algebraization-lemma-connected

Let $I \subset \mathfrak a$ be ideals of a Noetherian ring $A$. Assume \begin{enumerate} \item $A$ is $I$-adically complete and has a dualizing complex, \item if $\mathfrak p \subset A$ is a minimal prime not contained in $V(I)$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$, then $\dim((A/\mathfrak p)_\mathfrak q) > \text{cd}(A, I) + 1$, \item any nonempty open $V \subset \Spec(A)$ which contains $V(I) \setminus V(\mathfrak a)$ is connected\footnote{For example if $A$ is a domain.}. \end{enumerate} Then $V(I) \setminus V(\mathfrak a)$ is either empty or connected.


algebraization-lemma-completion-fully-faithful

Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme. Let $Y_n \subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$. Consider the following conditions \begin{enumerate} \item $X$ is quasi-affine and $\Gamma(X, \mathcal{O}_X) \to \lim \Gamma(Y_n, \mathcal{O}_{Y_n})$ is an isomorphism, \item $X$ has an ample invertible module $\mathcal{L}$ and $\Gamma(X, \mathcal{L}^{\otimes m}) \to \lim \Gamma(Y_n, \mathcal{L}^{\otimes m}|_{Y_n})$ is an isomorphism for all $m \gg 0$, \item for every finite locally free $\mathcal{O}_X$-module $\mathcal{E}$ the map $\Gamma(X, \mathcal{E}) \to \lim \Gamma(Y_n, \mathcal{E}|_{Y_n})$ is an isomorphism, and \item the completion functor $\textit{Coh}(\mathcal{O}_X) \to \textit{Coh}(X, \mathcal{I})$ is fully faithful on the full subcategory of finite locally free objects. \end{enumerate} Then (1) $\Rightarrow$ (2) $\Rightarrow$ (3) $\Rightarrow$ (4) and (4) $\Rightarrow$ (3).


algebraization-lemma-completion-fully-faithful-general

Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. Let $Y_n \subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$. Let $\mathcal{V}$ be the set of open subschemes $V \subset X$ containing $Y$ ordered by reverse inclusion. \begin{enumerate} \item $X$ is quasi-affine and $$ \colim_\mathcal{V} \Gamma(V, \mathcal{O}_V) \longrightarrow \lim \Gamma(Y_n, \mathcal{O}_{Y_n}) $$ is an isomorphism, \item $X$ has an ample invertible module $\mathcal{L}$ and $$ \colim_\mathcal{V} \Gamma(V, \mathcal{L}^{\otimes m}) \longrightarrow \lim \Gamma(Y_n, \mathcal{L}^{\otimes m}|_{Y_n}) $$ is an isomorphism for all $m \gg 0$, \item for every $V \in \mathcal{V}$ and every finite locally free $\mathcal{O}_V$-module $\mathcal{E}$ the map $$ \colim_{V' \geq V} \Gamma(V', \mathcal{E}|_{V'}) \longrightarrow \lim \Gamma(Y_n, \mathcal{E}|_{Y_n}) $$ is an isomorphism, and \item the completion functor $$ \colim_\mathcal{V} \textit{Coh}(\mathcal{O}_V) \longrightarrow \textit{Coh}(X, \mathcal{I}), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge $$ is fully faithful on the full subcategory of finite locally free objects (see explanation above). \end{enumerate} Then (1) $\Rightarrow$ (2) $\Rightarrow$ (3) $\Rightarrow$ (4) and (4) $\Rightarrow$ (3).


algebraization-lemma-recognize-formal-coherent-modules

Let $X$ be a Noetherian scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. The functor $$ \textit{Coh}(X, \mathcal{I}) \longrightarrow \text{Pro-}\QCoh(\mathcal{O}_X) $$ is fully faithful, see Categories, Remark \ref{categories-remark-pro-category}.


algebraization-lemma-fully-faithful

Let $I \subset \mathfrak a$ be ideals of a Noetherian ring $A$. Let $U = \Spec(A) \setminus V(\mathfrak a)$. Assume \begin{enumerate} \item $A$ is $I$-adically complete and has a dualizing complex, \item for any associated prime $\mathfrak p \subset A$, $I \not \subset \mathfrak p$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$ we have $\dim((A/\mathfrak p)_\mathfrak q) > \text{cd}(A, I) + 1$. \item for $\mathfrak p \subset A$, $I \not \subset \mathfrak p$ with with $V(\mathfrak p) \cap V(I) \subset V(\mathfrak a)$ we have $\text{depth}(A_\mathfrak p) \geq 2$. \end{enumerate} Then the completion functor $$ \textit{Coh}(\mathcal{O}_U) \longrightarrow \textit{Coh}(U, I\mathcal{O}_U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge $$ is fully faithful on the full subcategory of finite locally free objects.


algebraization-lemma-fully-faithful-simple-one

Let $A$ be a Noetherian ring. Let $f \in \mathfrak a$ be an element of an ideal of $A$. Let $U = \Spec(A) \setminus V(\mathfrak a)$. Assume \begin{enumerate} \item $A$ has a dualizing complex and is complete with respect to $f$, \item $A_f$ is $(S_2)$ and for every minimal prime $\mathfrak p \subset A$, $f \not \in \mathfrak p$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$ we have $\dim((A/\mathfrak p)_\mathfrak q) \geq 3$. \end{enumerate} Then the completion functor $$ \textit{Coh}(\mathcal{O}_U) \longrightarrow \textit{Coh}(U, I\mathcal{O}_U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge $$ is fully faithful on the full subcategory of finite locally free objects.


algebraization-lemma-fully-faithful-alternative

Let $A$ be a Noetherian ring. Let $f \in \mathfrak a \subset A$ be an element of an ideal of $A$. Let $U = \Spec(A) \setminus V(\mathfrak a)$. Assume \begin{enumerate} \item $A$ is $f$-adically complete, \item $H^1_\mathfrak a(A)$ and $H^2_\mathfrak a(A)$ are annihilated by a power of $f$. \end{enumerate} Then the completion functor $$ \textit{Coh}(\mathcal{O}_U) \longrightarrow \textit{Coh}(U, I\mathcal{O}_U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge $$ is fully faithful on the full subcategory of finite locally free objects.


algebraization-lemma-fully-faithful-simple-two

Let $A$ be a Noetherian ring. Let $f \in \mathfrak a$ be an element of an ideal of $A$. Let $U = \Spec(A) \setminus V(\mathfrak a)$. Assume \begin{enumerate} \item $A$ has a dualizing complex and is complete with respect to $f$, \item for every prime $\mathfrak p \subset A$, $f \not \in \mathfrak p$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$ we have $\text{depth}(A_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > 2$. \end{enumerate} Then the completion functor $$ \textit{Coh}(\mathcal{O}_U) \longrightarrow \textit{Coh}(U, I\mathcal{O}_U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge $$ is fully faithful on the full subcategory of finite locally free objects.


algebraization-lemma-fully-faithful-general

Let $I \subset \mathfrak a \subset A$ be ideals of a Noetherian ring $A$. Let $U = \Spec(A) \setminus V(\mathfrak a)$. Let $\mathcal{V}$ be the set of open subschemes of $U$ containing $U \cap V(I)$ ordered by reverse inclusion. Assume \begin{enumerate} \item $A$ is $I$-adically complete and has a dualizing complex, \item for any associated prime $\mathfrak p \subset A$ with $I \not \subset \mathfrak p$ and $V(\mathfrak p) \cap V(I) \not \subset V(\mathfrak a)$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$ we have $\dim((A/\mathfrak p)_\mathfrak q) > \text{cd}(A, I) + 1$. \end{enumerate} Then the completion functor $$ \colim_\mathcal{V} \textit{Coh}(\mathcal{O}_V) \longrightarrow \textit{Coh}(U, I\mathcal{O}_U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge $$ is fully faithful on the full subcategory of finite locally free objects.


algebraization-lemma-fully-faithful-general-alternative

Let $A$ be a Noetherian ring. Let $f \in \mathfrak a \subset A$ be an element of an ideal of $A$. Let $U = \Spec(A) \setminus V(\mathfrak a)$. Let $\mathcal{V}$ be the set of open subschemes of $U$ containing $U \cap V(f)$ ordered by reverse inclusion. Assume \begin{enumerate} \item $A$ is $f$-adically complete, \item $f$ is a nonzerodivisor, \item $H^1_\mathfrak a(A/fA)$ is a finite $A$-module. \end{enumerate} Then the completion functor $$ \colim_\mathcal{V} \textit{Coh}(\mathcal{O}_V) \longrightarrow \textit{Coh}(U, f\mathcal{O}_U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge $$ is fully faithful on the full subcategory of finite locally free objects.


algebraization-lemma-system-of-modules

In Situation \ref{situation-algebraize}. Consider an inverse system $(M_n)$ of $A$-modules such that \begin{enumerate} \item $M_n$ is a finite $A$-module, \item $M_n$ is annihilated by $I^n$, \item the kernel and cokernel of $M_{n + 1}/I^nM_{n + 1} \to M_n$ are $\mathfrak a$-power torsion. \end{enumerate} Then $(\widetilde{M}_n|_U)$ is in $\textit{Coh}(U, I\mathcal{O}_U)$. Conversely, every object of $\textit{Coh}(U, I\mathcal{O}_U)$ arises in this manner.


algebraization-lemma-essential-image-completion

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Consider the following conditions: \begin{enumerate} \item $(\mathcal{F}_n)$ is in the essential image of the functor (\ref{equation-completion}), \item $(\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_U$-module, \item $(\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_V$-module for $U \cap Y \subset V \subset U$ open, \item $(\mathcal{F}_n)$ is the completion of the restriction to $U$ of a coherent $\mathcal{O}_X$-module, \item $(\mathcal{F}_n)$ is the restriction to $U$ of the completion of a coherent $\mathcal{O}_X$-module, \item there exists an object $(\mathcal{G}_n)$ of $\textit{Coh}(X, I\mathcal{O}_X)$ whose restriction to $U$ is $(\mathcal{F}_n)$. \end{enumerate} Then conditions (1), (2), (3), (4), and (5) are equivalent and imply (6). If $A$ is $I$-adically complete then condition (6) implies the others.


algebraization-lemma-algebraizable

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Let $A', I', \mathfrak a'$ be the $I$-adic completions of $A, I, \mathfrak a$. Set $X' = \Spec(A')$ and $U' = X' \setminus V(\mathfrak a')$. The following are equivalent \begin{enumerate} \item $(\mathcal{F}_n)$ extends to $X$, and \item the pullback of $(\mathcal{F}_n)$ to $U'$ is the completion of a coherent $\mathcal{O}_{U'}$-module. \end{enumerate}


algebraization-lemma-canonically-algebraizable

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. If $(\mathcal{F}_n)$ canonically extends to $X$, then \begin{enumerate} \item $(\widetilde{H^0(U, \mathcal{F}_n)})$ is pro-isomorphic to an object $(\mathcal{G}_n)$ of $\textit{Coh}(X, I \mathcal{O}_X)$ unique up to unique isomorphism, \item the restriction of $(\mathcal{G}_n)$ to $U$ is isomorphic to $(\mathcal{F}_n)$, i.e., $(\mathcal{F}_n)$ extends to $X$, \item the inverse system $\{H^0(U, \mathcal{F}_n)\}$ satisfies the Mittag-Leffler condition, and \item the module $M$ in (\ref{equation-guess}) is finite over the $I$-adic completion of $A$ and the limit topology on $M$ is the $I$-adic topology. \end{enumerate}


algebraization-lemma-canonically-extend-base-change

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Let $A \to A'$ be a flat ring map. Set $X' = \Spec(A')$, let $U' \subset X'$ be the inverse image of $U$, and denote $g : U' \to U$ the induced morphism. Set $(\mathcal{F}'_n) = (g^*\mathcal{F}_n)$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-inverse-systems-pullback}. If $(\mathcal{F}_n)$ canonically extends to $X$, then $(\mathcal{F}'_n)$ canonically extends to $X'$. Moreover, the extension found in Lemma \ref{lemma-canonically-algebraizable} for $(\mathcal{F}_n)$ pulls back to the extension for $(\mathcal{F}'_n)$.


algebraization-lemma-when-done

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Let $M$ be as in (\ref{equation-guess}). Assume \begin{enumerate} \item[(a)] the inverse system $H^0(U, \mathcal{F}_n)$ has Mittag-Leffler, \item[(b)] the limit topology on $M$ agrees with the $I$-adic topology, and \item[(c)] the image of $M \to H^0(U, \mathcal{F}_n)$ is a finite $A$-module for all $n$. \end{enumerate} Then $(\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$ is $I$-adically complete, then $(\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_U$-module.


