topologies-lemma-change-alpha

Let $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$. Suppose given big sites $\Sch_\tau$ and $\Sch'_\tau$. Assume that $\Sch_\tau$ is contained in $\Sch'_\tau$. The inclusion functor $\Sch_\tau \to \Sch'_\tau$ satisfies the assumptions of Sites, Lemma \ref{sites-lemma-bigger-site}. There are morphisms of topoi \begin{eqnarray*} g : \Sh(\Sch_\tau) & \longrightarrow & \Sh(\Sch'_\tau) \\ f : \Sh(\Sch'_\tau) & \longrightarrow & \Sh(\Sch_\tau) \end{eqnarray*} such that $f \circ g \cong \text{id}$. For any object $S$ of $\Sch_\tau$ the inclusion functor $(\Sch/S)_\tau \to (\Sch'/S)_\tau$ satisfies the assumptions of Sites, Lemma \ref{sites-lemma-bigger-site} also. Hence similarly we obtain morphisms \begin{eqnarray*} g : \Sh((\Sch/S)_\tau) & \longrightarrow & \Sh((\Sch'/S)_\tau) \\ f : \Sh((\Sch'/S)_\tau) & \longrightarrow & \Sh((\Sch/S)_\tau) \end{eqnarray*} with $f \circ g \cong \text{id}$.


topologies-lemma-extend

Let $S$ be a scheme. Let $\mathcal{C}$ be a full subcategory of the category $\Sch/S$ of all schemes over $S$. Assume \begin{enumerate} \item if $X \to S$ is an object of $\mathcal{C}$ and $U \subset X$ is an affine open, then $U \to S$ is isomorphic to an object of $\mathcal{C}$, \item if $V$ is an affine scheme lying over an affine open $U \subset S$ such that $V \to U$ is of finite presentation, then $V \to S$ is isomorphic to an object of $\mathcal{C}$. \end{enumerate} Let $F : \mathcal{C}^{opp} \to \textit{Sets}$ be a functor. Assume \begin{enumerate} \item[(a)] for any Zariski covering $\{f_i : X_i \to X\}_{i \in I}$ with $X, X_i$ objects of $\mathcal{C}$ we have the sheaf condition for $F$ and this family\footnote{As we do not know that $X_i \times_X X_j$ is in $\mathcal{C}$ this has to be interpreted as follows: by property (1) there exist Zariski coverings $\{U_{ijk} \to X_i \times_X X_j\}_{k \in K_{ij}}$ with $U_{ijk}$ an object of $\mathcal{C}$. Then the sheaf condition says that $F(X)$ is the equalizer of the two maps from $\prod F(X_i)$ to $\prod F(U_{ijk})$.}, \item[(b)] if $X = \lim X_i$ is a directed limit of affine schemes over $S$ with $X, X_i$ objects of $\mathcal{C}$, then $F(X) = \colim F(X_i)$. \end{enumerate} Then there is a unique way to extend $F$ to a functor $F' : (\Sch/S)^{opp} \to \textit{Sets}$ satisfying the analogues of (a) and (b), i.e., $F'$ satisfies the sheaf condition for any Zariski covering and $F'(X) = \colim F'(X_i)$ whenever $X = \lim X_i$ is a directed limit of affine schemes over $S$.


topologies-lemma-limit-fppf-topology

Let $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$. Let $T$ be an affine scheme which is written as a limit $T = \lim_{i \in I} T_i$ of a directed inverse system of affine schemes. \begin{enumerate} \item Let $\mathcal{V} = \{V_j \to T\}_{j = 1, \ldots, m}$ be a standard $\tau$-covering of $T$, see Definitions \ref{definition-standard-Zariski}, \ref{definition-standard-etale}, \ref{definition-standard-smooth}, \ref{definition-standard-syntomic}, and \ref{definition-standard-fppf}. Then there exists an index $i$ and a standard $\tau$-covering $\mathcal{V}_i = \{V_{i, j} \to T_i\}_{j = 1, \ldots, m}$ whose base change $T \times_{T_i} \mathcal{V}_i$ to $T$ is isomorphic to $\mathcal{V}$. \item Let $\mathcal{V}_i$, $\mathcal{V}'_i$ be a pair of standard $\tau$-coverings of $T_i$. If $f : T \times_{T_i} \mathcal{V}_i \to T \times_{T_i} \mathcal{V}'_i$ is a morphism of coverings of $T$, then there exists an index $i' \geq i$ and a morphism $f_{i'} : T_{i'} \times_{T_i} \mathcal{V} \to T_{i'} \times_{T_i} \mathcal{V}'_i$ whose base change to $T$ is $f$. \item If $f, g : \mathcal{V} \to \mathcal{V}'_i$ are morphisms of standard $\tau$-coverings of $T_i$ whose base changes $f_T, g_T$ to $T$ are equal then there exists an index $i' \geq i$ such that $f_{T_{i'}} = g_{T_{i'}}$. \end{enumerate} In other words, the category of standard $\tau$-coverings of $T$ is the colimit over $I$ of the categories of standard $\tau$-coverings of $T_i$.


topologies-lemma-extend-sheaf-general

Let $S$, $\mathcal{C}$, $F$ satisfy conditions (1), (2), (a), and (b) of Lemma \ref{lemma-extend} and denote $F' : (\Sch/S)^{opp} \to \textit{Sets}$ the unique extension constructed in the lemma. Let $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$. Assume \begin{enumerate} \item[(c)] for any standard $\tau$-covering $\{V_i \to V\}_{i = 1, \ldots, n}$ of affines in $\Sch/S$ such that $V \to S$ factors through an affine open $U \subset S$ and $V \to U$ is of finite presentation, the sheaf condition hold for $F$ and $\{V_i \to V\}_{i = 1, \ldots, n}$\footnote{This makes sense as $V$, $V_i$, and $V_i \times_V V_j$ are isomorphic to objects of $\mathcal{C}$ by (2).}. \end{enumerate} Then $F'$ satisfies the sheaf condition for all $\tau$-coverings.


topologies-lemma-extend-sheaf

Let $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$. Let $S$ be a scheme contained in a big site $\Sch_\tau$. Let $F : (\Sch/S)_\tau^{opp} \to \textit{Sets}$ be a $\tau$-sheaf satisfying property (b) of Lemma \ref{lemma-extend} with $\mathcal{C} = (\Sch/S)_\tau$. Then the extension $F'$ of $F$ to the category of all schemes over $S$ satisfies the sheaf condition for all $\tau$-coverings.


pic-lemma-hilb-d-sheaf

Let $X \to S$ be a morphism of schemes. The functor $\Hilbfunctor^d_{X/S}$ satisfies the sheaf property for the fpqc topology (Topologies, Definition \ref{topologies-definition-sheaf-property-fpqc}).


pic-lemma-hilb-d-of-closed

Let $S$ be a scheme. Let $i : X \to Y$ be a closed immersion of schemes. If $\Hilbfunctor^d_{Y/S}$ is representable by a scheme, so is $\Hilbfunctor^d_{X/S}$ and the corresponding morphism of schemes $\underline{\Hilbfunctor}^d_{X/S} \to \underline{\Hilbfunctor}^d_{Y/S}$ is a closed immersion.


pic-lemma-hilb-d-separated

Let $X \to S$ be a morphism of schemes. If $X \to S$ is separated and $\Hilbfunctor^d_{X/S}$ is representable, then $\underline{\Hilbfunctor}^d_{X/S} \to S$ is separated.


pic-lemma-hilb-d-An

Let $X \to S$ be a morphism of affine schemes. Let $d \geq 0$. Then $\Hilbfunctor^d_{X/S}$ is representable.


pic-lemma-divisors-on-curves

Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$. Let $D \subset X$ be a closed subscheme. Consider the following conditions \begin{enumerate} \item $D \to S$ is finite locally free, \item $D$ is a relative effective Cartier divisor on $X/S$, \item $D \to S$ is locally quasi-finite, flat, and locally of finite presentation, and \item $D \to S$ is locally quasi-finite and flat. \end{enumerate} We always have the implications $$ (1) \Rightarrow (2) \Leftrightarrow (3) \Rightarrow (4) $$ If $S$ is locally Noetherian, then the last arrow is an if and only if. If $X \to S$ is proper (and $S$ arbitrary), then the first arrow is an if and only if.


pic-lemma-sum-divisors-on-curves

Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$. Let $D_1, D_2 \subset X$ be closed subschemes finite locally free of degrees $d_1$, $d_2$ over $S$. Then $D_1 + D_2$ is finite locally free of degree $d_1 + d_2$ over $S$.


pic-lemma-difference-divisors-on-curves

Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$. Let $D_1, D_2 \subset X$ be closed subschemes finite locally free of degrees $d_1$, $d_2$ over $S$. If $D_1 \subset D_2$ (as closed subschemes) then there is a closed subscheme $D \subset X$ finite locally free of degree $d_2 - d_1$ over $S$ such that $D_2 = D_1 + D$.


pic-lemma-universal-object

Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$ such that the functors $\Hilbfunctor^d_{X/S}$ are representable. The morphism $\underline{\Hilbfunctor}^d_{X/S} \times_S X \to \underline{\Hilbfunctor}^{d + 1}_{X/S}$ is finite locally free of degree $d + 1$.


pic-lemma-hilb-d-smooth

Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$ such that the functors $\Hilbfunctor^d_{X/S}$ are representable. The schemes $\underline{\Hilbfunctor}^d_{X/S}$ are smooth over $S$ of relative dimension $d$.


pic-lemma-flat-geometrically-connected-fibres

Let $f : X \to S$ be as in Definition \ref{definition-picard-functor}. If $\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}$ is an isomorphism for all $T \in \Ob((\Sch/S)_{fppf})$, then $$ 0 \to \Pic(T) \to \Pic(X_T) \to \Picardfunctor_{X/S}(T) $$ is an exact sequence for all $T$.


pic-lemma-criterion

Let $k$ be a field. Let $G : (\Sch/k)^{opp} \to \textit{Groups}$ be a functor. With terminology as in Schemes, Definition \ref{schemes-definition-representable-by-open-immersions}, assume that \begin{enumerate} \item $G$ satisfies the sheaf property for the Zariski topology, \item there exists a subfunctor $F \subset G$ such that \begin{enumerate} \item $F$ is representable, \item $F \subset G$ is representable by open immersion, \item for every field extension $K$ of $k$ and $g \in G(K)$ there exists a $g' \in G(k)$ such that $g'g \in F(K)$. \end{enumerate} \end{enumerate} Then $G$ is representable by a group scheme over $k$.


pic-lemma-check-conditions

Let $k$ be a field. Let $X$ be a smooth projective curve over $k$ which has a $k$-rational point. Then the hypotheses of Lemma \ref{lemma-flat-geometrically-connected-fibres-with-section} are satisfied.


pic-lemma-define-open

Let $k$ be a field. Let $X$ be a smooth projective curve over $k$ with a $k$-rational point $\sigma$. For a scheme $T$ over $k$, consider the subset $F(T) \subset \Picardfunctor_{X/k, \sigma}(T)$ consisting of $\mathcal{L}$ such that $Rf_{T, *}\mathcal{L}$ is isomorphic to an invertible $\mathcal{O}_T$-module placed in degree $0$. Then $F \subset \Picardfunctor_{X/k, \sigma}$ is a subfunctor and the inclusion is representable by open immersions.


pic-lemma-open-representable

Let $k$ be a field. Let $X$ be a smooth projective curve of genus $g$ over $k$ with a $k$-rational point $\sigma$. The open subfunctor $F$ defined in Lemma \ref{lemma-define-open} is representable by an open subscheme of $\underline{\Hilbfunctor}^g_{X/k}$.


pic-lemma-twist-with-general-divisor

Let $k$ be a separably closed field. Let $X$ be a smooth projective curve of genus $g$ over $k$. Let $K/k$ be a field extension and let $\mathcal{L}$ be an invertible sheaf on $X_K$. Then there exists an invertible sheaf $\mathcal{L}_0$ on $X$ such that $\dim_K H^0(X_K, \mathcal{L} \otimes_{\mathcal{O}_{X_K}} \mathcal{L}_0|_{X_K}) = 1$ and $\dim_K H^1(X_K, \mathcal{L} \otimes_{\mathcal{O}_{X_K}} \mathcal{L}_0|_{X_K}) = 0$.


pic-lemma-picard-pieces

Let $k$ be a separably closed field. Let $X$ be a smooth projective curve of genus $g$ over $k$. \begin{enumerate} \item $\underline{\Picardfunctor}_{X/k}$ is a disjoint union of $g$-dimensional smooth proper varieties $\underline{\Picardfunctor}^d_{X/k}$, \item $k$-points of $\underline{\Picardfunctor}^d_{X/k}$ correspond to invertible $\mathcal{O}_X$-modules of degree $d$, \item $\underline{\Picardfunctor}^0_{X/k}$ is an open and closed subgroup scheme, \item for $d \geq 0$ there is a canonical morphism $\gamma_d : \underline{\Hilbfunctor}^d_{X/k} \to \underline{\Picardfunctor}^d_{X/k}$ \item the morphisms $\gamma_d$ are surjective for $d \geq g$ and smooth for $d \geq 2g - 1$, \item the morphism $\underline{\Hilbfunctor}^g_{X/k} \to \underline{\Picardfunctor}^g_{X/k}$ is birational. \end{enumerate}


pic-lemma-torsion-descends

Let $k$ be a field of characteristic $p > 0$. Let $X$ be a quasi-compact and quasi-separated scheme over $k$ with $H^0(X, \mathcal{O}_X) = k$. Let $n$ be an integer prime to $p$. Then the map $$ \Pic(X)[n] \longrightarrow \Pic(X_{k'})[n] $$ is bijective for any purely inseparable extension $k'/k$.


pic-proposition-hilb-d-representable

Let $X \to S$ be a morphism of schemes. Let $d \geq 0$. Assume for all $(s, x_1, \ldots, x_d)$ where $s \in S$ and $x_1, \ldots, x_d \in X_s$ there exists an affine open $U \subset X$ with $x_1, \ldots, x_d \in U$. Then $\Hilbfunctor^d_{X/S}$ is representable by a scheme.


pic-proposition-hilb-d

Let $X$ be a geometrically irreducible smooth proper curve over a field $k$. \begin{enumerate} \item The functors $\Hilbfunctor^d_{X/k}$ are representable by smooth proper varieties $\underline{\Hilbfunctor}^d_{X/k}$ of dimension $d$ over $k$. \item For a field extension $k'/k$ the $k'$-rational points of $\underline{\Hilbfunctor}^d_{X/k}$ are in $1$-to-$1$ bijection with effective Cartier divisors of degree $d$ on $X_{k'}$. \item For $d_1, d_2 \geq 0$ there is a morphism $$ \underline{\Hilbfunctor}^{d_1}_{X/k} \times_k \underline{\Hilbfunctor}^{d_2}_{X/k} \longrightarrow \underline{\Hilbfunctor}^{d_1 + d_2}_{X/k} $$ which is finite locally free of degree ${d_1 + d_2 \choose d_1}$. \end{enumerate}


pic-proposition-pic-curve

Let $k$ be a separably closed field. Let $X$ be a smooth projective curve over $k$. The Picard functor $\Picardfunctor_{X/k}$ is representable.


