modules-lemma-simple-invert-module

Let $X$ be a topological space. Let $\mathcal{O}_X$ be a presheaf of rings. Let $\mathcal{S} \subset \mathcal{O}_X$ be a pre-sheaf of sets contained in $\mathcal{O}_X$. Suppose that for every open $U \subset X$ the set $\mathcal{S}(U) \subset \mathcal{O}_X(U)$ is a multiplicative subset. For any presheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ we have $$ \mathcal{S}^{-1}\mathcal{F} = \mathcal{S}^{-1}\mathcal{O}_X \otimes_{p, \mathcal{O}_X} \mathcal{F} $$ (see Sheaves, Section \ref{sheaves-section-presheaves-modules} for notation) and if $\mathcal{F}$ and $\mathcal{O}_X$ are sheaves then $$ (\mathcal{S}^{-1}\mathcal{F})^\# = (\mathcal{S}^{-1}\mathcal{O}_X)^\# \otimes_{\mathcal{O}_X} \mathcal{F} $$ (see Sheaves, Section \ref{sheaves-section-sheafification-presheaves-modules} for notation).


modules-lemma-universal-module

Let $X$ be a topological space. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. The functor $$ \textit{Mod}(\mathcal{O}_2) \longrightarrow \textit{Ab}, \quad \mathcal{F} \longmapsto \text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F}) $$ is representable.


modules-lemma-differentials-sheafify

Let $X$ be a topological space. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Then $\Omega_{\mathcal{O}_2/\mathcal{O}_1}$ is the sheaf associated to the presheaf $U \mapsto \Omega_{\mathcal{O}_2(U)/\mathcal{O}_1(U)}$.


modules-lemma-localize-differentials

Let $X$ be a topological space. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. For $U \subset X$ open there is a canonical isomorphism $$ \Omega_{\mathcal{O}_2/\mathcal{O}_1}|_U = \Omega_{(\mathcal{O}_2|_U)/(\mathcal{O}_1|_U)} $$ compatible with universal derivations.


modules-lemma-pullback-differentials

Let $f : Y \to X$ be a continuous map of topological spaces. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Then there is a canonical identification $f^{-1}\Omega_{\mathcal{O}_2/\mathcal{O}_1} = \Omega_{f^{-1}\mathcal{O}_2/f^{-1}\mathcal{O}_1}$ compatible with universal derivations.


modules-lemma-stalk-module-differentials

Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Let $x \in X$. Then we have $\Omega_{\mathcal{O}_2/\mathcal{O}_1, x} = \Omega_{\mathcal{O}_{2, x}/\mathcal{O}_{1, x}}$.


modules-lemma-functoriality-differentials

Let $X$ be a topological space. Let $$ \xymatrix{ \mathcal{O}_2 \ar[r]_\varphi & \mathcal{O}_2' \\ \mathcal{O}_1 \ar[r] \ar[u] & \mathcal{O}'_1 \ar[u] } $$ be a commutative diagram of sheaves of rings on $X$. The map $\mathcal{O}_2 \to \mathcal{O}'_2$ composed with the map $\text{d} : \mathcal{O}'_2 \to \Omega_{\mathcal{O}'_2/\mathcal{O}'_1}$ is a $\mathcal{O}_1$-derivation. Hence we obtain a canonical map of $\mathcal{O}_2$-modules $\Omega_{\mathcal{O}_2/\mathcal{O}_1} \to \Omega_{\mathcal{O}'_2/\mathcal{O}'_1}$. It is uniquely characterized by the property that $\text{d}(f) \mapsto \text{d}(\varphi(f))$ for any local section $f$ of $\mathcal{O}_2$. In this way $\Omega_{-/-}$ becomes a functor on the category of arrows of sheaves of rings.


modules-lemma-double-structure-gives-derivation

Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)$ be a morphism of ringed spaces. Consider a short exact sequence $$ 0 \to \mathcal{I} \to \mathcal{A} \to \mathcal{O}_X \to 0 $$ Here $\mathcal{A}$ is a sheaf of $f^{-1}\mathcal{O}_S$-algebras, $\pi : \mathcal{A} \to \mathcal{O}_X$ is a surjection of sheaves of $f^{-1}\mathcal{O}_S$-algebras, and $\mathcal{I} = \Ker(\pi)$ is its kernel. Assume $\mathcal{I}$ an ideal sheaf with square zero in $\mathcal{A}$. So $\mathcal{I}$ has a natural structure of an $\mathcal{O}_X$-module. A section $s : \mathcal{O}_X \to \mathcal{A}$ of $\pi$ is a $f^{-1}\mathcal{O}_S$-algebra map such that $\pi \circ s = \text{id}$. Given any section $s : \mathcal{O}_X \to \mathcal{A}$ of $\pi$ and any $S$-derivation $D : \mathcal{O}_X \to \mathcal{I}$ the map $$ s + D : \mathcal{O}_X \to \mathcal{A} $$ is a section of $\pi$ and every section $s'$ is of the form $s + D$ for a unique $S$-derivation $D$.


modules-lemma-functoriality-differentials-ringed-spaces

Let $$ \xymatrix{ X' \ar[d]_{h'} \ar[r]_f & X \ar[d]^h \\ S' \ar[r]^g & S } $$ be a commutative diagram of ringed spaces. \begin{enumerate} \item The canonical map $\mathcal{O}_X \to f_*\mathcal{O}_{X'}$ composed with $f_*\text{d}_{X'/S'} : f_*\mathcal{O}_{X'} \to f_*\Omega_{X'/S'}$ is a $S$-derivation and we obtain a canonical map of $\mathcal{O}_X$-modules $\Omega_{X/S} \to f_*\Omega_{X'/S'}$. \item The commutative diagram $$ \xymatrix{ f^{-1}\mathcal{O}_X \ar[r] & \mathcal{O}_{X'} \\ f^{-1}h^{-1}\mathcal{O}_S \ar[u] \ar[r] & (h')^{-1}\mathcal{O}_{S'} \ar[u] } $$ induces by Lemmas \ref{lemma-pullback-differentials} and \ref{lemma-functoriality-differentials} a canonical map $f^{-1}\Omega_{X/S} \to \Omega_{X'/S'}$. \end{enumerate} These two maps correspond (via adjointness of $f_*$ and $f^*$ and via $f^*\Omega_{X/S} = f^{-1}\Omega_{X/S} \otimes_{f^{-1}\mathcal{O}_X} \mathcal{O}_{X'}$ and Sheaves, Lemma \ref{sheaves-lemma-adjointness-tensor-restrict}) to the same $\mathcal{O}_{X'}$-module homomorphism $$ c_f : f^*\Omega_{X/S} \longrightarrow \Omega_{X'/S'} $$ which is uniquely characterized by the property that $f^*\text{d}_{X/S}(a)$ maps to $\text{d}_{X'/S'}(f^*a)$ for any local section $a$ of $\mathcal{O}_X$.


modules-lemma-check-functoriality-differentials

Let $$ \xymatrix{ X'' \ar[d] \ar[r]_g & X' \ar[d] \ar[r]_f & X \ar[d] \\ S'' \ar[r] & S' \ar[r] & S } $$ be a commutative diagram of ringed spaces. With notation as in Lemma \ref{lemma-functoriality-differentials-ringed-spaces} we have $$ c_{f \circ g} = c_g \circ g^* c_f $$ as maps $(f \circ g)^*\Omega_{X/S} \to \Omega_{X''/S''}$.


modules-lemma-composition-differential-operators

Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a map of sheaves of rings on $X$. Let $\mathcal{E}, \mathcal{F}, \mathcal{G}$ be sheaves of $\mathcal{O}_2$-modules. If $D : \mathcal{E} \to \mathcal{F}$ and $D' : \mathcal{F} \to \mathcal{G}$ are differential operators of order $k$ and $k'$, then $D' \circ D$ is a differential operator of order $k + k'$.


modules-lemma-module-principal-parts

Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a map of sheaves of rings on $X$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_2$-modules. Let $k \geq 0$. There exists a sheaf of $\mathcal{O}_2$-modules $\mathcal{P}^k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F})$ and a canonical isomorphism $$ \text{Diff}^k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}, \mathcal{G}) = \Hom_{\mathcal{O}_2}( \mathcal{P}^k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}), \mathcal{G}) $$ functorial in the $\mathcal{O}_2$-module $\mathcal{G}$.


modules-lemma-differential-operators-sheafify

Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of presheaves of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_2$-modules. Then $\mathcal{P}^k_{\mathcal{O}_2^\#/\mathcal{O}_1^\#}(\mathcal{F}^\#)$ is the sheaf associated to the presheaf $U \mapsto P^k_{\mathcal{O}_2(U)/\mathcal{O}_1(U)}(\mathcal{F}(U))$.


modules-lemma-sequence-of-principal-parts

Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_2$-modules. There is a canonical short exact sequence $$ 0 \to \Omega_{\mathcal{O}_2/\mathcal{O}_1} \otimes_{\mathcal{O}_2} \mathcal{F} \to \mathcal{P}^1_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}) \to \mathcal{F} \to 0 $$ functorial in $\mathcal{F}$ called the {\it sequence of principal parts}.


modules-lemma-differentials-de-rham-complex-order-1

Let $X$ be a topological space. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $X$. The differentials $\text{d} : \Omega^i_{\mathcal{B}/\mathcal{A}}  \to \Omega^{i + 1}_{\mathcal{B}/\mathcal{A}}$ are differential operators of order $1$.


modules-lemma-NL-up-to-qis

In the situation above there is a canonical isomorphism $\NL(\alpha) = \NL_{\mathcal{B}/\mathcal{A}}$ in $D(\mathcal{B})$.


modules-lemma-pullback-NL

Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $Y$. Then $f^{-1}\NL_{\mathcal{B}/\mathcal{A}} = \NL_{f^{-1}\mathcal{B}/f^{-1}\mathcal{A}}$.


modules-lemma-stalk-NL

Let $X$ be a topological space. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $X$. Let $x \in X$. Then we have $\NL_{\mathcal{B}/\mathcal{A}, x} = \NL_{\mathcal{B}_x/\mathcal{A}_x}$.


modules-lemma-exact-sequence-NL

Let $X$ be a topological space. Let $\mathcal{A} \to \mathcal{B} \to \mathcal{C}$ be maps of sheaves of rings. Let $C$ be the cone (Derived Categories, Definition \ref{derived-definition-cone}) of the map of complexes $\NL_{\mathcal{C}/\mathcal{A}} \to \NL_{\mathcal{C}/\mathcal{B}}$. There is a canonical map $$ c : \NL_{\mathcal{B}/\mathcal{A}} \otimes_\mathcal{B} \mathcal{C} \longrightarrow C[-1] $$ of complexes of $\mathcal{C}$-modules which produces a canonical six term exact sequence $$ \xymatrix{ H^0(\NL_{\mathcal{B}/\mathcal{A}} \otimes_\mathcal{B} \mathcal{C}) \ar[r] & H^0(\NL_{\mathcal{C}/\mathcal{A}}) \ar[r] & H^0(\NL_{\mathcal{C}/\mathcal{B}}) \ar[r] & 0 \\ H^{-1}(\NL_{\mathcal{B}/\mathcal{A}} \otimes_\mathcal{B} \mathcal{C}) \ar[r] & H^{-1}(\NL_{\mathcal{C}/\mathcal{A}}) \ar[r] & H^{-1}(\NL_{\mathcal{C}/\mathcal{B}}) \ar[llu] } $$ of cohomology sheaves.


modules-lemma-exact-sequence-NL-ringed-topoi

Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. Let $C$ be the cone of the map $\NL_{X/Z} \to \NL_{X/Y}$ of complexes of $\mathcal{O}_X$-modules. There is a canonical map $$ f^*\NL_{Y/Z} \to C[-1] $$ which produces a canonical six term exact sequence $$ \xymatrix{ H^0(f^*\NL_{Y/Z}) \ar[r] & H^0(\NL_{X/Z}) \ar[r] & H^0(\NL_{X/Y}) \ar[r] & 0 \\ H^{-1}(f^*\NL_{Y/Z}) \ar[r] & H^{-1}(\NL_{X/Z}) \ar[r] & H^{-1}(\NL_{X/Y}) \ar[llu] } $$ of cohomology sheaves.


defos-lemma-huge-diagram

Given a commutative diagram $$ \xymatrix{ & 0 \ar[r] & N_2 \ar[r] & B'_2 \ar[r] & B_2 \ar[r] & 0 \\ & 0 \ar[r]|\hole & I_2 \ar[u]_{c_2} \ar[r] & A'_2 \ar[u] \ar[r]|\hole & A_2 \ar[u] \ar[r] & 0 \\ 0 \ar[r] & N_1 \ar[ruu] \ar[r] & B'_1 \ar[r] & B_1 \ar[ruu] \ar[r] & 0 \\ 0 \ar[r] & I_1 \ar[ruu]|\hole \ar[u]^{c_1} \ar[r] & A'_1 \ar[ruu]|\hole \ar[u] \ar[r] & A_1 \ar[ruu]|\hole \ar[u] \ar[r] & 0 } $$ with front and back solutions to (\ref{equation-to-solve}) we have \begin{enumerate} \item There exist a canonical element in $\Ext^1_{B_1}(\NL_{B_1/A_1}, N_2)$ whose vanishing is a necessary and sufficient condition for the existence of a ring map $B'_1 \to B'_2$ fitting into the diagram. \item If there exists a map $B'_1 \to B'_2$ fitting into the diagram the set of all such maps is a principal homogeneous space under $\Hom_{B_1}(\Omega_{B_1/A_1}, N_2)$. \end{enumerate}


defos-lemma-choices

If there exists a solution to (\ref{equation-to-solve}), then the set of isomorphism classes of solutions is principal homogeneous under $\Ext^1_B(\NL_{B/A}, N)$.


defos-lemma-extensions-of-rings

Let $A$ be a ring and let $I$ be an $A$-module. \begin{enumerate} \item The set of extensions of rings $0 \to I \to A' \to A \to 0$ where $I$ is an ideal of square zero is canonically bijective to $\Ext^1_A(\NL_{A/\mathbf{Z}}, I)$. \item Given a ring map $A \to B$, a $B$-module $N$, an $A$-module map $c : I \to N$, and given extensions of rings with square zero kernels: \begin{enumerate} \item[(a)] $0 \to I \to A' \to A \to 0$ corresponding to $\alpha \in \Ext^1_A(\NL_{A/\mathbf{Z}}, I)$, and \item[(b)] $0 \to N \to B' \to B \to 0$ corresponding to $\beta \in \Ext^1_B(\NL_{B/\mathbf{Z}}, N)$ \end{enumerate} then there is a map $A' \to B'$ fitting into a diagram (\ref{equation-to-solve}) if and only if $\beta$ and $\alpha$ map to the same element of $\Ext^1_A(\NL_{A/\mathbf{Z}}, N)$. \end{enumerate}


defos-lemma-strict-morphism-thickenings

In Situation \ref{situation-morphism-thickenings} the morphism $(f, f')$ is a strict morphism of thickenings if and only if (\ref{equation-morphism-thickenings}) is cartesian in the category of ringed spaces.