algebraization-lemma-algebraization-principal-variant

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Assume \begin{enumerate} \item $I = (f)$ is a principal ideal for a nonzerodivisor $f \in \mathfrak a$, \item $\mathcal{F}_n$ is a finite locally free $\mathcal{O}_U/f^n\mathcal{O}_U$-module, \item $H^1_\mathfrak a(A/fA)$ and $H^2_\mathfrak a(A/fA)$ are finite $A$-modules. \end{enumerate} Then $(\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$ is complete, then $(\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_U$-module.


algebraization-lemma-map-kernel-cokernel-on-closed

In Situation \ref{situation-algebraize}. Let $(\mathcal{F}_n) \to (\mathcal{F}'_n)$ be a morphism of $\textit{Coh}(U, I\mathcal{O}_U)$ whose kernel and cokernel are annihilated by a power of $I$. Then \begin{enumerate} \item $(\mathcal{F}_n)$ extends to $X$ if and only if $(\mathcal{F}'_n)$ extends to $X$, and \item $(\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_U$-module if and only if $(\mathcal{F}'_n)$ is. \end{enumerate}


algebraization-lemma-discussion

Let $Y$ be a Noetherian scheme and let $Z \subset Y$ be a closed subset. \begin{enumerate} \item For $y \in Y$ we have $\delta_Z(y) = 0 \Leftrightarrow y \in Z$. \item The subsets $\{y \in Y \mid \delta_Z(y) \leq k\}$ are stable under specialization. \item For $y \in Y$ and $z \in \overline{\{y\}} \cap Z$ we have $\dim(\mathcal{O}_{\overline{\{y\}}, z}) \geq \delta_Z(y)$. \item If $\delta$ is a dimension function on $Y$, then $\delta(y) \leq \delta_Z(y) + \delta_{max}$ where $\delta_{max}$ is the maximum value of $\delta$ on $Z$. \item If $Y = \Spec(A)$ is the spectrum of a catenary Noetherian local ring with maximal ideal $\mathfrak m$ and $Z = \{\mathfrak m\}$, then $\delta_Z(y) = \dim(\overline{\{y\}})$. \item Given a pattern of specializations $$ \xymatrix{ & y'_0 \ar@{~>}[ld] \ar@{~>}[rd] & & y'_1 \ar@{~>}[ld] & \ldots & y'_{k - 1} \ar@{~>}[rd] & \\ y_0 & & y_1 & & \ldots & & y_k = y } $$ between points of $Y$ with $y_0 \in Z$ and $y_i' \leadsto y_i$ an immediate specialization, then $\delta_Z(y_k) \leq k$. \item If $Y' \subset Y$ is an open subscheme, then $\delta^{Y'}_{Y' \cap Z}(y') \geq \delta^Y_Z(y')$ for $y' \in Y'$. \end{enumerate}


algebraization-lemma-change-distance-function

Let $Y$ be a universally catenary Noetherian scheme. Let $Z \subset Y$ be a closed subscheme. Let $f : Y' \to Y$ be a finite type morphism all of whose fibres have dimension $\leq e$. Set $Z' = f^{-1}(Z)$. Then $$ \delta_Z(y) \leq \delta_{Z'}(y') + e - \text{trdeg}_{\kappa(y)}(\kappa(y')) $$ for $y' \in Y'$ with image $y \in Y$.


algebraization-lemma-elementary

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Let $a, b$ be integers. \begin{enumerate} \item If $(\mathcal{F}_n)$ is annihilated by a power of $I$, then $(\mathcal{F}_n)$ satisfies the $(a, b)$-inequalities for any $a, b$. \item If $(\mathcal{F}_n)$ satisfies the $(a + 1, b)$-inequalities, then $(\mathcal{F}_n)$ satisfies the strict $(a, b)$-inequalities. \end{enumerate} If $\text{cd}(A, I) \leq d$ and $A$ has a dualizing complex, then \begin{enumerate} \item[(3)] $(\mathcal{F}_n)$ satisfies the $(s, s + d)$-inequalities if and only if for all $y \in U \cap Y$ the tuple $\mathcal{O}_{X, y}^\wedge, I\mathcal{O}_{X, y}^\wedge, \{\mathfrak m_y^\wedge\}, \mathcal{F}_y^\wedge, s - \delta^Y_Z(y), d$ is as in Situation \ref{situation-bootstrap}. \item[(4)] If $(\mathcal{F}_n)$ satisfies the strict $(s, s + d)$-inequalities, then $(\mathcal{F}_n)$ satisfies the $(s, s + d)$-inequalities. \end{enumerate}


algebraization-lemma-explain-2-3-cd-1

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. If $\text{cd}(A, I) = 1$, then $\mathcal{F}$ satisfies the $(2, 3)$-inequalities if and only if $$ \text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) > 3 $$ for all $y \in U \cap Y$ and $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$ with $\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)$.


algebraization-lemma-sanity

In Situation \ref{situation-algebraize} let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module and $d \geq 1$. Assume \begin{enumerate} \item $A$ is $I$-adically complete, has a dualizing complex, and $\text{cd}(A, I) \leq d$, \item the completion $\mathcal{F}^\wedge$ of $\mathcal{F}$ satisfies the strict $(1, 1 + d)$-inequalities. \end{enumerate} Let $x \in X$ be a point. Let $W = \overline{\{x\}}$. If $W \cap Y$ has an irreducible component contained in $Z$ and one which is not, then $\text{depth}(\mathcal{F}_x) \geq 1$.


algebraization-lemma-recover

In Situation \ref{situation-algebraize} let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module and $d \geq 1$. Assume \begin{enumerate} \item $A$ is $I$-adically complete, has a dualizing complex, and $\text{cd}(A, I) \leq d$, \item the completion $\mathcal{F}^\wedge$ of $\mathcal{F}$ satisfies the strict $(1, 1+ d)$-inequalities, and \item for $x \in U$ with $\overline{\{x\}} \cap Y \subset Z$ we have $\text{depth}(\mathcal{F}_x) \geq 2$. \end{enumerate} Then $H^0(U, \mathcal{F}) \to \lim H^0(U, \mathcal{F}/I^n\mathcal{F})$ is an isomorphism.


algebraization-lemma-fully-faithful-inequalities

In Situation \ref{situation-algebraize} let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module and $d \geq 1$. Assume \begin{enumerate} \item $A$ is $I$-adically complete, has a dualizing complex, and $\text{cd}(A, I) \leq d$, \item the completion $\mathcal{F}^\wedge$ of $\mathcal{F}$ satisfies the strict $(1, 1 + d)$-inequalities, and \item for $x \in U$ with $\overline{\{x\}} \cap Y \subset Z$ we have $\text{depth}(\mathcal{F}_x) \geq 2$. \end{enumerate} Then the map $$ \Hom_U(\mathcal{G}, \mathcal{F}) \longrightarrow \Hom_{\textit{Coh}(U, I\mathcal{O}_U)}(\mathcal{G}^\wedge, \mathcal{F}^\wedge) $$ is bijective for every coherent $\mathcal{O}_U$-module $\mathcal{G}$.


algebraization-lemma-algebraization-principal

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Assume \begin{enumerate} \item $A$ is local and $\mathfrak a = \mathfrak m$ is the maximal ideal, \item $A$ has a dualizing complex, \item $I = (f)$ is a principal ideal for a nonzerodivisor $f \in \mathfrak m$, \item $\mathcal{F}_n$ is a finite locally free $\mathcal{O}_U/f^n\mathcal{O}_U$-module, \item if $\mathfrak p \in V(f) \setminus \{\mathfrak m\}$, then $\text{depth}((A/f)_\mathfrak p) + \dim(A/\mathfrak p) > 1$, and \item if $\mathfrak p \not \in V(f)$ and $V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}$, then $\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3$. \end{enumerate} Then $(\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$ is complete, then $(\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_U$-module.


algebraization-lemma-helper-algebraize

In Situation \ref{situation-algebraize} let $(M_n)$ be an inverse system of $A$-modules as in Lemma \ref{lemma-system-of-modules} and let $(\mathcal{F}_n)$ be the corresponding object of $\textit{Coh}(U, I\mathcal{O}_U)$. Let $d \geq \text{cd}(A, I)$ and $s \geq 0$ be integers. With notation as above assume \begin{enumerate} \item $A$ is local with maximal ideal $\mathfrak m = \mathfrak a$, \item $A$ has a dualizing complex, and \item $(\mathcal{F}_n)$ satisfies the $(s, s + d)$-inequalities (Definition \ref{definition-s-d-inequalities}). \end{enumerate} Let $E$ be an injective hull of the residue field of $A$. Then for $i \leq s$ there exists a finite $A$-module $N$ annihilated by a power of $I$ and for $n \gg 0$ compatible maps $$ H^i_\mathfrak m(M_n) \to \Hom_A(N, E) $$ whose cokernels are finite length $A$-modules and whose kernels $K_n$ form an inverse system such that $\Im(K_{n''} \to K_{n'})$ has finite length for $n'' \gg n' \gg 0$.


algebraization-lemma-algebraization-principal-bis

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Assume \begin{enumerate} \item $A$ is local and $\mathfrak a = \mathfrak m$ is the maximal ideal, \item $A$ has a dualizing complex, \item $I = (f)$ is a principal ideal, \item $(\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities. \end{enumerate} Then $(\mathcal{F}_n)$ extends to $X$. In particular, if $A$ is $I$-adically complete, then $(\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_U$-module.


algebraization-lemma-unwinding-conditions

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Assume \begin{enumerate} \item $A$ is local with maximal ideal $\mathfrak a = \mathfrak m$, \item $\text{cd}(A, I) = 1$. \end{enumerate} Then $(\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities if and only if for all $y \in U \cap Y$ with $\dim(\{y\}) = 1$ and every prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$, $\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)$ we have $$ \text{depth}((\mathcal{F}_y^\wedge)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) > 2 $$


algebraization-lemma-unwinding-conditions-bis

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Assume \begin{enumerate} \item $A$ is local with maximal ideal $\mathfrak a = \mathfrak m$, \item $A$ has a dualizing complex, \item $\text{cd}(A, I) = 1$, \item for $y \in U \cap Y$ the module $\mathcal{F}_y^\wedge$ is finite locally free outside $V(I\mathcal{O}_{X, y}^\wedge)$, for example if $\mathcal{F}_n$ is a finite locally free $\mathcal{O}_U/I^n\mathcal{O}_U$-module, and \item one of the following is true \begin{enumerate} \item $A_f$ is $(S_2)$ and every irreducible component of $X$ not contained in $Y$ has dimension $\geq 4$, or \item if $\mathfrak p \not \in V(f)$ and $V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}$, then $\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3$. \end{enumerate} \end{enumerate} Then $(\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities.


algebraization-lemma-divide-torsion-formal-coherent-module

In the situation above assume $X$ locally has a dualizing complex. Let $T \subset Y$ be a subset stable under specialization. Assume for $y \in T$ and for a nonmaximal prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$ with $V(\mathfrak p) \cap V(\mathcal{I}^\wedge_y) = \{\mathfrak m_y^\wedge\}$ we have $$ \text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge_y)_\mathfrak p) > 0 $$ Then there exists a canonical map $(\mathcal{F}_n) \to (\mathcal{F}_n')$ of inverse systems of coherent $\mathcal{O}_X$-modules with the following properties \begin{enumerate} \item for $y \in T$ we have $\text{depth}(\mathcal{F}'_{n, y}) \geq 1$, \item $(\mathcal{F}'_n)$ is isomorphic as a pro-system to an object $(\mathcal{G}_n)$ of $\textit{Coh}(X, \mathcal{I})$, \item the induced morphism $(\mathcal{F}_n) \to (\mathcal{G}_n)$ of $\textit{Coh}(X, \mathcal{I})$ is surjective with kernel annihilated by a power of $\mathcal{I}$. \end{enumerate}


