 0$ for $i > 0$ where $\mathcal{P}$ is the category of module-valued functors on $\textit{Alg}_A$.


adequate-lemma-RQ-zero

With notation as in Lemma \ref{lemma-enough-injectives} we have $R^pQ(F) = 0$ for all $p > 0$ and any adequate functor $F$.


adequate-lemma-adequate-local

Let $S$ be a scheme. Let $\mathcal{F}$ be an adequate $\mathcal{O}$-module on $(\Sch/S)_\tau$. For any affine scheme $\Spec(A)$ over $S$ the functor $F_{\mathcal{F}, A}$ is adequate.


adequate-lemma-adequate-affine

Let $S = \Spec(A)$ be an affine scheme. The category of adequate $\mathcal{O}$-modules on $(\Sch/S)_\tau$ is equivalent to the category of adequate module-valued functors on $\textit{Alg}_A$.


adequate-lemma-pullback-adequate

Let $f : T \to S$ be a morphism of schemes. The pullback $f^*\mathcal{F}$ of an adequate $\mathcal{O}$-module $\mathcal{F}$ on $(\Sch/S)_\tau$ is an adequate $\mathcal{O}$-module on $(\Sch/T)_\tau$.


adequate-lemma-adequate-characterize

Let $S$ be a scheme. Let $\mathcal{F}$ be an $\mathcal{O}$-module on $(\Sch/S)_\tau$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is adequate, \item there exists an affine open covering $S = \bigcup S_i$ and maps of quasi-coherent $\mathcal{O}_{S_i}$-modules $\mathcal{G}_i \to \mathcal{H}_i$ such that $\mathcal{F}|_{(\Sch/S_i)_\tau}$ is the kernel of $\mathcal{G}_i^a \to \mathcal{H}_i^a$ \item there exists a $\tau$-covering $\{S_i \to S\}_{i \in I}$ and maps of $\mathcal{O}_{S_i}$-quasi-coherent modules $\mathcal{G}_i \to \mathcal{H}_i$ such that $\mathcal{F}|_{(\Sch/S_i)_\tau}$ is the kernel of $\mathcal{G}_i^a \to \mathcal{H}_i^a$, \item there exists a $\tau$-covering $\{f_i : S_i \to S\}_{i \in I}$ such that each $f_i^*\mathcal{F}$ is adequate, \item for any affine scheme $U$ over $S$ the restriction $\mathcal{F}|_{(\Sch/U)_\tau}$ is the kernel of a map $\mathcal{G}^a \to \mathcal{H}^a$ of quasi-coherent $\mathcal{O}_U$-modules. \end{enumerate}


adequate-lemma-adequate-fpqc

Let $\mathcal{F}$ be an adequate $\mathcal{O}$-module on $(\Sch/S)_\tau$. For any surjective flat morphism $\Spec(B) \to \Spec(A)$ of affines over $S$ the extended {\v C}ech complex $$ 0 \to \mathcal{F}(\Spec(A)) \to \mathcal{F}(\Spec(B)) \to \mathcal{F}(\Spec(B \otimes_A B)) \to \ldots $$ is exact. In particular $\mathcal{F}$ satisfies the sheaf condition for fpqc coverings, and is a sheaf of $\mathcal{O}$-modules on $(\Sch/S)_{fppf}$.


adequate-lemma-same-cohomology-adequate

Let $S$ be a scheme. Let $\mathcal{F}$ be an adequate $\mathcal{O}$-module on $(\Sch/S)_\tau$. \begin{enumerate} \item The restriction $\mathcal{F}|_{S_{Zar}}$ is a quasi-coherent $\mathcal{O}_S$-module on the scheme $S$. \item The restriction $\mathcal{F}|_{S_\etale}$ is the quasi-coherent module associated to $\mathcal{F}|_{S_{Zar}}$. \item For any affine scheme $U$ over $S$ we have $H^q(U, \mathcal{F}) = 0$ for all $q > 0$. \item There is a canonical isomorphism $$ H^q(S, \mathcal{F}|_{S_{Zar}}) = H^q((\Sch/S)_\tau, \mathcal{F}). $$ \end{enumerate}


adequate-lemma-sheafification-adequate

Let $S$ be a scheme. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules on $(\Sch/S)_\tau$. If for every affine scheme $\Spec(A)$ over $S$ the functor $F_{\mathcal{F}, A}$ is adequate, then the sheafification of $\mathcal{F}$ is an adequate $\mathcal{O}$-module.


