properties-lemma-ample-on-product

Let $S$ be a quasi-separated scheme. Let $X$, $Y$ be schemes over $S$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module and let $\mathcal{N}$ be an ample invertible $\mathcal{O}_Y$-module. Then $\mathcal{M} = \text{pr}_1^*\mathcal{L} \otimes_{\mathcal{O}_{X \times_S Y}} \text{pr}_2^*\mathcal{N}$ is an ample invertible sheaf on $X \times_S Y$.


properties-lemma-quasi-affine-O-ample

Let $X$ be a scheme. Then $X$ is quasi-affine if and only if $\mathcal{O}_X$ is ample.


properties-lemma-quasi-affine-locally-closed

Let $X$ be a quasi-affine scheme. For any quasi-compact immersion $i : X' \to X$ the scheme $X'$ is quasi-affine.


properties-lemma-characterize-affine

Let $X$ be a scheme. Suppose that there exist finitely many elements $f_1, \ldots, f_n \in \Gamma(X, \mathcal{O}_X)$ such that \begin{enumerate} \item each $X_{f_i}$ is an affine open of $X$, and \item the ideal generated by $f_1, \ldots, f_n$ in $\Gamma(X, \mathcal{O}_X)$ is equal to the unit ideal. \end{enumerate} Then $X$ is affine.


properties-lemma-ample-gcd-is-one

In Situation \ref{situation-ample}. The canonical morphism $f : X \to Y$ maps $X$ into the open subscheme $W = W_1 \subset Y$ where $\mathcal{O}_Y(1)$ is invertible and where all multiplication maps $\mathcal{O}_Y(n) \otimes_{\mathcal{O}_Y} \mathcal{O}_Y(m) \to \mathcal{O}_Y(n + m)$ are isomorphisms (see Constructions, Lemma \ref{constructions-lemma-where-invertible}). Moreover, the maps $f^*\mathcal{O}_Y(n) \to \mathcal{L}^{\otimes n}$ are all isomorphisms.


properties-lemma-ample-quasi-coherent

In Situation \ref{situation-ample}. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Set $M = \Gamma_*(X, \mathcal{L}, \mathcal{F})$ as a graded $S$-module. There are isomorphisms $$ f^*\widetilde{M} \longrightarrow \mathcal{F} $$ functorial in $\mathcal{F}$ such that $M_0 \to \Gamma(\text{Proj}(S), \widetilde{M}) \to \Gamma(X, \mathcal{F})$ is the identity map.


properties-lemma-proj-quasi-coherent

Let $S$ be a graded ring such that $X = \text{Proj}(S)$ is quasi-compact. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Set $M = \bigoplus_{n \in \mathbf{Z}} \Gamma(X, \mathcal{F}(n))$ as a graded $S$-module, see Constructions, Section \ref{constructions-section-invertible-on-proj}. The map $$ \widetilde{M} \longrightarrow \mathcal{F} $$ of Constructions, Lemma \ref{constructions-lemma-comparison-proj-quasi-coherent} is an isomorphism. If $X$ is covered by standard opens $D_+(f)$ where $f$ has degree $1$, then the induced maps $M_n \to \Gamma(X, \mathcal{F}(n))$ are the identity maps.


properties-lemma-maximal-points-affine

Let $X$ be a quasi-separated scheme. Let $Z_1, \ldots, Z_n$ be pairwise distinct irreducible components of $X$, see Topology, Section \ref{topology-section-irreducible-components}. Let $\eta_i \in Z_i$ be their generic points, see Schemes, Lemma \ref{schemes-lemma-scheme-sober}. There exist affine open neighbourhoods $\eta_i \in U_i$ such that $U_i \cap U_j = \emptyset$ for all $i \not = j$. In particular, $U = U_1 \cup \ldots \cup U_n$ is an affine open containing all of the points $\eta_1, \ldots, \eta_n$.


properties-lemma-quasi-compact-dense-open-separated

Let $X$ be a quasi-compact scheme. There exists a dense open $V \subset X$ which is separated.


properties-lemma-point-and-maximal-points-affine

Let $X$ be a quasi-separated scheme. Let $Z_1, \ldots, Z_n$ be pairwise distinct irreducible components of $X$. Let $\eta_i \in Z_i$ be their generic points. Let $x \in X$ be arbitrary. There exists an affine open $U \subset X$ containing $x$ and all the $\eta_i$.


properties-lemma-ample-finite-set-in-affine

Let $X$ be a scheme. Assume either \begin{enumerate} \item The scheme $X$ is quasi-affine. \item The scheme $X$ is isomorphic to a locally closed subscheme of an affine scheme. \item There exists an ample invertible sheaf on $X$. \item The scheme $X$ is isomorphic to a locally closed subscheme of $\text{Proj}(S)$ for some graded ring $S$. \end{enumerate} Then for any finite subset $E \subset X$ there exists an affine open $U \subset X$ with $E \subset U$.


properties-lemma-ample-finite-set-in-principal-affine

Let $X$ be a scheme. Let $\mathcal{L}$ be an ample invertible sheaf on $X$. Let $$ E \subset W \subset X $$ with $E$ finite and $W$ open in $X$. Then there exists an $n > 0$ and a section $s \in \Gamma(X, \mathcal{L}^{\otimes n})$ such that $X_s$ is affine and $E \subset X_s \subset W$.


properties-lemma-ring-affine-open-injective-local-ring

Let $X$ be a scheme and $x \in X$ a point. There exists an affine open neighbourhood $U \subset X$ of $x$ such that the canonical map $\mathcal{O}_X(U) \to \mathcal{O}_{X, x}$ is injective in each of the following cases: \begin{enumerate} \item $X$ is integral, \item $X$ is locally Noetherian, \item $X$ is reduced and has a finite number of irreducible components. \end{enumerate}


properties-proposition-coherator

Let $X$ be a scheme. \begin{enumerate} \item The category $\QCoh(\mathcal{O}_X)$ is a Grothendieck abelian category. Consequently, $\QCoh(\mathcal{O}_X)$ has enough injectives and all limits. \item The inclusion functor $\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)$ has a right adjoint\footnote{This functor is sometimes called the {\it coherator}.} $$ Q : \textit{Mod}(\mathcal{O}_X) \longrightarrow \QCoh(\mathcal{O}_X) $$ such that for every quasi-coherent sheaf $\mathcal{F}$ the adjunction mapping $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism. \end{enumerate}


properties-proposition-characterize-ample

Let $X$ be a quasi-compact scheme. Let $\mathcal{L}$ be an invertible sheaf on $X$. Set $S = \Gamma_*(X, \mathcal{L})$. The following are equivalent: \begin{enumerate} \item \label{item-ample} $\mathcal{L}$ is ample, \item \label{item-immersion} the open sets $X_s$, with $s \in S_{+}$ homogeneous, cover $X$ and the associated morphism $X \to \text{Proj}(S)$ is an open immersion, \item \label{item-s-basis} the open sets $X_s$, with $s \in S_{+}$ homogeneous, form a basis for the topology of $X$, \item \label{item-s-affine-basis} the open sets $X_s$, with $s \in S_{+}$ homogeneous, which are affine form a basis for the topology of $X$, \item \label{item-qc-gg} for every quasi-coherent sheaf $\mathcal{F}$ on $X$ the sum of the images of the canonical maps $$ \Gamma(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) \otimes_{\mathbf{Z}} \mathcal{L}^{\otimes -n} \longrightarrow \mathcal{F} $$ with $n \geq 1$ equals $\mathcal{F}$, \item \label{item-qc-i-gg} same property as (\ref{item-qc-gg}) with $\mathcal{F}$ ranging over all quasi-coherent sheaves of ideals, \item \label{item-c-gg} $X$ is quasi-separated and for every quasi-coherent sheaf $\mathcal{F}$ of finite type on $X$ there exists an integer $n_0$ such that $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}$ is globally generated for all $n \geq n_0$, \item \label{item-c-q} $X$ is quasi-separated and for every quasi-coherent sheaf $\mathcal{F}$ of finite type on $X$ there exist integers $n > 0$, $k \geq 0$ such that $\mathcal{F}$ is a quotient of a direct sum of $k$ copies of $\mathcal{L}^{\otimes - n}$, and \item \label{item-c-i-q} same as in (\ref{item-c-q}) with $\mathcal{F}$ ranging over all sheaves of ideals of finite type on $X$. \end{enumerate}


criteria-theorem-bootstrap

Let $S$ be a scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\Sch/S)_{fppf}$. If \begin{enumerate} \item $\mathcal{X}$ is representable by an algebraic space, and \item $F$ is representable by algebraic spaces, surjective, flat and locally of finite presentation, \end{enumerate} then $\mathcal{Y}$ is an algebraic stack.


criteria-theorem-flat-groupoid-gives-algebraic-stack

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Then the quotient stack $[U/R]$ is an algebraic stack over $S$.


criteria-lemma-etale-permanence

Let $\mathcal{X} \to \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $\mathcal{X} \to \mathcal{Z}$ and $\mathcal{Y} \to \mathcal{Z}$ are representable by algebraic spaces and \'etale so is $\mathcal{X} \to \mathcal{Y}$.


criteria-lemma-flat-finite-presentation-surjective-diagonal

Let $S$ be a scheme. Let $u : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of stacks in groupoids over $(\Sch/S)_{fppf}$. If \begin{enumerate} \item $\mathcal{U}$ is representable by an algebraic space, and \item $u$ is representable by algebraic spaces, surjective, flat and locally of finite presentation, \end{enumerate} then $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ representable by algebraic spaces.


criteria-lemma-second-diagonal

Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. The following are equivalent \begin{enumerate} \item $\Delta_\Delta : \mathcal{X} \to \mathcal{X} \times_{\mathcal{X} \times \mathcal{X}} \mathcal{X}$ is representable by algebraic spaces, \item for every $1$-morphism $\mathcal{V} \to \mathcal{X} \times \mathcal{X}$ with $\mathcal{V}$ representable (by a scheme) the fibre product $\mathcal{Y} = \mathcal{X} \times_{\Delta, \mathcal{X} \times \mathcal{X}} \mathcal{V}$ has diagonal representable by algebraic spaces. \end{enumerate}


criteria-lemma-base-change-limit-preserving

Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $p : \mathcal{X} \to \mathcal{Y}$ is limit preserving on objects, then so is the base change $p' : \mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Z}$ of $p$ by $q$.


criteria-lemma-composition-limit-preserving

Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $p$ and $q$ are limit preserving on objects, then so is the composition $q \circ p$.


criteria-lemma-representable-by-spaces-limit-preserving

Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $p$ is representable by algebraic spaces, then the following are equivalent: \begin{enumerate} \item $p$ is limit preserving on objects, and \item $p$ is locally of finite presentation (see Algebraic Stacks, Definition \ref{algebraic-definition-relative-representable-property}). \end{enumerate}


criteria-lemma-open-immersion-limit-preserving

Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Assume $p$ is representable by algebraic spaces and an open immersion. Then $p$ is limit preserving on objects.


criteria-lemma-check-representable-limit-preserving

Let $S$ be a scheme. Let $\kappa = \text{size}(T)$ for some $T \in \Ob((\Sch/S)_{fppf})$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$ such that \begin{enumerate} \item $\mathcal{Y} \to (\Sch/S)_{fppf}$ is limit preserving on objects, \item for an affine scheme $V$ locally of finite presentation over $S$ and $y \in \Ob(\mathcal{Y}_V)$ the fibre product $(\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}$ is representable by an algebraic space of size $\leq \kappa$\footnote{The condition on size can be dropped by those ignoring set theoretic issues.}, \item $\mathcal{X}$ and $\mathcal{Y}$ are stacks for the Zariski topology. \end{enumerate} Then $f$ is representable by algebraic spaces.


criteria-lemma-check-property-limit-preserving

Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces as in Algebraic Stacks, Definition \ref{algebraic-definition-relative-representable-property}. If \begin{enumerate} \item $f$ is representable by algebraic spaces, \item $\mathcal{Y} \to (\Sch/S)_{fppf}$ is limit preserving on objects, \item for an affine scheme $V$ locally of finite presentation over $S$ and $y \in \mathcal{Y}_V$ the resulting morphism of algebraic spaces $f_y : F_y \to V$, see Algebraic Stacks, Equation (\ref{algebraic-equation-representable-by-algebraic-spaces}), has property $\mathcal{P}$. \end{enumerate} Then $f$ has property $\mathcal{P}$.


criteria-lemma-base-change-formally-smooth

Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $p : \mathcal{X} \to \mathcal{Y}$ is formally smooth on objects, then so is the base change $p' : \mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Z}$ of $p$ by $q$.


criteria-lemma-composition-formally-smooth

Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $p$ and $q$ are formally smooth on objects, then so is the composition $q \circ p$.


criteria-lemma-representable-by-spaces-formally-smooth

Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $p$ is representable by algebraic spaces, then the following are equivalent: \begin{enumerate} \item $p$ is formally smooth on objects, and \item $p$ is formally smooth (see Algebraic Stacks, Definition \ref{algebraic-definition-relative-representable-property}). \end{enumerate}


criteria-lemma-base-change-surjective

Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $p : \mathcal{X} \to \mathcal{Y}$ is surjective on objects, then so is the base change $p' : \mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Z}$ of $p$ by $q$.


criteria-lemma-composition-surjective

Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $p$ and $q$ are surjective on objects, then so is the composition $q \circ p$.