algebraization-lemma-improvement-formal-coherent-module-better

In the situation above assume $X$ locally has a dualizing complex. Let $T' \subset T \subset Y$ be subsets stable under specialization. Let $d \geq 0$ be an integer. Assume \begin{enumerate} \item[(a)] affine locally we have $X = \Spec(A_0)$ and $Y = V(I_0)$ and $\text{cd}(A_0, I_0) \leq d$, \item[(b)] for $y \in T$ and a nonmaximal prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$ with $V(\mathfrak p) \cap V(\mathcal{I}_y^\wedge) = \{\mathfrak m_y^\wedge\}$ we have $$ \text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge_y)_\mathfrak p) > 0 $$ \item[(c)] for $y \in T'$ and for a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$ with $\mathfrak p \not \in V(\mathcal{I}_y^\wedge)$ and $V(\mathfrak p) \cap V(\mathcal{I}_y^\wedge) \not = \{\mathfrak m_y^\wedge\}$ we have $$ \text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge_y)_\mathfrak p) \geq 1 \quad\text{or}\quad \text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge_y)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) > 1 + d $$ \item[(d)] for $y \in T'$ and a nonmaximal prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$ with $V(\mathfrak p) \cap V(\mathcal{I}_y^\wedge) = \{\mathfrak m_y^\wedge\}$ we have $$ \text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge_y)_\mathfrak p) > 1 $$ \item[(e)] if $y \leadsto y'$ is an immediate specialization and $y' \in T'$, then $y \in T$. \end{enumerate} Then there exists a canonical map $(\mathcal{F}_n) \to (\mathcal{F}_n'')$ of inverse systems of coherent $\mathcal{O}_X$-modules with the following properties \begin{enumerate} \item for $y \in T$ we have $\text{depth}(\mathcal{F}''_{n, y}) \geq 1$, \item for $y' \in T'$ we have $\text{depth}(\mathcal{F}''_{n, y'}) \geq 2$, \item $(\mathcal{F}''_n)$ is isomorphic as a pro-system to an object $(\mathcal{H}_n)$ of $\textit{Coh}(X, \mathcal{I})$, \item the induced morphism $(\mathcal{F}_n) \to (\mathcal{H}_n)$ of $\textit{Coh}(X, \mathcal{I})$ has kernel and cokernel annihilated by a power of $\mathcal{I}$. \end{enumerate}


algebraization-lemma-improvement-application

In Situation \ref{situation-algebraize} assume that $A$ has a dualizing complex. Let $d \geq \text{cd}(A, I)$. Let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Assume $(\mathcal{F}_n)$ satisfies the $(2, 2 + d)$-inequalities, see Definition \ref{definition-s-d-inequalities}. Then there exists a canonical map $(\mathcal{F}_n) \to (\mathcal{F}_n'')$ of inverse systems of coherent $\mathcal{O}_U$-modules with the following properties \begin{enumerate} \item if $\text{depth}(\mathcal{F}''_{n, y}) + \delta^Y_Z(y) \geq 3$ for all $y \in U \cap Y$, \item $(\mathcal{F}''_n)$ is isomorphic as a pro-system to an object $(\mathcal{H}_n)$ of $\textit{Coh}(U, I\mathcal{O}_U)$, \item the induced morphism $(\mathcal{F}_n) \to (\mathcal{H}_n)$ of $\textit{Coh}(U, I\mathcal{O}_U)$ has kernel and cokernel annihilated by a power of $I$, \item the modules $H^0(U, \mathcal{F}''_n)$ and $H^1(U, \mathcal{F}''_n)$ are finite $A$-modules for all $n$. \end{enumerate}


algebraization-lemma-cd-1-canonical

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Assume \begin{enumerate} \item $A$ has a dualizing complex and $\text{cd}(A, I) = 1$, \item $(\mathcal{F}_n)$ is pro-isomorphic to an inverse system $(\mathcal{F}_n'')$ of coherent $\mathcal{O}_U$-modules such that $\text{depth}(\mathcal{F}''_{n, y}) + \delta^Y_Z(y) \geq 3$ for all $y \in U \cap Y$. \end{enumerate} Then $(\mathcal{F}_n)$ extends canonically to $X$, see Definition \ref{definition-canonically-algebraizable}.


algebraization-lemma-blowup

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Assume \begin{enumerate} \item $A$ has a dualizing complex, \item all fibres of the blowing up $b : X' \to X$ of $I$ have dimension $\leq d - 1$, \item one of the following is true \begin{enumerate} \item $(\mathcal{F}_n)$ satisfies the $(d + 1, d + 2)$-inequalities (Definition \ref{definition-s-d-inequalities}), or \item for $y \in U \cap Y$ and a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$ with $\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)$ we have $$ \text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) > d + 2 $$ \end{enumerate} \end{enumerate} Then $(\mathcal{F}_n)$ extends to $X$.


algebraization-lemma-equivalence-better

In Situation \ref{situation-algebraize} assume \begin{enumerate} \item $A$ has a dualizing complex and is $I$-adically complete, \item $I = (f)$ generated by a single element, \item $A$ is local with maximal ideal $\mathfrak a = \mathfrak m$, \item one of the following is true \begin{enumerate} \item $A_f$ is $(S_2)$ and for $\mathfrak p \subset A$, $f \not \in \mathfrak p$ minimal we have $\dim(A/\mathfrak p) \geq 4$, or \item if $\mathfrak p \not \in V(f)$ and $V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}$, then $\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3$. \end{enumerate} \end{enumerate} Then with $U_0 = U \cap V(f)$ the completion functor $$ \colim_{U_0 \subset U' \subset U\text{ open}} \textit{Coh}(\mathcal{O}_{U'}) \longrightarrow \textit{Coh}(U, f\mathcal{O}_U) $$ is an equivalence on the full subcategories of finite locally free objects.


algebraization-lemma-equivalence

In Situation \ref{situation-algebraize} assume \begin{enumerate} \item $I = (f)$ is principal, \item $A$ is $f$-adically complete, \item $f$ is a nonzerodivisor, \item $H^1_\mathfrak a(A/fA)$ and $H^2_\mathfrak a(A/fA)$ are finite $A$-modules. \end{enumerate} Then with $U_0 = U \cap V(f)$ the completion functor $$ \colim_{U_0 \subset U' \subset U\text{ open}} \textit{Coh}(\mathcal{O}_{U'}) \longrightarrow \textit{Coh}(U, f\mathcal{O}_U) $$ is an equivalence on the full subcategories of finite locally free objects.


algebraization-lemma-prepare-chi-triple

For any coherent triple $(\mathcal{F}, \mathcal{F}_0, \alpha)$ there exists a coherent $\mathcal{O}_X$-module $\mathcal{F}'$ such that $f : \mathcal{F}' \to \mathcal{F}'$ is injective, an isomorphism $\alpha' : \mathcal{F}'|_U \to \mathcal{F}$, and a map $\alpha'_0 : \mathcal{F}'/f\mathcal{F}' \to \mathcal{F}_0$ such that $\alpha \circ (\alpha' \bmod f) = \alpha'_0|_{U_0}$.


algebraization-lemma-ses-chi-triple

We have $\chi(\mathcal{G}, \mathcal{G}_0, \beta) = \chi(\mathcal{F}, \mathcal{F}_0, \alpha) + \chi(\mathcal{H}, \mathcal{H}_0, \gamma)$ if $$ 0 \to (\mathcal{F}, \mathcal{F}_0, \alpha) \to (\mathcal{G}, \mathcal{G}_0, \beta) \to (\mathcal{H}, \mathcal{H}_0, \gamma) \to 0 $$ is a short exact sequence of coherent triples.


algebraization-lemma-nonnegative-chi-triple

Assume $\text{depth}(A) \geq 3$ or equivalently $\text{depth}(A/fA) \geq 2$. Let $(\mathcal{L}, \mathcal{L}_0, \lambda)$ be an invertible coherent triple. Then $$ \chi(\mathcal{L}, \mathcal{L}_0, \lambda) = \text{length}_A \Coker(\Gamma(U, \mathcal{L}) \to \Gamma(U_0, \mathcal{L}_0)) $$ and in particular this is $\geq 0$. Moreover,  $\chi(\mathcal{L}, \mathcal{L}_0, \lambda) = 0$ if and only if $\mathcal{L} \cong \mathcal{O}_U$.


algebraization-lemma-surjective-Pic-first

Let $(A, \mathfrak m)$ be a Noetherian local ring and $f \in \mathfrak m$. Assume \begin{enumerate} \item $A$ is $f$-adically complete, \item $f$ is a nonzerodivisor, \item $H^1_\mathfrak m(A/fA)$ and $H^2_\mathfrak m(A/fA)$ are finite $A$-modules, and \item $H^3_\mathfrak m(A/fA) = 0$\footnote{Observe that (3) and (4) hold if $\text{depth}(A/fA) \geq 4$, or equivalently $\text{depth}(A) \geq 5$.}. \end{enumerate} Let $U$, resp.\ $U_0$ be the punctured spectrum of $A$, resp.\ $A/fA$. Then $$ \colim_{U_0 \subset U' \subset U\text{ open}} \Pic(U') \longrightarrow \Pic(U_0) $$ is surjective.


algebraization-lemma-surjective-Pic-first-better

Let $(A, \mathfrak m)$ be a Noetherian local ring and $f \in \mathfrak m$. Assume \begin{enumerate} \item the conditions of Lemma \ref{lemma-surjective-Pic-first} hold, and \item for every maximal ideal $\mathfrak p \subset A_f$ the punctured spectrum of $(A_f)_\mathfrak p$ has trivial Picard group. \end{enumerate} Let $U$, resp.\ $U_0$ be the punctured spectrum of $A$, resp.\ $A/fA$. Then $$ \Pic(U) \longrightarrow \Pic(U_0) $$ is surjective.


algebraization-lemma-local-pic-to-completion

Let $(A, \mathfrak m)$ be a Noetherian local ring of depth $\geq 2$. Let $A^\wedge$ be its completion. Let $U$, resp.\ $U^\wedge$ be the punctured spectrum of $A$, resp.\ $A^\wedge$. Then $\Pic(U) \to \Pic(U^\wedge)$ is injective.


algebraization-lemma-trivial-local-pic-regular

Let $(A, \mathfrak m)$ be a regular local ring. Then the Picard group of the punctured spectrum of $A$ is trivial.


algebraization-lemma-lefschetz-addendum

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$ with $n$th infinitesimal neighbourhood $Y_n = Z(s^n)$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Assume that for all $x \in X \setminus Y$ we have $$ \text{depth}(\mathcal{F}_x) + \dim(\overline{\{x\}}) > 1 $$ Then $\Gamma(V, \mathcal{F}) \to \lim \Gamma(Y_n, \mathcal{F}|_{Y_n})$ is an isomorphism for any open subscheme $V \subset X$ containing $Y$.


algebraization-lemma-Gm-equivariant-extend-canonically

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Assume \begin{enumerate} \item $A$ is a graded ring, $\mathfrak a = A_+$, and $I$ is a homogeneous ideal, \item $(\mathcal{F}_n) = (\widetilde{M_n}|_U)$ where $(M_n)$ is an inverse system of graded $A$-modules, and \item $(\mathcal{F}_n)$ extends canonically to $X$. \end{enumerate} Then there is a finite graded $A$-module $N$ such that \begin{enumerate} \item[(a)] the inverse systems $(N/I^nN)$ and $(M_n)$ are pro-isomorphic in the category of graded $A$-modules modulo $A_+$-power torsion modules, and \item[(b)] $(\mathcal{F}_n)$ is the completion of of the coherent module associated to $N$. \end{enumerate}


algebraization-proposition-derived-completion

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 left adjoint to the inclusion functor $D_{comp}(\mathcal{O}) \to D(\mathcal{O})$.


algebraization-proposition-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 $x \in \text{Ass}(\mathcal{F})$, $x \not \in V(I)$, $\overline{\{x\}} \cap V(I) \not \subset V(\mathfrak a)$ and $z \in \overline{\{x\}} \cap V(\mathfrak a)$, then $\dim(\mathcal{O}_{\overline{\{x\}}, z}) > \text{cd}(A, I) + 1$. \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-proposition-cd-1

\begin{reference} The local case of this result is \cite[IV Corollaire 2.9]{MRaynaud-book}. \end{reference} In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Assume \begin{enumerate} \item $A$ has a dualizing complex and $\text{cd}(A, I) = 1$, \item $(\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities, see Definition \ref{definition-s-d-inequalities}. \end{enumerate} Then $(\mathcal{F}_n)$ extends to $X$. In particular, if $A$ is $I$-adically complete, then $(\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_U$-module.


algebraization-proposition-d-generators

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Assume \begin{enumerate} \item $A$ has a dualizing complex, \item $V(I) = V(f_1, \ldots, f_d)$ for some $d \geq 1$ and $f_1, \ldots, f_d \in A$, \item one of the following is true \begin{enumerate} \item $(\mathcal{F}_n)$ satisfies the $(d + 1, d + 2)$-inequalities (Definition \ref{definition-s-d-inequalities}), or \item for $y \in U \cap Y$ and a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$ with $\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)$ we have $$ \text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) > d + 2 $$ \end{enumerate} \end{enumerate} Then $(\mathcal{F}_n)$ extends to $X$. In particular, if $A$ is $I$-adically complete, then $(\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_U$-module.