adequate-lemma-abelian-adequate

Let $S$ be a scheme. \begin{enumerate} \item The category $\textit{Adeq}(\mathcal{O})$ is abelian. \item The functor $\textit{Adeq}(\mathcal{O}) \to \textit{Mod}((\Sch/S)_\tau, \mathcal{O})$ is exact. \item If $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ is a short exact sequence of $\mathcal{O}$-modules and $\mathcal{F}_1$ and $\mathcal{F}_3$ are adequate, then $\mathcal{F}_2$ is adequate. \item The category $\textit{Adeq}(\mathcal{O})$ has colimits and $\textit{Adeq}(\mathcal{O}) \to \textit{Mod}((\Sch/S)_\tau, \mathcal{O})$ commutes with them. \end{enumerate}


adequate-lemma-direct-image-adequate

Let $f : T \to S$ be a quasi-compact and quasi-separated morphism of schemes. For any adequate $\mathcal{O}_T$-module on $(\Sch/T)_\tau$ the pushforward $f_*\mathcal{F}$ and the higher direct images $R^if_*\mathcal{F}$ are adequate $\mathcal{O}_S$-modules on $(\Sch/S)_\tau$.


adequate-lemma-parasitic-adequate

Let $S$ be a scheme. Let $\mathcal{F}$ be an adequate $\mathcal{O}$-module on $(\Sch/S)_\tau$. The following are equivalent: \begin{enumerate} \item $v\mathcal{F} = 0$, \item $\mathcal{F}$ is parasitic, \item $\mathcal{F}$ is parasitic for the $\tau$-topology, \item $\mathcal{F}(U) = 0$ for all $U \subset S$ open, and \item there exists an affine open covering $S = \bigcup U_i$ such that $\mathcal{F}(U_i) = 0$ for all $i$. \end{enumerate}


adequate-lemma-right-adjoint-adequate

Let $U = \Spec(A)$ be an affine scheme. The inclusion functor $$ \textit{Adeq}(\mathcal{O}) \to \textit{Mod}((\Sch/U)_\tau, \mathcal{O}) $$ has a right adjoint $A$\footnote{This is the ``adequator''.}. Moreover, the adjunction mapping $A(\mathcal{F}) \to \mathcal{F}$ is an isomorphism for every adequate module $\mathcal{F}$.


adequate-lemma-RA-zero

Let $U = \Spec(A)$ be an affine scheme. For any object $\mathcal{F}$ of $\textit{Adeq}(\mathcal{O})$ we have $R^pA(\mathcal{F}) = 0$ for all $p > 0$ where $A$ is as in Lemma \ref{lemma-right-adjoint-adequate}.


adequate-lemma-bounded-below

If $U = \Spec(A)$ is an affine scheme, then the bounded below version \begin{equation} \label{equation-compare-bounded-adequate} D^+(\textit{Adeq}(\mathcal{O})) \longrightarrow D^+_{\textit{Adeq}}(\mathcal{O}) \end{equation} of the functor above is an equivalence.


adequate-lemma-ext-adequate

Let $U = \Spec(A)$ be an affine scheme. Let $\mathcal{F}$ and $\mathcal{G}$ be adequate $\mathcal{O}$-modules. For any $i \geq 0$ the natural map $$ \Ext^i_{\textit{Adeq}(\mathcal{O})}(\mathcal{F}, \mathcal{G}) \longrightarrow \Ext^i_{\textit{Mod}(\mathcal{O})}(\mathcal{F}, \mathcal{G}) $$ is an isomorphism.