constructions-lemma-relative-glueing

Let $S$ be a scheme. Let $\mathcal{B}$ be a basis for the topology of $S$. Suppose given the following data: \begin{enumerate} \item For every $U \in \mathcal{B}$ a scheme $f_U : X_U \to U$ over $U$. \item For $U, V \in \mathcal{B}$ with $V \subset U$ a morphism $\rho^U_V : X_V \to X_U$ over $U$. \end{enumerate} Assume that \begin{enumerate} \item[(a)] each $\rho^U_V$ induces an isomorphism $X_V \to f_U^{-1}(V)$ of schemes over $V$, \item[(b)] whenever $W, V, U \in \mathcal{B}$, with $W \subset V \subset U$ we have $\rho^U_W = \rho^U_V \circ \rho ^V_W$. \end{enumerate} Then there exists a morphism $f : X \to S$ of schemes and isomorphisms $i_U : f^{-1}(U) \to X_U$ over $U \in \mathcal{B}$ such that for $V, U \in \mathcal{B}$ with $V \subset U$ the composition $$ \xymatrix{ X_V \ar[r]^{i_V^{-1}} & f^{-1}(V) \ar[rr]^{inclusion} & & f^{-1}(U) \ar[r]^{i_U} & X_U } $$ is the morphism $\rho^U_V$. Moreover $X$ is unique up to unique isomorphism over $S$.


constructions-lemma-relative-glueing-sheaves

Let $S$ be a scheme. Let $\mathcal{B}$ be a basis for the topology of $S$. Suppose given the following data: \begin{enumerate} \item For every $U \in \mathcal{B}$ a scheme $f_U : X_U \to U$ over $U$. \item For every $U \in \mathcal{B}$ a quasi-coherent sheaf $\mathcal{F}_U$ over $X_U$. \item For every pair $U, V \in \mathcal{B}$ such that $V \subset U$ a morphism $\rho^U_V : X_V \to X_U$. \item  For every pair $U, V \in \mathcal{B}$ such that $V \subset U$ a morphism $\theta^U_V : (\rho^U_V)^*\mathcal{F}_U \to \mathcal{F}_V$. \end{enumerate} Assume that \begin{enumerate} \item[(a)] each $\rho^U_V$ induces an isomorphism $X_V \to f_U^{-1}(V)$ of schemes over $V$, \item[(b)] each $\theta^U_V$ is an isomorphism, \item[(c)] whenever $W, V, U \in \mathcal{B}$, with $W \subset V \subset U$ we have $\rho^U_W = \rho^U_V \circ \rho ^V_W$, \item[(d)] whenever $W, V, U \in \mathcal{B}$, with $W \subset V \subset U$ we have $\theta^U_W = \theta^V_W \circ (\rho^V_W)^*\theta^U_V$. \end{enumerate} Then there exists a morphism of schemes $f : X \to S$ together with a quasi-coherent sheaf $\mathcal{F}$ on $X$ and isomorphisms $i_U : f^{-1}(U) \to X_U$ and $\theta_U : i_U^*\mathcal{F}_U \to \mathcal{F}|_{f^{-1}(U)}$ over $U \in \mathcal{B}$ such that for $V, U \in \mathcal{B}$ with $V \subset U$ the composition $$ \xymatrix{ X_V \ar[r]^{i_V^{-1}} & f^{-1}(V) \ar[rr]^{inclusion} & & f^{-1}(U) \ar[r]^{i_U} & X_U } $$ is the morphism $\rho^U_V$, and the composition \begin{equation} \label{equation-glue} (\rho^U_V)^*\mathcal{F}_U = (i_V^{-1})^*((i_U^*\mathcal{F}_U)|_{f^{-1}(V)}) \xrightarrow{\theta_U|_{f^{-1}(V)}} (i_V^{-1})^*(\mathcal{F}|_{f^{-1}(V)}) \xrightarrow{\theta_V^{-1}} \mathcal{F}_V \end{equation} is equal to $\theta^U_V$. Moreover $(X, \mathcal{F})$ is unique up to unique isomorphism over $S$.


constructions-lemma-spec-inclusion

In Situation \ref{situation-relative-spec}. Suppose $U \subset U' \subset S$ are affine opens. Let $A = \mathcal{A}(U)$ and $A' = \mathcal{A}(U')$. The map of rings $A' \to A$ induces a morphism $\Spec(A) \to \Spec(A')$, and the diagram $$ \xymatrix{ \Spec(A) \ar[r] \ar[d] & \Spec(A') \ar[d] \\ U \ar[r] & U' } $$ is cartesian.


constructions-lemma-transitive-spec

In Situation \ref{situation-relative-spec}. Suppose $U \subset U' \subset U'' \subset S$ are affine opens. Let $A = \mathcal{A}(U)$, $A' = \mathcal{A}(U')$ and $A'' = \mathcal{A}(U'')$. The composition of the morphisms $\Spec(A) \to \Spec(A')$, and $\Spec(A') \to \Spec(A'')$ of Lemma \ref{lemma-spec-inclusion} gives the morphism $\Spec(A) \to \Spec(A'')$ of Lemma \ref{lemma-spec-inclusion}.


constructions-lemma-glue-relative-spec

In Situation \ref{situation-relative-spec}. There exists a morphism of schemes $$ \pi : \underline{\Spec}_S(\mathcal{A}) \longrightarrow S $$ with the following properties: \begin{enumerate} \item for every affine open $U \subset S$ there exists an isomorphism $i_U : \pi^{-1}(U) \to \Spec(\mathcal{A}(U))$, and \item for $U \subset U' \subset S$ affine open the composition $$ \xymatrix{ \Spec(\mathcal{A}(U)) \ar[r]^{i_U^{-1}} & \pi^{-1}(U) \ar[rr]^{inclusion} & & \pi^{-1}(U') \ar[r]^{i_{U'}} & \Spec(\mathcal{A}(U')) } $$ is the open immersion of Lemma \ref{lemma-spec-inclusion} above. \end{enumerate}


constructions-lemma-spec-base-change

In Situation \ref{situation-relative-spec}. Let $F$ be the functor associated to $(S, \mathcal{A})$ above. Let $g : S' \to S$ be a morphism of schemes. Set $\mathcal{A}' = g^*\mathcal{A}$. Let $F'$ be the functor associated to $(S', \mathcal{A}')$ above. Then there is a canonical isomorphism $$ F' \cong h_{S'} \times_{h_S} F $$ of functors.


constructions-lemma-spec-affine

In Situation \ref{situation-relative-spec}. Let $F$ be the functor associated to $(S, \mathcal{A})$ above. If $S$ is affine, then $F$ is representable by the affine scheme $\Spec(\Gamma(S, \mathcal{A}))$.


constructions-lemma-glueing-gives-functor-spec

In Situation \ref{situation-relative-spec}. The scheme $\pi : \underline{\Spec}_S(\mathcal{A}) \to S$ constructed in Lemma \ref{lemma-glue-relative-spec} and the scheme representing the functor $F$ are canonically isomorphic as schemes over $S$.


constructions-lemma-spec-properties

Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_S$-algebras. Let $\pi : \underline{\Spec}_S(\mathcal{A}) \to S$ be the relative spectrum of $\mathcal{A}$ over $S$. \begin{enumerate} \item For every affine open $U \subset S$ the inverse image $\pi^{-1}(U)$ is affine. \item For every morphism $g : S' \to S$ we have $S' \times_S \underline{\Spec}_S(\mathcal{A}) = \underline{\Spec}_{S'}(g^*\mathcal{A})$. \item The universal map $$ \mathcal{A} \longrightarrow \pi_*\mathcal{O}_{\underline{\Spec}_S(\mathcal{A})} $$ is an isomorphism of $\mathcal{O}_S$-algebras. \end{enumerate}


constructions-lemma-canonical-morphism

Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. By Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent} the sheaf $f_*\mathcal{O}_X$ is a quasi-coherent sheaf of $\mathcal{O}_S$-algebras. There is a canonical morphism $$ can : X \longrightarrow \underline{\Spec}_S(f_*\mathcal{O}_X) $$ of schemes over $S$. For any affine open $U \subset S$ the restriction $can|_{f^{-1}(U)}$ is identified with the canonical morphism $$ f^{-1}(U) \longrightarrow \Spec(\Gamma(f^{-1}(U), \mathcal{O}_X)) $$ coming from Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}.


constructions-lemma-category-vector-bundles

The category of vector bundles over a scheme $S$ is anti-equivalent to the category of quasi-coherent $\mathcal{O}_S$-modules.


constructions-lemma-standard-open

Let $S$ be a graded ring. Let $f \in S$ homogeneous of positive degree. \begin{enumerate} \item If $g\in S$ homogeneous of positive degree and $D_{+}(g) \subset D_{+}(f)$, then \begin{enumerate} \item $f$ is invertible in $S_g$, and $f^{\deg(g)}/g^{\deg(f)}$ is invertible in $S_{(g)}$, \item $g^e = af$ for some $e \geq 1$ and $a \in S$ homogeneous, \item there is a canonical $S$-algebra map $S_f \to S_g$, \item there is a canonical $S_0$-algebra map $S_{(f)} \to S_{(g)}$ compatible with the map $S_f \to S_g$, \item the map $S_{(f)} \to S_{(g)}$ induces an isomorphism $$ (S_{(f)})_{g^{\deg(f)}/f^{\deg(g)}} \cong S_{(g)}, $$ \item these maps induce a commutative diagram of topological spaces $$ \xymatrix{ D_{+}(g) \ar[d] & \{\mathbf{Z}\text{-graded primes of }S_g\} \ar[l] \ar[r] \ar[d] & \Spec(S_{(g)}) \ar[d] \\ D_{+}(f) & \{\mathbf{Z}\text{-graded primes of }S_f\} \ar[l] \ar[r] & \Spec(S_{(f)}) } $$ where the horizontal maps are homeomorphisms and the vertical maps are open immersions, \item there are compatible canonical $S_f$ and $S_{(f)}$-module maps $M_f \to M_g$ and $M_{(f)} \to M_{(g)}$ for any graded $S$-module $M$, and \item the map $M_{(f)} \to M_{(g)}$ induces an isomorphism $$ (M_{(f)})_{g^{\deg(f)}/f^{\deg(g)}} \cong M_{(g)}. $$ \end{enumerate} \item Any open covering of $D_{+}(f)$ can be refined to a finite open covering of the form $D_{+}(f) = \bigcup_{i = 1}^n D_{+}(g_i)$. \item Let $g_1, \ldots, g_n \in S$ be homogeneous of positive degree. Then $D_{+}(f) \subset \bigcup D_{+}(g_i)$ if and only if $g_1^{\deg(f)}/f^{\deg(g_1)}, \ldots, g_n^{\deg(f)}/f^{\deg(g_n)}$ generate the unit ideal in $S_{(f)}$. \end{enumerate}


constructions-lemma-proj-sheaves

Let $S$ be a graded ring. Let $M$ be a graded $S$-module. Let $\widetilde M$ be the sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules associated to $M$. \begin{enumerate} \item For every $f \in S$ homogeneous of positive degree we have $$ \Gamma(D_{+}(f), \mathcal{O}_{\text{Proj}(S)}) = S_{(f)}. $$ \item For every $f\in S$ homogeneous of positive degree we have $\Gamma(D_{+}(f), \widetilde M) = M_{(f)}$ as an $S_{(f)}$-module. \item Whenever $D_{+}(g) \subset D_{+}(f)$ the restriction mappings on $\mathcal{O}_{\text{Proj}(S)}$ and $\widetilde M$ are the maps $S_{(f)} \to S_{(g)}$ and $M_{(f)} \to M_{(g)}$ from Lemma \ref{lemma-standard-open}. \item Let $\mathfrak p$ be a homogeneous prime of $S$ not containing $S_{+}$, and let $x \in \text{Proj}(S)$ be the corresponding point. We have $\mathcal{O}_{\text{Proj}(S), x} = S_{(\mathfrak p)}$. \item Let $\mathfrak p$ be a homogeneous prime of $S$ not containing $S_{+}$, and let $x \in \text{Proj}(S)$ be the corresponding point. We have $\mathcal{F}_x = M_{(\mathfrak p)}$ as an $S_{(\mathfrak p)}$-module. \item \label{item-map} There is a canonical ring map $ S_0 \longrightarrow \Gamma(\text{Proj}(S), \widetilde S) $ and a canonical $S_0$-module map $ M_0 \longrightarrow \Gamma(\text{Proj}(S), \widetilde M) $ compatible with the descriptions of sections over standard opens and stalks above. \end{enumerate} Moreover, all these identifications are functorial in the graded $S$-module $M$. In particular, the functor $M \mapsto \widetilde M$ is an exact functor from the category of graded $S$-modules to the category of $\mathcal{O}_{\text{Proj}(S)}$-modules.


constructions-lemma-standard-open-proj

Let $S$ be a graded ring. Let $f \in S$ be homogeneous of positive degree. Suppose that $D(g) \subset \Spec(S_{(f)})$ is a standard open. Then there exists a $h \in S$ homogeneous of positive degree such that $D(g)$ corresponds to $D_{+}(h) \subset D_{+}(f)$ via the homeomorphism of Algebra, Lemma \ref{algebra-lemma-topology-proj}. In fact we can take $h$ such that $g = h/f^n$ for some $n$.


constructions-lemma-proj-scheme

Let $S$ be a graded ring. The locally ringed space $\text{Proj}(S)$ is a scheme. The standard opens $D_{+}(f)$ are affine opens. For any graded $S$-module $M$ the sheaf $\widetilde M$ is a quasi-coherent sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules.


constructions-lemma-proj-separated

Let $S$ be a graded ring. The scheme $\text{Proj}(S)$ is separated.


constructions-lemma-proj-quasi-compact

Let $S$ be a graded ring. The scheme $\text{Proj}(S)$ is quasi-compact if and only if there exist finitely many homogeneous elements $f_1, \ldots, f_n \in S_{+}$ such that $S_{+} \subset \sqrt{(f_1, \ldots, f_n)}$. In this case $\text{Proj}(S) = D_+(f_1) \cup \ldots \cup D_+(f_n)$.


constructions-lemma-proj-valuative-criterion

Let $S$ be a graded ring. If $S$ is finitely generated as an algebra over $S_0$, then the morphism $\text{Proj}(S) \to \Spec(S_0)$ satisfies the existence and uniqueness parts of the valuative criterion, see Schemes, Definition \ref{schemes-definition-valuative-criterion}.