algebraization-proposition-algebraization-regular-sequence

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Assume \begin{enumerate} \item there exist $f_1, \ldots, f_d \in I$ such that for $y \in U \cap Y$ the ideal $I\mathcal{O}_{X, y}$ is generated by $f_1, \ldots, f_d$ and $f_1, \ldots, f_d$ form a $\mathcal{F}_y^\wedge$-regular sequence, \item $H^0(U, \mathcal{F}_1)$ and $H^1(U, \mathcal{F}_1)$ are finite $A$-modules. \end{enumerate} Then $(\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$ is complete, then $(\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_U$-module.


algebraization-proposition-algebraization-flat

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Assume there is Noetherian local ring $(R, \mathfrak m)$ and a ring map $R \to A$ such that \begin{enumerate} \item $I = \mathfrak m A$, \item for $y \in U \cap Y$ the stalk $\mathcal{F}_y^\wedge$ is $R$-flat, \item $H^0(U, \mathcal{F}_1)$ and $H^1(U, \mathcal{F}_1)$ are finite $A$-modules. \end{enumerate} Then $(\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$ is complete, then $(\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_U$-module.


algebraization-proposition-hilbert-triple

Let $(\mathcal{F}, \mathcal{F}_0, \alpha)$ be a coherent triple. Let $(\mathcal{L}, \mathcal{L}_0, \lambda)$ be an invertible coherent triple. Then the function $$ \mathbf{Z} \longrightarrow \mathbf{Z},\quad n \longmapsto \chi((\mathcal{F}, \mathcal{F}_0, \alpha) \otimes (\mathcal{L}, \mathcal{L}_0, \lambda)^{\otimes n}) $$ is a polynomial of degree $\leq \dim(\text{Supp}(\mathcal{F}))$.


algebraization-proposition-injective-pic

\begin{reference} \cite[Theorem 1.9]{Kollar-map-pic} \end{reference} Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $f \in \mathfrak m$. Assume \begin{enumerate} \item $A$ has a dualizing complex, \item $f$ is a nonzerodivisor, \item $\text{depth}(A/fA) \geq 2$, or equivalently $\text{depth}(A) \geq 3$, \item if $f \in \mathfrak p \subset A$ is a prime ideal with $\dim(A/\mathfrak p) = 2$, then $\text{depth}(A_\mathfrak p) \geq 2$. \end{enumerate} Let $U$, resp.\ $U_0$ be the punctured spectrum of $A$, resp.\ $A/fA$. The map $$ \Pic(U) \to \Pic(U_0) $$ is injective. Finally, if (1), (2), (3), $A$ is $(S_2)$, and $\dim(A) \geq 4$, then (4) holds.


algebraization-proposition-lefschetz

In the situation above assume for all points $p \in P \setminus Q$ we have $$ \text{depth}(\mathcal{F}_p) + \dim(\overline{\{p\}}) > s $$ Then the map $$ H^i(P, \mathcal{F}) \longrightarrow \lim H^i(Q_n, \mathcal{F}_n) $$ is an isomorphism for $0 \leq i < s$.


algebraization-proposition-lefschetz-existence

In the situation above let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(P, \mathcal{I})$. Assume for all $q \in Q$ and for all primes $\mathfrak p \in \mathcal{O}_{P, q}^\wedge$, $\mathfrak p \not \in V(\mathcal{I}_q^\wedge)$ we have $$ \text{depth}((\mathcal{F}_q^\wedge)_\mathfrak p) + \dim(\mathcal{O}_{P, q}^\wedge/\mathfrak p) + \dim(\overline{\{q\}}) > 2 $$ Then $(\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_P$-module.


algebraization-proposition-lefschetz-equivalence

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module and let $s \in \Gamma(X, \mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$ and denote $\mathcal{I} \subset \mathcal{O}_X$ the corresponding sheaf of ideals. Let $\mathcal{V}$ be the set of open subschemes of $X$ containing $Y$ ordered by reverse inclusion. Assume that for all $x \in X \setminus Y$ we have $$ \text{depth}(\mathcal{O}_{X, x}) + \dim(\overline{\{x\}}) > 2 $$ Then the completion functor $$ \colim_\mathcal{V} \textit{Coh}(\mathcal{O}_V) \longrightarrow \textit{Coh}(X, \mathcal{I}) $$ is an equivalence on the full subcategories of finite locally free objects.


spaces-over-fields-lemma-quasi-finite-in-codim-1

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ is locally Noetherian. Let $y \in |Y|$ be a point of codimension $\leq 1$ on $Y$. Let $X^0 \subset |X|$ be the set of points of codimension $0$ on $X$. Assume in addition one of the following conditions is satisfied \begin{enumerate} \item for every $x \in X^0$ the transcendence degree of $x/f(x)$ is $0$, \item for every $x \in X^0$ with $f(x) \leadsto y$ the transcendence degree of $x/f(x)$ is $0$, \item $f$ is quasi-finite at every $x \in X^0$, \item $f$ is quasi-finite at a dense set of points of $|X|$, \item add more here. \end{enumerate} Then $f$ is quasi-finite at every point of $X$ lying over $y$.


spaces-over-fields-lemma-finite-in-codim-1

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is proper and $Y$ is locally Noetherian. Let $y \in Y$ be a point of codimension $\leq 1$ in $Y$. Let $X^0 \subset |X|$ be the set of points of codimension $0$ on $X$. Assume in addition one of the following conditions is satisfied \begin{enumerate} \item for every $x \in X^0$ the transcendence degree of $x/f(x)$ is $0$, \item for every $x \in X^0$ with $f(x) \leadsto y$ the transcendence degree of $x/f(x)$ is $0$, \item $f$ is quasi-finite at every $x \in X^0$, \item $f$ is quasi-finite at a dense set of points of $|X|$, \item add more here. \end{enumerate} Then there exists an open subspace $Y' \subset Y$ containing $y$ such that $Y' \times_Y X \to Y'$ is finite.


spaces-over-fields-lemma-modification-normal-iso-over-codimension-1

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $f : Y \to X$ be a birational proper morphism of algebraic spaces with $Y$ reduced. Let $U \subset X$ be the maximal open over which $f$ is an isomorphism. Then $U$ contains \begin{enumerate} \item every point of codimension $0$ in $X$, \item every $x \in |X|$ of codimension $1$ on $X$ such that the local ring of $X$ at $x$ is normal (Properties of Spaces, Remark \ref{spaces-properties-remark-list-properties-local-ring-local-etale-topology}), and \item every $x \in |X|$ such that the fibre of $|Y| \to |X|$ over $x$ is finite and such that the local ring of $X$ at $x$ is normal. \end{enumerate}


spaces-over-fields-lemma-integral-algebraic-space-rational-functions

Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$. Let $\eta \in |X|$ be the generic point of $X$. There are canonical identifications $$ R(X) = \mathcal{O}_{X, \eta}^h = \kappa(\eta) $$ where $R(X)$ is the ring of rational functions defined in Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-ring-of-rational-functions}, $\kappa(\eta)$ is the residue field defined in Decent Spaces, Definition \ref{decent-spaces-definition-residue-field}, and $\mathcal{O}_{X, \eta}^h$ is the henselian local ring defined in Decent Spaces, Definition \ref{decent-spaces-definition-elemenary-etale-neighbourhood}. In particular, these rings are fields.


spaces-over-fields-lemma-integral-sections

Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$. Then $\Gamma(X, \mathcal{O}_X)$ is a domain.


spaces-over-fields-lemma-normal-integral-cover-by-affines

Let $S$ be a scheme. Let $X$ be a normal integral algebraic space over $S$. For every $x \in |X|$ there exists a normal integral affine scheme $U$ and an \'etale morphism $U \to X$ such that $x$ is in the image.


spaces-over-fields-lemma-normal-integral-sections

Let $S$ be a scheme. Let $X$ be a normal integral algebraic space over $S$. Then $\Gamma(X, \mathcal{O}_X)$ is a normal domain.


spaces-over-fields-lemma-decent-irreducible-closed

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. There are canonical bijections between the following sets: \begin{enumerate} \item the set of points of $X$, i.e., $|X|$, \item the set of irreducible closed subsets of $|X|$, \item the set of integral closed subspaces of $X$. \end{enumerate} The bijection from (1) to (2) sends $x$ to $\overline{\{x\}}$. The bijection from (3) to (2) sends $Z$ to $|Z|$.


spaces-over-fields-lemma-finite-degree

Let $S$ be a scheme. Let $X$, $Y$ be integral algebraic spaces over $S$ Let $x \in |X|$ and $y \in |Y|$ be the generic points. Let $f : X \to Y$ be locally of finite type. Assume $f$ is dominant (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-dominant}). The following are equivalent: \begin{enumerate} \item the transcendence degree of $x/y$ is $0$, \item the extension $\kappa(x) \supset \kappa(y)$ (see proof) is finite, \item there exist nonempty affine opens $U \subset X$ and $V \subset Y$ such that $f(U) \subset V$ and $f|_U : U \to V$ is finite, \item $f$ is quasi-finite at $x$, and \item $x$ is the only point of $|X|$ mapping to $y$. \end{enumerate} If $f$ is separated or if $f$ is quasi-compact, then these are also equivalent to \begin{enumerate} \item[(6)] there exists a nonempty affine open $V \subset Y$ such that $f^{-1}(V) \to V$ is finite. \end{enumerate}


spaces-over-fields-lemma-degree-composition

Let $S$ be a scheme. Let $X$, $Y$, $Z$ be integral algebraic spaces over $S$. Let $f : X \to Y$ and $g : Y \to Z$ be dominant morphisms locally of finite type. Assume any of the equivalent conditions (1) -- (5) of Lemma \ref{lemma-finite-degree} hold for $f$ and $g$. Then $$ \deg(X/Z) = \deg(X/Y) \deg(Y/Z). $$


spaces-over-fields-lemma-components-locally-finite

Let $S$ be a scheme and let $X$ be a locally Noetherian algebraic space over $S$. If $T \subset |X|$ is a closed subset, then the collection of irreducible components of $T$ is locally finite.


spaces-over-fields-lemma-order-vanishing

Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $Z \subset X$ be a prime divisor and let $\xi \in |Z|$ be the generic point. Then the henselian local ring $\mathcal{O}_{X, \xi}^h$ is a reduced $1$-dimensional Noetherian local ring and there is a canonical injective map $$ R(X) \longrightarrow Q(\mathcal{O}_{X, \xi}^h) $$ from the function field $R(X)$ of $X$ into the total ring of fractions.


spaces-over-fields-lemma-divisor-locally-finite

Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $f \in R(X)^*$. Then the collections $$ \{Z \subset X \mid Z\text{ a prime divisor with generic point }\xi \text{ and }f\text{ not in }\mathcal{O}_{X, \xi}\} $$ and $$ \{Z \subset X \mid Z \text{ a prime divisor and }\text{ord}_Z(f) \not = 0\} $$ are locally finite in $X$.


spaces-over-fields-lemma-div-additive

Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $f, g \in R(X)^*$. Then $$ \text{div}_X(fg) = \text{div}_X(f) + \text{div}_X(g) $$ as Weil divisors on $X$.


spaces-over-fields-lemma-divisor-meromorphic-locally-finite

Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \mathcal{K}_X(\mathcal{L})$ be a regular (i.e., nonzero) meromorphic section of $\mathcal{L}$. Then the sets $$ \{Z \subset X \mid Z \text{ a prime divisor with generic point }\xi \text{ and }s\text{ not in }\mathcal{L}_\xi\} $$ and $$ \{Z \subset X \mid Z \text{ is a prime divisor and } \text{ord}_{Z, \mathcal{L}}(s) \not = 0\} $$ are locally finite in $X$.


spaces-over-fields-lemma-divisor-meromorphic-well-defined

Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$ Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s, s' \in \mathcal{K}_X(\mathcal{L})$ be nonzero meromorphic sections of $\mathcal{L}$. Then $f = s/s'$ is an element of $R(X)^*$ and we have $$ \sum \text{ord}_{Z, \mathcal{L}}(s)[Z] = \sum \text{ord}_{Z, \mathcal{L}}(s')[Z] + \text{div}(f) $$ as Weil divisors.


spaces-over-fields-lemma-c1-additive

Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $\mathcal{L}$, $\mathcal{N}$ be invertible $\mathcal{O}_X$-modules. Let $s$, resp.\ $t$ be a nonzero meromorphic section of $\mathcal{L}$, resp.\ $\mathcal{N}$. Then $st$ is a nonzero meromorphic section of $\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{N}$ and $$ \text{div}_{\mathcal{L} \otimes \mathcal{N}}(st) = \text{div}_\mathcal{L}(s) + \text{div}_\mathcal{N}(t) $$ in $\text{Div}(X)$. In particular, the Weil divisor class of $\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{N}$ is the sum of the Weil divisor classes of $\mathcal{L}$ and $\mathcal{N}$.