defos-lemma-inf-map

Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$ be a first order thickening of ringed spaces. Assume given extensions $$ 0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0 \quad\text{and}\quad 0 \to \mathcal{L} \to \mathcal{G}' \to \mathcal{G} \to 0 $$ as in (\ref{equation-extension}) and maps $\varphi : \mathcal{F} \to \mathcal{G}$ and $\psi : \mathcal{K} \to \mathcal{L}$. \begin{enumerate} \item If there exists an $\mathcal{O}_{X'}$-module map $\varphi' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi$ and $\psi$, then the diagram $$ \xymatrix{ \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi} & \mathcal{K} \ar[d]^\psi \\ \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{G} \ar[r]^-{c_{\mathcal{G}'}} & \mathcal{L} } $$ is commutative. \item The set of $\mathcal{O}_{X'}$-module maps $\varphi' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi$ and $\psi$ is, if nonempty, a principal homogeneous space under $\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{L})$. \end{enumerate}


defos-lemma-inf-obs-map

Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$ be a first order thickening of ringed spaces. Assume given extensions $$ 0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0 \quad\text{and}\quad 0 \to \mathcal{L} \to \mathcal{G}' \to \mathcal{G} \to 0 $$ as in (\ref{equation-extension}) and maps $\varphi : \mathcal{F} \to \mathcal{G}$ and $\psi : \mathcal{K} \to \mathcal{L}$. Assume the diagram $$ \xymatrix{ \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi} & \mathcal{K} \ar[d]^\psi \\ \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{G} \ar[r]^-{c_{\mathcal{G}'}} & \mathcal{L} } $$ is commutative. Then there exists an element $$ o(\varphi, \psi) \in \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{L}) $$ whose vanishing is a necessary and sufficient condition for the existence of a map $\varphi' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi$ and $\psi$.


defos-lemma-inf-ext

Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$ be a first order thickening of ringed spaces. Assume given $\mathcal{O}_X$-modules $\mathcal{F}$, $\mathcal{K}$ and an $\mathcal{O}_X$-linear map $c : \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}$. If there exists a sequence (\ref{equation-extension}) with $c_{\mathcal{F}'} = c$ then the set of isomorphism classes of these extensions is principal homogeneous under $\Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K})$.


defos-lemma-inf-obs-ext

Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$ be a first order thickening of ringed spaces. Assume given $\mathcal{O}_X$-modules $\mathcal{F}$, $\mathcal{K}$ and an $\mathcal{O}_X$-linear map $c : \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}$. Then there exists an element $$ o(\mathcal{F}, \mathcal{K}, c) \in \Ext^2_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}) $$ whose vanishing is a necessary and sufficient condition for the existence of a sequence (\ref{equation-extension}) with $c_{\mathcal{F}'} = c$.


defos-lemma-inf-map-special

Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$ be a first order thickening of ringed spaces. Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}_{X'}$-modules. Set $\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}_X$-linear map. The set of lifts of $\varphi$ to an $\mathcal{O}_{X'}$-linear map $\varphi' : \mathcal{F}' \to \mathcal{G}'$ is, if nonempty, a principal homogeneous space under $\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{I}\mathcal{G}')$.


defos-lemma-deform-module

Let $(f, f')$ be a morphism of first order thickenings of ringed spaces as in Situation \ref{situation-morphism-thickenings}. Let $\mathcal{F}'$ be an $\mathcal{O}_{X'}$-module and set $\mathcal{F} = i^*\mathcal{F}'$. Assume that $\mathcal{F}$ is flat over $S$ and that $(f, f')$ is a strict morphism of thickenings (Definition \ref{definition-strict-morphism-thickenings}). Then the following are equivalent \begin{enumerate} \item $\mathcal{F}'$ is flat over $S'$, and \item the canonical map $f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{I}\mathcal{F}'$ is an isomorphism. \end{enumerate} Moreover, in this case the maps $$ f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{I}\mathcal{F}' $$ are isomorphisms.


defos-lemma-inf-map-rel

Let $(f, f')$ be a morphism of first order thickenings as in Situation \ref{situation-morphism-thickenings}. Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}_{X'}$-modules and set $\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}_X$-linear map. Assume that $\mathcal{G}'$ is flat over $S'$ and that $(f, f')$ is a strict morphism of thickenings. The set of lifts of $\varphi$ to an $\mathcal{O}_{X'}$-linear map $\varphi' : \mathcal{F}' \to \mathcal{G}'$ is, if nonempty, a principal homogeneous space under $$ \Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{J}) $$


defos-lemma-inf-obs-map-special

Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$ be a first order thickening of ringed spaces. Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}_{X'}$-modules and set $\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}_X$-linear map. There exists an element $$ o(\varphi) \in \Ext^1_{\mathcal{O}_X}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}') $$ whose vanishing is a necessary and sufficient condition for the existence of a lift of $\varphi$ to an $\mathcal{O}_{X'}$-linear map $\varphi' : \mathcal{F}' \to \mathcal{G}'$.


defos-lemma-inf-ext-rel

Let $(f, f')$ be a morphism of first order thickenings as in Situation \ref{situation-morphism-thickenings}. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Assume $(f, f')$ is a strict morphism of thickenings and $\mathcal{F}$ flat over $S$. If there exists a pair $(\mathcal{F}', \alpha)$ consisting of an $\mathcal{O}_{X'}$-module $\mathcal{F}'$ flat over $S'$ and an isomorphism $\alpha : i^*\mathcal{F}' \to \mathcal{F}$, then the set of isomorphism classes of such pairs is principal homogeneous under $\Ext^1_{\mathcal{O}_X}( \mathcal{F}, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F})$.


defos-lemma-inf-obs-ext-rel

Let $(f, f')$ be a morphism of first order thickenings as in Situation \ref{situation-morphism-thickenings}. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Assume $(f, f')$ is a strict morphism of thickenings and $\mathcal{F}$ flat over $S$. There exists an $\mathcal{O}_{X'}$-module $\mathcal{F}'$ flat over $S'$ with $i^*\mathcal{F}' \cong \mathcal{F}$, if and only if \begin{enumerate} \item the canonical map $ f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}$ is an isomorphism, and \item the class $o(\mathcal{F}, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}, 1) \in \Ext^2_{\mathcal{O}_X}( \mathcal{F}, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F})$ of Lemma \ref{lemma-inf-obs-ext} is zero. \end{enumerate}


defos-lemma-flat

In the situation above. \begin{enumerate} \item There exists an $\mathcal{O}_{X'}$-module $\mathcal{F}'$ flat over $S'$ with $i^*\mathcal{F}' \cong \mathcal{F}$, if and only if the class $o(\mathcal{F}, f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F}, 1) \in \Ext^2_{\mathcal{O}_X}( \mathcal{F}, f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F})$ of Lemma \ref{lemma-inf-obs-ext} is zero. \item If such a module exists, then the set of isomorphism classes of lifts is principal homogeneous under $\Ext^1_{\mathcal{O}_X}( \mathcal{F}, f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F})$. \item Given a lift $\mathcal{F}'$, the set of automorphisms of $\mathcal{F}'$ which pull back to $\text{id}_\mathcal{F}$ is canonically isomorphic to $\Ext^0_{\mathcal{O}_X}( \mathcal{F}, f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F})$. \end{enumerate}


defos-lemma-huge-diagram-ringed-spaces

Assume given a commutative diagram of morphisms of ringed spaces \begin{equation} \label{equation-huge-1} \vcenter{ \xymatrix{ & (X_2, \mathcal{O}_{X_2}) \ar[r]_{i_2} \ar[d]_{f_2} \ar[ddl]_g & (X'_2, \mathcal{O}_{X'_2}) \ar[d]^{f'_2} \\ & (S_2, \mathcal{O}_{S_2}) \ar[r]^{t_2} \ar[ddl]|\hole & (S'_2, \mathcal{O}_{S'_2}) \ar[ddl] \\ (X_1, \mathcal{O}_{X_1}) \ar[r]_{i_1} \ar[d]_{f_1} & (X'_1, \mathcal{O}_{X'_1}) \ar[d]^{f'_1} \\ (S_1, \mathcal{O}_{S_1}) \ar[r]^{t_1} & (S'_1, \mathcal{O}_{S'_1}) } } \end{equation} whose horizontal arrows are first order thickenings. Set $\mathcal{G}_j = \Ker(i_j^\sharp)$ and assume given a $g$-map $\nu : \mathcal{G}_1 \to \mathcal{G}_2$ of modules giving rise to the commutative diagram \begin{equation} \label{equation-huge-2} \vcenter{ \xymatrix{ & 0 \ar[r] & \mathcal{G}_2 \ar[r] & \mathcal{O}_{X'_2} \ar[r] & \mathcal{O}_{X_2} \ar[r] & 0 \\ & 0 \ar[r]|\hole & \mathcal{J}_2 \ar[u]_{c_2} \ar[r] & \mathcal{O}_{S'_2} \ar[u] \ar[r]|\hole & \mathcal{O}_{S_2} \ar[u] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{G}_1 \ar[ruu] \ar[r] & \mathcal{O}_{X'_1} \ar[r] & \mathcal{O}_{X_1} \ar[ruu] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{J}_1 \ar[ruu]|\hole \ar[u]^{c_1} \ar[r] & \mathcal{O}_{S'_1} \ar[ruu]|\hole \ar[u] \ar[r] & \mathcal{O}_{S_1} \ar[ruu]|\hole \ar[u] \ar[r] & 0 } } \end{equation} with front and back solutions to (\ref{equation-to-solve-ringed-spaces}). \begin{enumerate} \item There exist a canonical element in $\Ext^1_{\mathcal{O}_{X_2}}(Lg^*\NL_{X_1/S_1}, \mathcal{G}_2)$ whose vanishing is a necessary and sufficient condition for the existence of a morphism of ringed spaces $X'_2 \to X'_1$ fitting into (\ref{equation-huge-1}) compatibly with $\nu$. \item If there exists a morphism $X'_2 \to X'_1$ fitting into (\ref{equation-huge-1}) compatibly with $\nu$ the set of all such morphisms is a principal homogeneous space under $$ \Hom_{\mathcal{O}_{X_1}}(\Omega_{X_1/S_1}, g_*\mathcal{G}_2) = \Hom_{\mathcal{O}_{X_2}}(g^*\Omega_{X_1/S_1}, \mathcal{G}_2) = \Ext^0_{\mathcal{O}_{X_2}}(Lg^*\NL_{X_1/S_1}, \mathcal{G}_2). $$ \end{enumerate}


defos-lemma-NL-represent-ext-class

Let $X$ be a topological space. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings. Let $\mathcal{G}$ be a $\mathcal{B}$-module. Let $\xi \in \Ext^1_\mathcal{B}(\NL_{\mathcal{B}/\mathcal{A}}, \mathcal{G})$.  There exists a map of sheaves of sets $\alpha : \mathcal{E} \to \mathcal{B}$ such that $\xi \in \Ext^1_\mathcal{B}(\NL(\alpha), \mathcal{G})$ is the class of a map $\mathcal{I}/\mathcal{I}^2 \to \mathcal{G}$ (see proof for notation).


defos-lemma-choices-ringed-spaces

If there exists a solution to (\ref{equation-to-solve-ringed-spaces}), then the set of isomorphism classes of solutions is principal homogeneous under $\Ext^1_{\mathcal{O}_X}(\NL_{X/S}, \mathcal{G})$.


defos-lemma-extensions-of-ringed-spaces

Let $(S, \mathcal{O}_S)$ be a ringed space and let $\mathcal{J}$ be an $\mathcal{O}_S$-module. \begin{enumerate} \item The set of extensions of sheaves of rings $0 \to \mathcal{J} \to \mathcal{O}_{S'} \to \mathcal{O}_S \to 0$ where $\mathcal{J}$ is an ideal of square zero is canonically bijective to $\Ext^1_{\mathcal{O}_S}(\NL_{S/\mathbf{Z}}, \mathcal{J})$. \item Given a morphism of ringed spaces $f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)$, an $\mathcal{O}_X$-module $\mathcal{G}$, an $f$-map $c : \mathcal{J} \to \mathcal{G}$, and given extensions of sheaves of rings with square zero kernels: \begin{enumerate} \item[(a)] $0 \to \mathcal{J} \to \mathcal{O}_{S'} \to \mathcal{O}_S \to 0$ corresponding to $\alpha \in \Ext^1_{\mathcal{O}_S}(\NL_{S/\mathbf{Z}}, \mathcal{J})$, \item[(b)] $0 \to \mathcal{G} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0$ corresponding to $\beta \in \Ext^1_{\mathcal{O}_X}(\NL_{X/\mathbf{Z}}, \mathcal{G})$ \end{enumerate} then there is a morphism $X' \to S'$ fitting into a diagram (\ref{equation-to-solve-ringed-spaces}) if and only if $\beta$ and $\alpha$ map to the same element of $\Ext^1_{\mathcal{O}_X}(Lf^*\NL_{S/\mathbf{Z}}, \mathcal{G})$. \end{enumerate}


defos-lemma-deform

Let $S \subset S'$ be a first order thickening of schemes. Let $f : X \to S$ be a flat morphism of schemes. If there exists a flat morphism $f' : X' \to S'$ of schemes and an isomorphism $a : X \to X' \times_{S'} S$ over $S$, then \begin{enumerate} \item the set of isomorphism classes of pairs $(f' : X' \to S', a)$ is principal homogeneous under $\Ext^1_{\mathcal{O}_X}(\NL_{X/S}, f^*\mathcal{C}_{S/S'})$, and \item the set of automorphisms of $\varphi : X' \to X'$ over $S'$ which reduce to the identity on $X' \times_{S'} S$ is $\Ext^0_{\mathcal{O}_X}(\NL_{X/S}, f^*\mathcal{C}_{S/S'})$. \end{enumerate}


defos-lemma-inf-map-ringed-topoi

Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$ be a first order thickening of ringed topoi. Assume given extensions $$ 0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0 \quad\text{and}\quad 0 \to \mathcal{L} \to \mathcal{G}' \to \mathcal{G} \to 0 $$ as in (\ref{equation-extension-ringed-topoi}) and maps $\varphi : \mathcal{F} \to \mathcal{G}$ and $\psi : \mathcal{K} \to \mathcal{L}$. \begin{enumerate} \item If there exists an $\mathcal{O}'$-module map $\varphi' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi$ and $\psi$, then the diagram $$ \xymatrix{ \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi} & \mathcal{K} \ar[d]^\psi \\ \mathcal{I} \otimes_\mathcal{O} \mathcal{G} \ar[r]^-{c_{\mathcal{G}'}} & \mathcal{L} } $$ is commutative. \item The set of $\mathcal{O}'$-module maps $\varphi' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi$ and $\psi$ is, if nonempty, a principal homogeneous space under $\Hom_\mathcal{O}(\mathcal{F}, \mathcal{L})$. \end{enumerate}


defos-lemma-inf-obs-map-ringed-topoi

Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$ be a first order thickening of ringed topoi. Assume given extensions $$ 0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0 \quad\text{and}\quad 0 \to \mathcal{L} \to \mathcal{G}' \to \mathcal{G} \to 0 $$ as in (\ref{equation-extension-ringed-topoi}) and maps $\varphi : \mathcal{F} \to \mathcal{G}$ and $\psi : \mathcal{K} \to \mathcal{L}$. Assume the diagram $$ \xymatrix{ \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi} & \mathcal{K} \ar[d]^\psi \\ \mathcal{I} \otimes_\mathcal{O} \mathcal{G} \ar[r]^-{c_{\mathcal{G}'}} & \mathcal{L} } $$ is commutative. Then there exists an element $$ o(\varphi, \psi) \in \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{L}) $$ whose vanishing is a necessary and sufficient condition for the existence of a map $\varphi' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi$ and $\psi$.