criteria-lemma-representable-by-spaces-surjective

Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $p$ is representable by algebraic spaces, then the following are equivalent: \begin{enumerate} \item $p$ is surjective on objects, and \item $p$ is surjective (see Algebraic Stacks, Definition \ref{algebraic-definition-relative-representable-property}). \end{enumerate}


criteria-lemma-diagonals-and-algebraic-morphisms

Let $S$ be a scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\Sch/S)_{fppf}$. If $F$ is algebraic and $\Delta : \mathcal{Y} \to \mathcal{Y} \times \mathcal{Y}$ is representable by algebraic spaces, then $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces.


criteria-lemma-surjection-space-of-sections

Let $Z \to U$ be a finite morphism of schemes. Let $W$ be an algebraic space and let $W \to Z$ be a surjective \'etale morphism. Then there exists a surjective \'etale morphism $U' \to U$ and a section $$ \sigma : Z_{U'} \to W_{U'} $$ of the morphism $W_{U'} \to Z_{U'}$.


criteria-lemma-space-of-sections

Let $Z \to U$ be a finite locally free morphism of schemes. Let $W$ be an algebraic space and let $W \to Z$ be an \'etale morphism. Then the functor $$ F : (\Sch/U)_{fppf}^{opp} \longrightarrow \textit{Sets}, $$ defined by the rule $$ U' \longmapsto F(U') = \{\sigma : Z_{U'} \to W_{U'}\text{ section of }W_{U'} \to Z_{U'}\} $$ is an algebraic space and the morphism $F \to U$ is \'etale.


criteria-lemma-hom-functor-sheaf

Let $S$ be a scheme. Let $Z \to B$ and $X \to B$ be morphisms of algebraic spaces over $S$. Then \begin{enumerate} \item $\mathit{Mor}_B(Z, X)$ is a sheaf on $(\Sch/S)_{fppf}$. \item If $T$ is an algebraic space over $S$, then there is a canonical bijection $$ \Mor_{\Sh((\Sch/S)_{fppf})}(T, \mathit{Mor}_B(Z, X)) = \{(a, b)\text{ as in }(\ref{equation-hom})\} $$ \end{enumerate}


criteria-lemma-base-change-hom-functor

Let $S$ be a scheme. Let $Z \to B$, $X \to B$, and $B' \to B$ be morphisms of algebraic spaces over $S$. Set $Z' = B' \times_B Z$ and $X' = B' \times_B X$. Then $$ \mathit{Mor}_{B'}(Z', X') = B' \times_B \mathit{Mor}_B(Z, X) $$ in $\Sh((\Sch/S)_{fppf})$.


criteria-lemma-etale-covering-hom-functor

Let $S$ be a scheme. Let $Z \to B$ and $X' \to X \to B$ be morphisms of algebraic spaces over $S$. Assume \begin{enumerate} \item $X' \to X$ is \'etale, and \item $Z \to B$ is finite locally free. \end{enumerate} Then $\mathit{Mor}_B(Z, X') \to \mathit{Mor}_B(Z, X)$ is representable by algebraic spaces and \'etale. If $X' \to X$ is also surjective, then $\mathit{Mor}_B(Z, X') \to \mathit{Mor}_B(Z, X)$ is surjective.


criteria-lemma-restriction-of-scalars-sheaf

Let $S$ be a scheme. Let $X \to Z \to B$ be morphisms of algebraic spaces over $S$. Then \begin{enumerate} \item $\text{Res}_{Z/B}(X)$ is a sheaf on $(\Sch/S)_{fppf}$. \item If $T$ is an algebraic space over $S$, then there is a canonical bijection $$ \Mor_{\Sh((\Sch/S)_{fppf})}(T, \text{Res}_{Z/B}(X)) = \{(a, b)\text{ as in }(\ref{equation-pairs})\} $$ \end{enumerate}


criteria-lemma-fibre-diagram

Let $S$ be a scheme. Let $X \to Z \to B$ be morphisms of algebraic spaces over $S$. The following diagram $$ \xymatrix{ \mathit{Mor}_B(Z, X) \ar[r] & \mathit{Mor}_B(Z, Z) \\ \text{Res}_{Z/B}(X) \ar[r] \ar[u] & B \ar[u]_{\text{id}_Z} } $$ is a cartesian diagram of sheaves on $(\Sch/S)_{fppf}$.


criteria-lemma-map-hilbert

Consider a $2$-commutative diagram $$ \xymatrix{ \mathcal{X}' \ar[r]_G \ar[d]_{F'} & \mathcal{X} \ar[d]^F \\ \mathcal{Y}' \ar[r]^H & \mathcal{Y} } $$ of stacks in groupoids over $(\Sch/S)_{fppf}$ with a given $2$-isomorphism $\gamma : H \circ F' \to F \circ G$. In this situation we obtain a canonical $1$-morphism $\mathcal{H}_d(\mathcal{X}'/\mathcal{Y}') \to \mathcal{H}_d(\mathcal{X}/\mathcal{Y})$. This morphism is compatible with the forgetful $1$-morphisms of Examples of Stacks, Equation (\ref{examples-stacks-equation-diagram-hilbert-d-stack}).


criteria-lemma-cartesian-map-hilbert

In the situation of Lemma \ref{lemma-map-hilbert} assume that the given square is $2$-cartesian. Then the diagram $$ \xymatrix{ \mathcal{H}_d(\mathcal{X}'/\mathcal{Y}') \ar[r] \ar[d] & \mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \ar[d] \\ \mathcal{Y}' \ar[r] & \mathcal{Y} } $$ is $2$-cartesian.


criteria-lemma-etale-covering-hilbert

In the situation of Lemma \ref{lemma-map-hilbert} assume \begin{enumerate} \item $\mathcal{Y}' = \mathcal{Y}$ and $H = \text{id}_\mathcal{Y}$, \item $G$ is representable by algebraic spaces and \'etale. \end{enumerate} Then $\mathcal{H}_d(\mathcal{X}'/\mathcal{Y}) \to \mathcal{H}_d(\mathcal{X}/\mathcal{Y})$ is representable by algebraic spaces and \'etale. If $G$ is also surjective, then $\mathcal{H}_d(\mathcal{X}'/\mathcal{Y}) \to \mathcal{H}_d(\mathcal{X}/\mathcal{Y})$ is surjective.


criteria-lemma-relative-hilbert

Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\Sch/S)_{fppf}$. Assume that $\Delta : \mathcal{Y} \to \mathcal{Y} \times \mathcal{Y}$ is representable by algebraic spaces. Then $$ \mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \longrightarrow \mathcal{H}_d(\mathcal{X}) \times \mathcal{Y} $$ see Examples of Stacks, Equation (\ref{examples-stacks-equation-diagram-hilbert-d-stack}) is representable by algebraic spaces.


criteria-lemma-representable-on-top

Let $F : \mathcal{X} \to \mathcal{Y}$ and $G : \mathcal{X}' \to \mathcal{X}$ be $1$-morphisms of stacks in groupoids over $(\Sch/S)_{fppf}$. If $G$ is representable by algebraic spaces, then the $1$-morphism $$ \mathcal{H}_d(\mathcal{X}'/\mathcal{Y}) \longrightarrow \mathcal{H}_d(\mathcal{X}/\mathcal{Y}) $$ is representable by algebraic spaces.


criteria-lemma-limit-preserving

Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\Sch/S)_{fppf}$. Assume $F$ is representable by algebraic spaces and locally of finite presentation. Then $$ p : \mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y} $$ is limit preserving on objects.


criteria-lemma-represent-FAd

The functor in groupoids $FA_d$ defined in (\ref{equation-FAd}) is isomorphic (!) to the functor in groupoids which associates to a scheme $T$ the category with \begin{enumerate} \item set of objects is $X(T)$, \item set of morphisms is $G(T) \times X(T)$, \item $s : G(T) \times X(T) \to X(T)$ is the projection map, \item $t : G(T) \times X(T) \to X(T)$ is $a(T)$, and \item composition $G(T) \times X(T) \times_{s, X(T), t} G(T) \times X(T) \to G(T) \times X(T)$ is given by $((g, m), (g', m')) \mapsto (gg', m')$. \end{enumerate}


criteria-lemma-hilbert-stack-of-space

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $\mathcal{H}_d(X)$ is an algebraic stack.


criteria-lemma-hilbert-stack-relative-space

Let $S$ be a scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\Sch/S)_{fppf}$ such that \begin{enumerate} \item $\mathcal{X}$ is representable by an algebraic space, and \item $F$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. \end{enumerate} Then $\mathcal{H}_d(\mathcal{X}/\mathcal{Y})$ is an algebraic stack.


criteria-lemma-lci-locus-stack-in-groupoids

Let $S$ be a scheme. Fix a $1$-morphism $F : \mathcal{X} \longrightarrow \mathcal{Y}$ of stacks in groupoids over $(\Sch/S)_{fppf}$. Assume $F$ is representable by algebraic spaces, flat, and locally of finite presentation. Then $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ is a stack in groupoids and the inclusion functor $$ \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \longrightarrow \mathcal{H}_d(\mathcal{X}/\mathcal{Y}) $$ is representable and an open immersion.


criteria-lemma-lci-unobstructed

Let $U \subset U'$ be a first order thickening of affine schemes. Let $X'$ be an algebraic space flat over $U'$. Set $X = U \times_{U'} X'$. Let $Z \to U$ be finite locally free of degree $d$. Finally, let $f : Z \to X$ be unramified and a local complete intersection morphism. Then there exists a commutative diagram $$ \xymatrix{ (Z \subset Z') \ar[rd] \ar[rr]_{(f, f')} & & (X \subset X') \ar[ld] \\ & (U \subset U') } $$ of algebraic spaces over $U'$ such that $Z' \to U'$ is finite locally free of degree $d$ and $Z = U \times_{U'} Z'$.


criteria-lemma-lci-formally-smooth

Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\Sch/S)_{fppf}$. Assume $F$ is representable by algebraic spaces, flat, and locally of finite presentation. Then $$ p : \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y} $$ is formally smooth on objects.


criteria-lemma-lci-surjective

Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\Sch/S)_{fppf}$. Assume $F$ is representable by algebraic spaces, flat, surjective, and locally of finite presentation. Then $$ \coprod\nolimits_{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \longrightarrow \mathcal{Y} $$ is surjective on objects.


criteria-lemma-flat-quotient-flat-presentation

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Then the morphism $\mathcal{S}_U \to [U/R]$ is flat, locally of finite presentation, and surjective.


criteria-lemma-quotient-algebraic

Let $S$ be a scheme and let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The quotient stack $[U/R]$ is an algebraic stack if and only if there exists a morphism of algebraic spaces $g : U' \to U$ such that \begin{enumerate} \item the composition $U' \times_{g, U, t} R \to R \xrightarrow{s} U$ is a surjection of sheaves, and \item the morphisms $s', t' : R' \to U'$ are flat and locally of finite presentation where $(U', R', s', t', c')$ is the restriction of $(U, R, s, t, c)$ via $g$. \end{enumerate}


criteria-lemma-group-quotient-algebraic

Let $S$ be a scheme and let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Let $X$ be an algebraic space over $B$ and let $a : G \times_B X \to X$ be an action of $G$ on $X$ over $B$. The quotient stack $[X/G]$ is an algebraic stack if and only if there exists a morphism of algebraic spaces $\varphi : X' \to X$ such that \begin{enumerate} \item $G \times_B X' \to X$, $(g, x') \mapsto a(g, \varphi(x'))$ is a surjection of sheaves, and \item the two projections $X'' \to X'$ of the algebraic space $X''$ given by the rule $$ T \longmapsto \{(x'_1, g, x'_2) \in (X' \times_B G \times_B X')(T) \mid \varphi(x'_1) = a(g, \varphi(x'_2))\} $$ are flat and locally of finite presentation. \end{enumerate}


criteria-lemma-BG-algebraic

\begin{slogan} Gerbes are algebraic if and only if the associated groups are flat and locally of finite presentation \end{slogan} Let $S$ be a scheme and let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Endow $B$ with the trivial action of $G$. Then the quotient stack $[B/G]$ is an algebraic stack if and only if $G$ is flat and locally of finite presentation over $B$.


criteria-lemma-stacks-etale

Denote the common underlying category of $\Sch_{fppf}$ and $\Sch_\etale$ by $\Sch_\alpha$ (see Sheaves on Stacks, Section \ref{stacks-sheaves-section-sheaves} and Topologies, Remark \ref{topologies-remark-choice-sites}). Let $S$ be an object of $\Sch_\alpha$. Let $$ p : \mathcal{X} \to \Sch_\alpha/S $$ be a category fibred in groupoids with the following properties: \begin{enumerate} \item $\mathcal{X}$ is a stack in groupoids over $(\Sch/S)_\etale$, \item the diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces\footnote{Here we can either mean sheaves in the \'etale topology whose diagonal is representable and which have an \'etale surjective covering by a scheme or algebraic spaces as defined in Algebraic Spaces, Definition \ref{spaces-definition-algebraic-space}. Namely, by Bootstrap, Lemma \ref{bootstrap-lemma-spaces-etale} there is no difference.}, and \item there exists $U \in \Ob(\Sch_\alpha/S)$ and a $1$-morphism $(\Sch/U)_\etale \to \mathcal{X}$ which is surjective and smooth. \end{enumerate} Then $\mathcal{X}$ is an algebraic stack in the sense of Algebraic Stacks, Definition \ref{algebraic-definition-algebraic-stack}.


criteria-proposition-hom-functor-algebraic-space

Let $S$ be a scheme. Let $Z \to B$ and $X \to B$ be morphisms of algebraic spaces over $S$. If $Z \to B$ is finite locally free then $\mathit{Mor}_B(Z, X)$ is an algebraic space.