spaces-over-fields-lemma-alteration-generically-finite

Let $S$ be a scheme. Let $f : X \to Y$ be a proper dominant morphism of integral algebraic spaces over $S$. Then $f$ is an alteration if and only if any of the equivalent conditions (1) -- (6) of Lemma \ref{lemma-finite-degree} hold.


spaces-over-fields-lemma-alteration-contained-in

Let $S$ be a scheme. Let $f : X \to Y$ be a proper surjective morphism of algebraic spaces over $S$. Assume $Y$ is integral. Then there exists an integral closed subspace $X' \subset X$ such that $f' = f|_{X'} : X' \to Y$ is an alteration.


spaces-over-fields-lemma-locally-finite-type-dim-zero

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. In each of the following cases $X$ is a scheme: \begin{enumerate} \item $X$ is quasi-compact and quasi-separated and $\dim(X) = 0$, \item $X$ is locally of finite type over a field $k$ and $\dim(X) = 0$, \item $X$ is Noetherian and $\dim(X) = 0$, and \item add more here. \end{enumerate}


spaces-over-fields-lemma-generic-point-in-schematic-locus

Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$. Let $x \in |X|$. The following are equivalent \begin{enumerate} \item $x$ is a point of codimension $0$ on $X$, \item the local ring of $X$ at $x$ has dimension $0$, and \item $x$ is a generic point of an irreducible component of $|X|$. \end{enumerate} If true, then there exists an open subspace of $X$ containing $x$ which is a scheme.


spaces-over-fields-lemma-codim-1-point-in-schematic-locus

\begin{slogan} Separated algebraic spaces are schemes in codimension 1. \end{slogan} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. If $X$ is separated, locally Noetherian, and the dimension of the local ring of $X$ at $x$ is $\leq 1$ (Properties of Spaces, Definition \ref{spaces-properties-definition-dimension-local-ring}), then there exists an open subspace of $X$ containing $x$ which is a scheme.


spaces-over-fields-lemma-scheme-after-purely-inseparable-base-change

Let $k$ be a field. Let $X$ be an algebraic space over $k$. If there exists a purely inseparable field extension $k \subset k'$ such that $X_{k'}$ is a scheme, then $X$ is a scheme.


spaces-over-fields-lemma-when-scheme-after-base-change

Let $k$ be a field with algebraic closure $\overline{k}$. Let $X$ be a quasi-separated algebraic space over $k$. \begin{enumerate} \item If there exists a field extension $k \subset K$ such that $X_K$ is a scheme, then $X_{\overline{k}}$ is a scheme. \item If $X$ is quasi-compact and there exists a field extension $k \subset K$ such that $X_K$ is a scheme, then $X_{k'}$ is a scheme for some finite separable extension $k'$ of $k$. \end{enumerate}


spaces-over-fields-lemma-base-change-by-Galois

Let $k \subset k'$ be a finite Galois extension with Galois group $G$. Let $X$ be an algebraic space over $k$. Then $G$ acts freely on the algebraic space $X_{k'}$ and $X = X_{k'}/G$ in the sense of Properties of Spaces, Lemma \ref{spaces-properties-lemma-quotient}.


spaces-over-fields-lemma-when-quotient-scheme-at-point

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ and let $G$ be a finite group acting freely on $X$. Set $Y = X/G$ as in Properties of Spaces, Lemma \ref{spaces-properties-lemma-quotient}. For $y \in |Y|$ the following are equivalent \begin{enumerate} \item $y$ is in the schematic locus of $Y$, and \item there exists an affine open $U \subset X$ containing the preimage of $y$. \end{enumerate}


spaces-over-fields-lemma-scheme-after-purely-transcendental-base-change

Let $k$ be a field. Let $X$ be a quasi-separated algebraic space over $k$. If there exists a purely transcendental field extension $k \subset K$ such that $X_K$ is a scheme, then $X$ is a scheme.


spaces-over-fields-lemma-scheme-over-algebraic-closure-enough-affines

Let $k$ be a field with algebraic closure $\overline{k}$. Let $X$ be an algebraic space over $k$ such that \begin{enumerate} \item $X$ is decent and locally of finite type over $k$, \item $X_{\overline{k}}$ is a scheme, and \item any finite set of $\overline{k}$-rational points of $X_{\overline{k}}$ are contained in an affine. \end{enumerate} Then $X$ is a scheme.


spaces-over-fields-lemma-locally-quasi-finite-over-field

Let $k$ be a field. Let $X$ be an algebraic space over $k$. The following are equivalent \begin{enumerate} \item $X$ is locally quasi-finite over $k$, \item $X$ is locally of finite type over $k$ and has dimension $0$, \item $X$ is a scheme and is locally quasi-finite over $k$, \item $X$ is a scheme and is locally of finite type over $k$ and has dimension $0$, and \item $X$ is a disjoint union of spectra of Artinian local $k$-algebras $A$ over $k$ with $\dim_k(A) < \infty$. \end{enumerate}


spaces-over-fields-lemma-mono-towards-locally-quasi-finite-over-field

Let $k$ be a field. Let $f : X \to Y$ be a monomorphism of algebraic spaces over $k$. If $Y$ is locally quasi-finite over $k$ so is $X$.


spaces-over-fields-lemma-geometrically-reduced-at-point

Let $k$ be a field. Let $X$ be an algebraic space over $k$. Let $x \in |X|$. The following are equivalent \begin{enumerate} \item $X$ is geometrically reduced at $x$, \item for some \'etale neighbourhood $(U, u) \to (X, x)$ where $U$ is a scheme, $U$ is geometrically reduced at $u$, \item for any \'etale neighbourhood $(U, u) \to (X, x)$ where $U$ is a scheme, $U$ is geometrically reduced at $u$. \end{enumerate}


spaces-over-fields-lemma-geometrically-reduced

Let $k$ be a field. Let $X$ be an algebraic space over $k$. The following are equivalent \begin{enumerate} \item $X$ is geometrically reduced, \item for some surjective \'etale morphism $U \to X$ where $U$ is a scheme, $U$ is geometrically reduced, \item for any \'etale morphism $U \to X$ where $U$ is a scheme, $U$ is geometrically reduced. \end{enumerate}


spaces-over-fields-lemma-perfect-reduced

Let $X$ be an algebraic space over a perfect field $k$ (for example $k$ has characteristic zero). \begin{enumerate} \item For $x \in |X|$, if $\mathcal{O}_{X, \overline{x}}$ is reduced, then $X$ is geometrically reduced at $x$. \item If $X$ is reduced, then $X$ is geometrically reduced over $k$. \end{enumerate}


spaces-over-fields-lemma-geometrically-reduced-positive-characteristic

Let $k$ be a field of characteristic $p > 0$. Let $X$ be an algebraic space over $k$. The following are equivalent \begin{enumerate} \item $X$ is geometrically reduced over $k$, \item $X_{k'}$ is reduced for every field extension $k'/k$, \item $X_{k'}$ is reduced for every finite purely inseparable field extension $k'/k$, \item $X_{k^{1/p}}$ is reduced, \item $X_{k^{perf}}$ is reduced, and \item $X_{\bar k}$ is reduced. \end{enumerate}


spaces-over-fields-lemma-geometrically-reduced-upstairs

Let $k$ be a field. Let $X$ be an algebraic space over $k$. Let $k'/k$ be a field extension. Let $x \in |X|$ be a point and let $x' \in |X_{k'}|$ be a point lying over $x$. The following are equivalent \begin{enumerate} \item $X$ is geometrically reduced at $x$, \item $X_{k'}$ is geometrically reduced at $x'$. \end{enumerate} In particular, $X$ is geometrically reduced over $k$ if and only if $X_{k'}$ is geometrically reduced over $k'$.


spaces-over-fields-lemma-geometrically-reduced-etale-local

Let $k$ be a field. Let $f : X \to Y$ be a morphism of algebraic spaces over $k$. Let $x \in |X|$ be a point with image $y \in |Y|$. \begin{enumerate} \item if $f$ is \'etale at $x$, then $X$ is geometrically reduced at $x$ $\Leftrightarrow$ $Y$ is geometrically reduced at $y$, \item if $f$ is surjective \'etale, then $X$ is geometrically reduced $\Leftrightarrow$ $Y$ is geometrically reduced. \end{enumerate}


spaces-over-fields-lemma-geometrically-connected-check-after-extension

Let $X$ be an algebraic space over the field $k$. Let $k \subset k'$ be a field extension. Then $X$ is geometrically connected over $k$ if and only if $X_{k'}$ is geometrically connected over $k'$.


spaces-over-fields-lemma-bijection-connected-components

Let $k$ be a field. Let $X$, $Y$ be algebraic spaces over $k$. Assume $X$ is geometrically connected over $k$. Then the projection morphism $$ p : X \times_k Y \longrightarrow Y $$ induces a bijection between connected components.


spaces-over-fields-lemma-separably-closed-field-connected-components

Let $k \subset k'$ be an extension of fields. Let $X$ be an algebraic space over $k$. Assume $k$ separably algebraically closed. Then the morphism $X_{k'} \to X$ induces a bijection of connected components. In particular, $X$ is geometrically connected over $k$ if and only if $X$ is connected.


spaces-over-fields-lemma-characterize-geometrically-connected

Let $k$ be a field. Let $X$ be an algebraic space over $k$. Let $\overline{k}$ be a separable algebraic closure of $k$. Then $X$ is geometrically connected if and only if the base change $X_{\overline{k}}$ is connected.


spaces-over-fields-lemma-Galois-action-quasi-compact-open

Let $k$ be a field. Let $X$ be an algebraic space over $k$. Let $\overline{k}$ be a (possibly infinite) Galois extension of $k$. Let $V \subset X_{\overline{k}}$ be a quasi-compact open. Then \begin{enumerate} \item there exists a finite subextension $k \subset k' \subset \overline{k}$ and a quasi-compact open $V' \subset X_{k'}$ such that $V = (V')_{\overline{k}}$, \item there exists an open subgroup $H \subset \text{Gal}(\overline{k}/k)$ such that $\sigma(V) = V$ for all $\sigma \in H$. \end{enumerate}


spaces-over-fields-lemma-closed-fixed-by-Galois

Let $k$ be a field. Let $k \subset \overline{k}$ be a (possibly infinite) Galois extension. Let $X$ be an algebraic space over $k$. Let $\overline{T} \subset |X_{\overline{k}}|$ have the following properties \begin{enumerate} \item $\overline{T}$ is a closed subset of $|X_{\overline{k}}|$, \item for every $\sigma \in \text{Gal}(\overline{k}/k)$ we have $\sigma(\overline{T}) = \overline{T}$. \end{enumerate} Then there exists a closed subset $T \subset |X|$ whose inverse image in $|X_{k'}|$ is $\overline{T}$.


spaces-over-fields-lemma-characterize-geometrically-disconnected

Let $k$ be a field. Let $X$ be an algebraic space over $k$. The following are equivalent \begin{enumerate} \item $X$ is geometrically connected, \item for every finite separable field extension $k \subset k'$ the algebraic space $X_{k'}$ is connected. \end{enumerate}


spaces-over-fields-lemma-geometrically-integral

Let $k$ be a field. Let $X$ be a decent algebraic space over $k$. Then $X$ is geometrically integral over $k$ if and only if $X$ is both geometrically reduced and geometrically irreducible over $k$.


spaces-over-fields-lemma-proper-geometrically-reduced-global-sections

Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. \begin{enumerate} \item $A = H^0(X, \mathcal{O}_X)$ is a finite dimensional $k$-algebra, \item $A = \prod_{i = 1, \ldots, n} A_i$ is a product of Artinian local $k$-algebras, one factor for each connected component of $|X|$, \item if $X$ is reduced, then $A = \prod_{i = 1, \ldots, n} k_i$ is a product of fields, each a finite extension of $k$, \item if $X$ is geometrically reduced, then $k_i$ is finite separable over $k$, \item if $X$ is geometrically connected, then $A$ is geometrically irreducible over $k$, \item if $X$ is geometrically irreducible, then $A$ is geometrically irreducible over $k$, \item if $X$ is geometrically reduced and connected, then $A = k$, and \item if $X$ is geometrically integral, then $A = k$. \end{enumerate}


spaces-over-fields-lemma-characterize-trivial-pic-integral

Let $k$ be a field. Let $X$ be a proper integral algebraic space over $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. If $H^0(X, \mathcal{L})$ and $H^0(X, \mathcal{L}^{\otimes - 1})$ are both nonzero, then $\mathcal{L} \cong \mathcal{O}_X$.