constructions-lemma-widetilde-tensor

Let $S$ be a graded ring. Let $(X, \mathcal{O}_X) = (\text{Proj}(S), \mathcal{O}_{\text{Proj}(S)})$ be the scheme of Lemma \ref{lemma-proj-scheme}. Let $f \in S_{+}$ be homogeneous. Let $x \in X$ be a point corresponding to the homogeneous prime $\mathfrak p \subset S$. Let $M$, $N$ be graded $S$-modules. There is a canonical map of $\mathcal{O}_{\text{Proj}(S)}$-modules $$ \widetilde M \otimes_{\mathcal{O}_X} \widetilde N \longrightarrow \widetilde{M \otimes_S N} $$ which induces the canonical map $ M_{(f)} \otimes_{S_{(f)}} N_{(f)} \to (M \otimes_S N)_{(f)} $ on sections over $D_{+}(f)$ and the canonical map $ M_{(\mathfrak p)} \otimes_{S_{(\mathfrak p)}} N_{(\mathfrak p)} \to (M \otimes_S N)_{(\mathfrak p)} $ on stalks at $x$. Moreover, the following diagram $$ \xymatrix{ M_0 \otimes_{S_0} N_0 \ar[r] \ar[d] & (M \otimes_S N)_0 \ar[d] \\ \Gamma(X, \widetilde M \otimes_{\mathcal{O}_X} \widetilde N) \ar[r] & \Gamma(X, \widetilde{M \otimes_R N}) } $$ is commutative where the vertical maps are given by (\ref{equation-map-global-sections}).


constructions-lemma-when-invertible

Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. Let $f \in S$ be homogeneous of degree $d > 0$. The sheaves $\mathcal{O}_X(nd)|_{D_{+}(f)}$ are invertible, and in fact trivial for all $n \in \mathbf{Z}$ (see Modules, Definition \ref{modules-definition-invertible}). The maps (\ref{equation-multiply}) restricted to $D_{+}(f)$ $$ \mathcal{O}_X(nd)|_{D_{+}(f)} \otimes_{\mathcal{O}_{D_{+}(f)}} \mathcal{O}_X(m)|_{D_{+}(f)} \longrightarrow \mathcal{O}_X(nd + m)|_{D_{+}(f)}, $$ the maps (\ref{equation-multiply-on-sheaf}) restricted to $D_+(f)$ $$ \mathcal{O}_X(nd)|_{D_{+}(f)} \otimes_{\mathcal{O}_{D_{+}(f)}} \mathcal{F}(m)|_{D_{+}(f)} \longrightarrow \mathcal{F}(nd + m)|_{D_{+}(f)}, $$ and the maps (\ref{equation-multiply-more-generally}) restricted to $D_{+}(f)$ $$ \widetilde M(nd)|_{D_{+}(f)} = \widetilde M|_{D_{+}(f)} \otimes_{\mathcal{O}_{D_{+}(f)}} \mathcal{O}_X(nd)|_{D_{+}(f)} \longrightarrow \widetilde{M(nd)}|_{D_{+}(f)} $$ are isomorphisms for all $n, m \in \mathbf{Z}$.


constructions-lemma-apply-modules

Let $S$ be a graded ring. Let $M$ be a graded $S$-module. Set $X = \text{Proj}(S)$. Assume $X$ is covered by the standard opens $D_+(f)$ with $f \in S_1$, e.g., if $S$ is generated by $S_1$ over $S_0$. Then the sheaves $\mathcal{O}_X(n)$ are invertible and the maps (\ref{equation-multiply}), (\ref{equation-multiply-on-sheaf}), and (\ref{equation-multiply-more-generally}) are isomorphisms. In particular, these maps induce isomorphisms $$ \mathcal{O}_X(1)^{\otimes n} \cong \mathcal{O}_X(n) \quad \text{and} \quad \widetilde{M} \otimes_{\mathcal{O}_X} \mathcal{O}_X(n) = \widetilde{M}(n) \cong \widetilde{M(n)} $$ Thus (\ref{equation-map-global-sections-degree-n}) becomes a map \begin{equation} \label{equation-map-global-sections-degree-n-simplified} M_n \longrightarrow \Gamma(X, \widetilde{M}(n)) \end{equation} and (\ref{equation-global-sections-more-generally}) becomes a map \begin{equation} \label{equation-global-sections-more-generally-simplified} M \longrightarrow \bigoplus\nolimits_{n \in \mathbf{Z}} \Gamma(X, \widetilde{M}(n)). \end{equation}


constructions-lemma-where-invertible

Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. Fix $d \geq 1$ an integer. The following open subsets of $X$ are equal: \begin{enumerate} \item The largest open subset $W = W_d \subset X$ such that each $\mathcal{O}_X(dn)|_W$ is invertible and all the multiplication maps $\mathcal{O}_X(nd)|_W \otimes_{\mathcal{O}_W} \mathcal{O}_X(md)|_W \to \mathcal{O}_X(nd + md)|_W$ (see \ref{equation-multiply}) are isomorphisms. \item The union of the open subsets $D_{+}(fg)$ with $f, g \in S$ homogeneous and $\deg(f) = \deg(g) + d$. \end{enumerate} Moreover, all the maps $\widetilde M(nd)|_W = \widetilde M|_W \otimes_{\mathcal{O}_W} \mathcal{O}_X(nd)|_W \to \widetilde{M(nd)}|_W$ (see \ref{equation-multiply-more-generally}) are isomorphisms.


constructions-lemma-principal-open

Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. Fix $d \geq 1$ an integer. Let $W = W_d \subset X$ be the open subscheme defined in Lemma \ref{lemma-where-invertible}. Let $n \geq 1$ and $f \in S_{nd}$. Denote $s \in \Gamma(W, \mathcal{O}_W(nd))$ the section which is the image of $f$ via (\ref{equation-global-sections}) restricted to $W$. Then $$ W_s = D_{+}(f) \cap W. $$


constructions-lemma-ample-on-proj

Let $S$ be a graded ring. Let $X = \text{Proj}(S)$. Let $Y \subset X$ be a quasi-compact open subscheme. Denote $\mathcal{O}_Y(n)$ the restriction of $\mathcal{O}_X(n)$ to $Y$. There exists an integer $d \geq 1$ such that \begin{enumerate} \item the subscheme $Y$ is contained in the open $W_d$ defined in Lemma \ref{lemma-where-invertible}, \item the sheaf $\mathcal{O}_Y(dn)$ is invertible for all $n \in \mathbf{Z}$, \item all the maps $\mathcal{O}_Y(nd) \otimes_{\mathcal{O}_Y} \mathcal{O}_Y(m) \longrightarrow \mathcal{O}_Y(nd + m)$ of Equation (\ref{equation-multiply}) are isomorphisms, \item all the maps $\widetilde M(nd)|_Y = \widetilde M|_Y \otimes_{\mathcal{O}_Y} \mathcal{O}_X(nd)|_Y \to \widetilde{M(nd)}|_Y$ (see \ref{equation-multiply-more-generally}) are isomorphisms, \item given $f \in S_{nd}$ denote $s \in \Gamma(Y, \mathcal{O}_Y(nd))$ the image of $f$ via (\ref{equation-global-sections}) restricted to $Y$, then $D_{+}(f) \cap Y = Y_s$, \item a basis for the topology on $Y$ is given by the collection of opens $Y_s$, where $s \in \Gamma(Y, \mathcal{O}_Y(nd))$, $n \geq 1$, and \item a basis for the topology of $Y$ is given by those opens $Y_s \subset Y$, for $s \in \Gamma(Y, \mathcal{O}_Y(nd))$, $n \geq 1$ which are affine. \end{enumerate}


constructions-lemma-comparison-proj-quasi-coherent

Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. 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, using (\ref{equation-global-sections-module}) and (\ref{equation-global-sections}). Then there is a canonical $\mathcal{O}_X$-module map $$ \widetilde{M} \longrightarrow \mathcal{F} $$ functorial in $\mathcal{F}$ such that the induced map $M_0 \to \Gamma(X, \mathcal{F})$ is the identity.


constructions-lemma-morphism-proj

Let $A$, $B$ be two graded rings. Set $X = \text{Proj}(A)$ and $Y = \text{Proj}(B)$. Let $\psi : A \to B$ be a graded ring map. Set $$ U(\psi) = \bigcup\nolimits_{f \in A_{+}\ \text{homogeneous}} D_{+}(\psi(f)) \subset Y. $$ Then there is a canonical morphism of schemes $$ r_\psi : U(\psi) \longrightarrow X $$ and a map of $\mathbf{Z}$-graded $\mathcal{O}_{U(\psi)}$-algebras $$ \theta = \theta_\psi : r_\psi^*\left( \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_X(d) \right) \longrightarrow \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_{U(\psi)}(d). $$ The triple $(U(\psi), r_\psi, \theta)$ is characterized by the following properties: \begin{enumerate} \item For every $d \geq 0$ the diagram $$ \xymatrix{ A_d \ar[d] \ar[rr]_{\psi} & & B_d \ar[d] \\ \Gamma(X, \mathcal{O}_X(d)) \ar[r]^-\theta & \Gamma(U(\psi), \mathcal{O}_Y(d)) & \Gamma(Y, \mathcal{O}_Y(d)) \ar[l] } $$ is commutative. \item For any $f \in A_{+}$ homogeneous we have $r_\psi^{-1}(D_{+}(f)) = D_{+}(\psi(f))$ and the restriction of $r_\psi$ to $D_{+}(\psi(f))$ corresponds to the ring map $A_{(f)} \to B_{(\psi(f))}$ induced by $\psi$. \end{enumerate}


constructions-lemma-morphism-proj-transitive

Let $A$, $B$, and $C$ be graded rings. Set $X = \text{Proj}(A)$, $Y = \text{Proj}(B)$ and $Z = \text{Proj}(C)$. Let $\varphi : A \to B$, $\psi : B \to C$ be graded ring maps. Then we have $$ U(\psi \circ \varphi) = r_\varphi^{-1}(U(\psi)) \quad \text{and} \quad r_{\psi \circ \varphi} = r_\varphi \circ r_\psi|_{U(\psi \circ \varphi)}. $$ In addition we have $$ \theta_\psi \circ r_\psi^*\theta_\varphi = \theta_{\psi \circ \varphi} $$ with obvious notation.


constructions-lemma-surjective-graded-rings-map-proj

With hypotheses and notation as in Lemma \ref{lemma-morphism-proj} above. Assume $A_d \to B_d$ is surjective for all $d \gg 0$. Then \begin{enumerate} \item $U(\psi) = Y$, \item $r_\psi : Y \to X$ is a closed immersion, and \item the maps $\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)$ are surjective but not isomorphisms in general (even if $A \to B$ is surjective). \end{enumerate}


constructions-lemma-eventual-iso-graded-rings-map-proj

With hypotheses and notation as in Lemma \ref{lemma-morphism-proj} above. Assume $A_d \to B_d$ is an isomorphism for all $d \gg 0$. Then \begin{enumerate} \item $U(\psi) = Y$, \item $r_\psi : Y \to X$ is an isomorphism, and \item the maps $\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)$ are isomorphisms. \end{enumerate}


constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj

With hypotheses and notation as in Lemma \ref{lemma-morphism-proj} above. Assume $A_d \to B_d$ is surjective for $d \gg 0$ and that $A$ is generated by $A_1$ over $A_0$. Then \begin{enumerate} \item $U(\psi) = Y$, \item $r_\psi : Y \to X$ is a closed immersion, and \item the maps $\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)$ are isomorphisms. \end{enumerate}


constructions-lemma-base-change-map-proj

With hypotheses and notation as in Lemma \ref{lemma-morphism-proj} above. Assume there exists a ring map $R \to A_0$ and a ring map $R \to R'$ such that $B = R' \otimes_R A$. Then \begin{enumerate} \item $U(\psi) = Y$, \item the diagram $$ \xymatrix{ Y = \text{Proj}(B) \ar[r]_{r_\psi} \ar[d] & \text{Proj}(A) = X \ar[d] \\ \Spec(R') \ar[r] & \Spec(R) } $$ is a fibre product square, and \item the maps $\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)$ are isomorphisms. \end{enumerate}


constructions-lemma-d-uple

Let $S$ be a graded ring. Let $d \geq 1$. Set $S' = S^{(d)}$ with notation as in Algebra, Section \ref{algebra-section-graded}. Set $X = \text{Proj}(S)$ and $X' = \text{Proj}(S')$. There is a canonical isomorphism $i : X \to X'$ of schemes such that \begin{enumerate} \item for any graded $S$-module $M$ setting $M' = M^{(d)}$, we have a canonical isomorphism $\widetilde{M} \to i^*\widetilde{M'}$, \item we have canonical isomorphisms $\mathcal{O}_{X}(nd) \to i^*\mathcal{O}_{X'}(n)$ \end{enumerate} and these isomorphisms are compatible with the multiplication maps of Lemma \ref{lemma-widetilde-tensor} and hence with the maps (\ref{equation-multiply}), (\ref{equation-multiply-on-sheaf}), (\ref{equation-global-sections}), (\ref{equation-global-sections-module}), (\ref{equation-multiply-more-generally}), and (\ref{equation-global-sections-more-generally}) (see proof for precise statements.


constructions-lemma-converse-construction

Let $S$ be a graded ring, and $X = \text{Proj}(S)$. Let $d \geq 1$ and $U_d \subset X$ as above. Let $Y$ be a scheme. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Let $\psi : S^{(d)} \to \Gamma_*(Y, \mathcal{L})$ be a graded ring homomorphism such that $\mathcal{L}$ is generated by the sections in the image of $\psi|_{S_d} : S_d \to \Gamma(Y, \mathcal{L})$. Then there exists a morphism $\varphi : Y \to X$ such that $\varphi(Y) \subset U_d$ and an isomorphism $\alpha : \varphi^*\mathcal{O}_{U_d}(d) \to \mathcal{L}$ such that $\psi_\varphi^d$ agrees with $\psi$ via $\alpha$: $$ \xymatrix{ \Gamma_*(Y, \mathcal{L}) & \Gamma_*(Y, \varphi^*\mathcal{O}_{U_d}(d)) \ar[l]^-\alpha & \Gamma_*(U_d, \mathcal{O}_{U_d}(d)) \ar[l]^-{\varphi^*} \\ S^{(d)} \ar[u]^\psi & & S^{(d)} \ar[u]^{\psi^d} \ar[ul]^{\psi^d_\varphi} \ar[ll]_{\text{id}} } $$ commutes. Moreover, the pair $(\varphi, \alpha)$ is unique.


constructions-lemma-proj-functor-strict

Let $S$ be a graded ring. Let $X = \text{Proj}(S)$. The open subscheme $U_d \subset X$ (\ref{equation-Ud}) represents the functor $F_d$ and the triple $(d, \mathcal{O}_{U_d}(d), \psi^d)$ defined above is the universal family (see Schemes, Section \ref{schemes-section-representable}).


constructions-lemma-apply

Let $S$ be a graded ring generated as an $S_0$-algebra by the elements of $S_1$. In this case the scheme $X = \text{Proj}(S)$ represents the functor which associates to a scheme $Y$ the set of pairs $(\mathcal{L}, \psi)$, where \begin{enumerate} \item $\mathcal{L}$ is an invertible $\mathcal{O}_Y$-module, and \item $\psi : S \to \Gamma_*(Y, \mathcal{L})$ is a graded ring homomorphism such that $\mathcal{L}$ is generated by the global sections $\psi(f)$, with $f \in S_1$ \end{enumerate} up to strict equivalence as above.