criteria-proposition-restriction-of-scalars-algebraic-space

Let $S$ be a scheme. Let $X \to Z \to B$ be morphisms of algebraic spaces over $S$. If $Z \to B$ is finite locally free then $\text{Res}_{Z/B}(X)$ is an algebraic space.


criteria-proposition-finite-hilbert-point

The stack $\mathcal{H}_d$ is equivalent to the quotient stack $[X/G]$ described above. In particular $\mathcal{H}_d$ is an algebraic stack.


quot-theorem-coherent-algebraic

Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation, separated, and flat\footnote{This assumption is not necessary. See Section \ref{section-not-flat}.}. Then $\Cohstack_{X/B}$ is an algebraic stack over $S$.


quot-theorem-coherent-algebraic-general

Let $S$ be a scheme. Let $f : X \to B$ be morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation and separated. Then $\Cohstack_{X/B}$ is an algebraic stack over $S$.


quot-lemma-hom-sheaf

In Situation \ref{situation-hom} the functor $\mathit{Hom}(\mathcal{F}, \mathcal{G})$  satisfies the sheaf property for the fpqc topology.


quot-lemma-extend-hom-to-spaces

In Situation \ref{situation-hom}. Let $T$ be an algebraic space over $S$. We have $$ \Mor_{\Sh((\Sch/S)_{fppf})}(T, \mathit{Hom}(\mathcal{F}, \mathcal{G})) = \{(h, u) \mid h : T \to B, u : \mathcal{F}_T \to \mathcal{G}_T\} $$ where $\mathcal{F}_T, \mathcal{G}_T$ denote the pullbacks of $\mathcal{F}$ and $\mathcal{G}$ to the algebraic space $X \times_{B, h} T$.


quot-lemma-hom-sheaf-in-X

In Situation \ref{situation-hom} let $\{X_i \to X\}_{i \in I}$ be an fppf covering and for each $i, j \in I$ let $\{X_{ijk} \to X_i \times_X X_j\}$ be an fppf covering. Denote $\mathcal{F}_i$, resp.\ $\mathcal{F}_{ijk}$ the pullback of $\mathcal{F}$ to $X_i$, resp.\ $X_{ijk}$. Similarly define $\mathcal{G}_i$ and $\mathcal{G}_{ijk}$. For every scheme $T$ over $B$ the diagram $$ \xymatrix{ \mathit{Hom}(\mathcal{F}, \mathcal{G})(T) \ar[r] & \prod\nolimits_i \mathit{Hom}(\mathcal{F}_i, \mathcal{G}_i)(T) \ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*} & \prod\nolimits_{i, j, k} \mathit{Hom}(\mathcal{F}_{ijk}, \mathcal{G}_{ijk})(T) } $$ presents the first arrow as the equalizer of the other two.


quot-lemma-cohomology-perfect-complex

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $K$ be a pseudo-coherent object of $D(\mathcal{O}_B)$. \begin{enumerate} \item If for all $g : T \to B$ in $(\Sch/B)$ the cohomology sheaf $H^{-1}(Lg^*K)$ is zero, then the functor $$ (\Sch/B)^{opp} \longrightarrow \textit{Sets},\quad (g : T \to B) \longmapsto H^0(T, H^0(Lg^*K)) $$ is an algebraic space affine and of finite presentation over $B$. \item If for all $g : T \to B$ in $(\Sch/B)$ the cohomology sheaves $H^i(Lg^*K)$ are zero for $i < 0$, then $K$ is perfect, $K$ locally has tor amplitude in $[0, b]$, and the functor $$ (\Sch/B)^{opp} \longrightarrow \textit{Sets},\quad (g : T \to B) \longmapsto H^0(T, Lg^*K) $$ is an algebraic space affine and of finite presentation over $B$. \end{enumerate}


quot-lemma-noetherian-hom

In Situation \ref{situation-hom} assume that \begin{enumerate} \item $B$ is a Noetherian algebraic space, \item $f$ is locally of finite type and quasi-separated, \item $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module, and \item $\mathcal{G}$ is a finite type $\mathcal{O}_X$-module, flat over $B$, with support proper over $B$. \end{enumerate} Then the functor $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ is an algebraic space affine and of finite presentation over $B$.


quot-lemma-coherent-diagonal

In Situation \ref{situation-coherent}. Denote $\mathcal{X} = \Cohstack_{X/B}$. Then $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces.


quot-lemma-coherent-stack

In Situation \ref{situation-coherent} the functor $p : \Cohstack_{X/B} \longrightarrow (\Sch/S)_{fppf}$ is a stack in groupoids.


quot-lemma-coherent-limits

In Situation \ref{situation-coherent} assume that $B \to S$ is locally of finite presentation. Then $p : \Cohstack_{X/B} \to (\Sch/S)_{fppf}$ is limit preserving (Artin's Axioms, Definition \ref{artin-definition-limit-preserving}).


quot-lemma-coherent-RS-star

In Situation \ref{situation-coherent}. Let $$ \xymatrix{ Z \ar[r] \ar[d] & Z' \ar[d] \\ Y \ar[r] & Y' } $$ be a pushout in the category of schemes over $S$ where $Z \to Z'$ is a thickening and $Z \to Y$ is affine, see More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-thickening}. Then the functor on fibre categories $$ \Cohstack_{X/B, Y'} \longrightarrow \Cohstack_{X/B, Y} \times_{\Cohstack_{X/B, Z}} \Cohstack_{X/B, Z'} $$ is an equivalence.


quot-lemma-coherent-over-first-order-thickening

Let $$ \xymatrix{ X \ar[d] \ar[r]_i & X' \ar[d] \\ T \ar[r] & T' } $$ be a cartesian square of algebraic spaces where $T \to T'$ is a first order thickening. Let $\mathcal{F}'$ be an $\mathcal{O}_{X'}$-module flat over $T'$. Set $\mathcal{F} = i^*\mathcal{F}'$. The following are equivalent \begin{enumerate} \item $\mathcal{F}'$ is a quasi-coherent $\mathcal{O}_{X'}$-module of finite presentation, \item $\mathcal{F}'$ is an $\mathcal{O}_{X'}$-module of finite presentation, \item $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module of finite presentation, \item $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation, \end{enumerate}


quot-lemma-coherent-tangent-space

In Situation \ref{situation-coherent} assume that $S$ is a locally Noetherian scheme and $B \to S$ is locally of finite presentation. Let $k$ be a finite type field over $S$ and let $x_0 = (\Spec(k), g_0, \mathcal{G}_0)$ be an object of $\mathcal{X} = \Cohstack_{X/B}$ over $k$. Then the spaces $T\mathcal{F}_{\mathcal{X}, k, x_0}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0})$ (Artin's Axioms, Section \ref{artin-section-tangent-spaces}) are finite dimensional.


quot-lemma-coherent-existence

In Situation \ref{situation-coherent} assume that $S$ is a locally Noetherian scheme and that $f : X \to B$ is separated. Let $\mathcal{X} = \Cohstack_{X/B}$. Then the functor Artin's Axioms, Equation (\ref{artin-equation-approximation}) is an equivalence.


quot-lemma-coherent-defo-thy

In Situation \ref{situation-coherent} assume that $S$ is a locally Noetherian scheme, $S = B$, and $f : X \to B$ is flat. Let $\mathcal{X} = \Cohstack_{X/B}$. Then we have openness of versality for $\mathcal{X}$ (see Artin's Axioms, Definition \ref{artin-definition-openness-versality}).


quot-lemma-q-sheaf

In Situation \ref{situation-q}. The functors $\text{Q}_{\mathcal{F}/X/B}$ and $\text{Q}^{fp}_{\mathcal{F}/X/B}$ satisfy the sheaf property for the fpqc topology.


quot-lemma-extend-q-to-spaces

In Situation \ref{situation-q}. Let $T$ be an algebraic space over $S$. We have $$ \Mor_{\Sh((\Sch/S)_{fppf})}(T,  \text{Q}_{\mathcal{F}/X/B}) = \left\{ \begin{matrix} (h, \mathcal{F}_T \to \mathcal{Q}) \text{ where } h : T \to B \text{ and}\\ \mathcal{Q}\text{ is quasi-coherent and flat over }T \end{matrix} \right\} $$ where $\mathcal{F}_T$ denotes the pullback of $\mathcal{F}$ to the algebraic space $X \times_{B, h} T$. Similarly, we have $$ \Mor_{\Sh((\Sch/S)_{fppf})}(T,  \text{Q}^{fp}_{\mathcal{F}/X/B}) = \left\{ \begin{matrix} (h, \mathcal{F}_T \to \mathcal{Q}) \text{ where } h : T \to B \text{ and}\\ \mathcal{Q}\text{ is of finite presentation and flat over }T \end{matrix} \right\} $$


quot-lemma-q-limit-preserving

In Situation \ref{situation-q} assume also that (a) $f$ is quasi-compact and quasi-separated and (b) $\mathcal{F}$ is of finite presentation. Then the functor $\text{Q}^{fp}_{\mathcal{F}/X/B}$ is limit preserving in the following sense: If $T = \lim T_i$ is a directed limit of affine schemes over $B$, then $\text{Q}^{fp}_{\mathcal{F}/X/B}(T) = \colim \text{Q}^{fp}_{\mathcal{F}/X/B}(T_i)$.


quot-lemma-q-RS-star

In Situation \ref{situation-q}. Let $$ \xymatrix{ Z \ar[r] \ar[d] & Z' \ar[d] \\ Y \ar[r] & Y' } $$ be a pushout in the category of schemes over $B$ where $Z \to Z'$ is a thickening and $Z \to Y$ is affine, see More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-thickening}. Then the natural map $$ Q_{\mathcal{F}/X/B}(Y') \longrightarrow Q_{\mathcal{F}/X/B}(Y) \times_{Q_{\mathcal{F}/X/B}(Z)} Q_{\mathcal{F}/X/B}(Z') $$ is bijective. If $X \to B$ is locally of finite presentation, then the same thing is true for $Q^{fp}_{\mathcal{F}/X/B}$.


quot-lemma-quot-sheaf

In Situation \ref{situation-quot}. The functor $\Quotfunctor_{\mathcal{F}/X/B}$ satisfies the sheaf property for the fpqc topology.


quot-lemma-extend-quot-to-spaces

In Situation \ref{situation-quot}. Let $T$ be an algebraic space over $S$. We have $$ \Mor_{\Sh((\Sch/S)_{fppf})}(T,  \Quotfunctor_{\mathcal{F}/X/B}) = \left\{ \begin{matrix} (h, \mathcal{F}_T \to \mathcal{Q}) \text{ where } h : T \to B \text{ and}\\ \mathcal{Q}\text{ is of finite presentation and}\\ \text{flat over }T\text{ with support proper over }T \end{matrix} \right\} $$ where $\mathcal{F}_T$ denotes the pullback of $\mathcal{F}$ to the algebraic space $X \times_{B, h} T$.


quot-lemma-hilb-is-quot

In Situation \ref{situation-hilb} we have $\Hilbfunctor_{X/B} = \Quotfunctor_{\mathcal{O}_X/X/B}$.


quot-lemma-extend-hilb-to-spaces

In Situation \ref{situation-hilb}. Let $T$ be an algebraic space over $S$. We have $$ \Mor_{\Sh((\Sch/S)_{fppf})}(T, \Hilbfunctor_{X/B}) = \left\{ \begin{matrix} (h, Z)\text{ where }h : T \to B,\ Z \subset X_T \\ \text{finite presentation, flat, proper over }T \end{matrix} \right\} $$ where $X_T = X \times_{B, h} T$.


quot-lemma-picard-stack-open-in-coh

Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$ which is flat, of finite presentation, and proper. The natural map $$ \Picardstack_{X/B} \longrightarrow \Cohstack_{X/B} $$ is representable by open immersions.


quot-lemma-pic-over-pic

In Situation \ref{situation-pic} the functor $\Picardfunctor_{X/B}$ is the sheafification of the functor $T \mapsto \Ob(\Picardstack_{X/B, T})/\cong$.


quot-lemma-flat-geometrically-connected-fibres

In Situation \ref{situation-pic}. If $\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}$ is an isomorphism for all schemes $T$ over $B$, then $$ 0 \to \Pic(T) \to \Pic(X_T) \to \Picardfunctor_{X/B}(T) $$ is an exact sequence for all $T$.


quot-lemma-flat-geometrically-connected-fibres-with-section-functor-stack

In Situation \ref{situation-pic} let $\sigma : B \to X$ be a section. If $\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}$ is an isomorphism for all $T$ over $B$, then $\Picardstack_{X/B, \sigma} \to (\Sch/S)_{fppf}$ is fibred in setoids with set of isomorphism classes over $T$ given by $$ \coprod\nolimits_{h : T \to B} \Ker(\sigma_T^* : \Pic(X \times_{B, h} T) \to \Pic(T)) $$


quot-lemma-diagonal-pic

With assumptions and notation as in Proposition \ref{proposition-pic-functor}. Then the diagonal $\Picardfunctor_{X/B} \to \Picardfunctor_{X/B} \times_B \Picardfunctor_{X/B}$ is representable by immersions. In other words, $\Picardfunctor_{X/B} \to B$ is locally separated.


quot-lemma-Mor-into-Hilb

Let $S$ be a scheme. Consider morphisms of algebraic spaces $Z \to B$ and $X \to B$ over $S$. If $X \to B$ is separated and $Z \to B$ is of finite presentation, flat, and proper, then there is a natural injective transformation of functors $$ \mathit{Mor}_B(Z, X) \longrightarrow \Hilbfunctor_{Z \times_B X/B} $$ which maps a morphism $f : Z_T \to X_T$ to its graph.