spaces-over-fields-lemma-integral-dimension

Let $S$ be a scheme. Let $f : X \to Y$ be an integral morphism of algebraic spaces. Then $\dim(X) \leq \dim(Y)$. If $f$ is surjective then $\dim(X) = \dim(Y)$.


spaces-over-fields-lemma-alteration-dimension

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that \begin{enumerate} \item $Y$ is locally Noetherian, \item $X$ and $Y$ are integral algebraic spaces, \item $f$ is dominant, and \item $f$ is locally of finite type. \end{enumerate} If $x \in |X|$ and $y \in |Y|$ are the generic points, then $$ \dim(X) \leq \dim(Y) + \text{transcendence degree of }x/y. $$ If $f$ is proper, then equality holds.


spaces-over-fields-lemma-smooth-regular

Let $k$ be a field. Let $X$ be an algebraic space smooth over $k$. Then $X$ is a regular algebraic space.


spaces-over-fields-lemma-smooth-separable-closed-points-dense

Let $k$ be a field. Let $X$ be an algebraic space smooth over $\Spec(k)$. The set of $x \in |X|$ which are image of morphisms $\Spec(k') \to X$ with $k' \supset k$ finite separable is dense in $|X|$.


spaces-over-fields-lemma-euler-characteristic-additive

Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ be a short exact sequence of coherent modules on $X$. Then $$ \chi(X, \mathcal{F}_2) = \chi(X, \mathcal{F}_1) + \chi(X, \mathcal{F}_3) $$


spaces-over-fields-lemma-euler-characteristic-morphism

Let $k$ be a field. Let $f : Y \to X$ be a morphism of algebraic spaces proper over $k$. Let $\mathcal{G}$ be a coherent $\mathcal{O}_Y$-module. Then $$ \chi(Y, \mathcal{G}) = \sum (-1)^i \chi(X, R^if_*\mathcal{G}) $$


spaces-over-fields-lemma-numerical-polynomial-from-euler

Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Let $\mathcal{L}_1, \ldots, \mathcal{L}_r$ be invertible $\mathcal{O}_X$-modules. The map $$ (n_1, \ldots, n_r) \longmapsto \chi(X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r}) $$ is a numerical polynomial in $n_1, \ldots, n_r$ of total degree at most the dimension of the scheme theoretic support of $\mathcal{F}$.


spaces-over-fields-lemma-numerical-polynomial-leading-term

Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Let $\mathcal{L}_1, \ldots, \mathcal{L}_r$ be invertible $\mathcal{O}_X$-modules. Let $d = \dim(\text{Supp}(\mathcal{F}))$. Let $Z_i \subset X$ be the irreducible components of $\text{Supp}(\mathcal{F})$ of dimension $d$. Let $\overline{x}_i$ be a geometric generic point of $Z_i$ and set $m_i = \text{length}_{\mathcal{O}_{X, \overline{x}_i}} (\mathcal{F}_{\overline{x}_i})$. Then $$ \chi(X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r}) - \sum\nolimits_i m_i\ \chi(Z_i, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r}|_{Z_i}) $$ is a numerical polynomial in $n_1, \ldots, n_r$ of total degree $< d$.


spaces-over-fields-lemma-intersection-number-integer

In the situation of Definition \ref{definition-intersection-number} the intersection number $(\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z)$ is an integer.


spaces-over-fields-lemma-intersection-number-and-pullback

Let $k$ be a field. Let $f : Y \to X$ be a morphism of algebraic spaces proper over $k$. Let $Z \subset Y$ be an integral closed subspace of dimension $d$ and let $\mathcal{L}_1, \ldots, \mathcal{L}_d$ be invertible $\mathcal{O}_X$-modules. Then $$ (f^*\mathcal{L}_1 \cdots f^*\mathcal{L}_d \cdot Z) = \deg(f|_Z : Z \to f(Z)) (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot f(Z)) $$ where $\deg(Z \to f(Z))$ is as in Definition \ref{definition-degree} or $0$ if $\dim(f(Z)) < d$.


spaces-divisors-lemma-associated

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in |X|$. The following are equivalent \begin{enumerate} \item for some \'etale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is weakly associated to $f^*\mathcal{F}$, \item for every \'etale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is weakly associated to $f^*\mathcal{F}$, \item the maximal ideal of $\mathcal{O}_{X, \overline{x}}$ is a weakly associated prime of the stalk $\mathcal{F}_{\overline{x}}$. \end{enumerate} If $X$ is locally Noetherian, then these are also equivalent to \begin{enumerate} \item[(4)] for some \'etale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is associated to $f^*\mathcal{F}$, \item[(5)] for every \'etale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is associated to $f^*\mathcal{F}$, \item[(6)] the maximal ideal of $\mathcal{O}_{X, \overline{x}}$ is an associated prime of the stalk $\mathcal{F}_{\overline{x}}$. \end{enumerate}


spaces-divisors-lemma-ses-weakly-ass

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ be a short exact sequence of quasi-coherent sheaves on $X$. Then $\text{WeakAss}(\mathcal{F}_2) \subset \text{WeakAss}(\mathcal{F}_1) \cup \text{WeakAss}(\mathcal{F}_3)$ and $\text{WeakAss}(\mathcal{F}_1) \subset \text{WeakAss}(\mathcal{F}_2)$.


spaces-divisors-lemma-weakly-ass-zero

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $$ \mathcal{F} = (0) \Leftrightarrow \text{WeakAss}(\mathcal{F}) = \emptyset $$


spaces-divisors-lemma-minimal-support-in-weakly-ass

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in |X|$. If \begin{enumerate} \item $x \in \text{Supp}(\mathcal{F})$ \item $x$ is a codimension $0$ point of $X$ (Properties of Spaces, Definition \ref{spaces-properties-definition-dimension-local-ring}). \end{enumerate} Then $x \in \text{WeakAss}(\mathcal{F})$. If $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module with scheme theoretic support $Z$ (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-scheme-theoretic-support}) and $x$ is a codimension $0$ point of $Z$, then $x \in \text{WeakAss}(\mathcal{F})$.


spaces-divisors-lemma-weakly-ass-change-fields

Let $K/k$ be a field extension. Let $X$ be an algebraic space over $k$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $y \in X_K$ with image $x \in X$. If $y$ is a weakly associated point of the pullback $\mathcal{F}_K$, then $x$ is a weakly associated point of $\mathcal{F}$.


spaces-divisors-lemma-etale-weak-assassin-up-down

Let $S$ be a scheme. Let $f : X \to Y$ be an \'etale morphism of algebraic spaces. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module. Let $x \in |X|$ be a point with image $y \in |Y|$. Then $$ x \in \text{WeakAss}(f^*\mathcal{G}) \Leftrightarrow y \in \text{WeakAss}(\mathcal{G}) $$


spaces-divisors-lemma-locally-noetherian-fibre

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $y \in |Y|$. The following are equivalent \begin{enumerate} \item for some scheme $V$, point $v \in V$, and \'etale morphism $V \to Y$ mapping $v$ to $y$, the algebraic space $X_v$ is locally Noetherian, \item for every scheme $V$, point $v \in V$, and \'etale morphism $V \to Y$ mapping $v$ to $y$, the algebraic space $X_v$ is locally Noetherian, and \item there exists a field $k$ and a morphism $\Spec(k) \to Y$ representing $y$ such that $X_k$ is locally Noetherian. \end{enumerate} If there exists a field $k_0$ and a monomorphism $\Spec(k_0) \to Y$ representing $y$, then these are also equivalent to \begin{enumerate} \item[(4)] the algebraic space $X_{k_0}$ is locally Noetherian. \end{enumerate}


spaces-divisors-lemma-relative-assassin

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$ and $y = f(x) \in |Y|$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Consider commutative diagrams $$ \xymatrix{ X \ar[d] & X \times_Y V \ar[d] \ar[l] & X_v \ar[d] \ar[l] \\ Y & V \ar[l] & v \ar[l] } \quad \xymatrix{ X \ar[d] & U \ar[d] \ar[l] & U_v \ar[d] \ar[l] \\ Y & V \ar[l] & v \ar[l] } \quad \xymatrix{ x \ar@{|->}[d] & x' \ar@{|->}[d] \ar@{|->}[l] & u \ar@{|->}[ld] \ar@{|->}[l] \\ y & v \ar@{|->}[l] } $$ where $V$ and $U$ are schemes, $V \to Y$ and $U \to X \times_Y V$ are \'etale, $v \in V$, $x' \in |X_v|$, $u \in U$ are points related as in the last diagram. Denote $\mathcal{F}|_{X_v}$ and $\mathcal{F}|_{U_v}$ the pullbacks of $\mathcal{F}$. The following are equivalent \begin{enumerate} \item for some $V, v, x'$ as above $x'$ is a weakly associated point of $\mathcal{F}|_{X_v}$, \item for every $V \to Y, v, x'$ as above $x'$ is a weakly associated point of $\mathcal{F}|_{X_v}$, \item for some $U, V, u, v$ as above $u$ is a weakly associated point of $\mathcal{F}|_{U_v}$, \item for every $U, V, u, v$ as above $u$ is a weakly associated point of $\mathcal{F}|_{U_v}$, \item for some field $k$ and morphism $\Spec(k) \to Y$ representing $y$ and some $t \in |X_k|$ mapping to $x$, the point $t$ is a weakly associated point of $\mathcal{F}|_{X_k}$. \end{enumerate} If there exists a field $k_0$ and a monomorphism $\Spec(k_0) \to Y$ representing $y$, then these are also equivalent to \begin{enumerate} \item[(6)] $x_0$ is a weakly associated point of $\mathcal{F}|_{X_{k_0}}$ where $x_0 \in |X_{k_0}|$ is the unique point mapping to $x$. \end{enumerate} If the fibre of $f$ over $y$ is locally Noetherian, then in conditions (1), (2), (3), (4), and (6) we may replace ``weakly associated'' with ``associated''.


spaces-divisors-lemma-base-change-relative-assassin

Let $S$ be a scheme. Let $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ be a cartesian diagram of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module and set $\mathcal{F}' = (g')^*\mathcal{F}$. If $f$ is locally of finite type, then \begin{enumerate} \item $x' \in \text{Ass}_{X'/Y'}(\mathcal{F}') \Rightarrow g'(x') \in \text{Ass}_{X/Y}(\mathcal{F})$ \item if $x \in \text{Ass}_{X/Y}(\mathcal{F})$, then given $y' \in |Y'|$ with $f(x) = g(y')$, there exists an $x' \in \text{Ass}_{X'/Y'}(\mathcal{F}')$ with $g'(x') = x$ and $f'(x') = y'$. \end{enumerate}


spaces-divisors-lemma-relative-weak-assassin-finite

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $i : Z \to X$ be a finite morphism. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Z$-module. Then $\text{WeakAss}_{X/Y}(i_*\mathcal{G}) = i(\text{WeakAss}_{Z/Y}(\mathcal{G}))$.


spaces-divisors-lemma-relative-assassin-constructible

Let $Y$ be a scheme. Let $X$ be an algebraic space of finite presentation over $Y$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite presentation. Let $U \subset X$ be an open subspace such that $U \to Y$ is quasi-compact. Then the set $$ E = \{y \in Y \mid \text{Ass}_{X_y}(\mathcal{F}_y) \subset |U_y|\} $$ is locally constructible in $Y$.


spaces-divisors-lemma-fitting-ideal-of-finitely-presented

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_X$-module. Then $\text{Fit}_r(\mathcal{F})$ is a quasi-coherent ideal of finite type.


spaces-divisors-lemma-on-subscheme-cut-out-by-Fit-0

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. Let $Z_0 \subset X$ be the closed subspace cut out by $\text{Fit}_0(\mathcal{F})$. Let $Z \subset X$ be the scheme theoretic support of $\mathcal{F}$. Then \begin{enumerate} \item $Z \subset Z_0 \subset X$ as closed subspaces, \item $|Z| = |Z_0| = \text{Supp}(\mathcal{F})$ as closed subsets of $|X|$, \item there exists a finite type, quasi-coherent $\mathcal{O}_{Z_0}$-module $\mathcal{G}_0$ with $$ (Z_0 \to X)_*\mathcal{G}_0 = \mathcal{F}. $$ \end{enumerate}


spaces-divisors-lemma-locally-free-rank-r-pullback

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. The closed subspaces $$ X = Z_{-1} \supset Z_0 \supset Z_1 \supset Z_2 \ldots $$ defined by the Fitting ideals of $\mathcal{F}$ have the following properties \begin{enumerate} \item The intersection $\bigcap Z_r$ is empty. \item The functor $(\Sch/X)^{opp} \to \textit{Sets}$ defined by the rule $$ T \longmapsto \left\{ \begin{matrix} \{*\} & \text{if }\mathcal{F}_T\text{ is locally generated by } \leq r\text{ sections} \\ \emptyset & \text{otherwise} \end{matrix} \right. $$ is representable by the open subspace $X \setminus Z_r$. \item The functor $F_r : (\Sch/X)^{opp} \to \textit{Sets}$ defined by the rule $$ T \longmapsto \left\{ \begin{matrix} \{*\} & \text{if }\mathcal{F}_T\text{ locally free rank }r\\ \emptyset & \text{otherwise} \end{matrix} \right. $$ is representable by the locally closed subspace $Z_{r - 1} \setminus Z_r$ of $X$. \end{enumerate} If $\mathcal{F}$ is of finite presentation, then $Z_r \to X$, $X \setminus Z_r \to X$, and $Z_{r - 1} \setminus Z_r \to X$ are of finite presentation.