constructions-lemma-equivalent

Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. Let $T$ be a scheme. Let $(d, \mathcal{L}, \psi)$ and $(d', \mathcal{L}', \psi')$ be two triples over $T$. The following are equivalent: \begin{enumerate} \item Let $n = \text{lcm}(d, d')$. Write $n = ad = a'd'$. There exists an isomorphism $\beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'}$ with the property that $\beta \circ \psi|_{S^{(n)}}$ and $\psi'|_{S^{(n)}}$ agree as graded ring maps $S^{(n)} \to \Gamma_*(Y, (\mathcal{L}')^{\otimes n})$. \item The triples $(d, \mathcal{L}, \psi)$ and $(d', \mathcal{L}', \psi')$ are equivalent. \item For some positive integer $n = ad = a'd'$ there exists an isomorphism $\beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'}$ with the property that $\beta \circ \psi|_{S^{(n)}}$ and $\psi'|_{S^{(n)}}$ agree as graded ring maps $S^{(n)} \to \Gamma_*(Y, (\mathcal{L}')^{\otimes n})$. \item The morphisms $\varphi : T \to X$ and $\varphi' : T \to X$ associated to $(d, \mathcal{L}, \psi)$ and $(d', \mathcal{L}', \psi')$ are equal. \end{enumerate}


constructions-lemma-proj-functor

Let $S$ be a graded ring. Let $X = \text{Proj}(S)$. The functor $F$ defined above is representable by the scheme $X$.


constructions-lemma-projective-space

Let $S = \mathbf{Z}[T_0, \ldots, T_n]$ with $\deg(T_i) = 1$. The scheme $$ \mathbf{P}^n_{\mathbf{Z}} = \text{Proj}(S) $$ represents the functor which associates to a scheme $Y$ the pairs $(\mathcal{L}, (s_0, \ldots, s_n))$ where \begin{enumerate} \item $\mathcal{L}$ is an invertible $\mathcal{O}_Y$-module, and \item $s_0, \ldots, s_n$ are global sections of $\mathcal{L}$ which generate $\mathcal{L}$ \end{enumerate} up to the following equivalence: $(\mathcal{L}, (s_0, \ldots, s_n)) \sim (\mathcal{N}, (t_0, \ldots, t_n))$ $\Leftrightarrow$ there exists an isomorphism $\beta : \mathcal{L} \to \mathcal{N}$ with $\beta(s_i) = t_i$ for $i = 0, \ldots, n$.


constructions-lemma-standard-covering-projective-space

Projective $n$-space over $\mathbf{Z}$ is covered by $n + 1$ standard opens $$ \mathbf{P}^n_{\mathbf{Z}} = \bigcup\nolimits_{i = 0, \ldots, n} D_{+}(T_i) $$ where each $D_{+}(T_i)$ is isomorphic to $\mathbf{A}^n_{\mathbf{Z}}$ affine $n$-space over $\mathbf{Z}$.


constructions-lemma-segre-embedding

Let $S$ be a scheme. There exists a closed immersion $$ \mathbf{P}^n_S \times_S \mathbf{P}^m_S \longrightarrow \mathbf{P}^{nm + n + m}_S $$ called the {\it Segre embedding}.


constructions-lemma-closed-in-projective-space

Let $R$ be a ring. Let $Z \subset \mathbf{P}^n_R$ be a closed subscheme. Let $$ I_d = \Ker\left( R[T_0, \ldots, T_n]_d \longrightarrow \Gamma(Z, \mathcal{O}_{\mathbf{P}^n_R}(d)|_Z)\right) $$ Then $I = \bigoplus I_d \subset R[T_0, \ldots, T_n]$ is a graded ideal and $Z = \text{Proj}(R[T_0, \ldots, T_n]/I)$.


constructions-lemma-quasi-coherent-projective-space

Let $R$ be a ring. Let $\mathcal{F}$ be a quasi-coherent sheaf on $\mathbf{P}^n_R$. For $d \geq 0$ set $$ M_d = \Gamma(\mathbf{P}^n_R, \mathcal{F} \otimes_{\mathcal{O}_{\mathbf{P}^n_R}} \mathcal{O}_{\mathbf{P}^n_R}(d)) = \Gamma(\mathbf{P}^n_R, \mathcal{F}(d)) $$ Then $M = \bigoplus_{d \geq 0} M_d$ is a graded $R[T_0, \ldots, R_n]$-module and there is a canonical isomorphism $\mathcal{F} = \widetilde{M}$.


constructions-lemma-globally-generated-omega-twist-1

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible sheaf and let $s_0, \ldots, s_n$ be global sections of $\mathcal{L}$ which generate it. Let $\mathcal{F}$ be the kernel of the induced map $\mathcal{O}_X^{\oplus n + 1} \to \mathcal{L}$. Then $\mathcal{F} \otimes \mathcal{L}$ is globally generated.


constructions-lemma-invertible-map-into-proj

Let $A$ be a graded ring. Set $X = \text{Proj}(A)$. Let $T$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_T$-module. Let $\psi : A \to \Gamma_*(T, \mathcal{L})$ be a homomorphism of graded rings. Set $$ U(\psi) = \bigcup\nolimits_{f \in A_{+}\text{ homogeneous}} T_{\psi(f)} $$ The morphism $\psi$ induces a canonical morphism of schemes $$ r_{\mathcal{L}, \psi} : U(\psi) \longrightarrow X $$ together with a map of $\mathbf{Z}$-graded $\mathcal{O}_T$-algebras $$ \theta : r_{\mathcal{L}, \psi}^*\left( \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_X(d) \right) \longrightarrow \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{L}^{\otimes d}|_{U(\psi)}. $$ The triple $(U(\psi), r_{\mathcal{L}, \psi}, \theta)$ is characterized by the following properties: \begin{enumerate} \item For $f \in A_{+}$ homogeneous we have $r_{\mathcal{L}, \psi}^{-1}(D_{+}(f)) = T_{\psi(f)}$. \item For every $d \geq 0$ the diagram $$ \xymatrix{ A_d \ar[d]_{(\ref{equation-global-sections})} \ar[r]_{\psi} & \Gamma(T, \mathcal{L}^{\otimes d}) \ar[d]^{restrict} \\ \Gamma(X, \mathcal{O}_X(d)) \ar[r]^{\theta} & \Gamma(U(\psi), \mathcal{L}^{\otimes d}) } $$ is commutative. \end{enumerate} Moreover, for any $d \geq 1$ and any open subscheme $V \subset T$ such that the sections in $\psi(A_d)$ generate $\mathcal{L}^{\otimes d}|_V$ the morphism $r_{\mathcal{L}, \psi}|_V$ agrees with the morphism $\varphi : V \to \text{Proj}(A)$ and the map $\theta|_V$ agrees with the map $\alpha : \varphi^*\mathcal{O}_X(d) \to \mathcal{L}^{\otimes d}|_V$ where $(\varphi, \alpha)$ is the pair of Lemma \ref{lemma-converse-construction} associated to $\psi|_{A^{(d)}} : A^{(d)} \to \Gamma_*(V, \mathcal{L}^{\otimes d})$.


constructions-lemma-proj-inclusion

In Situation \ref{situation-relative-proj}. Suppose $U \subset U' \subset S$ are affine opens. Let $A = \mathcal{A}(U)$ and $A' = \mathcal{A}(U')$. The map of graded rings $A' \to A$ induces a morphism $r : \text{Proj}(A) \to \text{Proj}(A')$, and the diagram $$ \xymatrix{ \text{Proj}(A) \ar[r] \ar[d] & \text{Proj}(A') \ar[d] \\ U \ar[r] & U' } $$ is cartesian. Moreover there are canonical isomorphisms $\theta : r^*\mathcal{O}_{\text{Proj}(A')}(n) \to \mathcal{O}_{\text{Proj}(A)}(n)$ compatible with multiplication maps.


constructions-lemma-transitive-proj

In Situation \ref{situation-relative-proj}. Suppose $U \subset U' \subset U'' \subset S$ are affine opens. Let $A = \mathcal{A}(U)$, $A' = \mathcal{A}(U')$ and $A'' = \mathcal{A}(U'')$. The composition of the morphisms $r : \text{Proj}(A) \to \text{Proj}(A')$, and $r' : \text{Proj}(A') \to \text{Proj}(A'')$ of Lemma \ref{lemma-proj-inclusion} gives the morphism $r'' : \text{Proj}(A) \to \text{Proj}(A'')$ of Lemma \ref{lemma-proj-inclusion}. A similar statement holds for the isomorphisms $\theta$.


constructions-lemma-glue-relative-proj

In Situation \ref{situation-relative-proj}. There exists a morphism of schemes $$ \pi : \underline{\text{Proj}}_S(\mathcal{A}) \longrightarrow S $$ with the following properties: \begin{enumerate} \item for every affine open $U \subset S$ there exists an isomorphism $i_U : \pi^{-1}(U) \to \text{Proj}(A)$ with $A = \mathcal{A}(U)$, and \item for $U \subset U' \subset S$ affine open the composition $$ \xymatrix{ \text{Proj}(A) \ar[r]^{i_U^{-1}} & \pi^{-1}(U) \ar[rr]^{inclusion} & & \pi^{-1}(U') \ar[r]^{i_{U'}} & \text{Proj}(A') } $$ with $A = \mathcal{A}(U)$, $A' = \mathcal{A}(U')$ is the open immersion of Lemma \ref{lemma-proj-inclusion} above. \end{enumerate}


constructions-lemma-glue-relative-proj-twists

In Situation \ref{situation-relative-proj}. The morphism $\pi : \underline{\text{Proj}}_S(\mathcal{A}) \to S$ of Lemma \ref{lemma-glue-relative-proj} comes with the following additional structure. There exists a quasi-coherent $\mathbf{Z}$-graded sheaf of $\mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}$-algebras $\bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n)$, and a morphism of graded $\mathcal{O}_S$-algebras $$ \psi : \mathcal{A} \longrightarrow \bigoplus\nolimits_{n \geq 0} \pi_*\left(\mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n)\right) $$ uniquely determined by the following property: For every affine open $U \subset S$ with $A = \mathcal{A}(U)$ there is an isomorphism $$ \theta_U : i_U^*\left( \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{O}_{\text{Proj}(A)}(n) \right) \longrightarrow \left( \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n) \right)|_{\pi^{-1}(U)} $$ of $\mathbf{Z}$-graded $\mathcal{O}_{\pi^{-1}(U)}$-algebras such that $$ \xymatrix{ A_n \ar[rr]_\psi \ar[dr]_-{(\ref{equation-global-sections})} & & \Gamma(\pi^{-1}(U), \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n)) \\ & \Gamma(\text{Proj}(A), \mathcal{O}_{\text{Proj}(A)}(n)) \ar[ru]_-{\theta_U} & } $$ is commutative.


constructions-lemma-proj-base-change

In Situation \ref{situation-relative-proj}. Let $d \geq 1$. Let $F_d$ be the functor associated to $(S, \mathcal{A})$ above. Let $g : S' \to S$ be a morphism of schemes. Set $\mathcal{A}' = g^*\mathcal{A}$. Let $F_d'$ be the functor associated to $(S', \mathcal{A}')$ above. Then there is a canonical isomorphism $$ F'_d \cong h_{S'} \times_{h_S} F_d $$ of functors.


constructions-lemma-relative-proj-affine

In Situation \ref{situation-relative-proj}. Let $F_d$ be the functor associated to $(d, S, \mathcal{A})$ above. If $S$ is affine, then $F_d$ is representable by the open subscheme $U_d$ (\ref{equation-Ud}) of the scheme $\text{Proj}(\Gamma(S, \mathcal{A}))$.


constructions-lemma-equivalent-relative

In Situation \ref{situation-relative-proj}. Let $T$ be a scheme. Let $(d, f, \mathcal{L}, \psi)$, $(d', f', \mathcal{L}', \psi')$ be two quadruples over $T$. The following are equivalent: \begin{enumerate} \item Let $m = \text{lcm}(d, d')$. Write $m = ad = a'd'$. We have $f = f'$ and there exists an isomorphism $\beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'}$ with the property that $\beta \circ \psi|_{f^*\mathcal{A}^{(m)}}$ and $\psi'|_{f^*\mathcal{A}^{(m)}}$ agree as graded ring maps $f^*\mathcal{A}^{(m)} \to \bigoplus_{n \geq 0} (\mathcal{L}')^{\otimes mn}$. \item The quadruples $(d, f, \mathcal{L}, \psi)$ and $(d', f', \mathcal{L}', \psi')$ are equivalent. \item We have $f = f'$ and for some positive integer $m = ad = a'd'$ there exists an isomorphism $\beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'}$ with the property that $\beta \circ \psi|_{f^*\mathcal{A}^{(m)}}$ and $\psi'|_{f^*\mathcal{A}^{(m)}}$ agree as graded ring maps $f^*\mathcal{A}^{(m)} \to \bigoplus_{n \geq 0} (\mathcal{L}')^{\otimes mn}$. \end{enumerate}


constructions-lemma-relative-proj

In Situation \ref{situation-relative-proj}. The functor $F$ above is representable by a scheme.


constructions-lemma-glueing-gives-functor-proj

In Situation \ref{situation-relative-proj}. The scheme $\pi : \underline{\text{Proj}}_S(\mathcal{A}) \to S$ constructed in Lemma \ref{lemma-glue-relative-proj} and the scheme representing the functor $F$ are canonically isomorphic as schemes over $S$.


constructions-lemma-relative-proj-separated

Let $S$ be a scheme and $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_S$-algebras. The morphism $\pi : \underline{\text{Proj}}_S(\mathcal{A}) \to S$ is separated.


constructions-lemma-relative-proj-base-change

Let $S$ be a scheme and $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_S$-algebras. Let $g : S' \to S$ be any morphism of schemes. Then there is a canonical isomorphism $$ r : \underline{\text{Proj}}_{S'}(g^*\mathcal{A}) \longrightarrow S' \times_S \underline{\text{Proj}}_S(\mathcal{A}) $$ as well as a corresponding isomorphism $$ \theta : r^*\text{pr}_2^*\left(\bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(d)\right) \longrightarrow \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_{S'}(g^*\mathcal{A})}(d) $$ of $\mathbf{Z}$-graded $\mathcal{O}_{\underline{\text{Proj}}_{S'}(g^*\mathcal{A})}$-algebras.


constructions-lemma-apply-relative

Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_S$-modules generated as an $\mathcal{A}_0$-algebra by $\mathcal{A}_1$. In this case the scheme $X = \underline{\text{Proj}}_S(\mathcal{A})$ represents the functor $F_1$ which associates to a scheme $f : T \to S$ over $S$ the set of pairs $(\mathcal{L}, \psi)$, where \begin{enumerate} \item $\mathcal{L}$ is an invertible $\mathcal{O}_T$-module, and \item $\psi : f^*\mathcal{A} \to \bigoplus_{n \geq 0} \mathcal{L}^{\otimes n}$ is a graded $\mathcal{O}_T$-algebra homomorphism such that $f^*\mathcal{A}_1 \to \mathcal{L}$ is surjective \end{enumerate} up to strict equivalence as above. Moreover, in this case all the quasi-coherent sheaves $\mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(n)$ are invertible $\mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}$-modules and the multiplication maps induce isomorphisms $ \mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(n) \otimes_{\mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}} \mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(m) = \mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(n + m)$.