adequate-lemma-pure-projective

Let $A$ be a ring. \begin{enumerate} \item A module is pure projective if and only if it is a direct summand of a direct sum of finitely presented $A$-modules. \item For any module $M$ there exists a universally exact sequence $0 \to N \to P \to M \to 0$ with $P$ pure projective. \end{enumerate}


adequate-lemma-pure-injective

Let $A$ be a ring. For any $A$-module $M$ set $M^\vee = \Hom_\mathbf{Z}(M, \mathbf{Q}/\mathbf{Z})$. \begin{enumerate} \item For any $A$-module $M$ the $A$-module $M^\vee$ is pure injective. \item An $A$-module $I$ is pure injective if and only if the map $I \to (I^\vee)^\vee$ splits. \item For any module $M$ there exists a universally exact sequence $0 \to M \to I \to N \to 0$ with $I$ pure injective. \end{enumerate}


adequate-lemma-split-universally-exact-sequence

Let $A$ be a ring. \begin{enumerate} \item Let $L \to M \to N$ be a universally exact sequence of $A$-modules. Let $K = \Im(M \to N)$. Then $K \to N$ is universally injective. \item Any universally exact complex can be split into universally exact short exact sequences. \end{enumerate}


adequate-lemma-pure-projective-resolutions

Let $A$ be a ring. \begin{enumerate} \item Any $A$-module has a pure projective resolution. \end{enumerate} Let $M \to N$ be a map of $A$-modules. Let $P_\bullet \to M$ be a pure projective resolution and let $N_\bullet \to N$ be a universally exact resolution. \begin{enumerate} \item[(2)] There exists a map of complexes $P_\bullet \to N_\bullet$ inducing the given map $$ M = \Coker(P_1 \to P_0) \to \Coker(N_1 \to N_0) = N $$ \item[(3)] two maps $\alpha, \beta : P_\bullet \to N_\bullet$ inducing the same map $M \to N$ are homotopic. \end{enumerate}


adequate-lemma-facts-pext

Let $A$ be a ring. \begin{enumerate} \item $\text{Pext}^i_A(M, N) = 0$ for $i > 0$ whenever $N$ is pure injective, \item $\text{Pext}^i_A(M, N) = 0$ for $i > 0$ whenever $M$ is pure projective, in particular if $M$ is an $A$-module of finite presentation, \item $\text{Pext}^i_A(M, N)$ is also the $i$th cohomology module of the complex $\Hom_A(P_\bullet, N)$ where $P_\bullet$ is a pure projective resolution of $M$. \end{enumerate}


adequate-lemma-pure-injective-injective-adequate

Let $A$ be a ring. Let $\mathcal{A}$ be the category of adequate functors on $\textit{Alg}_A$. The injective objects of $\mathcal{A}$ are exactly the functors $\underline{I}$ where $I$ is a pure injective $A$-module.


spaces-topologies-lemma-zariski

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item If $X' \to X$ is an isomorphism then $\{X' \to X\}$ is a Zariski covering of $X$. \item If $\{X_i \to X\}_{i\in I}$ is a Zariski covering and for each $i$ we have a Zariski covering $\{X_{ij} \to X_i\}_{j\in J_i}$, then $\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is a Zariski covering. \item If $\{X_i \to X\}_{i\in I}$ is a Zariski covering and $X' \to X$ is a morphism of algebraic spaces then $\{X' \times_X X_i \to X'\}_{i\in I}$ is a Zariski covering. \end{enumerate}


spaces-topologies-lemma-zariski-etale

Any Zariski covering is an \'etale covering.


spaces-topologies-lemma-etale

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item If $X' \to X$ is an isomorphism then $\{X' \to X\}$ is a \'etale covering of $X$. \item If $\{X_i \to X\}_{i\in I}$ is a \'etale covering and for each $i$ we have a \'etale covering $\{X_{ij} \to X_i\}_{j\in J_i}$, then $\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is a \'etale covering. \item If $\{X_i \to X\}_{i\in I}$ is a \'etale covering and $X' \to X$ is a morphism of algebraic spaces then $\{X' \times_X X_i \to X'\}_{i\in I}$ is a \'etale covering. \end{enumerate}


spaces-topologies-lemma-etale-dominates-smooth

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\{X_i \to X\}_{i \in I}$ be a smooth covering of $X$. Then there exists an \'etale covering $\{U_j \to X\}_{j \in J}$ of $X$ which refines $\{X_i \to X\}_{i \in I}$.