quot-lemma-Mor-into-Hilb-open

Assumption and notation as in Lemma \ref{lemma-Mor-into-Hilb}. The transformation $\mathit{Mor}_B(Z, X) \longrightarrow \Hilbfunctor_{Z \times_B X/B}$ is representable by open immersions.


quot-lemma-spaces-fibred-in-groupoids

The category $\Spacesstack'_{ft}$ is fibred in groupoids over $\Sch_{fppf}$. The same is true for $\Spacesstack'_{fp, flat, proper}$.


quot-lemma-spaces-diagonal

The diagonal $$ \Delta : \Spacesstack'_{fp, flat, proper} \longrightarrow \Spacesstack'_{fp, flat, proper} \times \Spacesstack'_{fp, flat, proper} $$ is representable by algebraic spaces.


quot-lemma-spaces-stack

The category $\Spacesstack'_{ft}$ is a stack in groupoids over $\Sch_{fppf}$. The same is true for $\Spacesstack'_{fp, flat, proper}$.


quot-lemma-extend-spaces-to-spaces

Let $T$ be an algebraic space over $\mathbf{Z}$. Let $\mathcal{S}_T$ denote the corresponding algebraic stack (Algebraic Stacks, Sections \ref{algebraic-section-split}, \ref{algebraic-section-representable-by-algebraic-spaces}, and \ref{algebraic-section-stacks-spaces}). We have an equivalence of categories $$ \left\{ \begin{matrix} \text{morphisms of algebraic spaces }\\ X \to T\text{ of finite type} \end{matrix} \right\} \longrightarrow \Mor_{\textit{Cat}/\Sch_{fppf}}(\mathcal{S}_T, \Spacesstack'_{ft}) $$ and an equivalence of categories $$ \left\{ \begin{matrix} \text{morphisms of algebraic spaces }X \to T\\ \text{of finite presentation, flat, and proper} \end{matrix} \right\} \longrightarrow \Mor_{\textit{Cat}/\Sch_{fppf}}(\mathcal{S}_T, \Spacesstack'_{fp, flat, proper}) $$


quot-lemma-spaces-limits

The stack $p'_{fp, flat, proper} : \Spacesstack'_{fp, flat, proper} \to \Sch_{fppf}$ is limit preserving (Artin's Axioms, Definition \ref{artin-definition-limit-preserving}).


quot-lemma-spaces-RS-star

Let $$ \xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' } $$ be a pushout in the category of schemes where $T \to T'$ is a thickening and $T \to S$ is affine, see More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-thickening}. Then the functor on fibre categories $$ \begin{matrix} \Spacesstack'_{fp, flat, proper, S'} \\ \downarrow \\ \Spacesstack'_{fp, flat, proper, S} \times_{\Spacesstack'_{fp, flat, proper, T}} \Spacesstack'_{fp, flat, proper, T'} \end{matrix} $$ is an equivalence.


quot-lemma-spaces-tangent-space

Let $k$ be a field and let $x = (X \to \Spec(k))$ be an object of $\mathcal{X} = \Spacesstack'_{fp, flat, proper}$ over $\Spec(k)$. \begin{enumerate} \item If $k$ is of finite type over $\mathbf{Z}$, then the vector spaces $T\mathcal{F}_{\mathcal{X}, k, x}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x})$ (see Artin's Axioms, Section \ref{artin-section-tangent-spaces}) are finite dimensional, and \item in general the vector spaces $T_x(k)$ and $\text{Inf}_x(k)$ (see Artin's Axioms, Section \ref{artin-section-inf}) are finite dimensional. \end{enumerate}


quot-lemma-spaces-defo-thy

The stack in groupoids $\mathcal{X} = \Spacesstack'_{fp, flat, proper}$ satisfies openness of versality over $\Spec(\mathbf{Z})$. Similarly, after base change (Remark \ref{remark-spaces-base-change}) openness of versality holds over any Noetherian base scheme $S$.


quot-lemma-polarized-fibred-in-groupoids

The category $\Polarizedstack$ is fibred in groupoids over $\Spacesstack'_{fp, flat, proper}$. The category $\Polarizedstack$ is fibred in groupoids over $\Sch_{fppf}$.


quot-lemma-polarized-stack

The category $\Polarizedstack$ is a stack in groupoids over $\Spacesstack'_{fp, flat, proper}$ (endowed with the inherited topology, see Stacks, Definition \ref{stacks-definition-topology-inherited}). The category $\Polarizedstack$ is a stack in groupoids over $\Sch_{fppf}$.


quot-lemma-extend-polarized-to-spaces

Let $T$ be an algebraic space over $\mathbf{Z}$. Let $\mathcal{S}_T$ denote the corresponding algebraic stack (Algebraic Stacks, Sections \ref{algebraic-section-split}, \ref{algebraic-section-representable-by-algebraic-spaces}, and \ref{algebraic-section-stacks-spaces}). We have an equivalence of categories $$ \left\{ \begin{matrix} (X \to T, \mathcal{L})\text{ where }X \to T\text{ is a morphism}\\ \text{of algebraic spaces, is proper, flat, and of}\\ \text{finite presentation and }\mathcal{L}\text{ ample on }X/T \end{matrix} \right\} \longrightarrow \Mor_{\textit{Cat}/\Sch_{fppf}}(\mathcal{S}_T, \Polarizedstack) $$


quot-lemma-polarized-to-spaces-algebraic

The functor (\ref{equation-over-proper-spaces}) defines a $1$-morphism $$ \Polarizedstack \to \Spacesstack'_{fp, flat, proper} $$ of stacks in groupoids over $\Sch_{fppf}$ which is algebraic in the sense of Criteria for Representability, Definition \ref{criteria-definition-algebraic}.


quot-lemma-polarized-diagonal

The diagonal $$ \Delta : \Polarizedstack \longrightarrow \Polarizedstack \times \Polarizedstack $$ is representable by algebraic spaces.


quot-lemma-polarized-limits

The stack in groupoids $\Polarizedstack$ is limit preserving (Artin's Axioms, Definition \ref{artin-definition-limit-preserving}).


quot-lemma-polarized-RS-star

In Situation \ref{situation-coherent}. Let $$ \xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' } $$ be a pushout in the category of schemes where $T \to T'$ is a thickening and $T \to S$ is affine, see More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-thickening}. Then the functor on fibre categories $$ \Polarizedstack_{S'} \longrightarrow \Polarizedstack_S \times_{\Polarizedstack_T} \Polarizedstack_{T'} $$ is an equivalence.


quot-lemma-polarized-strong-effectiveness

\begin{slogan} Grothendieck's algebraization theorem continues to hold in the non-Noetherian setting if one assumes flatness and finite presentation. \end{slogan} Let $(R_n)$ be an inverse system of rings with surjective transition maps whose kernels are locally nilpotent. Set $R = \lim R_n$. Set $S_n = \Spec(R_n)$ and $S = \Spec(R)$. Consider a commutative diagram $$ \xymatrix{ X_1 \ar[r]_{i_1} \ar[d] & X_2 \ar[r]_{i_2} \ar[d] & X_3 \ar[r] \ar[d] & \ldots \\ S_1 \ar[r] & S_2 \ar[r] & S_3 \ar[r] & \ldots } $$ of schemes with cartesian squares. Suppose given $(\mathcal{L}_n, \varphi_n)$ where each $\mathcal{L}_n$ is an invertible sheaf on $X_n$ and $\varphi_n : i_n^*\mathcal{L}_{n + 1} \to \mathcal{L}_n$ is an isomorphism. If \begin{enumerate} \item $X_n \to S_n$ is proper, flat, of finite presentation, and \item $\mathcal{L}_1$ is ample on $X_1$ \end{enumerate} then there exists a morphism of schemes $X \to S$ proper, flat, and of finite presentation and an ample invertible $\mathcal{O}_X$-module $\mathcal{L}$ and isomorphisms $X_n \cong X \times_S S_n$ and $\mathcal{L}_n \cong \mathcal{L}|_{X_n}$ compatible with the morphisms $i_n$ and $\varphi_n$.


quot-lemma-polarized-existence

Consider the stack $\Polarizedstack$ over the base scheme $\Spec(\mathbf{Z})$. Then every formal object is effective.


quot-lemma-polarized-defo-thy

The stack in groupoids $\Polarizedstack$ satisfies openness of versality over $\Spec(\mathbf{Z})$. Similarly, after base change (Remark \ref{remark-polarized-base-change}) openness of versality holds over any Noetherian base scheme $S$.


quot-lemma-curves-diagonal

The diagonal $$ \Delta : \Curvesstack \longrightarrow \Curvesstack \times \Curvesstack $$ is representable by algebraic spaces.


quot-lemma-curves-limits

The stack $\Curvesstack \to \Sch_{fppf}$ is limit preserving (Artin's Axioms, Definition \ref{artin-definition-limit-preserving}).


quot-lemma-curves-RS-star

Let $$ \xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' } $$ be a pushout in the category of schemes where $T \to T'$ is a thickening and $T \to S$ is affine, see More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-thickening}. Then the functor on fibre categories $$ \Curvesstack_{S'} \longrightarrow \Curvesstack_S \times_{\Curvesstack_T} \Curvesstack_{T'} $$ is an equivalence.


quot-lemma-curves-existence

Consider the stack $\Curvesstack$ over the base scheme $\Spec(\mathbf{Z})$. Then every formal object is effective.


quot-lemma-curves-defo-thy

The stack in groupoids $\mathcal{X} = \Curvesstack$ satisfies openness of versality over $\Spec(\mathbf{Z})$. Similarly, after base change (Remark \ref{remark-curves-base-change}) openness of versality holds over any Noetherian base scheme $S$.


quot-lemma-curves-open-and-closed-in-spaces

The $1$-morphism (\ref{equation-curves-over-proper-spaces}) $$ \Curvesstack \longrightarrow \Spacesstack'_{fp, flat, proper} $$ is representable by open and closed immersions.


quot-lemma-complexes-open-neg-exts-vanishing

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is proper, flat, and of finite presentation. Let $K, E \in D(\mathcal{O}_X)$. Assume $K$ is pseudo-coherent and $E$ is $Y$-perfect (More on Morphisms of Spaces, Definition \ref{spaces-more-morphisms-definition-relatively-perfect}). For a field $k$ and a morphism $y : \Spec(k) \to Y$ denote $K_y$, $E_y$ the pullback to the fibre $X_y$. \begin{enumerate} \item There is an open $W \subset Y$ characterized by the property $$ y \in |W| \Leftrightarrow \Ext^i_{\mathcal{O}_{X_y}}(K_y, E_y) = 0 \text{ for }i < 0. $$ \item For any morphism $V \to Y$ factoring through $W$ we have $$ \Ext^i_{\mathcal{O}_{X_V}}(K_V, E_V) = 0 \quad\text{for}\quad i < 0 $$ where $X_V$ is the base change of $X$ and $K_V$ and $E_V$ are the derived pullbacks of $K$ and $E$ to $X_V$. \item The functor $V \mapsto \Hom_{\mathcal{O}_{X_V}}(K_V, E_V)$ is a sheaf on $(\textit{Spaces}/W)_{fppf}$ representable by an algebraic space affine and of finite presentation over $W$. \end{enumerate}


quot-lemma-complexes

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is proper, flat, and of finite presentation. Let $E \in D(\mathcal{O}_X)$. Assume \begin{enumerate} \item $E$ is $S$-perfect (More on Morphisms of Spaces, Definition \ref{spaces-more-morphisms-definition-relatively-perfect}), and \item for every point $s \in S$ we have $$ \Ext^i_{\mathcal{O}_{X_s}}(E_s, E_s) = 0 \quad\text{for}\quad i < 0 $$ where $E_s$ is the pullback to the fibre $X_s$. \end{enumerate} Then \begin{enumerate} \item[(a)] (1) and (2) are preserved by arbitrary base change $V \to Y$, \item[(b)] $\Ext^i_{\mathcal{O}_{X_V}}(E_V, E_V) = 0$ for $i < 0$ and all $V$ over $Y$, \item[(c)] $V \mapsto \Hom_{\mathcal{O}_{X_V}}(E_V, E_V)$ is representable by an algebraic space affine and of finite presentation over $Y$. \end{enumerate} Here $X_V$ is the base change of $X$ and $E_V$ is the derived pullback of $E$ to $X_V$.


quot-lemma-complexes-diagonal

In Situation \ref{situation-complexes}. Denote $\mathcal{X} = \Complexesstack_{X/B}$. Then $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces.


quot-lemma-complexes-stack

In Situation \ref{situation-complexes} the functor $p : \Complexesstack_{X/B} \longrightarrow (\Sch/S)_{fppf}$ is a stack in groupoids.


quot-lemma-complexes-limits

In Situation \ref{situation-complexes} assume that $B \to S$ is locally of finite presentation. Then $p : \Complexesstack_{X/B} \to (\Sch/S)_{fppf}$ is limit preserving (Artin's Axioms, Definition \ref{artin-definition-limit-preserving}).


quot-lemma-complexes-RS-star

In Situation \ref{situation-complexes}. Let $$ \xymatrix{ Z \ar[r] \ar[d] & Z' \ar[d] \\ Y \ar[r] & Y' } $$ be a pushout in the category of schemes over $S$ where $Z \to Z'$ is a finite order thickening and $Z \to Y$ is affine, see More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-thickening}. Then the functor on fibre categories $$ \Complexesstack_{X/B, Y'} \longrightarrow \Complexesstack_{X/B, Y} \times_{\Complexesstack_{X/B, Z}} \Complexesstack_{X/B, Z'} $$ is an equivalence.