defos-lemma-inf-ext-ringed-topoi

Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$ be a first order thickening of ringed topoi. Assume given $\mathcal{O}$-modules $\mathcal{F}$, $\mathcal{K}$ and an $\mathcal{O}$-linear map $c : \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}$. If there exists a sequence (\ref{equation-extension-ringed-topoi}) with $c_{\mathcal{F}'} = c$ then the set of isomorphism classes of these extensions is principal homogeneous under $\Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{K})$.


defos-lemma-inf-obs-ext-ringed-topoi

Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$ be a first order thickening of ringed topoi. Assume given $\mathcal{O}$-modules $\mathcal{F}$, $\mathcal{K}$ and an $\mathcal{O}$-linear map $c : \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}$. Then there exists an element $$ o(\mathcal{F}, \mathcal{K}, c) \in \Ext^2_\mathcal{O}(\mathcal{F}, \mathcal{K}) $$ whose vanishing is a necessary and sufficient condition for the existence of a sequence (\ref{equation-extension-ringed-topoi}) with $c_{\mathcal{F}'} = c$.


defos-lemma-inf-map-special-ringed-topoi

Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$ be a first order thickening of ringed topoi. Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}'$-modules. Set $\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}$-linear map. The set of lifts of $\varphi$ to an $\mathcal{O}'$-linear map $\varphi' : \mathcal{F}' \to \mathcal{G}'$ is, if nonempty, a principal homogeneous space under $\Hom_\mathcal{O}(\mathcal{F}, \mathcal{I}\mathcal{G}')$.


defos-lemma-deform-module-ringed-topoi

Let $(f, f')$ be a morphism of first order thickenings of ringed topoi as in Situation \ref{situation-morphism-thickenings-ringed-topoi}. Let $\mathcal{F}'$ be an $\mathcal{O}'$-module and set $\mathcal{F} = i^*\mathcal{F}'$. Assume that $\mathcal{F}$ is flat over $\mathcal{O}_\mathcal{B}$ and that $(f, f')$ is a strict morphism of thickenings (Definition \ref{definition-strict-morphism-thickenings-ringed-topoi}). Then the following are equivalent \begin{enumerate} \item $\mathcal{F}'$ is flat over $\mathcal{O}_{\mathcal{B}'}$, and \item the canonical map $f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{I}\mathcal{F}'$ is an isomorphism. \end{enumerate} Moreover, in this case the maps $$ f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{I}\mathcal{F}' $$ are isomorphisms.


defos-lemma-deform-fp-module-ringed-topoi

Let $(f, f')$ be a morphism of first order thickenings of ringed topoi as in Situation \ref{situation-morphism-thickenings-ringed-topoi}. Let $\mathcal{F}'$ be an $\mathcal{O}'$-module and set $\mathcal{F} = i^*\mathcal{F}'$. Assume that $\mathcal{F}'$ is flat over $\mathcal{O}_{\mathcal{B}'}$ and that $(f, f')$ is a strict morphism of thickenings. Then the following are equivalent \begin{enumerate} \item $\mathcal{F}'$ is an $\mathcal{O}'$-module of finite presentation, and \item $\mathcal{F}$ is an $\mathcal{O}$-module of finite presentation. \end{enumerate}


defos-lemma-inf-map-rel-ringed-topoi

Let $(f, f')$ be a morphism of first order thickenings as in Situation \ref{situation-morphism-thickenings-ringed-topoi}. Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}'$-modules and set $\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}$-linear map. Assume that $\mathcal{G}'$ is flat over $\mathcal{O}_{\mathcal{B}'}$ and that $(f, f')$ is a strict morphism of thickenings. The set of lifts of $\varphi$ to an $\mathcal{O}'$-linear map $\varphi' : \mathcal{F}' \to \mathcal{G}'$ is, if nonempty, a principal homogeneous space under $$ \Hom_\mathcal{O}(\mathcal{F}, \mathcal{G} \otimes_\mathcal{O} f^*\mathcal{J}) $$


defos-lemma-inf-obs-map-special-ringed-topoi

Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$ be a first order thickening of ringed topoi. Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}'$-modules and set $\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}$-linear map. There exists an element $$ o(\varphi) \in \Ext^1_\mathcal{O}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}') $$ whose vanishing is a necessary and sufficient condition for the existence of a lift of $\varphi$ to an $\mathcal{O}'$-linear map $\varphi' : \mathcal{F}' \to \mathcal{G}'$.


defos-lemma-inf-ext-rel-ringed-topoi

Let $(f, f')$ be a morphism of first order thickenings as in Situation \ref{situation-morphism-thickenings-ringed-topoi}. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Assume $(f, f')$ is a strict morphism of thickenings and $\mathcal{F}$ flat over $\mathcal{O}_\mathcal{B}$. If there exists a pair $(\mathcal{F}', \alpha)$ consisting of an $\mathcal{O}'$-module $\mathcal{F}'$ flat over $\mathcal{O}_{\mathcal{B}'}$ and an isomorphism $\alpha : i^*\mathcal{F}' \to \mathcal{F}$, then the set of isomorphism classes of such pairs is principal homogeneous under $\Ext^1_\mathcal{O}( \mathcal{F}, \mathcal{I} \otimes_\mathcal{O} \mathcal{F})$.


defos-lemma-inf-obs-ext-rel-ringed-topoi

Let $(f, f')$ be a morphism of first order thickenings as in Situation \ref{situation-morphism-thickenings-ringed-topoi}. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Assume $(f, f')$ is a strict morphism of thickenings and $\mathcal{F}$ flat over $\mathcal{O}_\mathcal{B}$. There exists an $\mathcal{O}'$-module $\mathcal{F}'$ flat over $\mathcal{O}_{\mathcal{B}'}$ with $i^*\mathcal{F}' \cong \mathcal{F}$, if and only if \begin{enumerate} \item the canonical map $f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{I} \otimes_\mathcal{O} \mathcal{F}$ is an isomorphism, and \item the class $o(\mathcal{F}, \mathcal{I} \otimes_\mathcal{O} \mathcal{F}, 1) \in \Ext^2_\mathcal{O}( \mathcal{F}, \mathcal{I} \otimes_\mathcal{O} \mathcal{F})$ of Lemma \ref{lemma-inf-obs-ext-ringed-topoi} is zero. \end{enumerate}


defos-lemma-flat-ringed-topoi

In the situation above. \begin{enumerate} \item There exists an $\mathcal{O}'$-module $\mathcal{F}'$ flat over $\mathcal{O}_{\mathcal{B}'}$ with $i^*\mathcal{F}' \cong \mathcal{F}$, if and only if the class $o(\mathcal{F}, f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F}, 1) \in \Ext^2_\mathcal{O}( \mathcal{F}, f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F})$ of Lemma \ref{lemma-inf-obs-ext-ringed-topoi} is zero. \item If such a module exists, then the set of isomorphism classes of lifts is principal homogeneous under $\Ext^1_\mathcal{O}( \mathcal{F}, f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F})$. \item Given a lift $\mathcal{F}'$, the set of automorphisms of $\mathcal{F}'$ which pull back to $\text{id}_\mathcal{F}$ is canonically isomorphic to $\Ext^0_\mathcal{O}( \mathcal{F}, f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F})$. \end{enumerate}


defos-lemma-functorial-ringed-topoi

In Situation \ref{situation-morphism-flat-thickenings-ringed-topoi} the obstruction class $o(\mathcal{F}, f^*\mathcal{J}_2 \otimes_\mathcal{O} \mathcal{F}, 1)$ maps to the obstruction class $o(\mathcal{F}, f^*\mathcal{J}_1 \otimes_\mathcal{O} \mathcal{F}, 1)$ under the canonical map $$ \Ext^2_\mathcal{O}( \mathcal{F}, f^*\mathcal{J}_2 \otimes_\mathcal{O} \mathcal{F}) \to \Ext^2_\mathcal{O}( \mathcal{F}, f^*\mathcal{J}_1 \otimes_\mathcal{O} \mathcal{F}) $$


defos-lemma-verify-iv-ringed-topoi

In Situation \ref{situation-ses-flat-thickenings-ringed-topoi} the modules $\pi^*\mathcal{F}$ and $h^*\mathcal{F}'_2$ are $\mathcal{O}'_1$-modules flat over $\mathcal{O}_{\mathcal{B}'_1}$ restricting to $\mathcal{F}$ on $(\Sh(\mathcal{C}), \mathcal{O})$. Their difference (Lemma \ref{lemma-flat-ringed-topoi}) is an element $\theta$ of $\Ext^1_\mathcal{O}(\mathcal{F}, f^*\mathcal{J}_1 \otimes_\mathcal{O} \mathcal{F})$ whose boundary in $\Ext^2_\mathcal{O}(\mathcal{F}, f^*\mathcal{J}_3 \otimes_\mathcal{O} \mathcal{F})$ equals the obstruction (Lemma \ref{lemma-flat-ringed-topoi}) to lifting $\mathcal{F}$ to an $\mathcal{O}'_3$-module flat over $\mathcal{O}_{\mathcal{B}'_3}$.


defos-lemma-huge-diagram-ringed-topoi

Assume given a commutative diagram of morphisms ringed topoi \begin{equation} \label{equation-huge-1-ringed-topoi} \vcenter{ \xymatrix{ & (\Sh(\mathcal{C}_2), \mathcal{O}_2) \ar[r]_{i_2} \ar[d]_{f_2} \ar[ddl]_g & (\Sh(\mathcal{C}'_2), \mathcal{O}'_2) \ar[d]^{f'_2} \\ & (\Sh(\mathcal{B}_2), \mathcal{O}_{\mathcal{B}_2}) \ar[r]^{t_2} \ar[ddl]|\hole & (\Sh(\mathcal{B}'_2), \mathcal{O}_{\mathcal{B}'_2}) \ar[ddl] \\ (\Sh(\mathcal{C}_1), \mathcal{O}_1) \ar[r]_{i_1} \ar[d]_{f_1} & (\Sh(\mathcal{C}'_1), \mathcal{O}'_1) \ar[d]^{f'_1} \\ (\Sh(\mathcal{B}_1), \mathcal{O}_{\mathcal{B}_1}) \ar[r]^{t_1} & (\Sh(\mathcal{B}'_1), \mathcal{O}_{\mathcal{B}'_1}) } } \end{equation} whose horizontal arrows are first order thickenings. Set $\mathcal{G}_j = \Ker(i_j^\sharp)$ and assume given a map of $g^{-1}\mathcal{O}_1$-modules $\nu : g^{-1}\mathcal{G}_1 \to \mathcal{G}_2$ giving rise to the commutative diagram \begin{equation} \label{equation-huge-2-ringed-topoi} \vcenter{ \xymatrix{ & 0 \ar[r] & \mathcal{G}_2 \ar[r] & \mathcal{O}'_2 \ar[r] & \mathcal{O}_2 \ar[r] & 0 \\ & 0 \ar[r]|\hole & f_2^{-1}\mathcal{J}_2 \ar[u]_{c_2} \ar[r] & f_2^{-1}\mathcal{O}_{\mathcal{B}'_2} \ar[u] \ar[r]|\hole & f_2^{-1}\mathcal{O}_{\mathcal{B}_2} \ar[u] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{G}_1 \ar[ruu] \ar[r] & \mathcal{O}'_1 \ar[r] & \mathcal{O}_1 \ar[ruu] \ar[r] & 0 \\ 0 \ar[r] & f_1^{-1}\mathcal{J}_1 \ar[ruu]|\hole \ar[u]^{c_1} \ar[r] & f_1^{-1}\mathcal{O}_{\mathcal{B}'_1} \ar[ruu]|\hole \ar[u] \ar[r] & f_1^{-1}\mathcal{O}_{\mathcal{B}_1} \ar[ruu]|\hole \ar[u] \ar[r] & 0 } } \end{equation} with front and back solutions to (\ref{equation-to-solve-ringed-topoi}). (The north-north-west arrows are maps on $\mathcal{C}_2$ after applying $g^{-1}$ to the source.) \begin{enumerate} \item There exist a canonical element in $\Ext^1_{\mathcal{O}_2}( Lg^*\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2)$ whose vanishing is a necessary and sufficient condition for the existence of a morphism of ringed topoi $(\Sh(\mathcal{C}'_2), \mathcal{O}'_2) \to (\Sh(\mathcal{C}'_1), \mathcal{O}'_1)$ fitting into (\ref{equation-huge-1-ringed-topoi}) compatibly with $\nu$. \item If there exists a morphism $(\Sh(\mathcal{C}'_2), \mathcal{O}'_2) \to (\Sh(\mathcal{C}'_1), \mathcal{O}'_1)$ fitting into (\ref{equation-huge-1-ringed-topoi}) compatibly with $\nu$ the set of all such morphisms is a principal homogeneous space under $$ \Hom_{\mathcal{O}_1}( \Omega_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, g_*\mathcal{G}_2) = \Hom_{\mathcal{O}_2}( g^*\Omega_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2) = \Ext^0_{\mathcal{O}_2}( Lg^*\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2). $$ \end{enumerate}


defos-lemma-NL-represent-ext-class-ringed-topoi

Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. Let $\mathcal{G}$ be a $\mathcal{B}$-module. Let $\xi \in \Ext^1_\mathcal{B}(\NL_{\mathcal{B}/\mathcal{A}}, \mathcal{G})$.  There exists a map of sheaves of sets $\alpha : \mathcal{E} \to \mathcal{B}$ such that $\xi \in \Ext^1_\mathcal{B}(\NL(\alpha), \mathcal{G})$ is the class of a map $\mathcal{I}/\mathcal{I}^2 \to \mathcal{G}$ (see proof for notation).