constructions-lemma-relative-proj-modules

In Situation \ref{situation-relative-proj}. For any quasi-coherent sheaf of graded $\mathcal{A}$-modules $\mathcal{M}$ on $S$, there exists a canonical associated sheaf of $\mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}$-modules $\widetilde{\mathcal{M}}$ with the following properties: \begin{enumerate} \item Given a scheme $T$ and a quadruple $(T \to S, d, \mathcal{L}, \psi)$ over $T$ corresponding to a morphism $h : T \to \underline{\text{Proj}}_S(\mathcal{A})$ there is a canonical isomorphism $\widetilde{\mathcal{M}}_T = h^*\widetilde{\mathcal{M}}$ where $\widetilde{\mathcal{M}}_T$ is defined by (\ref{equation-widetilde-M}). \item The isomorphisms of (1) are compatible with pullbacks. \item There is a canonical map $$ \pi^*\mathcal{M}_0 \longrightarrow \widetilde{\mathcal{M}}. $$ \item The construction $\mathcal{M} \mapsto \widetilde{\mathcal{M}}$ is functorial in $\mathcal{M}$. \item The construction $\mathcal{M} \mapsto \widetilde{\mathcal{M}}$ is exact. \item There are canonical maps $$ \widetilde{\mathcal{M}} \otimes_{\mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}} \widetilde{\mathcal{N}} \longrightarrow \widetilde{\mathcal{M} \otimes_\mathcal{A} \mathcal{N}} $$ as in Lemma \ref{lemma-widetilde-tensor}. \item There exist canonical maps $$ \pi^*\mathcal{M} \longrightarrow \bigoplus\nolimits_{n \in \mathbf{Z}} \widetilde{\mathcal{M}(n)} $$ generalizing (\ref{equation-global-sections-more-generally}). \item The formation of $\widetilde{\mathcal{M}}$ commutes with base change. \end{enumerate}


constructions-lemma-morphism-relative-proj

Let $S$ be a scheme. Let $\mathcal{A}$, $\mathcal{B}$ be two graded quasi-coherent $\mathcal{O}_S$-algebras. Set $p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$ and $q : Y = \underline{\text{Proj}}_S(\mathcal{B}) \to S$. Let $\psi : \mathcal{A} \to \mathcal{B}$ be a homomorphism of graded $\mathcal{O}_S$-algebras. There is a canonical open $U(\psi) \subset Y$ and a canonical morphism of schemes $$ r_\psi : U(\psi) \longrightarrow X $$ over $S$ and a map of $\mathbf{Z}$-graded $\mathcal{O}_{U(\psi)}$-algebras $$ \theta = \theta_\psi : r_\psi^*\left( \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_X(d) \right) \longrightarrow \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_{U(\psi)}(d). $$ The triple $(U(\psi), r_\psi, \theta)$ is characterized by the property that for any affine open $W \subset S$ the triple $$ (U(\psi) \cap p^{-1}W,\quad r_\psi|_{U(\psi) \cap p^{-1}W} : U(\psi) \cap p^{-1}W \to q^{-1}W,\quad \theta|_{U(\psi) \cap p^{-1}W}) $$ is equal to the triple associated to $\psi : \mathcal{A}(W) \to \mathcal{B}(W)$ in Lemma \ref{lemma-morphism-proj} via the identifications $p^{-1}W = \text{Proj}(\mathcal{A}(W))$ and $q^{-1}W = \text{Proj}(\mathcal{B}(W))$ of Section \ref{section-relative-proj-via-glueing}.


constructions-lemma-morphism-relative-proj-transitive

Let $S$ be a scheme. Let $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ be quasi-coherent graded $\mathcal{O}_S$-algebras. Set $X = \underline{\text{Proj}}_S(\mathcal{A})$, $Y = \underline{\text{Proj}}_S(\mathcal{B})$ and $Z = \underline{\text{Proj}}_S(\mathcal{C})$. Let $\varphi : \mathcal{A} \to \mathcal{B}$, $\psi : \mathcal{B} \to \mathcal{C}$ be graded $\mathcal{O}_S$-algebra maps. Then we have $$ U(\psi \circ \varphi) = r_\varphi^{-1}(U(\psi)) \quad \text{and} \quad r_{\psi \circ \varphi} = r_\varphi \circ r_\psi|_{U(\psi \circ \varphi)}. $$ In addition we have $$ \theta_\psi \circ r_\psi^*\theta_\varphi = \theta_{\psi \circ \varphi} $$ with obvious notation.


constructions-lemma-surjective-graded-rings-map-relative-proj

With hypotheses and notation as in Lemma \ref{lemma-morphism-relative-proj} above. Assume $\mathcal{A}_d \to \mathcal{B}_d$ is surjective for $d \gg 0$. Then \begin{enumerate} \item $U(\psi) = Y$, \item $r_\psi : Y \to X$ is a closed immersion, and \item the maps $\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)$ are surjective but not isomorphisms in general (even if $\mathcal{A} \to \mathcal{B}$ is surjective). \end{enumerate}


constructions-lemma-eventual-iso-graded-rings-map-relative-proj

With hypotheses and notation as in Lemma \ref{lemma-morphism-relative-proj} above. Assume $\mathcal{A}_d \to \mathcal{B}_d$ is an isomorphism for all $d \gg 0$. Then \begin{enumerate} \item $U(\psi) = Y$, \item $r_\psi : Y \to X$ is an isomorphism, and \item the maps $\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)$ are isomorphisms. \end{enumerate}


constructions-lemma-surjective-generated-degree-1-map-relative-proj

With hypotheses and notation as in Lemma \ref{lemma-morphism-relative-proj} above. Assume $\mathcal{A}_d \to \mathcal{B}_d$ is surjective for $d \gg 0$ and that $\mathcal{A}$ is generated by $\mathcal{A}_1$ over $\mathcal{A}_0$. Then \begin{enumerate} \item $U(\psi) = Y$, \item $r_\psi : Y \to X$ is a closed immersion, and \item the maps $\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)$ are isomorphisms. \end{enumerate}


constructions-lemma-invertible-map-into-relative-proj

With assumptions and notation as above. The morphism $\psi$ induces a canonical morphism of schemes over $S$ $$ r_{\mathcal{L}, \psi} : U(\psi) \longrightarrow \underline{\text{Proj}}_S(\mathcal{A}) $$ together with a map of graded $\mathcal{O}_{U(\psi)}$-algebras $$ \theta : r_{\mathcal{L}, \psi}^*\left( \bigoplus\nolimits_{d \geq 0} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(d) \right) \longrightarrow \bigoplus\nolimits_{d \geq 0} \mathcal{L}^{\otimes d}|_{U(\psi)} $$ characterized by the following properties: \begin{enumerate} \item For every open $V \subset S$ and every $d \geq 0$ the diagram $$ \xymatrix{ \mathcal{A}_d(V) \ar[d]_{\psi} \ar[r]_{\psi} & \Gamma(f^{-1}(V), \mathcal{L}^{\otimes d}) \ar[d]^{restrict} \\ \Gamma(\pi^{-1}(V), \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(d)) \ar[r]^{\theta} & \Gamma(f^{-1}(V) \cap U(\psi), \mathcal{L}^{\otimes d}) } $$ is commutative. \item For any $d \geq 1$ and any open subscheme $W \subset X$ such that $\psi|_W : f^*\mathcal{A}_d|_W \to \mathcal{L}^{\otimes d}|_W$ is surjective the restriction of the morphism $r_{\mathcal{L}, \psi}$ agrees with the morphism $W \to \underline{\text{Proj}}_S(\mathcal{A})$ which exists by the construction of the relative homogeneous spectrum, see Definition \ref{definition-relative-proj}. \item For any affine open $V \subset S$, the restriction $$ (U(\psi) \cap f^{-1}(V), r_{\mathcal{L}, \psi}|_{U(\psi) \cap f^{-1}(V)}, \theta|_{U(\psi) \cap f^{-1}(V)}) $$ agrees via $i_V$ (see Lemma \ref{lemma-glue-relative-proj}) with the triple $(U(\psi'), r_{\mathcal{L}, \psi'}, \theta')$ of Lemma \ref{lemma-invertible-map-into-proj} associated to the map $\psi' : A = \mathcal{A}(V) \to \Gamma_*(f^{-1}(V), \mathcal{L}|_{f^{-1}(V)})$ induced by $\psi$. \end{enumerate}


constructions-lemma-twisting-and-proj

With notation $S$, $\mathcal{A}$, $\mathcal{L}$ and $\mathcal{B}$ as above. There is a canonical isomorphism $$ \xymatrix{ P = \underline{\text{Proj}}_S(\mathcal{A}) \ar[rr]_g \ar[rd]_\pi & & \underline{\text{Proj}}_S(\mathcal{B}) = P' \ar[ld]^{\pi'} \\ & S & } $$ with the following properties \begin{enumerate} \item There are isomorphisms $\theta_n : g^*\mathcal{O}_{P'}(n) \to \mathcal{O}_P(n) \otimes \pi^*\mathcal{L}^{\otimes n}$ which fit together to give an isomorphism of $\mathbf{Z}$-graded algebras $$ \theta : g^*\left( \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{O}_{P'}(n) \right) \longrightarrow \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{O}_P(n) \otimes \pi^*\mathcal{L}^{\otimes n} $$ \item For every open $V \subset S$ the diagrams $$ \xymatrix{ \mathcal{A}_n(V) \otimes \mathcal{L}^{\otimes n}(V) \ar[r]_{multiply} \ar[d]^{\psi \otimes \pi^*} & \mathcal{B}_n(V) \ar[dd]^\psi \\ \Gamma(\pi^{-1}V, \mathcal{O}_P(n)) \otimes \Gamma(\pi^{-1}V, \pi^*\mathcal{L}^{\otimes n}) \ar[d]^{multiply} \\ \Gamma(\pi^{-1}V, \mathcal{O}_P(n) \otimes \pi^*\mathcal{L}^{\otimes n}) & \Gamma(\pi'^{-1}V, \mathcal{O}_{P'}(n)) \ar[l]_-{\theta_n} } $$ are commutative. \item Add more here as necessary. \end{enumerate}


constructions-lemma-projective-bundle-separated

Let $S$ be a scheme. The structure morphism $\mathbf{P}(\mathcal{E}) \to S$ of a projective bundle over $S$ is separated.


constructions-lemma-projective-space-bundle

Let $S$ be a scheme. Let $n \geq 0$. Then $\mathbf{P}^n_S$ is a projective bundle over $S$.


algebraization-theorem-final-bootstrap

In Situation \ref{situation-bootstrap} the inverse system $\{H^i_T(M/I^nM)\}_{n \geq 0}$ satisfies the Mittag-Leffler condition for $i \leq s$, the map $$ H^i_T(M) \longrightarrow \lim H^i_T(M/I^nM) $$ is an isomorphism for $i \leq s$, and $H^i_T(M)$ is annihilated by a power of $I$ for $i \leq s$.


algebraization-theorem-algebraization-formal-sections

\begin{reference} The method of proof follows roughly the method of proof of \cite[Theorem 1]{Faltings-algebraisation} and \cite[Satz 2]{Faltings-uber}. The result is almost the same as \cite[Theorem 1.1]{MRaynaud-paper} (affine complement case) and \cite[Theorem 3.9]{MRaynaud-book} (complement is union of few affines). \end{reference} Let $(A, \mathfrak m)$ be a Noetherian local ring which has a dualizing complex and is complete with respect to an ideal $I$. Set $X = \Spec(A)$, $Y = V(I)$, and $U = X \setminus \{\mathfrak m\}$. Let $\mathcal{F}$ be a coherent sheaf on $U$. Assume \begin{enumerate} \item $\text{cd}(A, I) \leq d$, i.e., $H^i(X \setminus Y, \mathcal{G}) = 0$ for $i \geq d$ and quasi-coherent $\mathcal{G}$ on $X$, \item for any $x \in X \setminus Y$ whose closure $\overline{\{x\}}$ in $X$ meets $U \cap Y$ we have $$ \text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) \geq s \quad\text{or}\quad \text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) + \dim(\overline{\{x\}}) > d + s $$ \end{enumerate} Then there exists an open $V_0 \subset U$ containing $U \cap Y$ such that for any open $V \subset V_0$ containing $U \cap Y$ the map $$ H^i(V, \mathcal{F}) \to \lim H^i(U, \mathcal{F}/I^n\mathcal{F}) $$ is an isomorphism for $i < s$. If in addition $ \text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) + \dim(\overline{\{x\}}) > s $ for all $x \in U \cap Y$, then these cohomology groups are finite $A$-modules.


algebraization-lemma-ML-general

Let $I$ be an ideal of a ring $A$. Let $X$ be a scheme over $\Spec(A)$. Let $$ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 $$ be an inverse system of $\mathcal{O}_X$-modules such that $\mathcal{F}_n = \mathcal{F}_{n + 1}/I^n\mathcal{F}_{n + 1}$. Assume $$ \bigoplus\nolimits_{n \geq 0} H^1(X, I^n\mathcal{F}_{n + 1}) $$ satisfies the ascending chain condition as a graded $\bigoplus_{n \geq 0} I^n/I^{n + 1}$-module. Then the inverse system $M_n = \Gamma(X, \mathcal{F}_n)$ satisfies the Mittag-Leffler condition.


algebraization-lemma-ML-general-better

Let $I$ be an ideal of a ring $A$. Let $X$ be a scheme over $\Spec(A)$. Let $$ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 $$ be an inverse system of $\mathcal{O}_X$-modules such that $\mathcal{F}_n = \mathcal{F}_{n + 1}/I^n\mathcal{F}_{n + 1}$. Given $n$ define $$ H^1_n = \bigcap\nolimits_{m \geq n} \Im\left( H^1(X, I^n\mathcal{F}_{m + 1}) \to H^1(X, I^n\mathcal{F}_{n + 1}) \right) $$ If $\bigoplus H^1_n$ satisfies the ascending chain condition as a graded $\bigoplus_{n \geq 0} I^n/I^{n + 1}$-module, then the inverse system $M_n = \Gamma(X, \mathcal{F}_n)$ satisfies the Mittag-Leffler condition.


algebraization-lemma-topology-I-adic-general

Let $I$ be a finitely generated ideal of a ring $A$. Let $X$ be a scheme over $\Spec(A)$. Let $$ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 $$ be an inverse system of $\mathcal{O}_X$-modules such that $\mathcal{F}_n = \mathcal{F}_{n + 1}/I^n\mathcal{F}_{n + 1}$. Assume $$ \bigoplus\nolimits_{n \geq 0} H^0(X, I^n\mathcal{F}_{n + 1}) $$ satisfies the ascending chain condition as a graded $\bigoplus_{n \geq 0} I^n/I^{n + 1}$-module. Then the limit topology on $M = \lim \Gamma(X, \mathcal{F}_n)$ is the $I$-adic topology.