quot-lemma-complexes-tangent-space

In Situation \ref{situation-complexes} assume that $S$ is a locally Noetherian scheme and $B \to S$ is locally of finite presentation. Let $k$ be a finite type field over $S$ and let $x_0 = (\Spec(k), g_0, E_0)$ be an object of $\mathcal{X} = \Complexesstack_{X/B}$ over $k$. Then the spaces $T\mathcal{F}_{\mathcal{X}, k, x_0}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0})$ (Artin's Axioms, Section \ref{artin-section-tangent-spaces}) are finite dimensional.


quot-lemma-complexes-strong-effectiveness

In Situation \ref{situation-complexes} assume $B = S$ is locally Noetherian. Then strong formal effectiveness in the sense of Artin's Axioms, Remark \ref{artin-remark-strong-effectiveness} holds for $p : \Complexesstack_{X/S} \to (\Sch/S)_{fppf}$.


quot-proposition-hom

In Situation \ref{situation-hom} assume that \begin{enumerate} \item $f$ is of finite presentation, and \item $\mathcal{G}$ is a finitely presented $\mathcal{O}_X$-module, flat over $B$, with support proper over $B$. \end{enumerate} Then the functor $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ is an algebraic space affine over $B$. If $\mathcal{F}$ is of finite presentation, then $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ is of finite presentation over $B$.


quot-proposition-isom

In Situation \ref{situation-hom} assume that \begin{enumerate} \item $f$ is of finite presentation, and \item $\mathcal{F}$ and $\mathcal{G}$ are finitely presented $\mathcal{O}_X$-modules, flat over $B$, with support proper over $B$. \end{enumerate} Then the functor $\mathit{Isom}(\mathcal{F}, \mathcal{G})$ is an algebraic space affine of finite presentation over $B$.


quot-proposition-quot

Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. If $f$ is of finite presentation and separated, then $\Quotfunctor_{\mathcal{F}/X/B}$ is an algebraic space. If $\mathcal{F}$ is of finite presentation, then $\Quotfunctor_{\mathcal{F}/X/B} \to B$ is locally of finite presentation.


quot-proposition-hilb

Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. If $f$ is of finite presentation and separated, then $\Hilbfunctor_{X/B}$ is an algebraic space locally of finite presentation over $B$.


quot-proposition-pic

Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. If $f$ is flat, of finite presentation, and proper, then $\Picardstack_{X/B}$ is an algebraic stack.


quot-proposition-pic-functor

Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Assume that \begin{enumerate} \item $f$ is flat, of finite presentation, and proper, and \item $\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}$ is an isomorphism for all schemes $T$ over $B$. \end{enumerate} Then $\Picardfunctor_{X/B}$ is an algebraic space.


quot-proposition-Mor

Let $S$ be a scheme. Let $Z \to B$ and $X \to B$ be morphisms of algebraic spaces over $S$. Assume $X \to B$ is of finite presentation and separated and $Z \to B$ is of finite presentation, flat, and proper. Then $\mathit{Mor}_B(Z, X)$ is an algebraic space locally of finite presentation over $B$.


spaces-more-cohomology-theorem-proper-base-change

Let $S$ be a scheme. Let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ be a cartesian square of algebraic spaces over $S$. Assume $f$ is proper. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_\etale$. Then the base change map $$ g^{-1}Rf_*\mathcal{F} \longrightarrow Rf'_*(g')^{-1}\mathcal{F} $$ is an isomorphism.


spaces-more-cohomology-lemma-compare-cohomology-other-topologies

Let $S$ be a scheme. Let $\tau \in \{\etale, fppf, ph\}$ (add more here). The inclusion functor $$ (\Sch/S)_\tau \longrightarrow (\textit{Spaces}/S)_\tau $$ is a special cocontinuous functor (Sites, Definition \ref{sites-definition-special-cocontinuous-functor}) and hence identifies topoi.


spaces-more-cohomology-lemma-surjective-proper

Let $S$ be a scheme. Let $f : Y \to X$ be a surjective proper morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a sheaf on $X_\etale$. Then $\mathcal{F} \to f_*f^{-1}\mathcal{F}$ is injective with image the equalizer of the two maps $f_*f^{-1}\mathcal{F} \to g_*g^{-1}\mathcal{F}$ where $g$ is the structure morphism $g : Y \times_X Y \to X$.


spaces-more-cohomology-lemma-h0-proper-over-henselian-pair

Let $(A, I)$ be a henselian pair. Let $X$ be an algebraic space over $A$ such that the structure morphism $f : X \to \Spec(A)$ is proper. Let $i : X_0 \to X$ be the inclusion of $X \times_{\Spec(A)} \Spec(A/I)$. For any sheaf $\mathcal{F}$ on $X_\etale$ we have $\Gamma(X, \mathcal{F}) = \Gamma(Z, i^{-1}\mathcal{F})$.


spaces-more-cohomology-lemma-proper-base-change-f-star

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y' \to Y$ be a morphisms of algebraic spaces over $S$. Assume $f$ is proper. Set $X' = Y' \times_Y X$ with projections $f' : X' \to Y'$ and $g' : X' \to X$. Let $\mathcal{F}$ be any sheaf on $X_\etale$. Then $g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$.


spaces-more-cohomology-lemma-proper-base-change

Let $S$ be a scheme. Let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ be a cartesian square of algebraic spaces over $S$. Assume $f$ is proper. Let $E \in D^+(X_\etale)$ have torsion cohomology sheaves. Then the base change map $g^{-1}Rf_*E \to Rf'_*(g')^{-1}E$ is an isomorphism.


spaces-more-cohomology-lemma-describe-pullback

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a sheaf on $X_\etale$. Then $\pi_X^{-1}\mathcal{F}$ is given by the rule $$ (\pi_X^{-1}\mathcal{F})(Y) = \Gamma(Y_\etale, f_{small}^{-1}\mathcal{F}) $$ for $f : Y \to X$ in $(\textit{Spaces}/X)_\etale$. Moreover, $\pi_Y^{-1}\mathcal{F}$ satisfies the sheaf condition with respect to smooth, syntomic, fppf, fpqc, and ph coverings.


spaces-more-cohomology-lemma-compare-injectives

Let $S$ be a scheme. Let $Y \to X$ be a morphism of $(\textit{Spaces}/S)_\etale$. \begin{enumerate} \item If $\mathcal{I}$ is injective in $\textit{Ab}((\textit{Spaces}/X)_\etale)$, then \begin{enumerate} \item $i_f^{-1}\mathcal{I}$ is injective in $\textit{Ab}(Y_\etale)$, \item $\mathcal{I}|_{X_\etale}$ is injective in $\textit{Ab}(X_\etale)$, \end{enumerate} \item If $\mathcal{I}^\bullet$ is a K-injective complex in $\textit{Ab}((\textit{Spaces}/X)_\etale)$, then \begin{enumerate} \item $i_f^{-1}\mathcal{I}^\bullet$ is a K-injective complex in $\textit{Ab}(Y_\etale)$, \item $\mathcal{I}^\bullet|_{X_\etale}$ is a K-injective complex in $\textit{Ab}(X_\etale)$, \end{enumerate} \end{enumerate} The corresponding statements for modules do not hold.


spaces-more-cohomology-lemma-compare-higher-direct-image

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item For $K$ in $D((\textit{Spaces}/Y)_\etale)$ we have $ (Rf_{big, *}K)|_{X_\etale} = Rf_{small, *}(K|_{Y_\etale}) $ in $D(X_\etale)$. \item For $K$ in $D((\textit{Spaces}/Y)_\etale, \mathcal{O})$ we have $ (Rf_{big, *}K)|_{X_\etale} = Rf_{small, *}(K|_{Y_\etale}) $ in $D(\textit{Mod}(X_\etale, \mathcal{O}_X))$. \end{enumerate} More generally, let $g : X' \to X$ be an object of $(\textit{Spaces}/X)_\etale$. Consider the fibre product $$ \xymatrix{ Y' \ar[r]_{g'} \ar[d]_{f'} & Y \ar[d]^f \\ X' \ar[r]^g & X } $$ Then \begin{enumerate} \item[(3)] For $K$ in $D((\textit{Spaces}/Y)_\etale)$ we have $i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$ in $D(X'_\etale)$. \item[(4)] For $K$ in $D((\textit{Spaces}/Y)_\etale, \mathcal{O})$ we have $i_g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$ in $D(\textit{Mod}(X'_\etale, \mathcal{O}_{X'}))$. \item[(5)] For $K$ in $D((\textit{Spaces}/Y)_\etale)$ we have $g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)$ in $D((\textit{Spaces}/X')_\etale)$. \item[(6)] For $K$ in $D((\textit{Spaces}/Y)_\etale, \mathcal{O})$ we have $g_{big}^*(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^*K)$ in $D(\textit{Mod}(X'_\etale, \mathcal{O}_{X'}))$. \end{enumerate}


spaces-more-cohomology-lemma-cohomological-descent-etale

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $K \in D(X_\etale)$ the map $$ K \longrightarrow R\pi_{X, *}\pi_X^{-1}K $$ is an isomorphism where $\pi_X : \Sh((\textit{Spaces}/X)_\etale) \to \Sh(X_\etale)$ is as above.


spaces-more-cohomology-lemma-compare-higher-direct-image-proper

Let $S$ be a scheme. Let $f : Y \to X$ be a proper morphism of algebraic spaces over $S$. Then we have \begin{enumerate} \item $\pi_X^{-1} \circ f_{small, *} = f_{big, *} \circ \pi_Y^{-1}$ as functors $\Sh(Y_\etale) \to \Sh((\textit{Spaces}/X)_\etale)$, \item $\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\pi_Y^{-1}K$ for $K$ in $D^+(Y_\etale)$ whose cohomology sheaves are torsion, and \item $\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\pi_Y^{-1}K$ for all $K$ in $D(Y_\etale)$ if $f$ is finite. \end{enumerate}


spaces-more-cohomology-lemma-comparison-fppf-etale

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item For $\mathcal{F} \in \Sh(X_\etale)$ we have $\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}$ and $a_{X, *}a_X^{-1}\mathcal{F} = \mathcal{F}$. \item For $\mathcal{F} \in \textit{Ab}(X_\etale)$ we have $R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0$ for $i > 0$. \end{enumerate}


spaces-more-cohomology-lemma-cohomological-descent-etale-fppf

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $K \in D^+(X_\etale)$ the maps $$ \pi_X^{-1}K \longrightarrow R\epsilon_{X, *}a_X^{-1}K \quad\text{and}\quad K \longrightarrow Ra_{X, *}a_X^{-1}K $$ are isomorphisms with $a_X : \Sh((\textit{Spaces}/X)_{fppf}) \to \Sh(X_\etale)$ as above.


spaces-more-cohomology-lemma-push-pull-fppf-etale

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then there are commutative diagrams of topoi $$ \xymatrix{ \Sh((\textit{Spaces}/X)_{fppf}) \ar[rr]_{f_{big, fppf}} \ar[d]_{\epsilon_X} & & \Sh((\textit{Spaces}/Y)_{fppf}) \ar[d]^{\epsilon_Y} \\ \Sh((\textit{Spaces}/X)_\etale) \ar[rr]^{f_{big, \etale}} & & \Sh((\textit{Spaces}/Y)_\etale) } $$ and $$ \xymatrix{ \Sh((\textit{Spaces}/X)_{fppf}) \ar[rr]_{f_{big, fppf}} \ar[d]_{a_X} & & \Sh((\textit{Spaces}/Y)_{fppf}) \ar[d]^{a_Y} \\ \Sh(X_\etale) \ar[rr]^{f_{small}} & & \Sh(Y_\etale) } $$ with $a_X = \pi_X \circ \epsilon_X$ and $a_Y = \pi_X \circ \epsilon_X$.


spaces-more-cohomology-lemma-descent-sheaf-fppf-etale

In Lemma \ref{lemma-push-pull-fppf-etale} assume $f$ is flat, locally of finite presentation, and surjective. Then the functor $$ \Sh(Y_\etale) \longrightarrow \left\{ (\mathcal{G}, \mathcal{H}, \alpha) \middle| \begin{matrix} \mathcal{G} \in \Sh(X_\etale),\ \mathcal{H} \in \Sh((\Sch/Y)_{fppf}), \\ \alpha : a_X^{-1}\mathcal{G} \to f_{big, fppf}^{-1}\mathcal{H} \text{ an isomorphism} \end{matrix} \right\} $$ sending $\mathcal{F}$ to $(f_{small}^{-1}\mathcal{F}, a_Y^{-1}\mathcal{F}, can)$ is an equivalence.


spaces-more-cohomology-lemma-review-quasi-coherent

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. \begin{enumerate} \item The rule $$ \mathcal{F}^a : (\textit{Spaces}/X)_\etale \longrightarrow \textit{Ab},\quad (f : Y \to X) \longmapsto \Gamma(Y, f^*\mathcal{F}) $$ satisfies the sheaf condition for fpqc and a fortiori fppf and \'etale coverings, \item $\mathcal{F}^a = \pi_X^*\mathcal{F}$ on $(\textit{Spaces}/X)_\etale$, \item $\mathcal{F}^a = a_X^*\mathcal{F}$ on $(\textit{Spaces}/X)_{fppf}$, \item the rule $\mathcal{F} \mapsto \mathcal{F}^a$ defines an equivalence between quasi-coherent $\mathcal{O}_X$-modules and quasi-coherent modules on $((\textit{Spaces}/X)_\etale, \mathcal{O})$, \item the rule $\mathcal{F} \mapsto \mathcal{F}^a$ defines an equivalence between quasi-coherent $\mathcal{O}_X$-modules and quasi-coherent modules on $((\textit{Spaces}/X)_{fppf}, \mathcal{O})$, \item we have $\epsilon_{X, *}a_X^*\mathcal{F} = \pi_X^*\mathcal{F}$ and $a_{X, *}a_X^*\mathcal{F} = \mathcal{F}$, \item we have $R^i\epsilon_{X, *}(a_X^*\mathcal{F}) = 0$ and $R^ia_{X, *}(a_X^*\mathcal{F}) = 0$ for $i > 0$. \end{enumerate}


spaces-more-cohomology-lemma-cohomological-descent-etale-fppf-modules

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $\mathcal{F}$ a quasi-coherent $\mathcal{O}_X$-module the maps $$ \pi_X^*\mathcal{F} \longrightarrow R\epsilon_{X, *}(a_X^*\mathcal{F}) \quad\text{and}\quad \mathcal{F} \longrightarrow Ra_{X, *}(a_X^*\mathcal{F}) $$ are isomorphisms.