spaces-topologies-lemma-put-in-T-etale

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of $(\textit{Spaces}/S)_\etale$. The inclusion functor $Y_{spaces, \etale} \to (\textit{Spaces}/X)_\etale$ is cocontinuous and induces a morphism of topoi $$ i_f : \Sh(Y_\etale) \longrightarrow \Sh((\textit{Spaces}/X)_\etale) $$ For a sheaf $\mathcal{G}$ on $(\textit{Spaces}/X)_\etale$ we have the formula $(i_f^{-1}\mathcal{G})(U/Y) = \mathcal{G}(U/X)$. The functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes with fibre products and equalizers.


spaces-topologies-lemma-at-the-bottom-etale

Let $S$ be a scheme. Let $X$ be an object of $(\textit{Spaces}/S)_\etale$. The inclusion functor $X_{spaces, \etale} \to (\textit{Spaces}/X)_\etale$ satisfies the hypotheses of Sites, Lemma \ref{sites-lemma-bigger-site} and hence induces a morphism of sites $$ \pi_X : (\textit{Spaces}/X)_\etale \longrightarrow X_{spaces, \etale} $$ and a morphism of topoi $$ i_X : \Sh(X_\etale) \longrightarrow \Sh((\textit{Spaces}/X)_\etale) $$ such that $\pi_X \circ i_X = \text{id}$. Moreover, $i_X = i_{\text{id}_X}$ with $i_{\text{id}_X}$ as in Lemma \ref{lemma-put-in-T-etale}. In particular the functor $i_X^{-1} = \pi_{X, *}$ is described by the rule $i_X^{-1}(\mathcal{G})(U/X) = \mathcal{G}(U/X)$.


spaces-topologies-lemma-morphism-big-etale

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism in $(\textit{Spaces}/S)_\etale$. The functor $$ u : (\textit{Spaces}/Y)_\etale \longrightarrow (\textit{Spaces}/X)_\etale, \quad V/Y \longmapsto V/X $$ is cocontinuous, and has a continuous right adjoint $$ v : (\textit{Spaces}/X)_\etale \longrightarrow (\textit{Spaces}/Y)_\etale, \quad (U \to X) \longmapsto (U \times_X Y \to Y). $$ They induce the same morphism of topoi $$ f_{big} : \Sh((\textit{Spaces}/Y)_\etale) \longrightarrow \Sh((\textit{Spaces}/X)_\etale) $$ We have $f_{big}^{-1}(\mathcal{G})(U/Y) = \mathcal{G}(U/X)$. We have $f_{big, *}(\mathcal{F})(U/X) = \mathcal{F}(U \times_X Y/Y)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.


spaces-topologies-lemma-morphism-big-small-etale

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism in $(\textit{Spaces}/S)_\etale$. \begin{enumerate} \item We have $i_f = f_{big} \circ i_T$ with $i_f$ as in Lemma \ref{lemma-put-in-T-etale} and $i_T$ as in Lemma \ref{lemma-at-the-bottom-etale}. \item The functor $X_{spaces, \etale} \to T_{spaces, \etale}$, $(U \to X) \mapsto (U \times_X Y \to Y)$ is continuous and induces a morphism of sites $$ f_{spaces, \etale} : Y_{spaces, \etale} \longrightarrow X_{spaces, \etale} $$ The corresponding morphism of small \'etale topoi is denoted $$ f_{small} : \Sh(Y_\etale) \to \Sh(X_\etale) $$ We have $f_{small, *}(\mathcal{F})(U/X) = \mathcal{F}(U \times_X Y/Y)$. \item We have a commutative diagram of morphisms of sites $$ \xymatrix{ Y_{spaces, \etale} \ar[d]_{f_{spaces, \etale}} & (\textit{Spaces}/Y)_\etale \ar[d]^{f_{big}} \ar[l]^-{\pi_Y}\\ X_{spaces, \etale} & (\textit{Spaces}/X)_\etale \ar[l]_-{\pi_X} } $$ so that $f_{small} \circ \pi_Y = \pi_X \circ f_{big}$ as morphisms of topoi. \item We have $f_{small} = \pi_X \circ f_{big} \circ i_Y = \pi_X \circ i_f$. \end{enumerate}