defos-lemma-choices-ringed-topoi

If there exists a solution to (\ref{equation-to-solve-ringed-topoi}), then the set of isomorphism classes of solutions is principal homogeneous under $\Ext^1_\mathcal{O}( \NL_{\mathcal{O}/\mathcal{O}_\mathcal{B}}, \mathcal{G})$.


defos-lemma-extensions-of-ringed-topoi

Let $(\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})$ be a ringed topos and let $\mathcal{J}$ be an $\mathcal{O}_\mathcal{B}$-module. \begin{enumerate} \item The set of extensions of sheaves of rings $0 \to \mathcal{J} \to \mathcal{O}_{\mathcal{B}'} \to \mathcal{O}_\mathcal{B} \to 0$ where $\mathcal{J}$ is an ideal of square zero is canonically bijective to $\Ext^1_{\mathcal{O}_\mathcal{B}}( \NL_{\mathcal{O}_\mathcal{B}/\mathbf{Z}}, \mathcal{J})$. \item Given a morphism of ringed topoi $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})$, an $\mathcal{O}$-module $\mathcal{G}$, an $f^{-1}\mathcal{O}_\mathcal{B}$-module map $c : f^{-1}\mathcal{J} \to \mathcal{G}$, and given extensions of sheaves of rings with square zero kernels: \begin{enumerate} \item[(a)] $0 \to \mathcal{J} \to \mathcal{O}_{\mathcal{B}'} \to \mathcal{O}_\mathcal{B} \to 0$ corresponding to $\alpha \in \Ext^1_{\mathcal{O}_\mathcal{B}}( \NL_{\mathcal{O}_\mathcal{B}/\mathbf{Z}}, \mathcal{J})$, \item[(b)] $0 \to \mathcal{G} \to \mathcal{O}' \to \mathcal{O} \to 0$ corresponding to $\beta \in \Ext^1_\mathcal{O}(\NL_{\mathcal{O}/\mathbf{Z}}, \mathcal{G})$ \end{enumerate} then there is a morphism $(\Sh(\mathcal{C}), \mathcal{O}') \to (\Sh(\mathcal{B}, \mathcal{O}_{\mathcal{B}'})$ fitting into a diagram (\ref{equation-to-solve-ringed-topoi}) if and only if $\beta$ and $\alpha$ map to the same element of $\Ext^1_\mathcal{O}( Lf^*\NL_{\mathcal{O}_\mathcal{B}/\mathbf{Z}}, \mathcal{G})$. \end{enumerate}


defos-lemma-match-thickenings

Let $S$ be a scheme. Let $i : Z \to Z'$ be a morphism of algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $i$ is a thickening of algebraic spaces as defined in More on Morphisms of Spaces, Section \ref{spaces-more-morphisms-section-thickenings}, and \item the associated morphism $i_{small} : (\Sh(Z_\etale), \mathcal{O}_Z) \to (\Sh(Z'_\etale), \mathcal{O}_{Z'})$ of ringed topoi (Properties of Spaces, Lemma \ref{spaces-properties-lemma-morphism-ringed-topoi}) is a thickening in the sense of Section \ref{section-thickenings-ringed-topoi}. \end{enumerate}


defos-lemma-deform-spaces

Let $S$ be a scheme. Let $Y \subset Y'$ be a first order thickening of algebraic spaces over $S$. Let $f : X \to Y$ be a flat morphism of algebraic spaces over $S$. If there exists a flat morphism $f' : X' \to Y'$ of algebraic spaces over $S$ and an isomorphsm $a : X \to X' \times_{Y'} Y$ over $Y$, then \begin{enumerate} \item the set of isomorphism classes of pairs $(f' : X' \to Y', a)$ is principal homogeneous under $\Ext^1_{\mathcal{O}_X}(\NL_{X/Y}, f^*\mathcal{C}_{Y/Y'})$, and \item the set of automorphisms of $\varphi : X' \to X'$ over $Y'$ which reduce to the identity on $X' \times_{Y'} Y$ is $\Ext^0_{\mathcal{O}_X}(\NL_{X/Y}, f^*\mathcal{C}_{Y/Y'})$. \end{enumerate}


defos-lemma-thickening-space-quasi-coherent

In the situation above assume that $X$ is quasi-compact and quasi-separated and that $DQ_X(\mathcal{F}) \to DQ_X(\mathcal{G})$ (Derived Categories of Spaces, Section \ref{spaces-perfect-section-better-coherator}) is an isomorphism. Then the functor $F$ is an equivalence of categories.


defos-lemma-thickening-over-thickening-space-quasi-coherent

In the situation above assume that $X$ is quasi-compact and quasi-separated and that $DQ_X(\mathcal{F}) \to DQ_X(\mathcal{G})$ (Derived Categories of Spaces, Section \ref{spaces-perfect-section-better-coherator}) is an isomorphism. Then the functor $FT$ is an equivalence of categories.


defos-lemma-canonical-class-algebra

Let $R' \to R$ be a surjection of rings whose kernel is an ideal $I$ of square zero. For every $K \in D^-(R)$ there is a canonical map $$ \omega(K) : K \longrightarrow K \otimes_R^\mathbf{L} I[2] $$ in $D(R)$ with the following properties \begin{enumerate} \item $\omega(K) = 0$ if and only if there exists $K' \in D(R')$ with $K' \otimes_{R'}^\mathbf{L} R = K$, \item given $K \to L$ in $D^-(R)$ the diagram $$ \xymatrix{ K \ar[d] \ar[rr]_-{\omega(K)} & & K \otimes^\mathbf{L}_R I[2] \ar[d] \\ L \ar[rr]^-{\omega(L)} & & L \otimes^\mathbf{L}_R I[2] } $$ commutes, and \item formation of $\omega(K)$ is compatible with ring maps $R' \to S'$ (see proof for a precise statement). \end{enumerate}


defos-lemma-lift-complex

Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}_0$ be a surjection of sheaves of rings. Assume given the following data \begin{enumerate} \item flat $\mathcal{O}$-modules $\mathcal{G}^n$, \item maps of $\mathcal{O}$-modules $\mathcal{G}^n \to \mathcal{G}^{n + 1}$, \item a complex $\mathcal{K}_0^\bullet$ of $\mathcal{O}_0$-modules, \item maps of $\mathcal{O}$-modules $\mathcal{G}^n \to \mathcal{K}_0^n$ \end{enumerate} such that \begin{enumerate} \item[(a)] $H^n(\mathcal{K}_0^\bullet) = 0$ for $n \gg 0$, \item[(b)] $\mathcal{G}^n = 0$ for $n \gg 0$, \item[(c)] with $\mathcal{G}^n_0 = \mathcal{G}^n \otimes_\mathcal{O} \mathcal{O}_0$ the induced maps determine a complex $\mathcal{G}_0^\bullet$ and a map of complexes $\mathcal{G}_0^\bullet \to \mathcal{K}_0^\bullet$. \end{enumerate} Then there exist \begin{enumerate} \item[(\romannumeral1)] flat $\mathcal{O}$-modules $\mathcal{F}^n$, \item[(\romannumeral2)] maps of $\mathcal{O}$-modules $\mathcal{F}^n \to \mathcal{F}^{n + 1}$, \item[(\romannumeral3)] maps of $\mathcal{O}$-modules $\mathcal{F}^n \to \mathcal{K}_0^n$, \item[(\romannumeral4)] maps of $\mathcal{O}$-modules $\mathcal{G}^n \to \mathcal{F}^n$, \end{enumerate} such that $\mathcal{F}^n = 0$ for $n \gg 0$, such that the diagrams $$ \xymatrix{ \mathcal{G}^n \ar[r] \ar[d] & \mathcal{G}^{n + 1} \ar[d] \\ \mathcal{F}^n \ar[r] & \mathcal{F}^{n + 1} } $$ commute for all $n$, such that the composition $\mathcal{G}^n \to \mathcal{F}^n \to \mathcal{K}_0^n$ is the given map $\mathcal{G}^n \to \mathcal{K}_0^n$, and such that with $\mathcal{F}^n_0 = \mathcal{F}^n \otimes_\mathcal{O} \mathcal{O}_0$ we obtain a complex $\mathcal{F}_0^\bullet$ and map of complexes $\mathcal{F}_0^\bullet \to \mathcal{K}_0^\bullet$ which is a quasi-isomorphism.


defos-lemma-canonical-class

Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}_0$ be a surjection of sheaves of rings whose kernel is an ideal sheaf $\mathcal{I}$ of square zero. For every object $K_0$ in $D^-(\mathcal{O}_0)$ there is a canonical map $$ \omega(K_0) : K_0 \longrightarrow K_0 \otimes_{\mathcal{O}_0}^\mathbf{L} \mathcal{I}[2] $$ in $D(\mathcal{O}_0)$ such that for any map $K_0 \to L_0$ in $D^-(\mathcal{O}_0)$ the diagram $$ \xymatrix{ K_0 \ar[d] \ar[rr]_-{\omega(K_0)} & & (K_0 \otimes^\mathbf{L}_{\mathcal{O}_0} \mathcal{I})[2] \ar[d] \\ L_0 \ar[rr]^-{\omega(L_0)} & & (L_0 \otimes^\mathbf{L}_{\mathcal{O}_0} \mathcal{I})[2] } $$ commutes.


defos-lemma-induced-map

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\alpha : K \to L$ be a map of $D^-(\mathcal{O})$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Let $n \in \mathbf{Z}$. \begin{enumerate} \item If $H^i(\alpha)$ is an isomorphism for $i \geq n$, then $H^i(\alpha \otimes_\mathcal{O}^\mathbf{L} \text{id}_\mathcal{F})$ is an isomorphism for $i \geq n$. \item If $H^i(\alpha)$ is an isomorphism for $i > n$  and surjective for $i = n$, then $H^i(\alpha \otimes_\mathcal{O}^\mathbf{L} \text{id}_\mathcal{F})$ is an isomorphism for $i > n$ and surjective for $i = n$. \end{enumerate}


defos-lemma-canonical-class-obstruction

Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}_0$ be a surjection of sheaves of rings whose kernel is an ideal sheaf $\mathcal{I}$ of square zero. For every object $K_0$ in $D^-(\mathcal{O}_0)$ the following are equivalent \begin{enumerate} \item  the class $\omega(K_0) \in \Ext^2_{\mathcal{O}_0}(K_0, K_0 \otimes_{\mathcal{O}_0} \mathcal{I})$ constructed in Lemma \ref{lemma-canonical-class} is zero, \item there exists $K \in D^-(\mathcal{O})$ with $K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}_0 = K_0$ in $D(\mathcal{O}_0)$. \end{enumerate}


defos-lemma-lift-map-complexes

Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}_0$ be a surjection of sheaves of rings. Assume given the following data \begin{enumerate} \item a complex of $\mathcal{O}$-modules $\mathcal{F}^\bullet$, \item a complex $\mathcal{K}_0^\bullet$ of $\mathcal{O}_0$-modules, \item a quasi-isomorphism $\mathcal{K}_0^\bullet \to \mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{O}_0$, \end{enumerate} Then there exist a quasi-isomorphism $\mathcal{G}^\bullet \to \mathcal{F}^\bullet$ such that the map of complexes $\mathcal{G}^\bullet  \otimes_\mathcal{O} \mathcal{O}_0 \to \mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{O}_0$ factors through $\mathcal{K}_0^\bullet$ in the homotopy category of complexes of $\mathcal{O}_0$-modules.


defos-lemma-inf-obs-map-defo-complex

Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}_0$ be a surjection of sheaves of rings whose kernel is an ideal sheaf $\mathcal{I}$ of square zero. Let $K, L \in D^-(\mathcal{O})$. Set $K_0 = K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}_0$ and $L_0 = L \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}_0$ in $D^-(\mathcal{O}_0)$. Given $\alpha_0 : K_0 \to L_0$ in $D(\mathcal{O}_0)$ there is a canonical element $$ o(\alpha_0) \in \Ext^1_{\mathcal{O}_0}(K_0, L_0 \otimes_{\mathcal{O}_0}^\mathbf{L} \mathcal{I}) $$ whose vanishing is necessary and sufficient for the existence of a map $\alpha : K \to L$ in $D(\mathcal{O})$ with $\alpha_0 = \alpha \otimes_\mathcal{O}^\mathbf{L} \text{id}$.


defos-lemma-first-order-defos-complex

Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}_0$ be a surjection of sheaves of rings whose kernel is an ideal sheaf $\mathcal{I}$ of square zero. Let $K_0 \in D^-(\mathcal{O})$. A lift of $K_0$ is a pair $(K, \alpha_0)$ consisting of an object $K$ in $D^-(\mathcal{O})$ and an isomorphism $\alpha_0 : K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}_0 \to K_0$ in $D(\mathcal{O}_0)$. \begin{enumerate} \item Given a lift $(K, \alpha)$ the group of automorphism of the pair is canonically the cokernel of a map $$ \Ext^{-1}_{\mathcal{O}_0}(K_0, K_0) \longrightarrow \Hom_{\mathcal{O}_0}(K_0, K_0 \otimes_{\mathcal{O}_0}^\mathbf{L} \mathcal{I}) $$ \item If there is a lift, then the set of isomorphism classes of lifts is principal homogenenous under $\Ext^1_{\mathcal{O}_0}(K_0, K_0 \otimes_{\mathcal{O}_0}^\mathbf{L} \mathcal{I})$. \end{enumerate}


groupoids-quotients-lemma-invariant

Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j = (t, s) : R \to U \times_B U$ be a pre-relation of algebraic spaces over $B$. A morphism of algebraic spaces $\phi : U \to X$ is $R$-invariant if and only if it factors as $U \to U/R \to X$.


