sites-lemma-site-gives-topology

Let $\mathcal{C}$ be a site with coverings $\text{Cov}(\mathcal{C})$. For every object $U$ of $\mathcal{C}$, let $J(U)$ denote the set of sieves $S$ on $U$ with the following property: there exists a covering $\{f_i : U_i \to U\}_{i\in I} \in \text{Cov}(\mathcal{C})$ so that the sieve $S'$ generated by the $f_i$ (see Definition \ref{definition-sieve-generated}) is contained in $S$. \begin{enumerate} \item This $J$ is a topology on $\mathcal{C}$. \item A presheaf $\mathcal{F}$ is a sheaf for this topology (see Definition \ref{definition-sheaf-sets-topology}) if and only if it is a sheaf on the site (see Definition \ref{definition-sheaf-sets}). \end{enumerate}


sites-lemma-L-presheaf

In the situation above. \begin{enumerate} \item The assignment $U \mapsto L\mathcal{F}(U)$ combined with the restriction mappings defined above is a presheaf. \item The maps $\ell$ glue to give a morphism of presheaves $\ell : \mathcal{F} \to L\mathcal{F}$. \item The rule $\mathcal{F} \mapsto (\mathcal{F} \xrightarrow{\ell} L\mathcal{F})$ is a functor. \item If $\mathcal{F}$ is a subpresheaf of $\mathcal{G}$, then $L\mathcal{F}$ is a subpresheaf of $L\mathcal{G}$. \item The map $\ell : \mathcal{F} \to L\mathcal{F}$ has the following property: For every section $s \in L\mathcal{F}(U)$ there exists a covering sieve $S$ on $U$ and an element $\varphi \in \Mor_{\textit{PSh}(\mathcal{C})}(S, \mathcal{F})$ such that $\ell(\varphi)$ equals the restriction of $s$ to $S$. \end{enumerate}


sites-lemma-sieve-sheafification

Let $\mathcal{C}$ be a category endowed with a topology $J$. Let $U$ be an object of $\mathcal{C}$. Let $S$ be a sieve on $U$. The following are equivalent \begin{enumerate} \item The sieve $S$ is a covering sieve. \item The sheafification $S^\# \to h_U^\#$ of the map $S \to h_U$ is an isomorphism. \end{enumerate}


sites-proposition-sheafification-adjoint

The canonical map $\mathcal{F} \to \mathcal{F}^\#$ has the following universal property: For any map $\mathcal{F} \to \mathcal{G}$, where $\mathcal{G}$ is a sheaf of sets, there is a unique map $\mathcal{F}^\# \to \mathcal{G}$ such that $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$ equals the given map.


sites-proposition-get-morphism

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be continuous. Assume furthermore the following: \begin{enumerate} \item the category $\mathcal{C}$ has a final object $X$ and $u(X)$ is a final object of $\mathcal{D}$ , and \item the category $\mathcal{C}$ has fibre products and $u$ commutes with them. \end{enumerate} Then $u$ defines a morphism of sites $\mathcal{D} \to \mathcal{C}$, in other words $u_s$ is exact.


sites-proposition-point-limits

Let $\mathcal{C}$ be a site. Assume that finite limits exist in $\mathcal{C}$. (I.e., $\mathcal{C}$ has fibre products, and a final object.) A point $p$ of such a site $\mathcal{C}$ is given by a functor $u : \mathcal{C} \to \textit{Sets}$ such that \begin{enumerate} \item $u$ commutes with finite limits, and \item if $\{U_i \to U\}$ is a covering, then $\coprod_i u(U_i) \to u(U)$ is surjective. \end{enumerate}


sites-proposition-criterion-points

\begin{reference} \cite[Expos\'e VI, Appendix by Deligne, Proposition 9.0]{SGA4} \end{reference} Let $\mathcal{C}$ be a site. Assume that \begin{enumerate} \item finite limits exist in $\mathcal{C}$, and \item every covering $\{U_i \to U\}_{i \in I}$ has a refinement by a finite covering of $\mathcal{C}$. \end{enumerate} Then $\mathcal{C}$ has enough points.


sites-proposition-functoriality-algebraic-structures-topoi

\begin{slogan} Morphisms of topoi preserve algebraic structure. \end{slogan} Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $f = (f^{-1}, f_*)$ be a morphism of topoi from $\Sh(\mathcal{D}) \to \Sh(\mathcal{C})$. The method introduced above gives rise to an adjoint pair of functors $(f^{-1}, f_*)$ on sheaves of algebraic structures compatible with taking the underlying sheaves of sets for the following types of algebraic structures: \begin{enumerate} \item pointed sets, \item abelian groups, \item groups, \item monoids, \item rings, \item modules over a fixed ring, and \item lie algebras over a fixed field. \end{enumerate} Moreover, in each of these cases the results above labeled ($\alpha$), ($\beta$), ($\gamma$), ($\delta$), ($\epsilon$), and ($\zeta$) hold.