spaces-topologies-lemma-zariski-etale-smooth

Any \'etale covering is a smooth covering, and a fortiori, any Zariski covering is a smooth covering.


spaces-topologies-lemma-smooth

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item If $X' \to X$ is an isomorphism then $\{X' \to X\}$ is a smooth covering of $X$. \item If $\{X_i \to X\}_{i\in I}$ is a smooth covering and for each $i$ we have a smooth covering $\{X_{ij} \to X_i\}_{j\in J_i}$, then $\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is a smooth covering. \item If $\{X_i \to X\}_{i\in I}$ is a smooth covering and $X' \to X$ is a morphism of algebraic spaces then $\{X' \times_X X_i \to X'\}_{i\in I}$ is a smooth covering. \end{enumerate}


spaces-topologies-lemma-zariski-etale-smooth-syntomic

Any smooth covering is a syntomic covering, and a fortiori, any \'etale or Zariski covering is a syntomic covering.


spaces-topologies-lemma-syntomic

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item If $X' \to X$ is an isomorphism then $\{X' \to X\}$ is a syntomic covering of $X$. \item If $\{X_i \to X\}_{i\in I}$ is a syntomic covering and for each $i$ we have a syntomic covering $\{X_{ij} \to X_i\}_{j\in J_i}$, then $\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is a syntomic covering. \item If $\{X_i \to X\}_{i\in I}$ is a syntomic covering and $X' \to X$ is a morphism of algebraic spaces then $\{X' \times_X X_i \to X'\}_{i\in I}$ is a syntomic covering. \end{enumerate}


spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf

Any syntomic covering is an fppf covering, and a fortiori, any smooth, \'etale, or Zariski covering is an fppf covering.


spaces-topologies-lemma-fppf

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item If $X' \to X$ is an isomorphism then $\{X' \to X\}$ is an fppf covering of $X$. \item If $\{X_i \to X\}_{i\in I}$ is an fppf covering and for each $i$ we have an fppf covering $\{X_{ij} \to X_i\}_{j\in J_i}$, then $\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is an fppf covering. \item If $\{X_i \to X\}_{i\in I}$ is an fppf covering and $X' \to X$ is a morphism of algebraic spaces then $\{X' \times_X X_i \to X'\}_{i\in I}$ is an fppf covering. \end{enumerate}


spaces-topologies-lemma-refine-fppf-schemes

Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Suppose that $\mathcal{U} = \{f_i : X_i \to X\}_{i \in I}$ is an fppf covering of $X$. Then there exists a refinement $\mathcal{V} = \{g_i : T_i \to X\}$ of $\mathcal{U}$ which is an fppf covering such that each $T_i$ is a scheme.


spaces-topologies-lemma-fppf-covering-surjective

Let $S$ be a scheme. Let $\{f_i : X_i \to X\}_{i \in I}$ be an fppf covering of algebraic spaces over $S$. Then the map of sheaves $$ \coprod X_i \longrightarrow X $$ is surjective.


spaces-topologies-lemma-morphism-big-fppf

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. The functor $$ u : (\textit{Spaces}/Y)_{fppf} \longrightarrow (\textit{Spaces}/X)_{fppf}, \quad V/Y \longmapsto V/X $$ is cocontinuous, and has a continuous right adjoint $$ v : (\textit{Spaces}/X)_{fppf} \longrightarrow (\textit{Spaces}/Y)_{fppf}, \quad (U \to Y) \longmapsto (U \times_X Y \to Y). $$ They induce the same morphism of topoi $$ f_{big} : \Sh((\textit{Spaces}/Y)_{fppf}) \longrightarrow \Sh((\textit{Spaces}/X)_{fppf}) $$ We have $f_{big}^{-1}(\mathcal{G})(U/Y) = \mathcal{G}(U/X)$. We have $f_{big, *}(\mathcal{F})(U/X) = \mathcal{F}(U \times_X Y/Y)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.


spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-ph

Any fppf covering is a ph covering, and a fortiori, any syntomic, smooth, \'etale or Zariski covering is a ph covering.


spaces-topologies-lemma-surjective-proper-ph

Let $S$ be a scheme. Let $f : Y \to X$ be a surjective proper morphism of algebraic spaces over $S$. Then $\{Y \to X\}$ is a ph covering.


spaces-topologies-lemma-characterize-sheaf

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a presheaf on $(\textit{Spaces}/X)_{ph}$. Then $\mathcal{F}$ is a sheaf if and only if \begin{enumerate} \item $\mathcal{F}$ satisfies the sheaf condition for \'etale coverings, and \item if $f : V \to U$ is a proper surjective morphism of $(\textit{Spaces}/X)_{ph}$, then $\mathcal{F}(U)$ maps bijectively to the equalizer of the two maps $\mathcal{F}(V) \to \mathcal{F}(V \times_U V)$. \end{enumerate}


spaces-topologies-lemma-morphism-big-ph

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. The functor $$ u : (\textit{Spaces}/Y)_{ph} \longrightarrow (\textit{Spaces}/X)_{ph}, \quad V/Y \longmapsto V/X $$ is cocontinuous, and has a continuous right adjoint $$ v : (\textit{Spaces}/X)_{ph} \longrightarrow (\textit{Spaces}/Y)_{ph}, \quad (U \to Y) \longmapsto (U \times_X Y \to Y). $$ They induce the same morphism of topoi $$ f_{big} : \Sh((\textit{Spaces}/Y)_{ph}) \longrightarrow \Sh((\textit{Spaces}/X)_{ph}) $$ We have $f_{big}^{-1}(\mathcal{G})(U/Y) = \mathcal{G}(U/X)$. We have $f_{big, *}(\mathcal{F})(U/X) = \mathcal{F}(U \times_X Y/Y)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.


spaces-topologies-lemma-cech-enough

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $P$ be a property of objects in $(\textit{Spaces}/X)_{fppf}$ such that whenever $\{U_i \to U\}$ is a covering in $(\textit{Spaces}/X)_{fppf}$, then $$ P(U_{i_0} \times_U \ldots \times_U U_{i_p}) \text{ for all } p \geq 0,\ i_0, \ldots, i_p \in I \Rightarrow P(U) $$ If $P(U)$ for all $U$ affine and flat, locally of finite presentation over $X$, then $P(X)$.


spaces-topologies-lemma-fpqc

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item If $X' \to X$ is an isomorphism then $\{X' \to X\}$ is an fpqc covering of $X$. \item If $\{X_i \to X\}_{i\in I}$ is an fpqc covering and for each $i$ we have an fpqc covering $\{X_{ij} \to X_i\}_{j\in J_i}$, then $\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is an fpqc covering. \item If $\{X_i \to X\}_{i\in I}$ is an fpqc covering and $X' \to X$ is a morphism of algebraic spaces then $\{X' \times_X X_i \to X'\}_{i\in I}$ is an fpqc covering. \end{enumerate}


spaces-topologies-lemma-recognize-fpqc-covering

Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Suppose that $\{f_i : X_i \to X\}_{i \in I}$ is a family of morphisms of algebraic spaces with target $X$. Let $U \to X$ be a surjective \'etale morphism from a scheme towards $X$. Then $\{f_i : X_i \to X\}_{i \in I}$ is an fpqc covering of $X$ if and only if $\{U \times_X X_i \to U\}_{i \in I}$ is an fpqc covering of $U$.