spaces-more-cohomology-lemma-vanishing-adequate

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3$ be a complex of quasi-coherent $\mathcal{O}_X$-modules. Set $$ \mathcal{H}_\etale = \Ker(\pi_X^*\mathcal{F}_2 \to \pi_X^*\mathcal{F}_3)/ \Im(\pi_X^*\mathcal{F}_1 \to \pi_X^*\mathcal{F}_2) $$ on $(\textit{Spaces}/X)_\etale$ and set $$ \mathcal{H}_{fppf} = \Ker(a_X^*\mathcal{F}_2 \to a_X^*\mathcal{F}_3)/ \Im(a_X^*\mathcal{F}_1 \to a_X^*\mathcal{F}_2) $$ on $(\textit{Spaces}/X)_{fppf}$. Then $\mathcal{H}_\etale = \epsilon_{X, *}\mathcal{H}_{fppf}$ and $$ H^p_\etale(U, \mathcal{H}_\etale) = H^p_{fppf}(U, \mathcal{H}_{fppf}) = 0 $$ for $p > 0$ and any affine object $U$ of $(\textit{Spaces}/X)_\etale$.


spaces-more-cohomology-lemma-comparison-ph-etale

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item For $\mathcal{F} \in \Sh(X_\etale)$ we have $\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}$ and $a_{X, *}a_X^{-1}\mathcal{F} = \mathcal{F}$. \item For $\mathcal{F} \in \textit{Ab}(X_\etale)$ torsion we have $R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0$ for $i > 0$. \end{enumerate}


spaces-more-cohomology-lemma-cohomological-descent-etale-ph

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $K \in D^+(X_\etale)$ with torsion cohomology sheaves the maps $$ \pi_X^{-1}K \longrightarrow R\epsilon_{X, *}a_X^{-1}K \quad\text{and}\quad K \longrightarrow Ra_{X, *}a_X^{-1}K $$ are isomorphisms with $a_X : \Sh((\textit{Spaces}/X)_{ph}) \to \Sh(X_\etale)$ as above.


spaces-more-cohomology-lemma-push-pull-ph-etale

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then there are commutative diagrams of topoi $$ \xymatrix{ \Sh((\textit{Spaces}/X)_{ph}) \ar[rr]_{f_{big, ph}} \ar[d]_{\epsilon_X} & & \Sh((\textit{Spaces}/Y)_{ph}) \ar[d]^{\epsilon_Y} \\ \Sh((\textit{Spaces}/X)_\etale) \ar[rr]^{f_{big, \etale}} & & \Sh((\textit{Spaces}/Y)_\etale) } $$ and $$ \xymatrix{ \Sh((\textit{Spaces}/X)_{ph}) \ar[rr]_{f_{big, ph}} \ar[d]_{a_X} & & \Sh((\textit{Spaces}/Y)_{ph}) \ar[d]^{a_Y} \\ \Sh(X_\etale) \ar[rr]^{f_{small}} & & \Sh(Y_\etale) } $$ with $a_X = \pi_X \circ \epsilon_X$ and $a_Y = \pi_X \circ \epsilon_X$.


spaces-more-cohomology-lemma-proper-push-pull-ph-etale

In Lemma \ref{lemma-push-pull-ph-etale} if $f$ is proper, then we have \begin{enumerate} \item $a_Y^{-1} \circ f_{small, *} = f_{big, ph, *} \circ a_X^{-1}$, and \item $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, ph, *}(a_X^{-1}K)$ for $K$ in $D^+(X_\etale)$ with torsion cohomology sheaves. \end{enumerate}


coherent-theorem-formal-functions

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $f : X \to \Spec(A)$ be a proper morphism. Let $\mathcal{F}$ be a coherent sheaf on $X$. Fix $p \geq 0$. The system of maps $$ H^p(X, \mathcal{F})/I^nH^p(X, \mathcal{F}) \longrightarrow H^p(X, \mathcal{F}/I^n\mathcal{F}) $$ define an isomorphism of limits $$ H^p(X, \mathcal{F})^\wedge \longrightarrow \lim_n H^p(X, \mathcal{F}/I^n\mathcal{F}) $$ where the left hand side is the completion of the $A$-module $H^p(X, \mathcal{F})$ with respect to the ideal $I$, see Algebra, Section \ref{algebra-section-completion}. Moreover, this is in fact a homeomorphism for the limit topologies.


coherent-theorem-grothendieck-existence

\begin{reference} \cite[III Theorem 5.1.5]{EGA} \end{reference} Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Let $X$ be a separated, finite type scheme over $A$. Then the functor (\ref{equation-completion-functor-proper-over-A}) $$ \textit{Coh}_{\text{support proper over }A}(\mathcal{O}_X) \longrightarrow \textit{Coh}_{\text{support proper over }A}(X, \mathcal{I}) $$ is an equivalence.


coherent-theorem-algebraization

Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Set $S = \Spec(A)$ and $S_n = \Spec(A/I^n)$. Consider a commutative diagram $$ \xymatrix{ X_1 \ar[r]_{i_1} \ar[d] & X_2 \ar[r]_{i_2} \ar[d] & X_3 \ar[r] \ar[d] & \ldots \\ S_1 \ar[r] & S_2 \ar[r] & S_3 \ar[r] & \ldots } $$ of schemes with cartesian squares. Suppose given $(\mathcal{L}_n, \varphi_n)$ where each $\mathcal{L}_n$ is an invertible sheaf on $X_n$ and $\varphi_n : i_n^*\mathcal{L}_{n + 1} \to \mathcal{L}_n$ is an isomorphism. If \begin{enumerate} \item $X_1 \to S_1$ is proper, and \item $\mathcal{L}_1$ is ample on $X_1$ \end{enumerate} then there exists a proper morphism of schemes $X \to S$ and an ample invertible $\mathcal{O}_X$-module $\mathcal{L}$ and isomorphisms $X_n \cong X \times_S S_n$ and $\mathcal{L}_n \cong \mathcal{L}|_{X_n}$ compatible with the morphisms $i_n$ and $\varphi_n$.


coherent-lemma-cech-cohomology-quasi-coherent-trivial

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{U} : U = \bigcup_{i = 1}^n D(f_i)$ be a standard open covering of an affine open of $X$. Then $\check{H}^p(\mathcal{U}, \mathcal{F}) = 0$ for all $p > 0$.


coherent-lemma-quasi-coherent-affine-cohomology-zero

\begin{slogan} Serre vanishing: Higher cohomology vanishes on affine schemes for quasi-coherent modules. \end{slogan} Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. For any affine open $U \subset X$ we have $H^p(U, \mathcal{F}) = 0$ for all $p > 0$.


coherent-lemma-relative-affine-vanishing

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $f$ is affine then $R^if_*\mathcal{F} = 0$ for all $i > 0$.


coherent-lemma-relative-affine-cohomology

Let $f : X \to S$ be an affine morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $H^i(X, \mathcal{F}) = H^i(S, f_*\mathcal{F})$ for all $i \geq 0$.


coherent-lemma-affine-diagonal

Let $X$ be a scheme. The following are equivalent \begin{enumerate} \item $X$ has affine diagonal $\Delta : X \to X \times X$, \item for $U, V \subset X$ affine open, the intersection $U \cap V$ is affine, and \item there exists an open covering $\mathcal{U} : X = \bigcup_{i \in I} U_i$ such that $U_{i_0 \ldots i_p}$ is affine open for all $p \ge 0$ and all $i_0, \ldots, i_p \in I$. \end{enumerate} In particular this holds if $X$ is separated.


coherent-lemma-cech-cohomology-quasi-coherent

Let $X$ be a scheme. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be an open covering such that $U_{i_0 \ldots i_p}$ is affine open for all $p \ge 0$ and all $i_0, \ldots, i_p \in I$. In this case for any quasi-coherent sheaf $\mathcal{F}$ we have $$ \check{H}^p(\mathcal{U}, \mathcal{F}) = H^p(X, \mathcal{F}) $$ as $\Gamma(X, \mathcal{O}_X)$-modules for all $p$.


coherent-lemma-quasi-compact-h1-zero-covering

\begin{reference} \cite{Serre-criterion}, \cite[II, Theorem 5.2.1 (d') and IV (1.7.17)]{EGA} \end{reference} \begin{slogan} Serre's criterion for affineness. \end{slogan} Let $X$ be a scheme. Assume that \begin{enumerate} \item $X$ is quasi-compact, \item for every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ we have $H^1(X, \mathcal{I}) = 0$. \end{enumerate} Then $X$ is affine.


coherent-lemma-quasi-separated-h1-zero-covering

\begin{reference} \cite{Serre-criterion}, \cite[II, Theorem 5.2.1]{EGA} \end{reference} \begin{slogan} Serre's criterion for affineness. \end{slogan} Let $X$ be a scheme. Assume that \begin{enumerate} \item $X$ is quasi-compact, \item $X$ is quasi-separated, and \item $H^1(X, \mathcal{I}) = 0$ for every quasi-coherent sheaf of ideals $\mathcal{I}$ of finite type. \end{enumerate} Then $X$ is affine.


coherent-lemma-quasi-compact-h1-zero-invertible

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume that \begin{enumerate} \item $X$ is quasi-compact, \item for every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ there exists an $n \geq 1$ such that $H^1(X, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) = 0$. \end{enumerate} Then $\mathcal{L}$ is ample.


coherent-lemma-criterion-affine-morphism

Let $f : X \to Y$ be a quasi-compact morphism with $X$ and $Y$ quasi-separated. If $R^1f_*\mathcal{I} = 0$ for every quasi-coherent sheaf of ideals $\mathcal{I}$ on $X$, then $f$ is affine.


coherent-lemma-induction-principle

Let $X$ be a quasi-compact and quasi-separated scheme. Let $P$ be a property of the quasi-compact opens of $X$. Assume that \begin{enumerate} \item $P$ holds for every affine open of $X$, \item if $U$ is quasi-compact open, $V$ affine open, $P$ holds for $U$, $V$, and $U \cap V$, then $P$ holds for $U \cup V$. \end{enumerate} Then $P$ holds for every quasi-compact open of $X$ and in particular for $X$.


coherent-lemma-vanishing-nr-affines

\begin{slogan} For schemes with affine diagonal, the cohomology of quasi-coherent modules vanishes in degrees bigger than the number of affine opens needed in a covering. \end{slogan} Let $X$ be a quasi-compact scheme with affine diagonal (for example if $X$ is separated). Let $t = t(X)$ be the minimal number of affine opens needed to cover $X$. Then $H^n(X, \mathcal{F}) = 0$ for all $n \geq t$ and all quasi-coherent sheaves $\mathcal{F}$.


coherent-lemma-affine-diagonal-universal-delta-functor

Let $X$ be a quasi-compact scheme with affine diagonal (for example if $X$ is separated). Then \begin{enumerate} \item given a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ there exists an embedding $\mathcal{F} \to \mathcal{F}'$ of quasi-coherent $\mathcal{O}_X$-modules such that $H^p(X, \mathcal{F}') = 0$ for all $p \geq 1$, and \item $\{H^n(X, -)\}_{n \geq 0}$ is a universal $\delta$-functor from $\QCoh(\mathcal{O}_X)$ to $\textit{Ab}$. \end{enumerate}


coherent-lemma-vanishing-nr-affines-quasi-separated

Let $X$ be a quasi-compact quasi-separated scheme. Let $X = U_1 \cup \ldots \cup U_t$ be an affine open covering. Set $$ d = \max\nolimits_{I \subset \{1, \ldots, t\}} \left(|I| + t(\bigcap\nolimits_{i \in I} U_i)\right) $$ where $t(U)$ is the minimal number of affines needed to cover the scheme $U$. Then $H^n(X, \mathcal{F}) = 0$ for all $n \geq d$ and all quasi-coherent sheaves $\mathcal{F}$.


coherent-lemma-quasi-coherence-higher-direct-images

Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is quasi-separated and quasi-compact. \begin{enumerate} \item For any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the higher direct images $R^pf_*\mathcal{F}$ are quasi-coherent on $S$. \item If $S$ is quasi-compact, there exists an integer $n = n(X, S, f)$ such that $R^pf_*\mathcal{F} = 0$ for all $p \geq n$ and any quasi-coherent sheaf $\mathcal{F}$ on $X$. \item In fact, if $S$ is quasi-compact we can find $n = n(X, S, f)$ such that for every morphism of schemes $S' \to S$ we have $R^p(f')_*\mathcal{F}' = 0$ for $p \geq n$ and any quasi-coherent sheaf $\mathcal{F}'$ on $X'$. Here $f' : X' = S' \times_S X \to S'$ is the base change of $f$. \end{enumerate}


coherent-lemma-quasi-coherence-higher-direct-images-application

Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is quasi-separated and quasi-compact. Assume $S$ is affine. For any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ we have $$ H^q(X, \mathcal{F}) = H^0(S, R^qf_*\mathcal{F}) $$ for all $q \in \mathbf{Z}$.