groupoids-quotients-lemma-base-change-on-invariant

Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j = (t, s) : R \to U \times_B U$ be a pre-relation of algebraic spaces over $B$. Let $U \to X$ be an $R$-invariant morphism of algebraic spaces over $B$. Let $X' \to X$ be any morphism of algebraic spaces. \begin{enumerate} \item Setting $U' = X' \times_X U$, $R' = X' \times_X R$ we obtain a pre-relation $j' : R' \to U' \times_B U'$. \item If $j$ is a relation, then $j'$ is a relation. \item If $j$ is a pre-equivalence relation, then $j'$ is a pre-equivalence relation. \item If $j$ is an equivalence relation, then $j'$ is an equivalence relation. \item If $j$ comes from a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$, then \begin{enumerate} \item $(U, R, s, t, c)$ is a groupoid in algebraic spaces over $X$, and \item $j'$ comes from the base change $(U', R', s', t', c')$ of this groupoid to $X'$, see Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-base-change-groupoid}. \end{enumerate} \item If $j$ comes from the action of a group algebraic space $G/B$ on $U$ as in Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-groupoid-from-action} then $j'$ comes from the induced action of $G$ on $U'$. \end{enumerate}


groupoids-quotients-lemma-base-change-quotient-sheaf

In the situation of Lemma \ref{lemma-base-change-on-invariant} there is an isomorphism of sheaves $$ U'/R' = X' \times_X U/R $$ For the construction of quotient sheaves, see Groupoids in Spaces, Section \ref{spaces-groupoids-section-quotient-sheaves}.


groupoids-quotients-lemma-base-change-quotient-stack

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $U \to X$ be an $R$-invariant morphism of algebraic spaces over $B$. Let $g : X' \to X$ be a morphism of algebraic spaces over $B$ and let $(U', R', s', t', c')$ be the base change as in Lemma \ref{lemma-base-change-on-invariant}. Then $$ \xymatrix{ [U'/R'] \ar[r] \ar[d] & [U/R] \ar[d] \\ \mathcal{S}_{X'} \ar[r] & \mathcal{S}_X } $$ is a $2$-fibre product of stacks in groupoids over $(\Sch/S)_{fppf}$. For the construction of quotient stacks and the morphisms in this diagram, see Groupoids in Spaces, Section \ref{spaces-groupoids-section-stacks}.


groupoids-quotients-lemma-categorical-unique

Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \to U \times_B U$ be a pre-relation in algebraic spaces over $B$. If a categorical quotient in the category of algebraic spaces over $B$ exists, then it is unique up to unique isomorphism. Similarly for categorical quotients in full subcategories of $\textit{Spaces}/B$.


groupoids-quotients-lemma-categorical-reduced

In the situation of Definition \ref{definition-categorical}. If $\phi : U \to X$ is a categorical quotient and $U$ is reduced, then $X$ is reduced. The same holds for categorical quotients in a category of spaces $\mathcal{C}$ listed in Example \ref{example-categories}.


groupoids-quotients-lemma-invariant-map-constant-on-orbit

In the situation of Definition \ref{definition-orbit}. Let $\phi : U \to X$ be an $R$-invariant morphism of algebraic spaces over $B$. Then $|\phi| : |U| \to |X|$ is constant on the orbits.


groupoids-quotients-lemma-weak-orbit-pre-equivalence

Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $\Spec(k) \to B$ be a geometric point of $B$. Let $j : R \to U \times_B U$ be a pre-equivalence relation over $B$. In this case the weak orbit of $\overline{u} \in U(k)$ is simply $$ \{ \overline{u}' \in U(k) \text{ such that } \exists \overline{r} \in R(k), \ s(\overline{r}) = \overline{u}, \ t(\overline{r}) = \overline{u}' \} $$ and the orbit of $\overline{u} \in U(k)$ is $$ \{ \overline{u}' \in U(k) : \exists\text{ field extension }k \subset K, \ \exists\ r \in R(K), \ s(r) = \overline{u}, \ t(r) = \overline{u}'\} $$


groupoids-quotients-lemma-make-pre-equivalence

Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \to U \times_B U$ be a pre-relation over $B$. Then $j_\infty : R_\infty \to U \times_B U$ is a pre-equivalence relation over $B$. Moreover \begin{enumerate} \item $\phi : U \to X$ is $R$-invariant if and only if it is $R_\infty$-invariant, \item the canonical map of quotient sheaves $U/R \to U/R_\infty$ (see Groupoids in Spaces, Section \ref{spaces-groupoids-section-quotient-sheaves}) is an isomorphism, \item weak $R$-orbits agree with weak $R_\infty$-orbits, \item $R$-orbits agree with $R_\infty$-orbits, \item if $s, t$ are locally of finite type, then $s_\infty$, $t_\infty$ are locally of finite type, \item add more here as needed. \end{enumerate}


groupoids-quotients-lemma-geometric-orbits

Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \to U \times_B U$ be a pre-relation over $B$. Let $\Spec(k) \to B$ be a geometric point of $B$. \begin{enumerate} \item If $s, t : R \to U$ are locally of finite type then weak $R$-equivalence on $U(k)$ agrees with $R$-equivalence, and weak $R$-orbits agree with $R$-orbits on $U(k)$. \item If $k$ has sufficiently large cardinality then weak $R$-equivalence on $U(k)$ agrees with $R$-equivalence, and weak $R$-orbits agree with $R$-orbits on $U(k)$. \end{enumerate}


groupoids-quotients-lemma-set-theoretic-invariant

In the situation of Definition \ref{definition-set-theoretically-invariant}. A morphism $\phi : U \to X$ is set-theoretically $R$-invariant if and only if for any algebraically closed field $k$ over $B$ the map $U(k) \to X(k)$ is constant on orbits.


groupoids-quotients-lemma-invariant-set-theoretically-invariant

In the situation of Definition \ref{definition-set-theoretically-invariant}. An invariant morphism is set-theoretically invariant.


groupoids-quotients-lemma-set-theoretic-pre-equivalence-geometric

In the situation of Definition \ref{definition-set-theoretic-equivalence}. The following are equivalent: \begin{enumerate} \item The morphism $j$ is a set-theoretic pre-equivalence relation. \item The subset $j(|R|) \subset |U \times_B U|$ contains the image of $|j'|$ for any of the morphisms $j'$ as in Equation (\ref{equation-list}). \item For every algebraically closed field $k$ over $B$ of sufficiently large cardinality the subset $j(R(k)) \subset U(k) \times U(k)$ is an equivalence relation. \end{enumerate} If $s, t$ are locally of finite type these are also equivalent to \begin{enumerate} \item[(4)] For every algebraically closed field $k$ over $B$ the subset $j(R(k)) \subset U(k) \times U(k)$ is an equivalence relation. \end{enumerate}


groupoids-quotients-lemma-set-theoretic-equivalence

Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \to U \times_B U$ be a pre-relation over $B$. \begin{enumerate} \item If $j$ is a pre-equivalence relation, then $j$ is a set-theoretic pre-equivalence relation. This holds in particular when $j$ comes from a groupoid in algebraic spaces, or from an action of a group algebraic space on $U$. \item If $j$ is an equivalence relation, then $j$ is a set-theoretic equivalence relation. \end{enumerate}


groupoids-quotients-lemma-separates-orbits

Let $B \to S$ be as in Section \ref{section-conventions-notation}. Let $j : R \to U \times_B U$ be a pre-relation. Let $\phi : U \to X$ be a morphism of algebraic spaces over $B$. Consider the diagram $$ \xymatrix{ (U \times_X U) \times_{(U \times_B U)} R \ar[d]^q \ar[r]_-p & R \ar[d]^j \\ U \times_X U \ar[r]^c & U \times_B U } $$ Then we have: \begin{enumerate} \item The morphism $\phi$ is set-theoretically invariant if and only if $p$ is surjective. \item If $j$ is a set-theoretic pre-equivalence relation then $\phi$ separates orbits if and only if $p$ and $q$ are surjective. \item If $p$ and $q$ are surjective, then $j$ is a set-theoretic pre-equivalence relation (and $\phi$ separates orbits). \item If $\phi$ is $R$-invariant and $j$ is a set-theoretic pre-equivalence relation, then $\phi$ separates orbits if and only if the induced morphism $R \to U \times_X U$ is surjective. \end{enumerate}


spaces-resolve-lemma-dimension-special-fibre

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local domain of dimension $\geq 1$. Let $f : X \to \Spec(A)$ be a morphism of algebraic spaces. Assume at least one of the following conditions is satisfied \begin{enumerate} \item $f$ is a modification (Spaces over Fields, Definition \ref{spaces-over-fields-definition-modification}), \item $f$ is an alteration (Spaces over Fields, Definition \ref{spaces-over-fields-definition-alteration}), \item $f$ is locally of finite type, quasi-separated, $X$ is integral, and there is exactly one point of $|X|$ mapping to the generic point of $\Spec(A)$, \item $f$ is locally of finite type, $X$ is decent, and the points of $|X|$ mapping to the generic point of $\Spec(A)$ are the generic points of irreducible components of $|X|$, \item add more here. \end{enumerate} Then $\dim(X_\kappa) \leq \dim(A) - 1$.


spaces-resolve-lemma-equivalence

The functor $F$ (\ref{equation-equivalence}) is an equivalence.


spaces-resolve-lemma-equivalence-fibre

Let $X, x_i, U_i \to X, u_i$ be as in (\ref{equation-equivalence}). If $f : Y \to X$ corresponds to $g_i : Y_i \to U_i$ under $F$, then $Y_{x_i} \cong (Y_i)_{u_i}$ as algebraic spaces.


spaces-resolve-lemma-equivalence-sequence-blowups

Let $X, x_i, U_i \to X, u_i$ be as in (\ref{equation-equivalence}) and assume $f : Y \to X$ corresponds to $g_i : Y_i \to U_i$ under $F$. Then there exists a factorization $$ Y = Z_m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = X $$ of $f$ where $Z_{j + 1} \to Z_j$ is the blowing up of $Z_j$ at a closed point $z_j$ lying over $\{x_1, \ldots, x_n\}$ if and only if for each $i$ there exists a factorization $$ Y_i = Z_{i, m_i} \to Z_{i, m_i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = U_i $$ of $g_i$ where $Z_{i, j + 1} \to Z_{i, j}$ is the blowing up of $Z_{i, j}$ at a closed point $z_{i, j}$ lying over $u_i$.


spaces-resolve-lemma-make-ideal-principal

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $T \subset |X|$ be a finite set of closed points $x$ such that (1) $X$ is regular at $x$ and (2) the local ring of $X$ at $x$ has dimension $2$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals such that $\mathcal{O}_X/\mathcal{I}$ is supported on $T$. Then there exists a sequence $$ X_m \to X_{m - 1} \to \ldots \to X_1 \to X_0 = X $$ where $X_{j + 1} \to X_j$ is the blowing up of $X_j$ at a closed point $x_j$ lying above a point of $T$ such that $\mathcal{I}\mathcal{O}_{X_n}$ is an invertible ideal sheaf.


spaces-resolve-lemma-equivalence-sequence-normalized-blowups

Let $X, x_i, U_i \to X, u_i$ be as in (\ref{equation-equivalence}) and assume $f : Y \to X$ corresponds to $g_i : Y_i \to U_i$ under $F$. Assume $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-prepare-normalization}. Then there exists a factorization $$ Y = Z_m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = X $$ of $f$ where $Z_{j + 1} \to Z_j$ is the normalized blowing up of $Z_j$ at a closed point $z_j$ lying over $\{x_1, \ldots, x_n\}$ if and only if for each $i$ there exists a factorization $$ Y_i = Z_{i, m_i} \to Z_{i, m_i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = U_i $$ of $g_i$ where $Z_{i, j + 1} \to Z_{i, j}$ is the normalized blowing up of $Z_{i, j}$ at a closed point $z_{i, j}$ lying over $u_i$.


spaces-resolve-lemma-port-regularity-to-completion

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $X \to \Spec(A)$ be a morphism which is locally of finite type with $X$ a decent algebraic space. Set $Y = X \times_{\Spec(A)} \Spec(A^\wedge)$. Let $y \in |Y|$ with image $x \in |X|$. Then \begin{enumerate} \item if $\mathcal{O}_{Y, y}^h$ is regular, then $\mathcal{O}_{X, x}^h$ is regular, \item if $y$ is in the closed fibre, then $\mathcal{O}_{Y, y}^h$ is regular $\Leftrightarrow \mathcal{O}_{X, x}^h$ is regular, and \item If $X$ is proper over $A$, then $X$ is regular if and only if $Y$ is regular. \end{enumerate}


spaces-resolve-lemma-regular-alteration-implies

Let $S$ be a scheme. Let $Y$ be a Noetherian integral algebraic space over $S$. Assume there exists an alteration $f : X \to Y$ with $X$ regular. Then the normalization $Y^\nu \to Y$ is finite and $Y$ has a dense open which is regular.


spaces-resolve-lemma-regular-alteration-implies-local

Let $(A, \mathfrak m, \kappa)$ be a local Noetherian domain. Assume there exists an alteration $f : X \to \Spec(A)$ with $X$ regular. Then \begin{enumerate} \item there exists a nonzero $f \in A$ such that $A_f$ is regular, \item the integral closure $B$ of $A$ in its fraction field is finite over $A$, \item the $\mathfrak m$-adic completion of $B$ is a normal ring, i.e., the completions of $B$ at its maximal ideals are normal domains, and \item the generic formal fibre of $A$ is regular. \end{enumerate}


duality-lemma-equivalent-definitions

Let $X$ be a locally Noetherian scheme. Let $K$ be an object of $D(\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item For every affine open $U = \Spec(A) \subset X$ there exists a dualizing complex $\omega_A^\bullet$ for $A$ such that $K|_U$ is isomorphic to the image of $\omega_A^\bullet$ by the functor $\widetilde{} : D(A) \to D(\mathcal{O}_U)$. \item There is an affine open covering $X = \bigcup U_i$, $U_i = \Spec(A_i)$ such that for each $i$ there exists a dualizing complex $\omega_i^\bullet$ for $A_i$ such that $K|_{U_i}$ is isomorphic to the image of $\omega_i^\bullet$ by the functor $\widetilde{} : D(A_i) \to D(\mathcal{O}_{U_i})$. \end{enumerate}


duality-lemma-affine-duality

Let $A$ be a Noetherian ring and let $X = \Spec(A)$. Let $K, L$ be objects of $D(A)$. If $K \in D_{\textit{Coh}}(A)$ and $L$ has finite injective dimension, then $$ R\SheafHom_{\mathcal{O}_X}(\widetilde{K}, \widetilde{L}) = \widetilde{R\Hom_A(K, L)} $$ in $D(\mathcal{O}_X)$.