algebraization-lemma-properties-system

Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $$ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 $$ be an inverse system of quasi-coherent $\mathcal{O}_X$-modules such that $\mathcal{F}_n = \mathcal{F}_{n + 1}/\mathcal{I}^n\mathcal{F}_{n + 1}$. Set $\mathcal{F} = \lim \mathcal{F}_n$. Then \begin{enumerate} \item $\mathcal{F} = R\lim \mathcal{F}_n$, \item for any affine open $U \subset X$ we have $H^p(U, \mathcal{F}) = 0$ for $p > 0$, and \item for each $p$ there is a short exact sequence $0 \to R^1\lim H^{p - 1}(X, \mathcal{F}_n) \to H^p(X, \mathcal{F}) \to \lim H^p(X, \mathcal{F}_n) \to 0$. \end{enumerate} If moreover $\mathcal{I}$ is of finite type, then \begin{enumerate} \item[(4)] $\mathcal{F}_n = \mathcal{F}/\mathcal{I}^n\mathcal{F}$, and \item[(5)] $\mathcal{I}^n \mathcal{F} = \lim_{m \geq n} \mathcal{I}^n\mathcal{F}_m$. \end{enumerate}


algebraization-lemma-equivalent-f-good

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $f \in \Gamma(X, \mathcal{O}_X)$. Let $$ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 $$ be inverse system of $\mathcal{O}_X$-modules. The following are equivalent \begin{enumerate} \item for all $n \geq 1$ the map $f : \mathcal{F}_{n + 1} \to \mathcal{F}_{n + 1}$ factors through $\mathcal{F}_{n + 1} \to \mathcal{F}_n$ to give a short exact sequence $0 \to \mathcal{F}_n \to \mathcal{F}_{n + 1} \to \mathcal{F}_1 \to 0$, \item for all $n \geq 1$ the map $f^n : \mathcal{F}_{n + 1} \to \mathcal{F}_{n + 1}$ factors through $\mathcal{F}_{n + 1} \to \mathcal{F}_1$ to give a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_{n + 1} \to \mathcal{F}_n \to 0$ \item there exists an $\mathcal{O}_X$-module $\mathcal{G}$ which is $f$-divisible such that $\mathcal{F}_n = \mathcal{G}[f^n]$. \end{enumerate} If $X$ is a scheme and $\mathcal{F}_n$ is quasi-coherent, then these are also equivalent to \begin{enumerate} \item[(4)] there exists an $\mathcal{O}_X$-module $\mathcal{F}$ which is $f$-torsion free such that $\mathcal{F}_n = \mathcal{F}/f^n\mathcal{F}$. \end{enumerate}


algebraization-lemma-topology-I-adic-f

Suppose $X$, $f$, $(\mathcal{F}_n)$ is as in Lemma \ref{lemma-equivalent-f-good}. Then the limit topology on $H^p = \lim H^p(X, \mathcal{F}_n)$ is the $f$-adic topology.


algebraization-lemma-limit-finite

Let $A$ be a Noetherian ring complete with respect to a principal ideal $(f)$. Let $X$ be a scheme over $\Spec(A)$. Let $$ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 $$ be an inverse system of $\mathcal{O}_X$-modules. Assume \begin{enumerate} \item $\Gamma(X, \mathcal{F}_1)$ is a finite $A$-module, \item the equivalent conditions of Lemma \ref{lemma-equivalent-f-good} hold. \end{enumerate} Then $$ M = \lim \Gamma(X, \mathcal{F}_n) $$ is a finite $A$-module, $f$ is a nonzerodivisor on $M$, and $M/fM$ is the image of $M$ in $\Gamma(X, \mathcal{F}_1)$.


algebraization-lemma-ML

Let $A$ be a ring. Let $f \in A$. Let $X$ be a scheme over $\Spec(A)$. Let $$ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 $$ be an inverse system of $\mathcal{O}_X$-modules. Assume \begin{enumerate} \item either $H^1(X, \mathcal{F}_1)$ is an $A$-module of finite length or $A$ is Noetherian and $H^1(X, \mathcal{F}_1)$ is a finite $A$-module, \item the equivalent conditions of Lemma \ref{lemma-equivalent-f-good} hold. \end{enumerate} Then the inverse system $M_n = \Gamma(X, \mathcal{F}_n)$ satisfies the Mittag-Leffler condition.


algebraization-lemma-ML-better

Let $A$ be a ring. Let $f \in A$. Let $X$ be a scheme over $\Spec(A)$. Let $$ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 $$ be an inverse system of $\mathcal{O}_X$-modules. Assume \begin{enumerate} \item either there is an $m \geq 1$ such that the image of $H^1(X, \mathcal{F}_m) \to H^1(X, \mathcal{F}_1)$ is an $A$-module of finite length or $A$ is Noetherian and the intersection of the images of $H^1(X, \mathcal{F}_m) \to H^1(X, \mathcal{F}_1)$ is a finite $A$-module, \item the equivalent conditions of Lemma \ref{lemma-equivalent-f-good} hold. \end{enumerate} Then the inverse system $M_n = \Gamma(X, \mathcal{F}_n)$ satisfies the Mittag-Leffler condition.


algebraization-lemma-formal-functions-principal

\begin{reference} \cite[Lemma 1.6]{Bhatt-local} \end{reference} Let $A$ be a ring and $f \in A$. Let $X$ be a scheme over $A$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume that $\mathcal{F}[f^n] = \Ker(f^n : \mathcal{F} \to \mathcal{F})$ stabilizes. Then $$ R\Gamma(X, \lim \mathcal{F}/f^n\mathcal{F}) = R\Gamma(X, \mathcal{F})^\wedge $$ where the right hand side indicates the derived completion with respect to the ideal $(f) \subset A$. Let $H^p$ be the $p$th cohomology group of this complex. Then there are short exact sequences $$ 0 \to R^1\lim H^{p - 1}(X, \mathcal{F}/f^n\mathcal{F}) \to H^p \to \lim H^p(X, \mathcal{F}/f^n\mathcal{F}) \to 0 $$ and $$ 0 \to H^0(H^p(X, \mathcal{F})^\wedge) \to H^p \to T_f(H^{p + 1}(X, \mathcal{F})) \to 0 $$ where $T_f(-)$ denote the $f$-adic Tate module as in More on Algebra, Example \ref{more-algebra-example-spectral-sequence-principal}.


algebraization-lemma-cd-one

Let $I = (f_1, \ldots, f_r)$ be an ideal of a Noetherian ring $A$. If $\text{cd}(A, I) = 1$, then there exist $c \geq 1$ and maps $\varphi_j : I^c \to A$ such that $\sum f_j \varphi_j : I^c \to I$ is the inclusion map.


algebraization-lemma-cd-one-extend

Let $I = (f_1, \ldots, f_r)$ be an ideal of a Noetherian ring $A$ with $\text{cd}(A, I) = 1$. Let $c \geq 1$ and $\varphi_j : I^c \to A$, $j = 1, \ldots, r$ be as in Lemma \ref{lemma-cd-one}. Then there is a unique graded $A$-algebra map $$ \Phi : \bigoplus\nolimits_{n \geq 0} I^{nc} \to A[T_1, \ldots, T_r] $$ with $\Phi(g) = \sum \varphi_j(g) T_j$ for $g \in I^c$. Moreover, the composition of $\Phi$ with the map $A[T_1, \ldots, T_r] \to \bigoplus_{n \geq 0} I^n$, $T_j \mapsto f_j$ is the inclusion map $\bigoplus_{n \geq 0} I^{nc} \to \bigoplus_{n \geq 0} I^n$.


algebraization-lemma-cd-one-extend-to-module

Let $I = (f_1, \ldots, f_r)$ be an ideal of a Noetherian ring $A$ with $\text{cd}(A, I) = 1$. Let $c \geq 1$ and $\varphi_j : I^c \to A$, $j = 1, \ldots, r$ be as in Lemma \ref{lemma-cd-one}. Let $A \to B$ be a ring map with $B$ Noetherian and let $N$ be a finite $B$-module. Then, after possibly increasing $c$ and adjusting $\varphi_j$ accordingly, there is a unique unique graded $B$-module map $$ \Phi_N : \bigoplus\nolimits_{n \geq 0} I^{nc}N \to N[T_1, \ldots, T_r] $$ with $\Phi_N(g x) = \Phi(g) x$ for $g \in I^{nc}$ and $x \in N$ where $\Phi$ is as in Lemma \ref{lemma-cd-one-extend}. The composition of $\Phi_N$ with the map $N[T_1, \ldots, T_r] \to \bigoplus_{n \geq 0} I^nN$, $T_j \mapsto f_j$ is the inclusion map $\bigoplus_{n \geq 0} I^{nc}N \to \bigoplus_{n \geq 0} I^nN$.


algebraization-lemma-cd-is-one-for-system

Let $I = (f_1, \ldots, f_r)$ be an ideal of a Noetherian ring $A$ with $\text{cd}(A, I) = 1$. Let $c \geq 1$ and $\varphi_j : I^c \to A$, $j = 1, \ldots, r$ be as in Lemma \ref{lemma-cd-one}. Let $X$ be a Noetherian scheme over $\Spec(A)$. Let $$ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 $$ be an inverse system of coherent $\mathcal{O}_X$-modules such that $\mathcal{F}_n = \mathcal{F}_{n + 1}/I^n\mathcal{F}_{n + 1}$. Set $\mathcal{F} = \lim \mathcal{F}_n$. Then, after possibly increasing $c$ and adjusting $\varphi_j$ accordingly, there exists a unique graded $\mathcal{O}_X$-module map $$ \Phi_\mathcal{F} : \bigoplus\nolimits_{n \geq 0} I^{nc}\mathcal{F} \longrightarrow \mathcal{F}[T_1, \ldots, T_r] $$ with $\Phi_\mathcal{F}(g s) = \Phi(g) s$ for $g \in I^{nc}$ and $s$ a local section of $\mathcal{F}$ where $\Phi$ is as in Lemma \ref{lemma-cd-one-extend}. The composition of $\Phi_\mathcal{F}$ with the map $\mathcal{F}[T_1, \ldots, T_r] \to \bigoplus_{n \geq 0} I^n\mathcal{F}$, $T_j \mapsto f_j$ is the canonical inclusion $\bigoplus_{n \geq 0} I^{nc}\mathcal{F} \to \bigoplus_{n \geq 0} I^n\mathcal{F}$.


algebraization-lemma-topology-I-adic

Let $I$ be an ideal of a Noetherian ring $A$. Let $X$ be a Noetherian scheme over $\Spec(A)$. Let $$ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 $$ be an inverse system of coherent $\mathcal{O}_X$-modules such that $\mathcal{F}_n = \mathcal{F}_{n + 1}/I^n\mathcal{F}_{n + 1}$. If $\text{cd}(A, I) = 1$, then for all $p \in \mathbf{Z}$ the limit topology on $\lim H^p(X, \mathcal{F}_n)$ is $I$-adic.


algebraization-lemma-descending-chain

Let $(A, \mathfrak m)$ be a Noetherian local ring. \begin{enumerate} \item Let $M$ be a finite $A$-module. Then the $A$-module $H^i_\mathfrak m(M)$ satisfies the descending chain condition for any $i$. \item Let $U = \Spec(A) \setminus \{\mathfrak m\}$ be the punctured spectrum of $A$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module. Then the $A$-module $H^i(U, \mathcal{F})$ satisfies the descending chain condition for $i > 0$. \end{enumerate}


algebraization-lemma-ML-local

Let $(A, \mathfrak m)$ be a Noetherian local ring. \begin{enumerate} \item Let $(M_n)$ be an inverse system of finite $A$-modules. Then the inverse system $H^i_\mathfrak m(M_n)$ satisfies the Mittag-Leffler condition for any $i$. \item Let $U = \Spec(A) \setminus \{\mathfrak m\}$ be the punctured spectrum of $A$. Let $\mathcal{F}_n$ be an inverse system of coherent $\mathcal{O}_U$-modules. Then the inverse system $H^i(U, \mathcal{F}_n)$ satisfies the Mittag-Leffler condition for $i > 0$. \end{enumerate}


algebraization-lemma-terrific

Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $(M_n)$ be an inverse system of finite $A$-modules. Let $M \to \lim M_n$ be a map where $M$ is a finite $A$-module such that for some $i$ the map $H^i_\mathfrak m(M) \to \lim H^i_\mathfrak m(M_n)$ is an isomorphism. Then the inverse system $H^i_\mathfrak m(M_n)$ is essentially constant with value $H^i_\mathfrak m(M)$.


algebraization-lemma-local-cohomology-derived-completion

Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Then $$ H^i(R\Gamma_\mathfrak m(M)^\wedge) = \lim H^i_\mathfrak m(M/I^nM) $$ for all $i$ where $R\Gamma_\mathfrak m(M)^\wedge$ denotes the derived $I$-adic completion.


algebraization-lemma-map-twice-localize

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $f$ be a global section of $\mathcal{O}$. \begin{enumerate} \item For $L, N \in D(\mathcal{O}_f)$ we have $R\SheafHom_\mathcal{O}(L, N) = R\SheafHom_{\mathcal{O}_f}(L, N)$. In particular the two $\mathcal{O}_f$-structures on $R\SheafHom_\mathcal{O}(L, N)$ agree. \item For $K \in D(\mathcal{O})$ and $L \in D(\mathcal{O}_f)$ we have $$ R\SheafHom_\mathcal{O}(L, K) = R\SheafHom_{\mathcal{O}_f}(L, R\SheafHom_\mathcal{O}(\mathcal{O}_f, K)) $$ In particular $R\SheafHom_\mathcal{O}(\mathcal{O}_f, R\SheafHom_\mathcal{O}(\mathcal{O}_f, K)) = R\SheafHom_\mathcal{O}(\mathcal{O}_f, K)$. \item If $g$ is a second global section of $\mathcal{O}$, then $$ R\SheafHom_\mathcal{O}(\mathcal{O}_f, R\SheafHom_\mathcal{O}(\mathcal{O}_g, K)) = R\SheafHom_\mathcal{O}(\mathcal{O}_{gf}, K). $$ \end{enumerate}


algebraization-lemma-hom-from-Af

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $f$ be a global section of $\mathcal{O}$. Let $K \in D(\mathcal{O})$. The following are equivalent \begin{enumerate} \item $R\SheafHom_\mathcal{O}(\mathcal{O}_f, K) = 0$, \item $R\SheafHom_\mathcal{O}(L, K) = 0$ for all $L$ in $D(\mathcal{O}_f)$, \item $T(K, f) = 0$. \end{enumerate}


algebraization-lemma-ideal-of-elements-complete-wrt

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K \in D(\mathcal{O})$. The rule which associates to $U$ the set $\mathcal{I}(U)$ of sections $f \in \mathcal{O}(U)$ such that $T(K|_U, f) = 0$ is a sheaf of ideals in $\mathcal{O}$.