coherent-lemma-affine-base-change

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume $f$ is affine. In this case $f_*\mathcal{F} \cong Rf_*\mathcal{F}$ is a quasi-coherent sheaf, and for every base change diagram (\ref{equation-base-change-diagram}) we have $$ g^*f_*\mathcal{F} = f'_*(g')^*\mathcal{F}. $$


coherent-lemma-flat-base-change-cohomology

Consider a cartesian diagram of schemes $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module with pullback $\mathcal{F}' = (g')^*\mathcal{F}$. Assume that $g$ is flat and that $f$ is quasi-compact and quasi-separated. For any $i \geq 0$ \begin{enumerate} \item the base change map of Cohomology, Lemma \ref{cohomology-lemma-base-change-map-flat-case} is an isomorphism $$ g^*R^if_*\mathcal{F} \longrightarrow R^if'_*\mathcal{F}', $$ \item if $S = \Spec(A)$ and $S' = \Spec(B)$, then $H^i(X, \mathcal{F}) \otimes_A B = H^i(X', \mathcal{F}')$. \end{enumerate}


coherent-lemma-finite-locally-free-base-change-cohomology

Consider a cartesian diagram of schemes $$ \xymatrix{ Y \ar[d]_{g} \ar[r]_h & X \ar[d]^f \\ \Spec(B) \ar[r] & \Spec(A) } $$ Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module with pullback $\mathcal{G} = h^*\mathcal{F}$. If $B$ is a finite locally free $A$-module, then $H^i(X, \mathcal{F}) \otimes_A B = H^i(Y, \mathcal{G})$.


coherent-lemma-colimit-cohomology

Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F} = \colim \mathcal{F}_i$ be a filtered colimit of quasi-coherent sheaves on $X$. Then for any $p \geq 0$ we have $$ R^pf_*\mathcal{F} = \colim R^pf_*\mathcal{F}_i. $$


coherent-lemma-separated-case-relative-cech

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume $X$ is quasi-compact and $X$ and $S$ have affine diagonal (e.g., if $X$ and $S$ are separated). In this case we can compute $Rf_*\mathcal{F}$ as follows: \begin{enumerate} \item Choose a finite affine open covering $\mathcal{U} : X = \bigcup_{i = 1, \ldots, n} U_i$. \item For $i_0, \ldots, i_p \in \{1, \ldots, n\}$ denote $f_{i_0 \ldots i_p} : U_{i_0 \ldots i_p} \to S$ the restriction of $f$ to the intersection $U_{i_0 \ldots i_p} = U_{i_0} \cap \ldots \cap U_{i_p}$. \item Set $\mathcal{F}_{i_0 \ldots i_p}$ equal to the restriction of $\mathcal{F}$ to $U_{i_0 \ldots i_p}$. \item Set $$ \check{\mathcal{C}}^p(\mathcal{U}, f, \mathcal{F}) = \bigoplus\nolimits_{i_0 \ldots i_p} f_{i_0 \ldots i_p *} \mathcal{F}_{i_0 \ldots i_p} $$ and define differentials $d : \check{\mathcal{C}}^p(\mathcal{U}, f, \mathcal{F}) \to \check{\mathcal{C}}^{p + 1}(\mathcal{U}, f, \mathcal{F})$ as in Cohomology, Equation (\ref{cohomology-equation-d-cech}). \end{enumerate} Then the complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F})$ is a complex of quasi-coherent sheaves on $S$ which comes equipped with an isomorphism $$ \check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F}) \longrightarrow Rf_*\mathcal{F} $$ in $D^{+}(S)$. This isomorphism is functorial in the quasi-coherent sheaf $\mathcal{F}$.


coherent-lemma-base-change-complex

With notation as in diagram (\ref{equation-base-change-diagram}). Assume $f : X \to S$ and $\mathcal{F}$ satisfy the hypotheses of Lemma \ref{lemma-separated-case-relative-cech}. Choose a finite affine open covering $\mathcal{U} : X = \bigcup U_i$ of $X$. There is a canonical isomorphism $$ g^*\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F}) \longrightarrow Rf'_*\mathcal{F}' $$ in $D^{+}(S')$. Moreover, if $S' \to S$ is affine, then in fact $$ g^*\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F}) = \check{\mathcal{C}}^\bullet(\mathcal{U}', f', \mathcal{F}') $$ with $\mathcal{U}' : X' = \bigcup U_i'$ where $U_i' = (g')^{-1}(U_i) = U_{i, S'}$ is also affine.


coherent-lemma-hypercoverings

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Assume that $f$ is quasi-compact and quasi-separated and that $S$ is quasi-compact and separated. There exists a bounded below complex $\mathcal{K}^\bullet$ of quasi-coherent $\mathcal{O}_S$-modules with the following property: For every morphism $g : S' \to S$ the complex $g^*\mathcal{K}^\bullet$ is a representative for $Rf'_*\mathcal{F}'$ with notation as in diagram (\ref{equation-base-change-diagram}).


coherent-lemma-cohomology-projective-space-over-ring

\begin{reference} \cite[III Proposition 2.1.12]{EGA} \end{reference} Let $R$ be a ring. Let $n \geq 0$ be an integer. We have $$ H^q(\mathbf{P}^n, \mathcal{O}_{\mathbf{P}^n_R}(d)) = \left\{ \begin{matrix} (R[T_0, \ldots, T_n])_d & \text{if} & q = 0 \\ 0 & \text{if} & q \not = 0, n \\ \left(\frac{1}{T_0 \ldots T_n} R[\frac{1}{T_0}, \ldots, \frac{1}{T_n}]\right)_d & \text{if} & q = n \end{matrix} \right. $$ as $R$-modules.


coherent-lemma-identify-functorially

The identifications of Equation (\ref{equation-identify}) are compatible with base change w.r.t.\ ring maps $R \to R'$. Moreover, for any $f \in R[T_0, \ldots, T_n]$ homogeneous of degree $m$ the map multiplication by $f$ $$ \mathcal{O}_{\mathbf{P}^n_R}(d) \longrightarrow \mathcal{O}_{\mathbf{P}^n_R}(d + m) $$ induces the map on the cohomology group via the identifications of Equation (\ref{equation-identify}) which is multiplication by $f$ for $H^0$ and the contragredient of multiplication by $f$ $$ (R[T_0, \ldots, T_n])_{-n - 1 - (d + m)} \longrightarrow (R[T_0, \ldots, T_n])_{-n - 1 - d} $$ on $H^n$.


coherent-lemma-cohomology-projective-space-over-base

Let $S$ be a scheme. Let $n \geq 0$ be an integer. Consider the structure morphism $$ f : \mathbf{P}^n_S \longrightarrow S. $$ We have $$ R^qf_*(\mathcal{O}_{\mathbf{P}^n_S}(d)) = \left\{ \begin{matrix} (\mathcal{O}_S[T_0, \ldots, T_n])_d & \text{if} & q = 0 \\ 0 & \text{if} & q \not = 0, n \\ \SheafHom_{\mathcal{O}_S}( (\mathcal{O}_S[T_0, \ldots, T_n])_{- n - 1 - d}, \mathcal{O}_S) & \text{if} & q = n \end{matrix} \right. $$


coherent-lemma-cohomology-projective-bundle

Let $S$ be a scheme. Let $n \geq 1$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_S$-module of constant rank $n + 1$. Consider the structure morphism $$ \pi : \mathbf{P}(\mathcal{E}) \longrightarrow S. $$ We have $$ R^q\pi_*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)) = \left\{ \begin{matrix} \text{Sym}^d(\mathcal{E}) & \text{if} & q = 0 \\ 0 & \text{if} & q \not = 0, n \\ \SheafHom_{\mathcal{O}_S}( \text{Sym}^{- n - 1 - d}(\mathcal{E}) \otimes_{\mathcal{O}_S} \wedge^{n + 1}\mathcal{E}, \mathcal{O}_S) & \text{if} & q = n \end{matrix} \right. $$ These identifications are compatible with base change and isomorphism between locally free sheaves.


coherent-lemma-coherent-Noetherian

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is coherent, \item $\mathcal{F}$ is a quasi-coherent, finite type $\mathcal{O}_X$-module, \item $\mathcal{F}$ is a finitely presented $\mathcal{O}_X$-module, \item for any affine open $\Spec(A) = U \subset X$ we have $\mathcal{F}|_U = \widetilde M$ with $M$ a finite $A$-module, and \item there exists an affine open covering $X = \bigcup U_i$, $U_i = \Spec(A_i)$ such that each $\mathcal{F}|_{U_i} = \widetilde M_i$ with $M_i$ a finite $A_i$-module. \end{enumerate} In particular $\mathcal{O}_X$ is coherent, any invertible $\mathcal{O}_X$-module is coherent, and more generally any finite locally free $\mathcal{O}_X$-module is coherent.


coherent-lemma-coherent-abelian-Noetherian

Let $X$ be a locally Noetherian scheme. The category of coherent $\mathcal{O}_X$-modules is abelian. More precisely, the kernel and cokernel of a map of coherent $\mathcal{O}_X$-modules are coherent. Any extension of coherent sheaves is coherent.


coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Any quasi-coherent submodule of $\mathcal{F}$ is coherent. Any quasi-coherent quotient module of $\mathcal{F}$ is coherent.


coherent-lemma-tensor-hom-coherent

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. The $\mathcal{O}_X$-modules $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$ and $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ are coherent.


coherent-lemma-local-isomorphism

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be a homomorphism of $\mathcal{O}_X$-modules. Let $x \in X$. \begin{enumerate} \item If $\mathcal{F}_x = 0$ then there exists an open neighbourhood $U \subset X$ of $x$ such that $\mathcal{F}|_U = 0$. \item If $\varphi_x : \mathcal{G}_x \to \mathcal{F}_x$ is injective, then there exists an open neighbourhood $U \subset X$ of $x$ such that $\varphi|_U$ is injective. \item If $\varphi_x : \mathcal{G}_x \to \mathcal{F}_x$ is surjective, then there exists an open neighbourhood $U \subset X$ of $x$ such that $\varphi|_U$ is surjective. \item If $\varphi_x : \mathcal{G}_x \to \mathcal{F}_x$ is bijective, then there exists an open neighbourhood $U \subset X$ of $x$ such that $\varphi|_U$ is an isomorphism. \end{enumerate}


coherent-lemma-map-stalks-local-map

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. Let $x \in X$. Suppose $\psi : \mathcal{G}_x \to \mathcal{F}_x$ is a map of $\mathcal{O}_{X, x}$-modules. Then there exists an open neighbourhood $U \subset X$ of $x$ and a map $\varphi : \mathcal{G}|_U \to \mathcal{F}|_U$ such that $\varphi_x = \psi$.


coherent-lemma-coherent-support-closed

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $\text{Supp}(\mathcal{F})$ is closed, and $\mathcal{F}$ comes from a coherent sheaf on the scheme theoretic support of $\mathcal{F}$, see Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-support}.


coherent-lemma-i-star-equivalence

Let $i : Z \to X$ be a closed immersion of locally Noetherian schemes. Let $\mathcal{I} \subset \mathcal{O}_X$ be the quasi-coherent sheaf of ideals cutting out $Z$. The functor $i_*$ induces an equivalence between the category of coherent $\mathcal{O}_X$-modules annihilated by $\mathcal{I}$ and the category of coherent $\mathcal{O}_Z$-modules.


coherent-lemma-finite-pushforward-coherent

Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume $f$ is finite and $Y$ locally Noetherian. Then $R^pf_*\mathcal{F} = 0$ for $p > 0$ and $f_*\mathcal{F}$ is coherent if $\mathcal{F}$ is coherent.


coherent-lemma-coherent-support-dimension-0

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf with $\dim(\text{Supp}(\mathcal{F})) \leq 0$. Then $\mathcal{F}$ is generated by global sections and $H^i(X, \mathcal{F}) = 0$ for $i > 0$.


coherent-lemma-pushforward-coherent-on-open

Let $X$ be a scheme. Let $j : U \to X$ be the inclusion of an open. Let $T \subset X$ be a closed subset contained in $U$. If $\mathcal{F}$ is a coherent $\mathcal{O}_U$-module with $\text{Supp}(\mathcal{F}) \subset T$, then $j_*\mathcal{F}$ is a coherent $\mathcal{O}_X$-module.


coherent-lemma-acc-coherent

Let $X$ be a Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. The ascending chain condition holds for quasi-coherent submodules of $\mathcal{F}$. In other words, given any sequence $$ \mathcal{F}_1 \subset \mathcal{F}_2 \subset \ldots \subset \mathcal{F} $$ of quasi-coherent submodules, then $\mathcal{F}_n = \mathcal{F}_{n + 1} = \ldots $ for some $n \geq 0$.


coherent-lemma-power-ideal-kills-sheaf

Let $X$ be a Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals corresponding to a closed subscheme $Z \subset X$. Then there is some $n \geq 0$ such that $\mathcal{I}^n\mathcal{F} = 0$ if and only if $\text{Supp}(\mathcal{F}) \subset Z$ (set theoretically).