examples-defos-lemma-finite-projective-modules-RS

Example \ref{example-finite-projective-modules} satisfies the Rim-Schlessinger condition (RS). In particular, $\Deformationcategory_V$ is a deformation category for any finite dimensional vector space $V$ over $k$.


examples-defos-lemma-finite-projective-modules-TI

In Example \ref{example-finite-projective-modules} let $V$ be a finite dimensional $k$-vector space. Then $$ T\Deformationcategory_V = (0) \quad\text{and}\quad \text{Inf}(\Deformationcategory_V) = \text{End}_k(V) $$ are finite dimensional.


examples-defos-lemma-representations-RS

Example \ref{example-representations} satisfies the Rim-Schlessinger condition (RS). In particular, $\Deformationcategory_{V, \rho_0}$ is a deformation category for any finite dimensional representation $\rho_0 : \Gamma \to \text{GL}_k(V)$.


examples-defos-lemma-representations-TI

In Example \ref{example-representations} let  $\rho_0 : \Gamma \to \text{GL}_k(V)$ be a finite dimensional representation. Then $$ T\Deformationcategory_{V, \rho_0} = \Ext^1_{k[\Gamma]}(V, V) = H^1(\Gamma, \text{End}_k(V)) \quad\text{and}\quad \text{Inf}(\Deformationcategory_{V, \rho_0}) = H^0(\Gamma, \text{End}_k(V)) $$ Thus $\text{Inf}(\Deformationcategory_{V, \rho_0})$ is always finite dimensional and $T\Deformationcategory_{V, \rho_0}$ is finite dimensional if $\Gamma$ is finitely generated.


examples-defos-lemma-representations-hull

In Example \ref{example-representations} assume $\Gamma$ finitely generated. Let $\rho_0 : \Gamma \to \text{GL}_k(V)$ be a finite dimensional representation. Assume $\Lambda$ is a complete local ring with residue field $k$ (the classical case). Then the functor $$ F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \Ob(\Deformationcategory_{V, \rho_0}(A))/\cong $$ of isomorphism classes of objects has a hull. If $H^0(\Gamma, \text{End}_k(V)) = k$, then $F$ is prorepresentable.


examples-defos-lemma-continuous-representations-RS

Example \ref{example-continuous-representations} satisfies the Rim-Schlessinger condition (RS). In particular, $\Deformationcategory_{V, \rho_0}$ is a deformation category for any finite dimensional continuous representation $\rho_0 : \Gamma \to \text{GL}_k(V)$.


examples-defos-lemma-continuous-representations-TI

In Example \ref{example-continuous-representations} let $\rho_0 : \Gamma \to \text{GL}_k(V)$ be a finite dimensional continuous representation. Then $$ T\Deformationcategory_{V, \rho_0} = H^1(\Gamma, \text{End}_k(V)) \quad\text{and}\quad \text{Inf}(\Deformationcategory_{V, \rho_0}) = H^0(\Gamma, \text{End}_k(V)) $$ Thus $\text{Inf}(\Deformationcategory_{V, \rho_0})$ is always finite dimensional and $T\Deformationcategory_{V, \rho_0}$ is finite dimensional if $\Gamma$ is topologically finitely generated.


examples-defos-lemma-graded-algebras-RS

Example \ref{example-graded-algebras} satisfies the Rim-Schlessinger condition (RS). In particular, $\Deformationcategory_P$ is a deformation category for any graded $k$-algebra $P$.


examples-defos-lemma-graded-algebras-TI

In Example \ref{example-graded-algebras} let $P$ be a graded $k$-algebra. Then $$ T\Deformationcategory_P \quad\text{and}\quad \text{Inf}(\Deformationcategory_P) = \text{Der}_k(P, P) $$ are finite dimensional if $P$ is finitely generated over $k$.


examples-defos-lemma-rings-RS

Example \ref{example-rings} satisfies the Rim-Schlessinger condition (RS). In particular, $\Deformationcategory_P$ is a deformation category for any $k$-algebra $P$.