spaces-divisors-lemma-characterize-effective-Cartier-divisor

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \subset X$ be a closed subspace. The following are equivalent: \begin{enumerate} \item The subspace $D$ is an effective Cartier divisor on $X$. \item For some scheme $U$ and surjective \'etale morphism $U \to X$ the inverse image $D \times_X U$ is an effective Cartier divisor on $U$. \item For every scheme $U$ and every \'etale morphism $U \to X$ the inverse image $D \times_X U$ is an effective Cartier divisor on $U$. \item For every $x \in |D|$ there exists an \'etale morphism $(U, u) \to (X, x)$ of pointed algebraic spaces such that $U = \Spec(A)$ and $D \times_X U = \Spec(A/(f))$ with $f \in A$ not a zerodivisor. \end{enumerate}


spaces-divisors-lemma-complement-locally-principal-closed-subscheme

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \subset X$ be a locally principal closed subspace. Let $U = X \setminus Z$. Then $U \to X$ is an affine morphism.


spaces-divisors-lemma-complement-effective-Cartier-divisor

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \subset X$ be an effective Cartier divisor. Let $U = X \setminus D$. Then $U \to X$ is an affine morphism and $U$ is scheme theoretically dense in $X$.


spaces-divisors-lemma-sum-effective-Cartier-divisors

The sum of two effective Cartier divisors is an effective Cartier divisor.


spaces-divisors-lemma-sum-closed-subschemes-effective-Cartier

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z, Y$ be two closed subspaces of $X$ with ideal sheaves $\mathcal{I}$ and $\mathcal{J}$. If $\mathcal{I}\mathcal{J}$ defines an effective Cartier divisor $D \subset X$, then $Z$ and $Y$ are effective Cartier divisors and $D = Z + Y$.


spaces-divisors-lemma-pullback-locally-principal

Let $S$ be a scheme. Let $f : X' \to X$ be a morphism of algebraic spaces over $S$. Let $Z \subset X$ be a locally principal closed subspace. Then the inverse image $f^{-1}(Z)$ is a locally principal closed subspace of $X'$.


spaces-divisors-lemma-pullback-effective-Cartier-divisors-additive

Let $S$ be a scheme. Let $f : X' \to X$ be a morphism of algebraic spaces over $S$. Let $D_1$, $D_2$ be effective Cartier divisors on $X$. If the pullbacks of $D_1$ and $D_2$ are defined then the pullback of $D = D_1 + D_2$ is defined and $f^*D = f^*D_1 + f^*D_2$.


spaces-divisors-lemma-conormal-effective-Cartier-divisor

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \subset X$ be an effective Cartier divisor. Then for the conormal sheaf we have $\mathcal{C}_{D/X} = \mathcal{I}_D|D = \mathcal{O}_X(D)^{\otimes -1}|_D$.


spaces-divisors-lemma-invertible-sheaf-sum-effective-Cartier-divisors

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D_1$, $D_2$ be effective Cartier divisors on $X$. Let $D = D_1 + D_2$. Then there is a unique isomorphism $$ \mathcal{O}_X(D_1) \otimes_{\mathcal{O}_X} \mathcal{O}_X(D_2) \longrightarrow \mathcal{O}_X(D) $$ which maps $1_{D_1} \otimes 1_{D_2}$ to $1_D$.


spaces-divisors-lemma-regular-section-structure-sheaf

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $f \in \Gamma(X, \mathcal{O}_X)$. The following are equivalent: \begin{enumerate} \item $f$ is a regular section, and \item for any $x \in X$ the image $f \in \mathcal{O}_{X, \overline{x}}$ is not a zerodivisor. \item for any affine $U = \Spec(A)$ \'etale over $X$ the restriction $f|_U$ is a nonzerodivisor of $A$, and \item there exists a scheme $U$ and a surjective \'etale morphism $U \to X$ such that $f|_U$ is a regular section of $\mathcal{O}_U$. \end{enumerate}


spaces-divisors-lemma-zero-scheme

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$. \begin{enumerate} \item Consider closed immersions $i : Z \to X$ such that $i^*s \in \Gamma(Z, i^*\mathcal{L}))$ is zero ordered by inclusion. The zero scheme $Z(s)$ is the maximal element of this ordered set. \item For any morphism of algebraic spaces $f : Y \to X$ over $S$ we have $f^*s = 0$ in $\Gamma(Y, f^*\mathcal{L})$ if and only if $f$ factors through $Z(s)$. \item The zero scheme $Z(s)$ is a locally principal closed subspace of $X$. \item The zero scheme $Z(s)$ is an effective Cartier divisor on $X$ if and only if $s$ is a regular section of $\mathcal{L}$. \end{enumerate}


spaces-divisors-lemma-characterize-OD

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item If $D \subset X$ is an effective Cartier divisor, then the canonical section $1_D$ of $\mathcal{O}_X(D)$ is regular. \item Conversely, if $s$ is a regular section of the invertible sheaf $\mathcal{L}$, then there exists a unique effective Cartier divisor $D = Z(s) \subset X$ and a unique isomorphism $\mathcal{O}_X(D) \to \mathcal{L}$ which maps $1_D$ to $s$. \end{enumerate} The constructions $D \mapsto (\mathcal{O}_X(D), 1_D)$ and $(\mathcal{L}, s) \mapsto Z(s)$ give mutually inverse maps $$ \left\{ \begin{matrix} \text{effective Cartier divisors on }X \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{pairs }(\mathcal{L}, s)\text{ consisting of an invertible}\\ \mathcal{O}_X\text{-module and a regular global section} \end{matrix} \right\} $$


spaces-divisors-lemma-Noetherian-regular-separated-pic-effective-Cartier

Let $S$ be a scheme. Let $X$ be a regular Noetherian separated algebraic space over $S$. Then every invertible $\mathcal{O}_X$-module is isomorphic to $$ \mathcal{O}_X(D - D') = \mathcal{O}_X(D) \otimes_{\mathcal{O}_X} \mathcal{O}_X(D')^{\otimes -1} $$ for some effective Cartier divisors $D, D'$ in $X$.


spaces-divisors-lemma-smooth-over-valuation-ring-effective-Cartier

Let $R$ be a valuation ring with fraction field $K$. Let $X$ be an algebraic space over $R$ such that $X \to \Spec(R)$ is smooth. For every effective Cartier divisor $D \subset X_K$ there exists an effective Cartier divisor $D' \subset X$ with $D'_K = D$.


spaces-divisors-lemma-relative-Cartier

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $D \subset X$ be a closed subspace. Assume \begin{enumerate} \item $D$ is an effective Cartier divisor, and \item $D \to Y$ is a flat morphism. \end{enumerate} Then for every morphism of schemes $g : Y' \to Y$ the pullback $(g')^{-1}D$ is an effective Cartier divisor on $X' = Y' \times_Y X$ where $g' : X' \to X$ is the projection.


spaces-divisors-lemma-describe-S

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $U$ affine and \'etale over $X$ the set $\mathcal{S}_X(U)$ is the set of nonzerodivisors in $\mathcal{O}_X(U)$.


spaces-divisors-lemma-meromorphic-quasi-coherent

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume \begin{enumerate} \item[(a)] every weakly associated point of $X$ is a point of codimension $0$, and \item[(b)] $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-prepare-normalization}. \end{enumerate} Then \begin{enumerate} \item $\mathcal{K}_X$ is a quasi-coherent sheaf of $\mathcal{O}_X$-algebras, \item for $U \in X_\etale$ affine $\mathcal{K}_X(U)$ is the total ring of fractions of $\mathcal{O}_X(U)$, \item for a geometric point $\overline{x}$ the set $\mathcal{S}_{\overline{x}}$ the set of nonzerodivisors of $\mathcal{O}_{X, \overline{x}}$, and \item for a geometric point $\overline{x}$ the ring $\mathcal{K}_{X, \overline{x}}$ is the total ring of fractions of $\mathcal{O}_{X, \overline{x}}$. \end{enumerate}


spaces-divisors-lemma-meromorphic-quasi-coherent-agree

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume \begin{enumerate} \item[(a)] every weakly associated point of $X$ is a point of codimension $0$, and \item[(b)] $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-prepare-normalization}. \item[(c)] $X$ is representable by a scheme $X_0$ (awkward but temporary notation). \end{enumerate} Then the sheaf of meromorphic functions $\mathcal{K}_X$ is the quasi-coherent sheaf of $\mathcal{O}_X$-algebras associated to the quasi-coherent sheaf of meromorphic functions $\mathcal{K}_{X_0}$.


spaces-divisors-lemma-pullback-meromorphic-sections-defined

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Pullbacks of meromorphic sections are defined in each of the following cases \begin{enumerate} \item weakly associated points of $X$ are mapped to points of codimension $0$ on $Y$, \item $f$ is flat, \item add more here as needed. \end{enumerate}


spaces-divisors-lemma-compute-meromorphic

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume \begin{enumerate} \item[(a)] every weakly associated point of $X$ is a point of codimension $0$, and \item[(b)] $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-prepare-normalization}, \item[(c)] every codimension $0$ point of $X$ can be represented by a monomorphism $\Spec(k) \to X$. \end{enumerate} Let $X^0 \subset |X|$ be the set of codimension $0$ points of $X$. Then we have $$ \mathcal{K}_X = \bigoplus\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{O}_{X, \eta} = \prod\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{O}_{X, \eta} $$ where $j_\eta : \Spec(\mathcal{O}_{X, \eta}) \to X$ is the canonical map of Schemes, Section \ref{schemes-section-points}; this makes sense because $X^0$ is contained in the schematic locus of $X$. Similarly, for every quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ we obtain the formula $$ \mathcal{K}_X(\mathcal{F}) = \bigoplus\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{F}_\eta = \prod\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{F}_\eta $$ for the sheaf of meromorphic sections of $\mathcal{F}$. Finally, the ring of rational functions of $X$ is the ring of meromorphic functions on $X$, in a formula: $R(X) = \Gamma(X, \mathcal{K}_X)$.


spaces-divisors-lemma-meromorphic-sections-pullback

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that pullbacks of meromorphic functions are defined for $f$ (see Definition \ref{definition-pullback-meromorphic-sections}). \begin{enumerate} \item Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_Y$-modules. There is a canonical pullback map $f^* : \Gamma(Y, \mathcal{K}_Y(\mathcal{F})) \to \Gamma(X, \mathcal{K}_X(f^*\mathcal{F}))$ for meromorphic sections of $\mathcal{F}$. \item Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. A regular meromorphic section $s$ of $\mathcal{L}$ pulls back to a regular meromorphic section $f^*s$ of $f^*\mathcal{L}$. \end{enumerate}


spaces-divisors-lemma-relative-proj

In Situation \ref{situation-relative-proj}. The functor $F$ above is an algebraic space. For any morphism $g : Z \to X$ where $Z$ is a scheme there is a canonical isomorphism $\underline{\text{Proj}}_Z(g^*\mathcal{A}) = Z \times_X F$ compatible with further base change.


spaces-divisors-lemma-relative-proj-base-change

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


spaces-divisors-lemma-relative-proj-separated

In Situation \ref{situation-relative-proj} the morphism $\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X$ is separated.