duality-lemma-internal-hom-evaluate-isom

Let $X$ be a Noetherian scheme. Let $K, L, M \in D_\QCoh(\mathcal{O}_X)$.  Then the map $$ R\SheafHom(L, M) \otimes_{\mathcal{O}_X}^\mathbf{L} K \longrightarrow R\SheafHom(R\SheafHom(K, L), M) $$ of Cohomology, Lemma \ref{cohomology-lemma-internal-hom-evaluate} is an isomorphism in the following two cases \begin{enumerate} \item $K \in D^-_{\textit{Coh}}(\mathcal{O}_X)$, $L \in D^+_{\textit{Coh}}(\mathcal{O}_X)$, and $M$ affine locally has finite injective dimension (see proof), or \item $K$ and $L$ are in $D_{\textit{Coh}}(\mathcal{O}_X)$, the object $R\SheafHom(L, M)$ has finite tor dimension, and $L$ and $M$ affine locally have finite injective dimension (in particular $L$ and $M$ are bounded). \end{enumerate}


duality-lemma-dualizing-schemes

Let $K$ be a dualizing complex on a locally Noetherian scheme $X$. Then $K$ is an object of $D_{\textit{Coh}}(\mathcal{O}_X)$ and $D = R\SheafHom_{\mathcal{O}_X}(-, K)$ induces an anti-equivalence $$ D : D_{\textit{Coh}}(\mathcal{O}_X) \longrightarrow D_{\textit{Coh}}(\mathcal{O}_X) $$ which comes equipped with a canonical isomorphism $\text{id} \to D \circ D$. If $X$ is quasi-compact, then $D$ exchanges $D^+_{\textit{Coh}}(\mathcal{O}_X)$ and $D^-_{\textit{Coh}}(\mathcal{O}_X)$ and induces an equivalence $D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_X)$.


duality-lemma-dualizing-unique-schemes

Let $X$ be a locally Noetherian scheme. If $K$ and $K'$ are dualizing complexes on $X$, then $K'$ is isomorphic to $K \otimes_{\mathcal{O}_X}^\mathbf{L} L$ for some invertible object $L$ of $D(\mathcal{O}_X)$.


duality-lemma-dimension-function-scheme

Let $X$ be a locally Noetherian scheme. Let $\omega_X^\bullet$ be a dualizing complex on $X$. Then $X$ is universally catenary and the function $X \to \mathbf{Z}$ defined by $$ x \longmapsto \delta(x)\text{ such that } \omega_{X, x}^\bullet[-\delta(x)] \text{ is a normalized dualizing complex over } \mathcal{O}_{X, x} $$ is a dimension function.


duality-lemma-sitting-in-degrees

Let $X$ be a locally Noetherian scheme. Let $\omega_X^\bullet$ be a dualizing complex on $X$ with associated dimension function $\delta$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Set $\mathcal{E}^i = \SheafExt^{-i}_{\mathcal{O}_X}(\mathcal{F}, \omega_X^\bullet)$. Then $\mathcal{E}^i$ is a coherent $\mathcal{O}_X$-module and for $x \in X$ we have \begin{enumerate} \item $\mathcal{E}^i_x$ is nonzero only for $\delta(x) \leq i \leq \delta(x) + \dim(\text{Supp}(\mathcal{F}_x))$, \item $\dim(\text{Supp}(\mathcal{E}^{i + \delta(x)}_x)) \leq i$, \item $\text{depth}(\mathcal{F}_x)$ is the smallest integer $i \geq 0$ such that $\mathcal{E}^{i + \delta(x)} \not = 0$, and \item we have $x \in \text{Supp}(\bigoplus_{j \leq i} \mathcal{E}^j) \Leftrightarrow \text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) + \delta(x) \leq i$. \end{enumerate}


duality-lemma-twisted-inverse-image

\begin{reference} This is almost the same as \cite[Example 4.2]{Neeman-Grothendieck}. \end{reference} Let $f : X \to Y$ be a morphism between quasi-separated and quasi-compact schemes. The functor $Rf_* : D_\QCoh(X) \to D_\QCoh(Y)$ has a right adjoint.


duality-lemma-twisted-inverse-image-bounded-below

Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $a : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_X)$ be the right adjoint to $Rf_*$ of Lemma \ref{lemma-twisted-inverse-image}. Then $a$ maps $D^+_\QCoh(\mathcal{O}_Y)$ into $D^+_\QCoh(\mathcal{O}_X)$. In fact, there exists an integer $N$ such that $H^i(K) = 0$ for $i \leq c$ implies $H^i(a(K)) = 0$ for $i \leq c - N$.


duality-lemma-iso-on-RSheafHom

Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $a$ be the right adjoint to $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$. Let $L \in D_\QCoh(\mathcal{O}_X)$ and $K \in D_\QCoh(\mathcal{O}_Y)$. Then the map (\ref{equation-sheafy-trace}) $$ Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K)) \longrightarrow R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K) $$ becomes an isomorphism after applying the functor $DQ_Y : D(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_Y)$ discussed in Derived Categories of Schemes, Section \ref{perfect-section-better-coherator}.


duality-lemma-iso-global-hom

Let $f : X \to Y$ be a morphism of quasi-separated and quasi-compact schemes. For all $L \in D_\QCoh(\mathcal{O}_X)$ and $K \in D_\QCoh(\mathcal{O}_Y)$ (\ref{equation-sheafy-trace}) induces an isomorphism $R\Hom_X(L, a(K)) \to R\Hom_Y(Rf_*L, K)$ of global derived homs.


duality-lemma-flat-precompose-pus

In diagram (\ref{equation-base-change}) assume that $g$ is flat or more generally that $f$ and $g$ are Tor independent. Then $a \circ Rg_* \leftarrow Rg'_* \circ a'$ is an isomorphism.


duality-lemma-when-sheafy

Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $a$ be the right adjoint to $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$. Let $V \subset Y$ be quasi-compact open with inverse image $U \subset X$. \begin{enumerate} \item For every $Q \in D_\QCoh^+(\mathcal{O}_Y)$ supported on $Y \setminus V$ the image $a(Q)$ is supported on $X \setminus U$ if and only if (\ref{equation-sheafy}) is an isomorphism on all $K$ in $D_\QCoh^+(\mathcal{O}_Y)$. \item For every $Q \in D_\QCoh(\mathcal{O}_Y)$ supported on $Y \setminus V$ the image $a(Q)$ is supported on $X \setminus U$ if and only if (\ref{equation-sheafy}) is an isomorphism on all $K$ in $D_\QCoh(\mathcal{O}_Y)$. \item If $a$ commutes with direct sums, then the equivalent conditions of (1) imply the equivalent conditions of (2). \end{enumerate}


duality-lemma-proper-noetherian

Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a proper morphism. If\footnote{This proof works for those morphisms of quasi-compact and quasi-separated schemes such that $Rf_*P$ is pseudo-coherent for all $P$ perfect on $X$. It follows easily from a theorem of Kiehl \cite{Kiehl} that this holds if $f$ is proper and pseudo-coherent. This is the correct generality for this lemma and some of the other results in this chapter.} \begin{enumerate} \item $f$ is flat and of finite presentation, or \item $Y$ is Noetherian \end{enumerate} then the equivalent conditions of Lemma \ref{lemma-when-sheafy} part (1) hold for all quasi-compact opens $V$ of $Y$.


duality-lemma-compose-base-change-maps

Consider a commutative diagram $$ \xymatrix{ X' \ar[r]_k \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^l \ar[d]_{g'} & Y \ar[d]^g \\ Z' \ar[r]^m & Z } $$ of quasi-compact and quasi-separated schemes where both diagrams are cartesian and where $f$ and $l$ as well as $g$ and $m$ are Tor independent. Then the maps (\ref{equation-base-change-map}) for the two squares compose to give the base change map for the outer rectangle (see proof for a precise statement).


duality-lemma-compose-base-change-maps-horizontal

Consider a commutative diagram $$ \xymatrix{ X'' \ar[r]_{g'} \ar[d]_{f''} & X' \ar[r]_g \ar[d]_{f'} & X \ar[d]^f \\ Y'' \ar[r]^{h'} & Y' \ar[r]^h & Y } $$ of quasi-compact and quasi-separated schemes where both diagrams are cartesian and where $f$ and $h$ as well as $f'$ and $h'$ are Tor independent. Then the maps (\ref{equation-base-change-map}) for the two squares compose to give the base change map for the outer rectangle (see proof for a precise statement).


duality-lemma-more-base-change

In diagram (\ref{equation-base-change}) assume \begin{enumerate} \item $g : Y' \to Y$ is a morphism of affine schemes, \item $f : X \to Y$ is proper, and \item $f$ and $g$ are Tor independent. \end{enumerate} Then the base change map (\ref{equation-base-change-map}) induces an isomorphism $$ L(g')^*a(K) \longrightarrow a'(Lg^*K) $$ in the following cases \begin{enumerate} \item for all $K \in D_\QCoh(\mathcal{O}_X)$ if $f$ is flat of finite presentation, \item for all $K \in D_\QCoh(\mathcal{O}_X)$ if $f$ is perfect and $Y$ Noetherian, \item for $K \in D_\QCoh^+(\mathcal{O}_X)$ if $g$ has finite Tor dimension and $Y$ Noetherian. \end{enumerate}


duality-lemma-trace-map-and-base-change

Suppose we have a diagram (\ref{equation-base-change}) where $f$ and $g$ are tor independent. Then the maps $1 \star \text{Tr}_f : Lg^* \circ Rf_* \circ a \to Lg^*$ and $\text{Tr}_{f'} \star 1 : Rf'_* \circ a' \circ Lg^* \to Lg^*$ agree via the base change maps $\beta : Lg^* \circ Rf_* \to Rf'_* \circ L(g')^*$ (Cohomology, Remark \ref{cohomology-remark-base-change}) and $\alpha : L(g')^* \circ a \to a' \circ Lg^*$ (\ref{equation-base-change-map}). More precisely, the diagram $$ \xymatrix{ Lg^* \circ Rf_* \circ a \ar[d]_{\beta \star 1} \ar[r]_-{1 \star \text{Tr}_f} & Lg^* \\ Rf'_* \circ L(g')^* \circ a \ar[r]^{1 \star \alpha} & Rf'_* \circ a' \circ Lg^* \ar[u]_{\text{Tr}_{f'} \star 1} } $$ of transformations of functors commutes.


duality-lemma-unit-and-base-change

Suppose we have a diagram (\ref{equation-base-change}) where $f$ and $g$ are tor independent. Then the maps $1 \star \eta_f : L(g')^* \to L(g')^* \circ a \circ Rf_*$ and $\eta_{f'} \star 1 : L(g')^* \to a' \circ Rf'_* \circ L(g')^*$ agree via the base change maps $\beta : Lg^* \circ Rf_* \to Rf'_* \circ L(g')^*$ (Cohomology, Remark \ref{cohomology-remark-base-change}) and $\alpha : L(g')^* \circ a \to a' \circ Lg^*$ (\ref{equation-base-change-map}). More precisely, the diagram $$ \xymatrix{ L(g')^* \ar[r]_-{1 \star \eta_f} \ar[d]_{\eta_{f'} \star 1} & L(g')^* \circ a \circ Rf_* \ar[d]^\alpha \\ a' \circ Rf'_* \circ L(g')^* & a' \circ Lg^* \circ Rf_* \ar[l]_-\beta } $$ of transformations of functors commutes.


duality-lemma-compare-with-pullback-perfect

Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes. The map $Lf^*K \otimes^\mathbf{L}_{\mathcal{O}_X} a(L) \to a(K \otimes_{\mathcal{O}_Y}^\mathbf{L} L)$ defined above for $K, L \in D_\QCoh(\mathcal{O}_Y)$ is an isomorphism if $K$ is perfect. In particular, (\ref{equation-compare-with-pullback}) is an isomorphism if $K$ is perfect.


duality-lemma-restriction-compare-with-pullback

Suppose we have a diagram (\ref{equation-base-change}) where $f$ and $g$ are tor independent. Let $K \in D_\QCoh(\mathcal{O}_Y)$. The diagram $$ \xymatrix{ L(g')^*(Lf^*K \otimes^\mathbf{L}_{\mathcal{O}_X} a(\mathcal{O}_Y)) \ar[r] \ar[d] & L(g')^*a(K) \ar[d] \\ L(f')^*Lg^*K \otimes_{\mathcal{O}_{X'}}^\mathbf{L} a'(\mathcal{O}_{Y'}) \ar[r] & a'(Lg^*K) } $$ commutes where the horizontal arrows are the maps (\ref{equation-compare-with-pullback}) for $K$ and $Lg^*K$ and the vertical maps are constructed using Cohomology, Remark \ref{cohomology-remark-base-change} and (\ref{equation-base-change-map}).


duality-lemma-compare-on-open

Let $f : X \to Y$ be a proper morphism of Noetherian schemes. Let $V \subset Y$ be an open such that $f^{-1}(V) \to V$ is an isomorphism. Then for $K \in D_\QCoh^+(\mathcal{O}_Y)$ the map (\ref{equation-compare-with-pullback}) restricts to an isomorphism over $f^{-1}(V)$.


duality-lemma-transitivity-compare-with-pullback

Let $f : X \to Y$ and $g : Y \to Z$ be composable morphisms of quasi-compact and quasi-separated schemes and set $h = g \circ f$. Let $a, b, c$ be the adjoints of Lemma \ref{lemma-twisted-inverse-image} for $f, g, h$. For any $K \in D_\QCoh(\mathcal{O}_Z)$ the diagram $$ \xymatrix{ Lf^*(Lg^*K \otimes_{\mathcal{O}_Y}^\mathbf{L} b(\mathcal{O}_Z)) \otimes_{\mathcal{O}_X}^\mathbf{L} a(\mathcal{O}_Y) \ar@{=}[d] \ar[r] & a(Lg^*K \otimes_{\mathcal{O}_Y}^\mathbf{L} b(\mathcal{O}_Z)) \ar[r] & a(b(K)) \ar@{=}[d] \\ Lh^*K \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*b(\mathcal{O}_Z) \otimes_{\mathcal{O}_X}^\mathbf{L} a(\mathcal{O}_Y) \ar[r] & Lh^*K \otimes_{\mathcal{O}_X}^\mathbf{L} c(\mathcal{O}_Z) \ar[r] & c(K) } $$ is commutative where the arrows are (\ref{equation-compare-with-pullback}) and we have used $Lh^* = Lf^* \circ Lg^*$ and $c = a \circ b$.