examples-defos-lemma-rings-TI

In Example \ref{example-rings} let $P$ be a $k$-algebra. Then $$ T\Deformationcategory_P = \text{Ext}^1_P(\NL_{P/k}, P) \quad\text{and}\quad \text{Inf}(\Deformationcategory_P) = \text{Der}_k(P, P) $$


examples-defos-lemma-smooth

In Example \ref{example-rings} let $P$ be a smooth $k$-algebra. Then $T\Deformationcategory_P = (0)$.


examples-defos-lemma-finite-type-rings-TI

In Lemma \ref{lemma-rings-TI} if $P$ is a finite type $k$-algebra, then \begin{enumerate} \item $\text{Inf}(\Deformationcategory_P)$ is finite dimensional if and only if $\dim(P) = 0$, and \item $T\Deformationcategory_P$ is finite dimensional if $\Spec(P) \to \Spec(k)$ is smooth except at a finite number of points. \end{enumerate}


examples-defos-lemma-localization

In Example \ref{example-rings} let $P$ be a $k$-algebra. Let $S \subset P$ be a multiplicative subset. There is a natural functor $$ \Deformationcategory_P \longrightarrow \Deformationcategory_{S^{-1}P} $$ of deformation categories.


examples-defos-lemma-henselization

In Example \ref{example-rings} let $P$ be a $k$-algebra. Let $J \subset P$ be an ideal. Denote $(P^h, J^h)$ the henselization of the pair $(P, J)$. There is a natural functor $$ \Deformationcategory_P \longrightarrow \Deformationcategory_{P^h} $$ of deformation categories.


examples-defos-lemma-completion

In Example \ref{example-rings} let $P$ be a $k$-algebra. Assume $P$ Noetherian and let $J \subset P$ be an ideal. Denote $P^\wedge$ the $J$-adic completion. There is a natural functor $$ \Deformationcategory_P \longrightarrow \Deformationcategory_{P^\wedge} $$ of deformation categories.


examples-defos-lemma-schemes-RS

Example \ref{example-schemes} satisfies the Rim-Schlessinger condition (RS). In particular, $\Deformationcategory_X$ is a deformation category for any scheme $X$ over $k$.


examples-defos-lemma-schemes-TI

In Example \ref{example-schemes} let $X$ be a scheme over $k$. Then $$ \text{Inf}(\Deformationcategory_X) = \text{Ext}^0_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X) = \Hom_{\mathcal{O}_X}(\Omega_{X/k}, \mathcal{O}_X) = \text{Der}_k(\mathcal{O}_X, \mathcal{O}_X) $$ and $$ T\Deformationcategory_X = \text{Ext}^1_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X) $$


examples-defos-lemma-proper-schemes-TI

In Lemma \ref{lemma-schemes-TI} if $X$ is proper over $k$, then $\text{Inf}(\Deformationcategory_X)$ and $T\Deformationcategory_X$ are finite dimensional.


examples-defos-lemma-open

In Example \ref{example-schemes} let $X$ be a scheme over $k$. Let $U \subset X$ be an open subscheme. There is a natural functor $$ \Deformationcategory_X \longrightarrow \Deformationcategory_U $$ of deformation categories.


examples-defos-lemma-affine

In Example \ref{example-schemes} let $X = \Spec(P)$ be an affine scheme over $k$. With $\Deformationcategory_P$ as in Example \ref{example-rings} there is a natural equivalence $$ \Deformationcategory_X \longrightarrow \Deformationcategory_P $$ of deformation categories.


examples-defos-lemma-glueing

In Situation \ref{situation-glueing} there is an equivalence $$ \Deformationcategory_X = \Deformationcategory_{P_1} \times_{\Deformationcategory_{P_{12}}} \Deformationcategory_{P_2} $$ of deformation categories, see Examples \ref{example-schemes} and \ref{example-rings}.


examples-defos-lemma-schemes-morphisms-RS

Example \ref{example-schemes-morphisms} satisfies the Rim-Schlessinger condition (RS). In particular, $\Deformationcategory_{X \to Y}$ is a deformation category for any morphism of schemes $X \to Y$ over $k$.