spaces-topologies-lemma-refine-fpqc-schemes

Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Suppose that $\mathcal{U} = \{f_i : X_i \to X\}_{i \in I}$ is an fpqc covering of $X$. Then there exists a refinement $\mathcal{V} = \{g_i : T_i \to X\}$ of $\mathcal{U}$ which is an fpqc covering such that each $T_i$ is a scheme.


proetale-lemma-spectral-split

Let $X$ be a spectral space. Let $X_0 \subset X$ be the set of closed points. The following are equivalent \begin{enumerate} \item Every open covering of $X$ can be refined by a finite disjoint union decomposition $X = \coprod U_i$ with $U_i$ open and closed in $X$. \item The composition $X_0 \to X \to \pi_0(X)$ is bijective. \end{enumerate} Moreover, if $X_0$ is closed in $X$ and every point of $X$ specializes to a unique point of $X_0$, then these conditions are satisfied.


proetale-lemma-closed-subspace-w-local

Let $X$ be a w-local spectral space. If $Y \subset X$ is closed, then $Y$ is w-local.


proetale-lemma-silly

Let $X$ be a spectral space. Let $$ \xymatrix{ Y \ar[r] \ar[d] & T \ar[d] \\ X \ar[r] & \pi_0(X) } $$ be a cartesian diagram in the category of topological spaces with $T$ profinite. Then $Y$ is spectral and $T = \pi_0(Y)$. If moreover $X$ is w-local, then $Y$ is w-local, $Y \to X$ is w-local, and the set of closed points of $Y$ is the inverse image of the set of closed points of $X$.


proetale-lemma-base-change-local-isomorphism

Let $A \to B$ and $A \to A'$ be ring maps. Let $B' = B \otimes_A A'$ be the base change of $B$. \begin{enumerate} \item If $A \to B$ is a local isomorphism, then $A' \to B'$ is a local isomorphism. \item If $A \to B$ identifies local rings, then $A' \to B'$ identifies local rings. \end{enumerate}


proetale-lemma-composition-local-isomorphism

Let $A \to B$ and $B \to C$ be ring maps. \begin{enumerate} \item If $A \to B$ and $B \to C$ are local isomorphisms, then $A \to C$ is a local isomorphism. \item If $A \to B$ and $B \to C$ identify local rings, then $A \to C$ identifies local rings. \end{enumerate}


proetale-lemma-local-isomorphism-permanence

Let $A$ be a ring. Let $B \to C$ be an $A$-algebra homomorphism. \begin{enumerate} \item If $A \to B$ and $A \to C$ are local isomorphisms, then $B \to C$ is a local isomorphism. \item If $A \to B$ and $A \to C$ identify local rings, then $B \to C$ identifies local rings. \end{enumerate}


proetale-lemma-local-isomorphism-implies

Let $A \to B$ be a local isomorphism. Then \begin{enumerate} \item $A \to B$ is \'etale, \item $A \to B$ identifies local rings, \item $A \to B$ is quasi-finite. \end{enumerate}


proetale-lemma-structure-local-isomorphism

Let $A \to B$ be a local isomorphism. Then there exist $n \geq 0$, $g_1, \ldots, g_n \in B$, $f_1, \ldots, f_n \in A$ such that $(g_1, \ldots, g_n) = B$ and $A_{f_i} \cong B_{g_i}$.


proetale-lemma-fully-faithful-spaces-over-X

Let $p : (Y, \mathcal{O}_Y) \to (X, \mathcal{O}_X)$ and $q : (Z, \mathcal{O}_Z) \to (X, \mathcal{O}_X)$ be morphisms of locally ringed spaces. If $\mathcal{O}_Y = p^{-1}\mathcal{O}_X$, then $$ \Mor_{\text{LRS}/(X, \mathcal{O}_X)}((Z, \mathcal{O}_Z), (Y, \mathcal{O}_Y)) \longrightarrow \Mor_{\textit{Top}/X}(Z, Y),\quad (f, f^\sharp) \longmapsto f $$ is bijective. Here $\text{LRS}/(X, \mathcal{O}_X)$ is the category of locally ringed spaces over $X$ and $\textit{Top}/X$ is the category of topological spaces over $X$.