duality-lemma-compute-sheaf-with-exact-support

With notation as above. The functor $\SheafHom(\mathcal{O}_Z, -)$ is a right adjoint to the functor $i_* : \textit{Mod}(\mathcal{O}_Z) \to \textit{Mod}(\mathcal{O}_X)$. For $V \subset Z$ open we have $$ \Gamma(V, \SheafHom(\mathcal{O}_Z, \mathcal{F})) = \{s \in \Gamma(U, \mathcal{F}) \mid \mathcal{I}s = 0\} $$ where $U \subset X$ is an open whose intersection with $Z$ is $V$.


duality-lemma-sheaf-with-exact-support-adjoint

With notation as above. The functor $R\SheafHom(\mathcal{O}_Z, -)$ is the right adjoint of the functor $Ri_* : D(\mathcal{O}_Z) \to D(\mathcal{O}_X)$.


duality-lemma-sheaf-with-exact-support-ext

With notation as above. We have $$ Ri_*R\SheafHom(\mathcal{O}_Z, K) = R\SheafHom_{\mathcal{O}_X}(i_*\mathcal{O}_Z, K) $$ in $D(\mathcal{O}_X)$ for all $K$ in $D(\mathcal{O}_X)$.


duality-lemma-sheaf-with-exact-support-internal-home

With notation as above. For $M \in D(\mathcal{O}_Z)$ we have $$ R\SheafHom_{\mathcal{O}_X}(Ri_*M, K) = Ri_*R\SheafHom_{\mathcal{O}_Z}(M, R\SheafHom(\mathcal{O}_Z, K)) $$ in $D(\mathcal{O}_Z)$ for all $K$ in $D(\mathcal{O}_X)$.


duality-lemma-sheaf-with-exact-support-quasi-coherent

Let $i : Z \to X$ be a pseudo-coherent closed immersion of schemes (any closed immersion if $X$ is locally Noetherian). Then \begin{enumerate} \item $R\SheafHom(\mathcal{O}_Z, -)$ maps $D^+_\QCoh(\mathcal{O}_X)$ into $D^+_\QCoh(\mathcal{O}_Z)$, and \item if $X = \Spec(A)$ and $Z = \Spec(B)$, then the diagram $$ \xymatrix{ D^+(B) \ar[r] & D_\QCoh^+(\mathcal{O}_Z) \\ D^+(A) \ar[r] \ar[u]^{R\Hom(B, -)} & D_\QCoh^+(\mathcal{O}_X) \ar[u]_{R\SheafHom(\mathcal{O}_Z, -)} } $$ is commutative. \end{enumerate}


duality-lemma-sheaf-with-exact-support-coherent

Let $i : Z \to X$ be a closed immersion of schemes. Assume $X$ is a locally Noetherian. Then $R\SheafHom(\mathcal{O}_Z, -)$ maps $D^+_{\textit{Coh}}(\mathcal{O}_X)$ into $D^+_{\textit{Coh}}(\mathcal{O}_Z)$.


duality-lemma-twisted-inverse-image-closed

Let $X$ be a quasi-compact and quasi-separated scheme. Let $i : Z \to X$ be a pseudo-coherent closed immersion (if $X$ is Noetherian, then any closed immersion is pseudo-coherent). Let $a : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Z)$ be the right adjoint to $Ri_*$. Then there is a functorial isomorphism $$ a(K) = R\SheafHom(\mathcal{O}_Z, K) $$ for $K \in D_\QCoh^+(\mathcal{O}_X)$.


duality-lemma-check-base-change-is-iso

In the situation above, the map (\ref{equation-base-change-exact-support}) is an isomorphism if and only if the base change map $$ Lf^*R\SheafHom_{\mathcal{O}_X}(\mathcal{O}_Z, K) \longrightarrow R\SheafHom_{\mathcal{O}_{X'}}(\mathcal{O}_{Z'}, Lf^*K) $$ of Cohomology, Remark \ref{cohomology-remark-prepare-fancy-base-change} is an isomorphism.


duality-lemma-flat-bc-sheaf-with-exact-support

In the situation above, assume $f$ is flat and $i$ pseudo-coherent. Then (\ref{equation-base-change-exact-support}) is an isomorphism for $K$ in $D^+_\QCoh(\mathcal{O}_X)$.


duality-lemma-sheaf-with-exact-support-tensor

Let $i : Z \to X$ be a pseudo-coherent closed immersion of schemes. Let $M \in D_\QCoh(\mathcal{O}_X)$ locally have tor-amplitude in $[a, \infty)$. Let $K \in D_\QCoh^+(\mathcal{O}_X)$. Then there is a canonical isomorphism $$ R\SheafHom(\mathcal{O}_Z, K) \otimes_{\mathcal{O}_Z}^\mathbf{L} Li^*M = R\SheafHom(\mathcal{O}_Z, K \otimes_{\mathcal{O}_X}^\mathbf{L} M) $$ in $D(\mathcal{O}_Z)$.


duality-lemma-compute-sheafhom-affine

With notation as above. The functor $\SheafHom(f_*\mathcal{O}_Y, -)$ is a right adjoint to the restriction functor $\textit{Mod}(f_*\mathcal{O}_Y) \to \textit{Mod}(\mathcal{O}_X)$. For an affine open $U \subset X$ we have $$ \Gamma(U, \SheafHom(f_*\mathcal{O}_Y, \mathcal{F})) = \Hom_A(B, \mathcal{F}(U)) $$ where $A = \mathcal{O}_X(U)$ and $B = \mathcal{O}_Y(f^{-1}(U))$.


duality-lemma-sheafhom-affine-adjoint

With notation as above. The functor $R\SheafHom(f_*\mathcal{O}_Y, -)$ is the right adjoint of the functor $D(f_*\mathcal{O}_Y) \to D(\mathcal{O}_X)$.


duality-lemma-sheafhom-affine-ext

With notation as above. The composition $$ D(\mathcal{O}_X) \xrightarrow{R\SheafHom(f_*\mathcal{O}_Y, -)} D(f_*\mathcal{O}_Y) \to D(\mathcal{O}_X) $$ is the functor $K \mapsto R\SheafHom_{\mathcal{O}_X}(f_*\mathcal{O}_Y, K)$.


duality-lemma-finite-twisted

Let $f : Y \to X$ be a finite pseudo-coherent morphism of schemes (a finite morphism of Noetherian schemes is pseudo-coherent). The functor $R\SheafHom(f_*\mathcal{O}_Y, -)$ maps $D_\QCoh^+(\mathcal{O}_X)$ into $D_\QCoh^+(f_*\mathcal{O}_Y)$. If $X$ is quasi-compact and quasi-separated, then the diagram $$ \xymatrix{ D_\QCoh^+(\mathcal{O}_X) \ar[rr]_a \ar[rd]_{R\SheafHom(f_*\mathcal{O}_Y, -)} & & D_\QCoh^+(\mathcal{O}_Y) \ar[ld]^\Phi \\ & D_\QCoh^+(f_*\mathcal{O}_Y) } $$ is commutative, where $a$ is the right adjoint of Lemma \ref{lemma-twisted-inverse-image} for $f$ and $\Phi$ is the equivalence of Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-morphism-equivalence}.


duality-lemma-proper-flat

Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a morphism of schemes which is proper, flat, and of finite presentation. Let $a$ be the right adjoint for $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ of Lemma \ref{lemma-twisted-inverse-image}. Then $a$ commutes with direct sums.


duality-lemma-proper-flat-relative

Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a morphism of schemes which is proper, flat, and of finite presentation. Let $a$ be the right adjoint for $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ of Lemma \ref{lemma-twisted-inverse-image}. Then \begin{enumerate} \item for every closed $T \subset Y$ if $Q \in D_\QCoh(Y)$ is supported on $T$, then $a(Q)$ is supported on $f^{-1}(T)$, \item for every open $V \subset Y$ and any $K \in D_\QCoh(\mathcal{O}_Y)$ the map (\ref{equation-sheafy}) is an isomorphism, and \end{enumerate}


duality-lemma-compare-with-pullback-flat-proper

Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a morphism of schemes which is proper, flat, and of finite presentation. The map (\ref{equation-compare-with-pullback}) is an isomorphism for every object $K$ of $D_\QCoh(\mathcal{O}_Y)$.


duality-lemma-proper-flat-base-change

Let $g : Y' \to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $f : X \to Y$ be a proper, flat morphism of finite presentation. Then the base change map (\ref{equation-base-change-map}) is an isomorphism for all $K \in D_\QCoh(\mathcal{O}_Y)$.


duality-lemma-properties-relative-dualizing

Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a morphism of schemes which is proper, flat, and of finite presentation with relative dualizing complex $\omega_{X/Y}^\bullet$ (Remark \ref{remark-relative-dualizing-complex}). Then \begin{enumerate} \item $\omega_{X/Y}^\bullet$ is a $Y$-perfect object of $D(\mathcal{O}_X)$, \item $Rf_*\omega_{X/Y}^\bullet$ has vanishing cohomology sheaves in positive degrees, \item $\mathcal{O}_X \to R\SheafHom_{\mathcal{O}_X}(\omega_{X/Y}^\bullet, \omega_{X/Y}^\bullet)$ is an isomorphism. \end{enumerate}


duality-lemma-van-den-bergh

Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a proper, flat morphism of finite presentation with relative dualizing complex $\omega_{X/Y}^\bullet$ (Remark \ref{remark-relative-dualizing-complex}). There is a canonical isomorphism \begin{equation} \label{equation-pre-rigid} \mathcal{O}_X = c(L\text{pr}_1^*\omega_{X/Y}^\bullet) = c(L\text{pr}_2^*\omega_{X/Y}^\bullet) \end{equation} and a canonical isomorphism \begin{equation} \label{equation-rigid} \omega_{X/Y}^\bullet = c\left(L\text{pr}_1^*\omega_{X/Y}^\bullet \otimes_{\mathcal{O}_{X \times_Y X}}^\mathbf{L} L\text{pr}_2^*\omega_{X/Y}^\bullet\right) \end{equation} where $c$ is the right adjoint of Lemma \ref{lemma-twisted-inverse-image} for the diagonal $\Delta : X \to X \times_Y X$.


duality-lemma-proper-flat-noetherian

Let $f : X \to Y$ be a perfect proper morphism of Noetherian schemes. Let $a$ be the right adjoint for $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ of Lemma \ref{lemma-twisted-inverse-image}. Then $a$ commutes with direct sums.


duality-lemma-proper-flat-noetherian-relative

Let $f : X \to Y$ be a perfect proper morphism of Noetherian schemes. Let $a$ be the right adjoint for $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ of Lemma \ref{lemma-twisted-inverse-image}. Then \begin{enumerate} \item for every closed $T \subset Y$ if $Q \in D_\QCoh(Y)$ is supported on $T$, then $a(Q)$ is supported on $f^{-1}(T)$, \item for every open $V \subset Y$ and any $K \in D_\QCoh(\mathcal{O}_Y)$ the map (\ref{equation-sheafy}) is an isomorphism, and \end{enumerate}


duality-lemma-compare-with-pullback-flat-proper-noetherian

Let $f : X \to Y$ be a perfect proper morphism of Noetherian schemes. The map (\ref{equation-compare-with-pullback}) is an isomorphism for every object $K$ of $D_\QCoh(\mathcal{O}_Y)$.


duality-lemma-compute-for-effective-Cartier

As above, let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor. There is a canonical isomorphism $R\SheafHom(\mathcal{O}_D, \mathcal{O}_X) = \mathcal{N}[-1]$ in $D(\mathcal{O}_D)$.


duality-lemma-sheaf-with-exact-support-effective-Cartier

As above, let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor. Then (\ref{equation-map-effective-Cartier}) combined with Lemma \ref{lemma-compute-for-effective-Cartier} defines an isomorphism $$ Li^*K \otimes_{\mathcal{O}_D}^\mathbf{L} \mathcal{N}[-1] \longrightarrow R\SheafHom(\mathcal{O}_D, K) $$ functorial in $K$ in $D(\mathcal{O}_X)$.


duality-lemma-upper-shriek-P1

Let $Y$ be a Noetherian scheme. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_Y$-module of rank $n + 1$ with determinant $\mathcal{L} = \wedge^{n + 1}(\mathcal{E})$. Let $f : X = \mathbf{P}(\mathcal{E}) \to Y$ be the projection. Let $a$ be the right adjoint for $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ of Lemma \ref{lemma-twisted-inverse-image}. Then there is an isomorphism $$ c : f^*\mathcal{L}(-n - 1)[n] \longrightarrow a(\mathcal{O}_Y) $$ In particular, if $\mathcal{E} = \mathcal{O}_Y^{\oplus n + 1}$, then $X = \mathbf{P}^n_Y$ and we obtain $a(\mathcal{O}_Y) = \mathcal{O}_X(-n - 1)[n]$.


duality-lemma-ext

Let $Y$ be a ringed space. Let $\mathcal{I} \subset \mathcal{O}_Y$ be a sheaf of ideals. Set $\mathcal{O}_X = \mathcal{O}_Y/\mathcal{I}$ and $\mathcal{N} = \SheafHom_{\mathcal{O}_Y}(\mathcal{I}/\mathcal{I}^2, \mathcal{O}_X)$. There is a canonical isomorphism $c : \mathcal{N} \to \SheafExt^1_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X) $.


duality-lemma-regular-ideal-ext

Let $Y$ be a ringed space. Let $\mathcal{I} \subset \mathcal{O}_Y$ be a sheaf of ideals. Set $\mathcal{O}_X = \mathcal{O}_Y/\mathcal{I}$. If $\mathcal{I}$ is Koszul-regular (Divisors, Definition \ref{divisors-definition-regular-ideal-sheaf}) then composition on $R\SheafHom_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)$ defines isomorphisms $$ \wedge^i(\SheafExt^1_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)) \longrightarrow \SheafExt^i_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X) $$ for all $i$.