spaces-divisors-lemma-relative-proj-generated-in-degree-1

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_X$-modules generated as an $\mathcal{A}_0$-algebra by $\mathcal{A}_1$. With $P = \underline{\text{Proj}}_X(\mathcal{A})$ we have \begin{enumerate} \item $P$ represents the functor $F_1$ which associates to $T$ over $S$ the set of isomorphism classes of triples $(f, \mathcal{L}, \psi)$, where $f : T \to X$ is a morphism over $S$, $\mathcal{L}$ is an invertible $\mathcal{O}_T$-module, and $\psi : f^*\mathcal{A} \to \bigoplus_{n \geq 0} \mathcal{L}^{\otimes n}$ is a map of graded $\mathcal{O}_T$-algebras inducing a surjection $f^*\mathcal{A}_1 \to \mathcal{L}$, \item the canonical map $\pi^*\mathcal{A}_1 \to \mathcal{O}_P(1)$ is surjective, and \item each $\mathcal{O}_P(n)$ is invertible and the multiplication maps induce isomorphisms $\mathcal{O}_P(n) \otimes_{\mathcal{O}_P} \mathcal{O}_P(m) = \mathcal{O}_P(n + m)$. \end{enumerate}


spaces-divisors-lemma-morphism-relative-proj

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\psi : \mathcal{A} \to \mathcal{B}$ be a map of quasi-coherent graded $\mathcal{O}_X$-algebras. Set $P = \underline{\text{Proj}}_X(\mathcal{A}) \to X$ and $Q = \underline{\text{Proj}}_X(\mathcal{B}) \to X$. There is a canonical open subspace $U(\psi) \subset Q$ and a canonical morphism of algebraic spaces $$ r_\psi : U(\psi) \longrightarrow P $$ over $X$ and a map of $\mathbf{Z}$-graded $\mathcal{O}_{U(\psi)}$-algebras $$ \theta = \theta_\psi : r_\psi^*\left( \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_P(d) \right) \longrightarrow \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_{U(\psi)}(d). $$ The triple $(U(\psi), r_\psi, \theta)$ is characterized by the property that for any scheme $W$ \'etale over $X$ the triple $$ (U(\psi) \times_X W,\quad r_\psi|_{U(\psi) \times_X W} : U(\psi) \times_X W \to  P \times_X W,\quad \theta|_{U(\psi) \times_X W}) $$ is equal to the triple associated to $\psi : \mathcal{A}|_W \to \mathcal{B}|_W$ of Constructions, Lemma \ref{constructions-lemma-morphism-relative-proj}.


spaces-divisors-lemma-morphism-relative-proj-transitive

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ be quasi-coherent graded $\mathcal{O}_X$-algebras. Set $P = \underline{\text{Proj}}_X(\mathcal{A})$, $Q = \underline{\text{Proj}}_X(\mathcal{B})$ and $R = \underline{\text{Proj}}_X(\mathcal{C})$. Let $\varphi : \mathcal{A} \to \mathcal{B}$, $\psi : \mathcal{B} \to \mathcal{C}$ be graded $\mathcal{O}_X$-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.


spaces-divisors-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 $U(\psi) = Q$, \item $r_\psi : Q \to R$ is a closed immersion, and \item the maps $\theta : r_\psi^*\mathcal{O}_P(n) \to \mathcal{O}_Q(n)$ are surjective but not isomorphisms in general (even if $\mathcal{A} \to \mathcal{B}$ is surjective). \end{enumerate}


spaces-divisors-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 $U(\psi) = Q$, \item $r_\psi : Q \to P$ is a closed immersion, and \item the maps $\theta : r_\psi^*\mathcal{O}_P(n) \to \mathcal{O}_Q(n)$ are isomorphisms. \end{enumerate}


spaces-divisors-lemma-invertible-map-into-relative-proj

With assumptions and notation as above. The morphism $\psi$ induces a canonical morphism of algebraic spaces over $Y$ $$ r_{\mathcal{L}, \psi} : U(\psi) \longrightarrow \underline{\text{Proj}}_Y(\mathcal{A}) $$ together with a map of graded $\mathcal{O}_{U(\psi)}$-algebras $$ \theta : r_{\mathcal{L}, \psi}^*\left( \bigoplus\nolimits_{d \geq 0} \mathcal{O}_{\underline{\text{Proj}}_Y(\mathcal{A})}(d) \right) \longrightarrow \bigoplus\nolimits_{d \geq 0} \mathcal{L}^{\otimes d}|_{U(\psi)} $$ characterized by the following properties: \begin{enumerate} \item For $V \to Y$ \'etale and $d \geq 0$ the diagram $$ \xymatrix{ \mathcal{A}_d(V) \ar[d]_{\psi} \ar[r]_{\psi} & \Gamma(V \times_Y X, \mathcal{L}^{\otimes d}) \ar[d]^{restrict} \\ \Gamma(V \times_Y \underline{\text{Proj}}_Y(\mathcal{A}), \mathcal{O}_{\underline{\text{Proj}}_Y(\mathcal{A})}(d)) \ar[r]^-\theta & \Gamma(V \times_Y U(\psi), \mathcal{L}^{\otimes d}) } $$ is commutative. \item For any $d \geq 1$ and any morphism $W \to X$ where $W$ is a scheme such that $\psi|_W : f^*\mathcal{A}_d|_W \to \mathcal{L}^{\otimes d}|_W$ is surjective we have (a) $W \to X$ factors through $U(\psi)$ and (b) composition of $W \to U(\psi)$ with $r_{\mathcal{L}, \psi}$ agrees with the morphism $W \to \underline{\text{Proj}}_Y(\mathcal{A})$ which exists by the construction of $\underline{\text{Proj}}_Y(\mathcal{A})$, see Definition \ref{definition-relative-proj}. \item Consider a commutative diagram $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ where $X'$ and $Y'$ are schemes, set $\mathcal{A}' = g^*\mathcal{A}$ and $\mathcal{L}' = (g')^*\mathcal{L}$ and denote $\psi' : (f')^*\mathcal{A} \to \bigoplus_{d \geq 0} (\mathcal{L}')^{\otimes d}$ the pullback of $\psi$. Let $U(\psi')$, $r_{\psi', \mathcal{L}'}$, and $\theta'$ be the open, morphism, and homomorphism constructed in Constructions, Lemma \ref{lemma-invertible-map-into-relative-proj}. Then $U(\psi') = (g')^{-1}(U(\psi))$ and $r_{\psi', \mathcal{L}'}$ agrees with the base change of $r_{\psi, \mathcal{L}}$ via the isomorphism $\underline{\text{Proj}}_{Y'}(\mathcal{A}') = Y' \times_Y \underline{\text{Proj}}_Y(\mathcal{A})$ of Lemma \ref{lemma-relative-proj-base-change}. Moreover, $\theta'$ is the pullback of $\theta$. \end{enumerate}


spaces-divisors-lemma-ample-base-change

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $Y' \to Y$ be a morphism of algebraic spaces over $S$. Let $f' : X' \to Y'$ be the base change of $f$ and denote $\mathcal{L}'$ the pullback of $\mathcal{L}$ to $X'$. If $\mathcal{L}$ is $f$-ample, then $\mathcal{L}'$ is $f'$-ample.


spaces-divisors-lemma-relatively-ample-properties

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If there exists an $f$-ample invertible sheaf, then $f$ is representable, quasi-compact, and separated.


spaces-divisors-lemma-descend-relatively-ample

Let $V \to U$ be a surjective \'etale morphism of affine schemes. Let $X$ be an algebraic space over $U$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $Y = V \times_U X$ and let $\mathcal{N}$ be the pullback of $\mathcal{L}$ to $Y$. The following are equivalent \begin{enumerate} \item $\mathcal{L}$ is ample on $X/U$, and \item $\mathcal{N}$ is ample on $Y/V$. \end{enumerate}


spaces-divisors-lemma-relatively-ample-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{L}$ is ample on $X/Y$, \item for every scheme $Z$ and every morphism $Z \to Y$ the algebraic space $X_Z = Z \times_Y X$ is a scheme and the pullback $\mathcal{L}_Z$ is ample on $X_Z/Z$, \item for every affine scheme $Z$ and every morphism $Z \to Y$ the algebraic space $X_Z = Z \times_Y X$ is a scheme and the pullback $\mathcal{L}_Z$ is ample on $X_Z/Z$, \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that the algebraic space $X_V = V \times_Y X$ is a scheme and the pullback $\mathcal{L}_V$ is ample on $X_V/V$. \end{enumerate}


spaces-divisors-lemma-ample-on-fibre

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


spaces-divisors-lemma-ample-in-neighbourhood

(For a more general version see Descent on Spaces, Lemma \ref{spaces-descent-lemma-ample-in-neighbourhood}). Let $Y$ be a Noetherian scheme. Let $X$ be an algebraic space over $Y$ such that the structure morphism $f : X \to Y$ is proper. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $y \in Y$ be a point such that $X_y$ is a scheme and $\mathcal{L}_y$ is ample on $X_y$. Then there is an open neighbourhood $V \subset Y$ of $y$ such that $\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$ (as in Definition \ref{definition-relatively-ample}).


spaces-divisors-lemma-closed-subscheme-proj

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_X$-algebra. Let $\pi : P = \underline{\text{Proj}}_X(\mathcal{A}) \to X$ be the relative Proj of $\mathcal{A}$. Let $i : Z \to P$ be a closed subspace. Denote $\mathcal{I} \subset \mathcal{A}$ the kernel of the canonical map $$ \mathcal{A} \longrightarrow \bigoplus\nolimits_{d \geq 0} \pi_*\left((i_*\mathcal{O}_Z)(d)\right) $$ If $\pi$ is quasi-compact, then there is an isomorphism $Z = \underline{\text{Proj}}_X(\mathcal{A}/\mathcal{I})$.


spaces-divisors-lemma-closed-subscheme-proj-finite

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_X$-algebra. Let $\pi : P = \underline{\text{Proj}}_X(\mathcal{A}) \to X$ be the relative Proj of $\mathcal{A}$. Let $i : Z \to P$ be a closed subscheme. If $\pi$ is quasi-compact and $i$ of finite presentation, then there exists a $d > 0$ and a quasi-coherent finite type $\mathcal{O}_X$-submodule $\mathcal{F} \subset \mathcal{A}_d$ such that $Z = \underline{\text{Proj}}_X(\mathcal{A}/\mathcal{F}\mathcal{A})$.


spaces-divisors-lemma-closed-subscheme-proj-finite-type

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_X$-algebra. Let $\pi : P = \underline{\text{Proj}}_X(\mathcal{A}) \to X$ be the relative Proj of $\mathcal{A}$. Let $i : Z \to X$ be a closed subspace. Let $U \subset X$ be an open. Assume that \begin{enumerate} \item $\pi$ is quasi-compact, \item $i$ of finite presentation, \item $|U| \cap |\pi|(|i|(|Z|)) = \emptyset$, \item $U$ is quasi-compact, \item $\mathcal{A}_n$ is a finite type $\mathcal{O}_X$-module for all $n$. \end{enumerate} Then there exists a $d > 0$ and a quasi-coherent finite type $\mathcal{O}_X$-submodule $\mathcal{F} \subset \mathcal{A}_d$ with (a) $Z = \underline{\text{Proj}}_X(\mathcal{A}/\mathcal{F}\mathcal{A})$ and (b) the support of $\mathcal{A}_d/\mathcal{F}$ is disjoint from $U$.


spaces-divisors-lemma-conormal-sheaf-section-projective-bundle

Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_X$-module. There is a bijection $$ \left\{ \begin{matrix} \text{sections }\sigma\text{ of the } \\ \text{morphism } \mathbf{P}(\mathcal{E}) \to X \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{surjections }\mathcal{E} \to \mathcal{L}\text{ where} \\ \mathcal{L}\text{ is an invertible }\mathcal{O}_X\text{-module} \end{matrix} \right\} $$ In this case $\sigma$ is a closed immersion and there is a canonical isomorphism $$ \Ker(\mathcal{E} \to \mathcal{L}) \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes -1} \longrightarrow \mathcal{C}_{\sigma(X)/\mathbf{P}(\mathcal{E})} $$ Both the bijection and isomorphism are compatible with base change.


spaces-divisors-lemma-blowing-up-affine

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $U = \Spec(A)$ be an affine scheme \'etale over $X$ and let $I \subset A$ be the ideal corresponding to $\mathcal{I}|_U$. If $X' \to X$ is the blowup of $X$ in $\mathcal{I}$, then there is a canonical isomorphism $$ U \times_X X' = \text{Proj}(\bigoplus\nolimits_{d \geq 0} I^d) $$ of schemes over $U$, where the right hand side is the homogeneous spectrum of the Rees algebra of $I$ in $A$. Moreover, $U \times_X X'$ has an affine open covering by spectra of the affine blowup algebras $A[\frac{I}{a}]$.


spaces-divisors-lemma-flat-base-change-blowing-up

Let $S$ be a scheme. Let $X_1 \to X_2$ be a flat morphism of algebraic spaces over $S$. Let $Z_2 \subset X_2$ be a closed subspace. Let $Z_1$ be the inverse image of $Z_2$ in $X_1$. Let $X'_i$ be the blowup of $Z_i$ in $X_i$. Then there exists a cartesian diagram $$ \xymatrix{ X_1' \ar[r] \ar[d] & X_2' \ar[d] \\ X_1 \ar[r] & X_2 } $$ of algebraic spaces over $S$.