coherent-lemma-Artin-Rees

Let $X$ be a Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $\mathcal{G} \subset \mathcal{F}$ be a quasi-coherent subsheaf. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Then there exists a $c \geq 0$ such that for all $n \geq c$ we have $$ \mathcal{I}^{n - c}(\mathcal{I}^c\mathcal{F} \cap \mathcal{G}) = \mathcal{I}^n\mathcal{F} \cap \mathcal{G} $$


coherent-lemma-homs-over-open

Let $X$ be a Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{G}$ be coherent $\mathcal{O}_X$-module. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Denote $Z \subset X$ the corresponding closed subscheme and set $U = X \setminus Z$. There is a canonical isomorphism $$ \colim_n \Hom_{\mathcal{O}_X}(\mathcal{I}^n\mathcal{G}, \mathcal{F}) \longrightarrow \Hom_{\mathcal{O}_U}(\mathcal{G}|_U, \mathcal{F}|_U). $$ In particular we have an isomorphism $$ \colim_n \Hom_{\mathcal{O}_X}( \mathcal{I}^n, \mathcal{F}) \longrightarrow \Gamma(U, \mathcal{F}). $$


coherent-lemma-extend-coherent

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. Let $U \subset X$ be open and let $\varphi : \mathcal{F}|_U \to \mathcal{G}|_U$ be an $\mathcal{O}_U$-module map. Then there exists a coherent submodule $\mathcal{F}' \subset \mathcal{F}$ agreeing with $\mathcal{F}$ over $U$ such that $\varphi$ extends to $\varphi' : \mathcal{F}' \to \mathcal{G}$.


coherent-lemma-hom-into-depth

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules and $x \in X$. \begin{enumerate} \item If $\mathcal{G}_x$ has depth $\geq 1$, then $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})_x$ has depth $\geq 1$. \item If $\mathcal{G}_x$ has depth $\geq 2$, then $\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})_x$ has depth $\geq 2$. \end{enumerate}


coherent-lemma-prepare-filter-support

Let $X$ be a Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf on $X$. Suppose that $\text{Supp}(\mathcal{F}) = Z \cup Z'$ with $Z$, $Z'$ closed. Then there exists a short exact sequence of coherent sheaves $$ 0 \to \mathcal{G}' \to \mathcal{F} \to \mathcal{G} \to 0 $$ with $\text{Supp}(\mathcal{G}') \subset Z'$ and $\text{Supp}(\mathcal{G}) \subset Z$.


coherent-lemma-prepare-filter-irreducible

Let $X$ be a Noetherian scheme. Let $i : Z \to X$ be an integral closed subscheme. Let $\xi \in Z$ be the generic point. Let $\mathcal{F}$ be a coherent sheaf on $X$. Assume that $\mathcal{F}_\xi$ is annihilated by $\mathfrak m_\xi$. Then there exists an integer $r \geq 0$ and a sheaf of ideals $\mathcal{I} \subset \mathcal{O}_Z$ and an injective map of coherent sheaves $$ i_*\left(\mathcal{I}^{\oplus r}\right) \to \mathcal{F} $$ which is an isomorphism in a neighbourhood of $\xi$.


coherent-lemma-coherent-filter

Let $X$ be a Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf on $X$. There exists a filtration $$ 0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \mathcal{F} $$ by coherent subsheaves such that for each $j = 1, \ldots, m$ there exists an integral closed subscheme $Z_j \subset X$ and a sheaf of ideals $\mathcal{I}_j \subset \mathcal{O}_{Z_j}$ such that $$ \mathcal{F}_j/\mathcal{F}_{j - 1} \cong (Z_j \to X)_* \mathcal{I}_j $$


coherent-lemma-property-initial

Let $X$ be a Noetherian scheme. Let $\mathcal{P}$ be a property of coherent sheaves on $X$. Assume \begin{enumerate} \item For any short exact sequence of coherent sheaves $$ 0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0 $$ if $\mathcal{F}_i$, $i = 1, 2$ have property $\mathcal{P}$ then so does $\mathcal{F}$. \item For every integral closed subscheme $Z \subset X$ and every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_Z$ we have $\mathcal{P}$ for $i_*\mathcal{I}$. \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf on $X$.


coherent-lemma-property-irreducible

Let $X$ be a Noetherian scheme. Let $Z_0 \subset X$ be an irreducible closed subset with generic point $\xi$. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ with support contained in $Z_0$ such that \begin{enumerate} \item For any short exact sequence of coherent sheaves if two out of three of them have property $\mathcal{P}$ then so does the third. \item For every integral closed subscheme $Z \subset Z_0 \subset X$, $Z \not = Z_0$ and every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_Z$ we have $\mathcal{P}$ for $(Z \to X)_*\mathcal{I}$. \item There exists some coherent sheaf $\mathcal{G}$ on $X$ such that \begin{enumerate} \item $\text{Supp}(\mathcal{G}) = Z_0$, \item $\mathcal{G}_\xi$ is annihilated by $\mathfrak m_\xi$, \item $\dim_{\kappa(\xi)} \mathcal{G}_\xi = 1$, and \item property $\mathcal{P}$ holds for $\mathcal{G}$. \end{enumerate} \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z_0$.


coherent-lemma-property

Let $X$ be a Noetherian scheme. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that \begin{enumerate} \item For any short exact sequence of coherent sheaves if two out of three of them have property $\mathcal{P}$ then so does the third. \item For every integral closed subscheme $Z \subset X$ with generic point $\xi$ there exists some coherent sheaf $\mathcal{G}$ such that \begin{enumerate} \item $\text{Supp}(\mathcal{G}) = Z$, \item $\mathcal{G}_\xi$ is annihilated by $\mathfrak m_\xi$, \item $\dim_{\kappa(\xi)} \mathcal{G}_\xi = 1$, and \item property $\mathcal{P}$ holds for $\mathcal{G}$. \end{enumerate} \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf on $X$.


coherent-lemma-property-irreducible-higher-rank-cohomological

Let $X$ be a Noetherian scheme. Let $Z_0 \subset X$ be an irreducible closed subset with generic point $\xi$. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that \begin{enumerate} \item For any short exact sequence of coherent sheaves $$ 0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0 $$ if $\mathcal{F}_i$, $i = 1, 2$ have property $\mathcal{P}$ then so does $\mathcal{F}$. \item If $\mathcal{P}$ holds for $\mathcal{F}^{\oplus r}$ for some $r \geq 1$, then it holds for $\mathcal{F}$. \item For every integral closed subscheme $Z \subset Z_0 \subset X$, $Z \not = Z_0$ and every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_Z$ we have $\mathcal{P}$ for $(Z \to X)_*\mathcal{I}$. \item There exists some coherent sheaf $\mathcal{G}$ such that \begin{enumerate} \item $\text{Supp}(\mathcal{G}) = Z_0$, \item $\mathcal{G}_\xi$ is annihilated by $\mathfrak m_\xi$, and \item for every quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_X$ such that $\mathcal{J}_\xi = \mathcal{O}_{X, \xi}$ there exists a quasi-coherent subsheaf $\mathcal{G}' \subset \mathcal{J}\mathcal{G}$ with $\mathcal{G}'_\xi = \mathcal{G}_\xi$ and such that $\mathcal{P}$ holds for $\mathcal{G}'$. \end{enumerate} \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z_0$.


coherent-lemma-property-higher-rank-cohomological

Let $X$ be a Noetherian scheme. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that \begin{enumerate} \item For any short exact sequence of coherent sheaves $$ 0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0 $$ if $\mathcal{F}_i$, $i = 1, 2$ have property $\mathcal{P}$ then so does $\mathcal{F}$. \item If $\mathcal{P}$ holds  for $\mathcal{F}^{\oplus r}$ for some $r \geq 1$, then it holds for $\mathcal{F}$. \item For every integral closed subscheme $Z \subset X$ with generic point $\xi$ there exists some coherent sheaf $\mathcal{G}$ such that \begin{enumerate} \item $\text{Supp}(\mathcal{G}) = Z$, \item $\mathcal{G}_\xi$ is annihilated by $\mathfrak m_\xi$, and \item for every quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_X$ such that $\mathcal{J}_\xi = \mathcal{O}_{X, \xi}$ there exists a quasi-coherent subsheaf $\mathcal{G}' \subset \mathcal{J}\mathcal{G}$ with $\mathcal{G}'_\xi = \mathcal{G}_\xi$ and such that $\mathcal{P}$ holds for $\mathcal{G}'$. \end{enumerate} \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf on $X$.


coherent-lemma-finite-morphism-Noetherian

Let $f : Y \to X$ be a morphism of schemes. Assume $f$ is finite, surjective and $X$ locally Noetherian. Let $Z \subset X$ be an integral closed subscheme with generic point $\xi$. Then there exists a coherent sheaf $\mathcal{F}$ on $Y$ such that the support of $f_*\mathcal{F}$ is equal to $Z$ and $(f_*\mathcal{F})_\xi$ is annihilated by $\mathfrak m_\xi$.


coherent-lemma-affine-morphism-projection-ideal

Let $f : Y \to X$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $Y$. Let $\mathcal{I}$ be a quasi-coherent sheaf of ideals on $X$. If the morphism $f$ is affine then $\mathcal{I}f_*\mathcal{F} = f_*(f^{-1}\mathcal{I}\mathcal{F})$.


coherent-lemma-image-affine-finite-morphism-affine-Noetherian

Let $f : Y \to X$ be a morphism of schemes. Assume \begin{enumerate} \item $f$ finite, \item $f$ surjective, \item $Y$ affine, and \item $X$ Noetherian. \end{enumerate} Then $X$ is affine.


coherent-lemma-coherent-projective

Let $R$ be a Noetherian ring. Let $n \geq 0$ be an integer. For every coherent sheaf $\mathcal{F}$ on $\mathbf{P}^n_R$ we have the following: \begin{enumerate} \item There exists an $r \geq 0$ and $d_1, \ldots, d_r \in \mathbf{Z}$ and a surjection $$ \bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{O}_{\mathbf{P}^n_R}(d_j) \longrightarrow \mathcal{F}. $$ \item We have $H^i(\mathbf{P}^n_R, \mathcal{F}) = 0$ unless $0 \leq i \leq n$. \item For any $i$ the cohomology group $H^i(\mathbf{P}^n_R, \mathcal{F})$ is a finite $R$-module. \item If $i > 0$, then $H^i(\mathbf{P}^n_R, \mathcal{F}(d)) = 0$ for all $d$ large enough. \item For any $k \in \mathbf{Z}$ the graded $R[T_0, \ldots, T_n]$-module $$ \bigoplus\nolimits_{d \geq k} H^0(\mathbf{P}^n_R, \mathcal{F}(d)) $$ is a finite $R[T_0, \ldots, T_n]$-module. \end{enumerate}


coherent-lemma-coherent-on-proj

Let $A$ be a graded ring such that $A_0$ is Noetherian and $A$ is generated by finitely many elements of $A_1$ over $A_0$. Set $X = \text{Proj}(A)$. Then $X$ is a Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. \begin{enumerate} \item There exists an $r \geq 0$ and $d_1, \ldots, d_r \in \mathbf{Z}$ and a surjection $$ \bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{O}_X(d_j) \longrightarrow \mathcal{F}. $$ \item For any $p$ the cohomology group $H^p(X, \mathcal{F})$ is a finite $A_0$-module. \item If $p > 0$, then $H^p(X, \mathcal{F}(d)) = 0$ for all $d$ large enough. \item For any $k \in \mathbf{Z}$ the graded $A$-module $$ \bigoplus\nolimits_{d \geq k} H^0(X, \mathcal{F}(d)) $$ is a finite $A$-module. \end{enumerate}


coherent-lemma-recover-tail-graded-module

Let $A$ be a graded ring such that $A_0$ is Noetherian and $A$ is generated by finitely many elements of $A_1$ over $A_0$. Let $M$ be a finite graded $A$-module. Set $X = \text{Proj}(A)$ and let $\widetilde{M}$ be the quasi-coherent $\mathcal{O}_X$-module on $X$ associated to $M$. The maps $$ M_n \longrightarrow \Gamma(X, \widetilde{M}(n)) $$ from Constructions, Lemma \ref{constructions-lemma-apply-modules} are isomorphisms for all sufficiently large $n$.


coherent-lemma-coherent-on-proj-general

Let $A$ be a Noetherian graded ring. Set $X = \text{Proj}(A)$. Then $X$ is a Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. \begin{enumerate} \item There exists an $r \geq 0$ and $d_1, \ldots, d_r \in \mathbf{Z}$ and a surjection $$ \bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{O}_X(d_j) \longrightarrow \mathcal{F}. $$ \item For any $p$ the cohomology group $H^p(X, \mathcal{F})$ is a finite $A_0$-module. \item If $p > 0$, then $H^p(X, \mathcal{F}(d)) = 0$ for all $d$ large enough. \item For any $k \in \mathbf{Z}$ the graded $A$-module $$ \bigoplus\nolimits_{d \geq k} H^0(X, \mathcal{F}(d)) $$ is a finite $A$-module. \end{enumerate}


coherent-lemma-recover-tail-graded-module-general

Let $A$ be a Noetherian graded ring and let $d$ be the lcm of generators of $A$ over $A_0$. Let $M$ be a finite graded $A$-module. Set $X = \text{Proj}(A)$ and let $\widetilde{M}$ be the quasi-coherent $\mathcal{O}_X$-module on $X$ associated to $M$. Let $k \in \mathbf{Z}$. \begin{enumerate} \item $N' = \bigoplus_{n \geq k} H^0(X, \widetilde{M(n)})$ is a finite $A$-module, \item $N = \bigoplus_{n \geq k} H^0(X, \widetilde{M}(n))$ is a finite $A$-module, \item there is a canonical map $N \to N'$, \item if $k$ is small enough there is a canonical map $M \to N'$, \item the map $M_n \to N'_n$ is an isomorphism for $n \gg 0$, \item $N_n \to N'_n$ is an isomorphism for $d | n$. \end{enumerate}