algebraization-lemma-derived-complete-internal-hom

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a sheaf of ideals. If $K \in D(\mathcal{O})$ and $L \in D_{comp}(\mathcal{O})$, then $R\SheafHom_\mathcal{O}(K, L) \in D_{comp}(\mathcal{O})$.


algebraization-lemma-pushforward-derived-complete

Let $f : (\Sh(\mathcal{D}), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})$ be a morphism of ringed topoi. Let $\mathcal{I} \subset \mathcal{O}$ and $\mathcal{I}' \subset \mathcal{O}'$ be sheaves of ideals such that $f^\sharp$ sends $f^{-1}\mathcal{I}$ into $\mathcal{I}'$. Then $Rf_*$ sends $D_{comp}(\mathcal{O}', \mathcal{I}')$ into $D_{comp}(\mathcal{O}, \mathcal{I})$.


algebraization-lemma-derived-completion

Let $(\mathcal{C}, \mathcal{O})$ be a ringed on a site. Let $f_1, \ldots, f_r$ be global sections of $\mathcal{O}$. Let $\mathcal{I} \subset \mathcal{O}$ be the ideal sheaf generated by $f_1, \ldots, f_r$. Then the inclusion functor $D_{comp}(\mathcal{O}) \to D(\mathcal{O})$ has a left adjoint, i.e., given any object $K$ of $D(\mathcal{O})$ there exists a map $K \to K^\wedge$ with $K^\wedge$ in $D_{comp}(\mathcal{O})$ such that the map $$ \Hom_{D(\mathcal{O})}(K^\wedge, E) \longrightarrow \Hom_{D(\mathcal{O})}(K, E) $$ is bijective whenever $E$ is in $D_{comp}(\mathcal{O})$. In fact we have $$ K^\wedge = R\SheafHom_\mathcal{O} (\mathcal{O} \to \prod\nolimits_{i_0} \mathcal{O}_{f_{i_0}} \to \prod\nolimits_{i_0 < i_1} \mathcal{O}_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{f_1\ldots f_r}, K) $$ functorially in $K$.


algebraization-lemma-derived-completion-koszul

Let $(\mathcal{C}, \mathcal{O})$ be a ringed on a site. Let $f_1, \ldots, f_r$ be global sections of $\mathcal{O}$. Let $\mathcal{I} \subset \mathcal{O}$ be the ideal sheaf generated by $f_1, \ldots, f_r$. Let $K \in D(\mathcal{O})$. The derived completion $K^\wedge$ of Lemma \ref{lemma-derived-completion} is given by the formula $$ K^\wedge = R\lim K \otimes^\mathbf{L}_\mathcal{O} K_n $$ where $K_n = K(\mathcal{O}, f_1^n, \ldots, f_r^n)$ is the Koszul complex on $f_1^n, \ldots, f_r^n$ over $\mathcal{O}$.


algebraization-lemma-all-rings

There exist a way to construct \begin{enumerate} \item for every pair $(A, I)$ consisting of a ring $A$ and a finitely generated ideal $I \subset A$ a complex $K(A, I)$ of $A$-modules, \item a map $K(A, I) \to A$ of complexes of $A$-modules, \item for every ring map $A \to B$ and finitely generated ideal $I \subset A$ a map of complexes $K(A, I) \to K(B, IB)$, \end{enumerate} such that \begin{enumerate} \item[(a)] for $A \to B$ and $I \subset A$ finitely generated the diagram $$ \xymatrix{ K(A, I) \ar[r] \ar[d] & A \ar[d] \\ K(B, IB) \ar[r] & B } $$ commutes, \item[(b)] for $A \to B \to C$ and $I \subset A$ finitely generated the composition of the maps $K(A, I) \to K(B, IB) \to K(C, IC)$ is the map $K(A, I) \to K(C, IC)$. \item[(c)] for $A \to B$ and a finitely generated ideal $I \subset A$ the induced map $K(A, I) \otimes_A^\mathbf{L} B \to K(B, IB)$ is an isomorphism in $D(B)$, and \item[(d)] if $I = (f_1, \ldots, f_r) \subset A$ then there is a commutative diagram $$ \xymatrix{ (A \to \prod\nolimits_{i_0} A_{f_{i_0}} \to \prod\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_r}) \ar[r] \ar[d] &  K(A, I) \ar[d] \\ A \ar[r]^1 & A } $$ in $D(A)$ whose horizontal arrows are isomorphisms. \end{enumerate}


algebraization-lemma-global-extended-cech-complex

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. There exists a map $K \to \mathcal{O}$ in $D(\mathcal{O})$ such that for every $U \in \Ob(\mathcal{C})$ such that $\mathcal{I}|_U$ is generated by $f_1, \ldots, f_r \in \mathcal{I}(U)$ there is an isomorphism $$ (\mathcal{O}_U \to \prod\nolimits_{i_0} \mathcal{O}_{U, f_{i_0}} \to \prod\nolimits_{i_0 < i_1} \mathcal{O}_{U, f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{U, f_1\ldots f_r}) \longrightarrow K|_U $$ compatible with maps to $\mathcal{O}_U$.


algebraization-lemma-map-identifies-koszul-and-cech-complexes

Let $\mathcal{C}$ be a site. Assume $\varphi : \mathcal{O} \to \mathcal{O}'$ is a flat homomorphism of sheaves of rings. Let $f_1, \ldots, f_r$ be global sections of $\mathcal{O}$ such that $\mathcal{O}/(f_1, \ldots, f_r) \cong \mathcal{O}'/(f_1, \ldots, f_r)$. Then the map of extended alternating {\v C}ech complexes $$ \xymatrix{ \mathcal{O} \to \prod_{i_0} \mathcal{O}_{f_{i_0}} \to \prod_{i_0 < i_1} \mathcal{O}_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{f_1\ldots f_r} \ar[d] \\ \mathcal{O}' \to \prod_{i_0} \mathcal{O}'_{f_{i_0}} \to \prod_{i_0 < i_1} \mathcal{O}'_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}'_{f_1\ldots f_r} } $$ is a quasi-isomorphism.


algebraization-lemma-pushforward-derived-complete-adjoint

Let $f : (\Sh(\mathcal{D}), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})$ be a morphism of ringed topoi. Let $\mathcal{I} \subset \mathcal{O}$ and $\mathcal{I}' \subset \mathcal{O}'$  be finite type sheaves of ideals such that $f^\sharp$ sends $f^{-1}\mathcal{I}$ into $\mathcal{I}'$. Then $Rf_*$ sends $D_{comp}(\mathcal{O}', \mathcal{I}')$ into $D_{comp}(\mathcal{O}, \mathcal{I})$ and has a left adjoint $Lf_{comp}^*$ which is $Lf^*$ followed by derived completion.


algebraization-lemma-pushforward-commutes-with-derived-completion

\begin{reference} Generalization of \cite[Lemma 6.5.9 (2)]{BS}. Compare with \cite[Theorem 6.5]{HL-P} in the setting of quasi-coherent modules and morphisms of (derived) algebraic stacks. \end{reference} Let $f : (\Sh(\mathcal{D}), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})$ be a morphism of ringed topoi. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. Let $\mathcal{I}' \subset \mathcal{O}'$ be the ideal generated by $f^\sharp(f^{-1}\mathcal{I})$. Then $Rf_*$ commutes with derived completion, i.e., $Rf_*(K^\wedge) = (Rf_*K)^\wedge$.


algebraization-lemma-formal-functions-general

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Let $\mathcal{C}$ be a site and let $\mathcal{O}$ be a sheaf of $A$-algebras. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Then we have $$ R\Gamma(\mathcal{C}, \mathcal{F})^\wedge = R\Gamma(\mathcal{C}, \mathcal{F}^\wedge) $$ in $D(A)$ where $\mathcal{F}^\wedge$ is the derived completion of $\mathcal{F}$ with respect to $I\mathcal{O}$ and on the left hand wide we have the derived completion with respect to $I$. This produces two spectral sequences $$ E_2^{i, j} = H^i(H^j(\mathcal{C}, \mathcal{F})^\wedge) \quad\text{and}\quad E_2^{p, q} = H^p(\mathcal{C}, H^q(\mathcal{F}^\wedge)) $$ both converging to $H^*(R\Gamma(\mathcal{C}, \mathcal{F})^\wedge) = H^*(\mathcal{C}, \mathcal{F}^\wedge)$


algebraization-lemma-sections-derived-completion-pseudo-coherent

Let $X$ be a locally Noetherian scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $K$ be a pseudo-coherent object of $D(\mathcal{O}_X)$ with derived completion $K^\wedge$. Then $$ H^p(U, K^\wedge) = \lim H^p(U, K)/I^nH^p(U, K) = H^p(U, K)^\wedge $$ for any affine open $U \subset X$ where $I = \mathcal{I}(U)$ and where on the right we have the derived completion with respect to $I$.


algebraization-lemma-derived-completion-pseudo-coherent

Let $X$ be a locally Noetherian scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $K$ be an object of $D(\mathcal{O}_X)$. Then \begin{enumerate} \item the derived completion $K^\wedge$ is equal to $R\lim (K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_X/\mathcal{I}^n)$. \end{enumerate} Let $K$ is a pseudo-coherent object of $D(\mathcal{O}_X)$. Then \begin{enumerate} \item[(2)] the cohomology sheaf $H^q(K^\wedge)$ is equal to $\lim H^q(K)/\mathcal{I}^nH^q(K)$. \end{enumerate} Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module\footnote{For example $H^q(K)$ for $K$ pseudo-coherent on our locally Noetherian $X$.}. Then \begin{enumerate} \item[(3)] the derived completion $\mathcal{F}^\wedge$ is equal to $\lim \mathcal{F}/\mathcal{I}^n\mathcal{F}$, \item[(4)] $\lim \mathcal{F}/I^n \mathcal{F} = R\lim \mathcal{F}/I^n \mathcal{F}$, \item[(5)] $H^p(U, \mathcal{F}^\wedge) = 0$ for $p \not = 0$ for all affine opens $U \subset X$. \end{enumerate}


algebraization-lemma-kill-completion-general

Let $I, J$ be ideals of a Noetherian ring $A$. Let $M$ be a finite $A$-module. Let $\mathfrak p \subset A$ be a prime. Let $s$ and $d$ be integers. Assume \begin{enumerate} \item $A$ has a dualizing complex, \item $\mathfrak p \not \in V(J) \cap V(I)$, \item $\text{cd}(A, I) \leq d$, and \item for all primes $\mathfrak p' \subset \mathfrak p$ we have $\text{depth}_{A_{\mathfrak p'}}(M_{\mathfrak p'}) + \dim((A/\mathfrak p')_\mathfrak q) > d + s$ for all $\mathfrak q \in V(\mathfrak p') \cap V(J) \cap V(I)$. \end{enumerate} Then there exists an $f \in A$, $f \not \in \mathfrak p$ which annihilates $H^i(R\Gamma_J(M)^\wedge)$ for $i \leq s$ where ${}^\wedge$ indicates $I$-adic completion.


algebraization-lemma-kill-colimit-weak-general

Let $I, J$ be ideals of a Noetherian ring. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. With $T$ as in (\ref{equation-associated-subset}) assume \begin{enumerate} \item $A$ has a dualizing complex, \item if $\mathfrak p \in V(I)$, then no condition, \item if $\mathfrak p \not \in V(I)$, $\mathfrak p \in T$, then $\dim((A/\mathfrak p)_\mathfrak q) \leq d$ for some $\mathfrak q \in V(\mathfrak p) \cap V(J) \cap V(I)$, \item if $\mathfrak p \not \in V(I)$, $\mathfrak p \not \in T$, then $$ \text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad\text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > d + s $$ for all $\mathfrak q \in V(\mathfrak p) \cap V(J) \cap V(I)$. \end{enumerate} Then there exists an ideal $J_0 \subset J$ with $V(J_0) \cap V(I) = V(J) \cap V(I)$ such that for any $J' \subset J_0$ with $V(J') \cap V(I) = V(J) \cap V(I)$ the map $$ R\Gamma_{J'}(M) \longrightarrow R\Gamma_{J_0}(M) $$ induces an isomorphism in cohomology in degrees $\leq s$ and moreover these modules are annihilated by a power of $J_0I$.


algebraization-lemma-kill-colimit-general

In Lemma \ref{lemma-kill-colimit-weak-general} if instead of the empty condition (2) we assume \begin{enumerate} \item[(2')] if $\mathfrak p \in V(I)$, $\mathfrak p \not \in V(J) \cap V(I)$, then $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > s$ for all $\mathfrak q \in V(\mathfrak p) \cap V(J) \cap V(I)$, \end{enumerate} then the conditions also imply that $H^i_{J_0}(M)$ is a finite $A$-module for $i \leq s$.


algebraization-lemma-kill-colimit-support-general

If in Lemma \ref{lemma-kill-colimit-weak-general} we additionally assume \begin{enumerate} \item[(6)] if $\mathfrak p \not \in V(I)$, $\mathfrak p \in T$, then $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) > s$, \end{enumerate} then $H^i_{J_0}(M) = H^i_J(M) = H^i_{J + I}(M)$ for $i \leq s$ and these modules are annihilated by a power of $I$.


algebraization-lemma-algebraize-local-cohomology-general

Let $I, J$ be ideals of a Noetherian ring $A$. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. With $T$ as in (\ref{equation-associated-subset}) assume \begin{enumerate} \item $A$ is $I$-adically complete and has a dualizing complex, \item if $\mathfrak p \in V(I)$ no condition, \item $\text{cd}(A, I) \leq d$, \item if $\mathfrak p \not \in V(I)$, $\mathfrak p \not \in T$ then $$ \text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad\text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > d + s $$ for all $\mathfrak q \in V(\mathfrak p) \cap V(J) \cap V(I)$, \item if $\mathfrak p \not \in V(I)$, $\mathfrak p \not \in T$, $V(\mathfrak p) \cap V(J) \cap V(I) \not = \emptyset$, and $\text{depth}(M_\mathfrak p) < s$, then one of the following holds\footnote{Our method forces this additional condition. We will return to this (insert future reference).}: \begin{enumerate} \item $\dim(\text{Supp}(M_\mathfrak p)) < s + 2$\footnote{For example if $M$ satisfies Serre's condition $(S_s)$ on the complement of $V(I) \cup T$.}, or \item  $\delta(\mathfrak p) > d + \delta_{max} - 1$ where $\delta$ is a dimension function and $\delta_{max}$ is the maximum of $\delta$ on $V(J) \cap V(I)$, or \item $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > d + s + \delta_{max} - \delta_{min} - 2$ for all $\mathfrak q \in V(\mathfrak p) \cap V(J) \cap V(I)$. \end{enumerate} \end{enumerate} Then there exists an ideal $J_0 \subset J$ with $V(J_0) \cap V(I) = V(J) \cap V(I)$ such that for any $J' \subset J_0$ with $V(J') \cap V(I) = V(J) \cap V(I)$ the map $$ R\Gamma_{J'}(M) \longrightarrow R\Gamma_J(M)^\wedge $$ induces an isomorphism on cohomology in degrees $\leq s$. Here ${}^\wedge$ denotes derived $I$-adic completion.