duality-lemma-regular-immersion-ext

Let $Y$ be a ringed space. Let $\mathcal{I} \subset \mathcal{O}_Y$ be a sheaf of ideals. Set $\mathcal{O}_X = \mathcal{O}_Y/\mathcal{I}$ and $\mathcal{N} = \SheafHom_{\mathcal{O}_Y}(\mathcal{I}/\mathcal{I}^2, \mathcal{O}_X)$. If $\mathcal{I}$ is Koszul-regular (Divisors, Definition \ref{divisors-definition-regular-ideal-sheaf}) then $$ R\SheafHom_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_Y) = \wedge^r \mathcal{N}[r] $$ where $r : Y \to \{1, 2, 3, \ldots \}$ sends $y$ to the minimal number of generators of $\mathcal{I}$ needed in a neighbourhood of $y$.


duality-lemma-regular-immersion

Let $Y$ be a quasi-compact and quasi-separated scheme. Let $i : X \to Y$ be a Koszul-regular closed immersion. Let $a$ be the right adjoint of $Ri_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ of Lemma \ref{lemma-twisted-inverse-image}. Then there is an isomorphism $$ \wedge^r\mathcal{N}[-r] \longrightarrow a(\mathcal{O}_Y) $$ where $\mathcal{N} = \SheafHom_{\mathcal{O}_X}(\mathcal{C}_{X/Y}, \mathcal{O}_X)$ is the normal sheaf of $i$ (Morphisms, Section \ref{morphisms-section-conormal-sheaf}) and $r$ is its rank viewed as a locally constant function on $X$.


duality-lemma-smooth-proper

Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a smooth proper morphism of relative dimension $d$. Let $a$ be the right adjoint of $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_S)$ as in Lemma \ref{lemma-twisted-inverse-image}. Then there is an isomorphism $$ \wedge^d \Omega_{X/S}[d] \longrightarrow a(\mathcal{O}_S) $$ in $D(\mathcal{O}_X)$.


duality-lemma-shriek-well-defined

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. The functor $f^!$ is, up to canonical isomorphism, independent of the choice of the compactification.


duality-lemma-upper-shriek-composition

In Situation \ref{situation-shriek} let $f : X \to Y$ and $g : Y \to Z$ be composable morphisms of $\textit{FTS}_S$. Then there is a canonical isomorphism $(g \circ f)^! \to f^! \circ g^!$.


duality-lemma-map-pullback-to-shriek-well-defined

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. There are canonical maps $$ \mu_{f, K} : Lf^*K \otimes_{\mathcal{O}_X}^\mathbf{L} f^!\mathcal{O}_Y \longrightarrow f^!K $$ functorial in $K$ in $D^+_\QCoh(\mathcal{O}_Y)$. If $g : Y \to Z$ is another morphism of $\textit{FTS}_S$, then the diagram $$ \xymatrix{ Lf^*(Lg^*K \otimes_{\mathcal{O}_Y}^\mathbf{L} g^!\mathcal{O}_Z) \otimes_{\mathcal{O}_X}^\mathbf{L} f^!\mathcal{O}_Y \ar@{=}[d] \ar[r]_-{\mu_f} & f^!(Lg^*K \otimes_{\mathcal{O}_Y}^\mathbf{L} g^!\mathcal{O}_Z) \ar[r]_-{f^!\mu_g} & f^!g^!K \ar@{=}[d] \\ Lf^*Lg^*K \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^* g^!\mathcal{O}_Z \otimes_{\mathcal{O}_X}^\mathbf{L} f^!\mathcal{O}_Y \ar[r]^-{\mu_f} & Lf^*Lg^*K \otimes_{\mathcal{O}_X}^\mathbf{L} f^!g^!\mathcal{O}_Z \ar[r]^-{\mu_{g \circ f}} & f^!g^!K } $$ commutes for all $K \in D^+_\QCoh(\mathcal{O}_Z)$.


duality-lemma-shriek-open-immersion

In Situation \ref{situation-shriek} let $Y$ be an object of $\textit{FTS}_S$ and let $j : X \to Y$ be an open immersion. Then there is a canonical isomorphism $j^! = j^*$ of functors.


duality-lemma-shriek-affine-line

In Situation \ref{situation-shriek} let $Y$ be an object of $\textit{FTS}_S$ and let $f : X = \mathbf{A}^1_Y \to Y$ be the projection. Then there is a (noncanonical) isomorphism $f^!(-) \cong Lf^*(-) [1]$ of functors.


duality-lemma-shriek-closed-immersion

In Situation \ref{situation-shriek} let $Y$ be an object of $\textit{FTS}_S$ and let $i : X \to Y$ be a closed immersion. Then there is a canonical isomorphism $i^!(-) = R\SheafHom(\mathcal{O}_X, -)$ of functors.


duality-lemma-shriek-coherent

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. Then $f^!$ maps $D_{\textit{Coh}}^+(\mathcal{O}_Y)$ into $D_{\textit{Coh}}^+(\mathcal{O}_X)$.


duality-lemma-shriek-dualizing

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. If $K$ is a dualizing complex for $Y$, then $f^!K$ is a dualizing complex for $X$.


duality-lemma-shriek-via-duality

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. Let $K$ be a dualizing complex on $Y$. Set $D_Y(M) = R\SheafHom_{\mathcal{O}_Y}(M, K)$ for $M \in D_{\textit{Coh}}(\mathcal{O}_Y)$ and $D_X(E) = R\SheafHom_{\mathcal{O}_X}(E, f^!K)$ for $E \in D_{\textit{Coh}}(\mathcal{O}_X)$. Then there is a canonical isomorphism $$ f^!M \longrightarrow D_X(Lf^*D_Y(M)) $$ for $M \in D_{\textit{Coh}}^+(\mathcal{O}_Y)$.


duality-lemma-perfect-comparison-shriek

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. Assume $f$ is perfect (e.g., flat). Then \begin{enumerate} \item[(a)] $f^!$ maps $D_{\textit{Coh}}^b(\mathcal{O}_Y)$ into $D_{\textit{Coh}}^b(\mathcal{O}_X)$, \item[(b)] the map $\mu_{f,  K} : Lf^*K \otimes_{\mathcal{O}_X}^\mathbf{L} f^!\mathcal{O}_Y \to f^!K$ of Lemma \ref{lemma-map-pullback-to-shriek-well-defined} is an isomorphism for all $K \in D_\QCoh^+(\mathcal{O}_Y)$. \end{enumerate}


duality-lemma-flat-shriek-relatively-perfect

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. If $f$ is flat, then $f^!\mathcal{O}_Y$ is a $Y$-perfect object of $D(\mathcal{O}_X)$ and $\mathcal{O}_X \to R\SheafHom_{\mathcal{O}_X}(f^!\mathcal{O}_Y, f^!\mathcal{O}_Y)$ is an isomorphism.


duality-lemma-lci-shriek

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. Assume $f : X \to Y$ is a local complete intersection morphism. Then \begin{enumerate} \item $f^!\mathcal{O}_Y$ is an invertible object of $D(\mathcal{O}_X)$, and \item $f^!$ maps perfect complexes to perfect complexes. \end{enumerate}


duality-lemma-base-change-shriek-flat

In Situation \ref{situation-shriek} let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ be a cartesian diagram of $\textit{FTS}_S$ with $g$ flat. Then there is an isomorphism $L(g')^* \circ f^! \to (f')^! \circ Lg^*$ on $D_\QCoh^+(\mathcal{O}_Y)$.


duality-lemma-shriek-etale

In Situation \ref{situation-shriek} let $f : X \to Y$ be an \'etale morphism of $\textit{FTS}_S$. Then $f^! \cong f^*$ as functors on $D^+_\QCoh(\mathcal{O}_Y)$.


duality-lemma-base-change-locally

In Situation \ref{situation-shriek} let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ be a cartesian diagram of $\textit{FTS}_S$. Let $E \in D^+_\QCoh(\mathcal{O}_Y)$ be an object such that $Lg^*E$ is in $D^+(\mathcal{O}_Y)$. If $f$ is flat, then $L(g')^*f^!E$ and $(f')^!Lg^*E$ restrict to isomorphic objects of $D(\mathcal{O}_{U'})$ for $U' \subset X'$ affine open mapping into affine opens of $Y$, $Y'$, and $X$.


duality-lemma-relative-dualizing-fibres

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. Assume $f$ is flat. Set $\omega_{X/Y}^\bullet = f^!\mathcal{O}_Y$ in $D^b_{\textit{Coh}}(X)$. Let $y \in Y$ and $h : X_y \to X$ the projection. Then $Lh^*\omega_{X/Y}^\bullet$ is a dualizing complex on $X_y$.


duality-lemma-good-dualizing-unique

In Situation \ref{situation-dualizing} let $X$ be a scheme of finite type over $S$ and let $\mathcal{U}$ be a finite open covering of $X$ by schemes separated over $S$. If there exists a dualizing complex normalized relative to $\omega_S^\bullet$ and $\mathcal{U}$, then it is unique up to unique isomorphism.


duality-lemma-good-over-both

Let $(S, \omega_S^\bullet)$ be as in Situation \ref{situation-dualizing}. Let $f : X \to Y$ be a morphism of finite type schemes over $S$. Let $\omega_X^\bullet$ and $\omega_Y^\bullet$ be dualizing complexes normalized relative to $\omega_S^\bullet$. Then $\omega_X^\bullet$ is a dualizing complex normalized relative to $\omega_Y^\bullet$.


duality-lemma-proper-map-good-dualizing-complex

Let $(S, \omega_S^\bullet)$ be as in Situation \ref{situation-dualizing}. Let $f : X \to Y$ be a proper morphism of schemes of finite type over $S$. Let $\omega_X^\bullet$ and $\omega_Y^\bullet$ be dualizing complexes normalized relative to $\omega_S^\bullet$. Let $a$ be the right adjoint of Lemma \ref{lemma-twisted-inverse-image} for $f$. Then there is a canonical isomorphism $a(\omega_Y^\bullet) = \omega_X^\bullet$.


duality-lemma-duality-bootstrap

Let $(S, \omega_S^\bullet)$ be as in Situation \ref{situation-dualizing}. With $f^!_{new}$ and $\omega_X^\bullet$ defined for all (morphisms of) schemes of finite type over $S$ as above: \begin{enumerate} \item the functors $f^!_{new}$ and the arrows $(g \circ f)^!_{new} \to f^!_{new} \circ g^!_{new}$ turn $D_{\textit{Coh}}^+$ into a pseudo functor from the category of schemes of finite type over $S$ into the $2$-category of categories, \item $\omega_X^\bullet = (X \to S)^!_{new} \omega_S^\bullet$, \item the functor $D_X$ defines an involution of $D_{\textit{Coh}}(\mathcal{O}_X)$ switching $D_{\textit{Coh}}^+(\mathcal{O}_X)$ and $D_{\textit{Coh}}^-(\mathcal{O}_X)$ and fixing $D_{\textit{Coh}}^b(\mathcal{O}_X)$, \item $\omega_X^\bullet = f^!_{new}\omega_Y^\bullet$ for $f : X \to Y$ a morphism of finite type schemes over $S$, \item $f^!_{new}M = D_X(Lf^*D_Y(M))$ for $M \in D_{\textit{Coh}}^+(\mathcal{O}_Y)$, and \item if in addition $f$ is proper, then $f^!_{new}$ is isomorphic to the restriction of the right adjoint of $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ to $D_{\textit{Coh}}^+(\mathcal{O}_Y)$ and there is a canonical isomorphism $$ Rf_*R\SheafHom_{\mathcal{O}_X}(K, f^!_{new}M) \to R\SheafHom_{\mathcal{O}_Y}(Rf_*K, M) $$ for $K \in D^-_{\textit{Coh}}(\mathcal{O}_X)$ and $M \in D_{\textit{Coh}}^+(\mathcal{O}_Y)$, and $$ Rf_*R\SheafHom_{\mathcal{O}_X}(K, \omega_X^\bullet) = R\SheafHom_{\mathcal{O}_Y}(Rf_*K, \omega_Y^\bullet) $$ for $K \in D^-_{\textit{Coh}}(\mathcal{O}_X)$ and \end{enumerate} If $X$ is separated over $S$, then $\omega_X^\bullet$ is canonically isomorphic to $(X \to S)^!\omega_S^\bullet$ and if $f$ is a morphism between schemes separated over $S$, then there is a canonical isomorphism\footnote{We haven't checked that these are compatible with the isomorphisms $(g \circ f)^! \to f^! \circ g^!$ and $(g \circ f)^!_{new} \to f^!_{new} \circ g^!_{new}$. We will do this here if we need this later.} $f_{new}^!K = f^!K$ for $K$ in $D_{\textit{Coh}}^+$.


duality-lemma-good-dualizing-normalized

Let $S$ be a Noetherian scheme and let $\omega_S^\bullet$ be a dualizing complex. Let $X$ be a scheme of finite type over $S$ and let $\omega_X^\bullet$ be the dualizing complex normalized relative to $\omega_S^\bullet$. If $x \in X$ is a closed point lying over a closed point $s$ of $S$, then $\omega_{X, x}^\bullet$ is a normalized dualizing complex over $\mathcal{O}_{X, x}$ provided that $\omega_{S, s}^\bullet$ is a normalized dualizing complex over $\mathcal{O}_{S, s}$.


duality-lemma-good-dualizing-dimension-function

Let $S$ be a Noetherian scheme and let $\omega_S^\bullet$ be a dualizing complex. Let $f : X \to S$ be of finite type and let $\omega_X^\bullet$ be the dualizing complex normalized relative to $\omega_S^\bullet$. For all $x \in X$ we have $$ \delta_X(x) - \delta_S(f(x)) = \text{trdeg}_{\kappa(f(x))}(\kappa(x)) $$ where $\delta_S$, resp.\ $\delta_X$ is the dimension function of $\omega_S^\bullet$, resp.\ $\omega_X^\bullet$, see Lemma \ref{lemma-dimension-function-scheme}.


duality-lemma-shriek

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. Let $x \in X$ with image $y \in Y$. Then $$ H^i(f^!\mathcal{O}_Y)_x \not = 0 \Rightarrow - \dim_x(X_y) \leq i. $$


duality-lemma-flat-shriek

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. Let $x \in X$ with image $y \in Y$. If $f$ is flat, then $$ H^i(f^!\mathcal{O}_Y)_x \not = 0 \Rightarrow - \dim_x(X_y) \leq i \leq 0. $$ In fact, if all fibres of $f$ have dimension $\leq d$, then $f^!\mathcal{O}_Y$ has tor-amplitude in $[-d, 0]$ as an object of $D(X, f^{-1}\mathcal{O}_Y)$.