algebraization-lemma-kill-completion

Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module and let $\mathfrak p \subset A$ be a prime. Let $s$ and $d$ be integers. Assume \begin{enumerate} \item $A$ has a dualizing complex, \item $\text{cd}(A, I) \leq d$, and \item $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim(A/\mathfrak p) > d + s$. \end{enumerate} Then there exists an $f \in A \setminus \mathfrak p$ which annihilates $H^i(R\Gamma_\mathfrak m(M)^\wedge)$ for $i \leq s$ where ${}^\wedge$ indicates $I$-adic completion.


algebraization-lemma-kill-colimit-weak

Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. Assume \begin{enumerate} \item $A$ has a dualizing complex, \item if $\mathfrak p \in V(I)$, then no condition, \item if $\mathfrak p \not \in V(I)$ and $V(\mathfrak p) \cap V(I) = \{\mathfrak m\}$, then $\dim(A/\mathfrak p) \leq d$, \item if $\mathfrak p \not \in V(I)$ and $V(\mathfrak p) \cap V(I) \not = \{\mathfrak m\}$, then $$ \text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad\text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim(A/\mathfrak p) > d + s $$ \end{enumerate} Then there exists an ideal $J_0 \subset A$ with $V(J_0) \cap V(I) = \{\mathfrak m\}$ such that for any $J \subset J_0$ with $V(J) \cap V(I) = \{\mathfrak m\}$ the map $$ R\Gamma_J(M) \longrightarrow R\Gamma_{J_0}(M) $$ induces an isomorphism in cohomology in degrees $\leq s$ and moreover these modules are annihilated by a power of $J_0I$.


algebraization-lemma-kill-colimit

In Lemma \ref{lemma-kill-colimit-weak} if instead of the empty condition (2) we assume \begin{enumerate} \item[(2')] if $\mathfrak p \in V(I)$ and $\mathfrak p \not = \mathfrak m$, then $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim(A/\mathfrak p) > s$, \end{enumerate} then the conditions also imply that $H^i_{J_0}(M)$ is a finite $A$-module for $i \leq s$.


algebraization-lemma-kill-colimit-support

If in Lemma \ref{lemma-kill-colimit-weak} we additionally assume \begin{enumerate} \item[(6)] if $\mathfrak p \not \in V(I)$ and $V(\mathfrak p) \cap V(I) = \{\mathfrak m\}$, then $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) > s$, \end{enumerate} then $H^i_{J_0}(M) = H^i_J(M) = H^i_\mathfrak m(M)$ for $i \leq s$ and these modules are annihilated by a power of $I$.


algebraization-lemma-algebraize-local-cohomology

Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. Assume \begin{enumerate} \item $A$ is $I$-adically complete and has a dualizing complex, \item if $\mathfrak p \in V(I)$, no condition, \item $\text{cd}(A, I) \leq d$, \item if $\mathfrak p \not \in V(I)$ and $V(\mathfrak p) \cap V(I) \not = \{\mathfrak m\}$ then $$ \text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad\text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim(A/\mathfrak p) > d + s $$ \end{enumerate} Then there exists an ideal $J_0 \subset A$ with $V(J_0) \cap V(I) = \{\mathfrak m\}$ such that for any $J \subset J_0$ with $V(J) \cap V(I) = \{\mathfrak m\}$ the map $$ R\Gamma_J(M) \longrightarrow R\Gamma_J(M)^\wedge = R\Gamma_\mathfrak m(M)^\wedge $$ induces an isomorphism in cohomology in degrees $\leq s$. Here ${}^\wedge$ denotes derived $I$-adic completion.


algebraization-lemma-helper-bootstrap

In Situation \ref{situation-bootstrap} let $\mathfrak p \subset \mathfrak q$ be primes of $A$ with $\mathfrak p \not \in V(I)$ and $\mathfrak q \in T$. If there does not exist an $\mathfrak r \in V(I) \setminus T$ with $\mathfrak p \subset \mathfrak r \subset \mathfrak q$ then $\text{depth}(M_\mathfrak p) > s$.


algebraization-lemma-bootstrap-inherited

In Situation \ref{situation-bootstrap} we have \begin{enumerate} \item[(E)] if $T' \subset T$ is a smaller specialization stable subset, then $A, I, T', M$ satisfies the assumptions of Situation \ref{situation-bootstrap}, \item[(F)] if $S \subset A$ is a multiplicative subset, then $S^{-1}A, S^{-1}I, T', S^{-1}M$ satisfies the assumptions of Situation \ref{situation-bootstrap} where $T' \subset V(S^{-1}I)$ is the inverse image of $T$, \item[(G)] the quadruple $A', I', T', M'$ satisfies the assumptions of Situation \ref{situation-bootstrap} where $A', I', M'$ are the usual $I$-adic completions of $A, I, M$ and $T' \subset V(I')$ is the inverse image of $T$. \end{enumerate} Let $I \subset \mathfrak a \subset A$ be an ideal such that $V(\mathfrak a) \subset T$. Then \begin{enumerate} \item[(A)] if $I$ is contained in the Jacobson radical of $A$, then all hypotheses of Lemmas \ref{lemma-kill-colimit-weak-general} and \ref{lemma-kill-colimit-support-general} are satisfied for $A, I, \mathfrak a, M$, \item[(B)] if $A$ is complete with respect to $I$, then all hypotheses except for possibly (5) of Lemma \ref{lemma-algebraize-local-cohomology-general} are satisfied for $A, I, \mathfrak a, M$, \item[(C)] if $A$ is local with maximal ideal $\mathfrak m = \mathfrak a$, then all hypotheses of Lemmas \ref{lemma-kill-colimit-weak} and \ref{lemma-kill-colimit-support} hold for $A, \mathfrak m, I, M$, \item[(D)] if $A$ is local with maximal ideal $\mathfrak m = \mathfrak a$ and $I$-adically complete, then all hypotheses of Lemma \ref{lemma-algebraize-local-cohomology} hold for $A, \mathfrak m, I, M$, \end{enumerate}


algebraization-lemma-algebraize-local-cohomology-bis

In Situation \ref{situation-bootstrap} assume $A$ is local with maximal ideal $\mathfrak m$ and $T = \{\mathfrak m\}$. Then $H^i_\mathfrak m(M) \to \lim H^i_\mathfrak m(M/I^nM)$ is an isomorphism for $i \leq s$ and these modules are annihilated by a power of $I$.


algebraization-lemma-bootstrap-bis-bis

Let $I \subset \mathfrak a$ be ideals of a Noetherian ring $A$. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. If we assume \begin{enumerate} \item[(a)] $A$ has a dualizing complex, \item[(b)] $\text{cd}(A, I) \leq d$, \item[(c)] if $\mathfrak p \not \in V(I)$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$ then $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) > s$ or $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > d + s$. \end{enumerate} Then $A, I, V(\mathfrak a), M, s, d$ are as in Situation \ref{situation-bootstrap}.


algebraization-lemma-bootstrap

In Situation \ref{situation-bootstrap} the inverse systems $\{H^i_T(I^nM)\}_{n \geq 0}$ are pro-zero for $i \leq s$. Moreover, there exists an integer $m_0$ such that for all $m \geq m_0$ there exists an integer $m'(m) \geq m$ such that for $k \geq m'(m)$ the image of $H^{s + 1}_T(I^kM) \to H^{s + 1}_T(I^mM)$ maps injectively to $H^{s + 1}_T(I^{m_0}M)$.


algebraization-lemma-final-bootstrap

In Situation \ref{situation-bootstrap} there exists an integer $m_0 \geq 0$ such that \begin{enumerate} \item $\{H^i_T(M/I^nM)\}_{n \geq 0}$ satisfies the Mittag-Leffler condition for $i < s$. \item $\{H^i_T(I^{m_0}M/I^nM)\}_{n \geq m_0}$ satisfies the Mittag-Leffler condition for $i \leq s$, \item $H^i_T(M) \to \lim H^i_T(M/I^nM)$ is an isomorphism for $i < s$, \item $H^s_T(I^{m_0}M) \to \lim H^s_T(I^{m_0}M/I^nM)$ is an isomorphism for $i \leq s$, \item $H^s_T(M) \to \lim H^s_T(M/I^nM)$ is injective with cokernel killed by $I^{m_0}$, and \item $R^1\lim H^s_T(M/I^nM)$ is killed by $I^{m_0}$. \end{enumerate}


algebraization-lemma-combine-two

Let $I \subset \mathfrak a \subset A$ be ideals of a Noetherian ring $A$ and let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. Suppose that \begin{enumerate} \item $A, I, V(\mathfrak a), M$ satisfy the assumptions of Situation \ref{situation-bootstrap} for $s$ and $d$, and \item $A, I, \mathfrak a, M$ satisfy the conditions of Lemma \ref{lemma-algebraize-local-cohomology-general} for $s + 1$ and $d$ with $J = \mathfrak a$. \end{enumerate} Then there exists an ideal $J_0 \subset \mathfrak a$ with $V(J_0) \cap V(I) = V(\mathfrak a)$ such that for any $J \subset J_0$ with $V(J) \cap V(I) = V(\mathfrak a)$ the map $$ H^{s + 1}_J(M) \longrightarrow \lim H^{s + 1}_\mathfrak a(M/I^nM) $$ is an isomorphism.


algebraization-lemma-compare-with-derived-completion

Let $U$ be the punctured spectrum of a Noetherian local ring $A$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module. Let $I \subset A$ be an ideal. Then $$ H^i(R\Gamma(U, \mathcal{F})^\wedge) = \lim H^i(U, \mathcal{F}/I^n\mathcal{F}) $$ for all $i$ where $R\Gamma(U, \mathcal{F})^\wedge$ denotes the derived $I$-adic completion.


algebraization-lemma-application-theorem

Let $(A, \mathfrak m)$ be a Noetherian local ring which has a dualizing complex and is complete with respect to an ideal $I$. Set $X = \Spec(A)$, $Y = V(I)$, and $U = X \setminus \{\mathfrak m\}$. Let $\mathcal{F}$ be a coherent sheaf on $U$. Assume for any associated point $x \in U$ of $\mathcal{F}$ we have $\dim(\overline{\{x\}}) > \text{cd}(A, I) + 1$ where $\overline{\{x\}}$ is the closure in $X$. Then the map $$ \colim H^0(V, \mathcal{F}) \longrightarrow \lim H^0(U, \mathcal{F}/I^n\mathcal{F}) $$ is an isomorphism of finite $A$-modules where the colimit is over opens $V \subset U$ containing $U \cap Y$.


algebraization-lemma-application-H0-pre

Let $I \subset \mathfrak a$ be ideals of a Noetherian ring $A$. Let $\mathcal{F}$ be a coherent module on $U = \Spec(A) \setminus V(\mathfrak a)$. Assume \begin{enumerate} \item $A$ is $I$-adically complete and has a dualizing complex, \item if $x \in \text{Ass}(\mathcal{F})$, $x \not \in V(I)$, $\overline{\{x\}} \cap V(I) \not \subset V(\mathfrak a)$ and $z \in \overline{\{x\}} \cap V(\mathfrak a)$, then $\dim(\mathcal{O}_{\overline{\{x\}}, z}) > \text{cd}(A, I) + 1$, \item one of the following holds: \begin{enumerate} \item the restriction of $\mathcal{F}$ to $U \setminus V(I)$ is $(S_1)$ \item the dimension of $V(\mathfrak a)$ is at most $2$\footnote{In the sense that the difference of the maximal and minimal values on $V(\mathfrak a)$ of a dimension function on $\Spec(A)$ is at most $2$.}. \end{enumerate} \end{enumerate} Then we obtain an isomorphism $$ \colim H^0(V, \mathcal{F}) \longrightarrow \lim H^0(U, \mathcal{F}/I^n\mathcal{F}) $$ where the colimit is over opens $V \subset U$ containing $U \cap V(I)$.


algebraization-lemma-application-H0

Let $I \subset \mathfrak a$ be ideals of a Noetherian ring $A$. Let $\mathcal{F}$ be a coherent module on $U = \Spec(A) \setminus V(\mathfrak a)$. Assume \begin{enumerate} \item $A$ is $I$-adically complete and has a dualizing complex, \item if $x \in \text{Ass}(\mathcal{F})$, $x \not \in V(I)$, $z \in V(\mathfrak a) \cap \overline{\{x\}}$, then $\dim(\mathcal{O}_{\overline{\{x\}}, z}) > \text{cd}(A, I) + 1$, \item for $x \in U$ with $\overline{\{x\}} \cap V(I) \subset V(\mathfrak a)$ we have $\text{depth}(\mathcal{F}_x) \geq 2$, \end{enumerate} Then we obtain an isomorphism $$ H^0(U, \mathcal{F}) \longrightarrow \lim H^0(U, \mathcal{F}/I^n\mathcal{F}) $$


algebraization-lemma-alternative-colim-H0

Let $A$ be a Noetherian ring. Let $f \in \mathfrak a \subset A$ be an element of an ideal of $A$. Let $M$ be a finite $A$-module. Assume \begin{enumerate} \item $A$ is $f$-adically complete, \item $f$ is a nonzerodivisor on $M$, \item $H^1_\mathfrak a(M/fM)$ is a finite $A$-module. \end{enumerate} Then with $U = \Spec(A) \setminus V(\mathfrak a)$ the map $$ \colim_V \Gamma(V, \widetilde{M}) \longrightarrow \lim \Gamma(U, \widetilde{M/f^nM}) $$ is an isomorphism where the colimit is over opens $V \subset U$ containing $U \cap V(f)$.


algebraization-lemma-alternative-H0

Let $A$ be a Noetherian ring. Let $f \in \mathfrak a \subset A$ be an element of an ideal of $A$. Let $M$ be a finite $A$-module. Assume \begin{enumerate} \item $A$ is $f$-adically complete, \item $H^1_\mathfrak a(M)$ and $H^2_\mathfrak a(M)$ are annihilated by a power of $f$. \end{enumerate} Then with $U = \Spec(A) \setminus V(\mathfrak a)$ the map $$ \Gamma(U, \widetilde{M}) \longrightarrow \lim \Gamma(U, \widetilde{M/f^nM}) $$ is an isomorphism.


algebraization-lemma-alternative-higher

Let $A$ be a Noetherian ring. Let $f \in \mathfrak a \subset A$ be an element of an ideal of $A$. Let $M$ be a finite $A$-module. Let $s \geq 0$. Assume \begin{enumerate} \item $A$ is $f$-adically complete, \item $H^i_\mathfrak a(M)$ is annihilated by a power of $f$ for $i \leq s + 1$. \end{enumerate} Then with $U = \Spec(A) \setminus V(\mathfrak a)$ the map $$ H^i(U, \widetilde{M}) \longrightarrow \lim H^i(U, \widetilde{M/f^nM}) $$ is an isomorphism for $i < s$.