examples-defos-lemma-schemes-morphisms-TI

In Example \ref{example-schemes} let $f : X \to Y$ be a morphism of schemes over $k$. There is a canonical exact sequence of $k$-vector spaces $$ \xymatrix{ 0 \ar[r] & \text{Inf}(\Deformationcategory_{X \to Y}) \ar[r] & \text{Inf}(\Deformationcategory_X \times \Deformationcategory_Y) \ar[r] & \text{Der}_k(\mathcal{O}_Y, f_*\mathcal{O}_X) \ar[lld] \\ & T\Deformationcategory_{X \to Y} \ar[r] & T(\Deformationcategory_X \times \Deformationcategory_Y) \ar[r] & \text{Ext}^1_{\mathcal{O}_X}(Lf^*\NL_{Y/k}, \mathcal{O}_X) } $$


examples-defos-lemma-proper-schemes-morphisms-TI

In Lemma \ref{lemma-schemes-morphisms-TI} if $X$ and $Y$ are both proper over $k$, then $\text{Inf}(\Deformationcategory_{X \to Y})$ and $T\Deformationcategory_{X \to Y}$ are finite dimensional.


examples-defos-lemma-schemes-morphisms-smooth-to-base

\begin{reference} This is discussed in \cite[Section 5.3]{Ravi-Murphys-Law} and \cite[Theorem 3.3]{Ran-deformations}. \end{reference} In Example \ref{example-schemes} let $f : X \to Y$ be a morphism of schemes over $k$. If $f_*\mathcal{O}_X = \mathcal{O}_Y$ and $R^1f_*\mathcal{O}_X = 0$, then the morphism of deformation categories $$ \Deformationcategory_{X \to Y} \to \Deformationcategory_X $$ is an equivalence.


examples-defos-lemma-spaces-RS

Example \ref{example-spaces} satisfies the Rim-Schlessinger condition (RS). In particular, $\Deformationcategory_X$ is a deformation category for any algebraic space $X$ over $k$.


examples-defos-lemma-spaces-TI

In Example \ref{example-spaces} let $X$ be an algebraic space over $k$. Then $$ \text{Inf}(\Deformationcategory_X) = \text{Ext}^0_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X) = \Hom_{\mathcal{O}_X}(\Omega_{X/k}, \mathcal{O}_X) = \text{Der}_k(\mathcal{O}_X, \mathcal{O}_X) $$ and $$ T\Deformationcategory_X = \text{Ext}^1_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X) $$


examples-defos-lemma-proper-spaces-TI

In Lemma \ref{lemma-spaces-TI} if $X$ is proper over $k$, then $\text{Inf}(\Deformationcategory_X)$ and $T\Deformationcategory_X$ are finite dimensional.


examples-defos-lemma-lift-equivalence-module-derived

Let $A' \to A$ be a surjection of rings with nilpotent kernel. Let $A' \to P'$ be a flat ring map. Set $P = P' \otimes_{A'} A$. Let $M$ be an $A$-flat $P$-module. Then the following are equivalent \begin{enumerate} \item there is an $A'$-flat $P'$-module $M'$ with $M' \otimes_{P'} P = M$, and \item there is an object $K' \in D^-(P')$ with $K' \otimes_{P'}^\mathbf{L} P = M$. \end{enumerate}


examples-defos-lemma-lift-equivalence-module

Consider a commutative diagram of Noetherian rings $$ \xymatrix{ A' \ar[d] \ar[r] & P' \ar[d] \ar[r] & Q' \ar[d] \\ A \ar[r] & P \ar[r] & Q } $$ with cartesian squares, with flat horizontal arrows, and with surjective vertial arrows whose kernels are nilpotent. Let $J' \subset P'$ be an ideal such that $P'/J' = Q'/J'Q'$. Let $M$ be an $A$-flat $P$-module. Assume for all $g \in J'$ there exists an $A'$-flat $(P')_g$-module lifting $M_g$. Then the following are equivalent \begin{enumerate} \item $M$ has an $A'$-flat lift to a $P'$-module, and \item $M \otimes_P Q$ has an $A'$-flat lift to a $Q'$-module. \end{enumerate}


examples-defos-lemma-lift-equivalence

Let $A' \to A$ be a surjective map of Noetherian rings with nilpotent kernel. Let $A \to B$ be a finite type flat ring map. Let $\mathfrak b \subset B$ be an ideal such that $\Spec(B) \to \Spec(A)$ is syntomic on the complement of $V(\mathfrak b)$. Then $B$ has a flat lift to $A'$ if and only if the $\mathfrak b$-adic completion $B^\wedge$ has a flat lift to $A'$.


examples-defos-lemma-first-order-completion

Let $k$ be a field. Let $B$ be a finite type $k$-algebra. Let $J \subset B$ be an ideal such that $\Spec(B) \to \Spec(k)$ is smooth on the complement of $V(J)$. Let $N$ be a finite $B$-module. Then there is a canonical bijection $$ \text{Exal}_k(B, N) \to \text{Exal}_k(B^\wedge, N^\wedge) $$ Here $B^\wedge$ and $N^\wedge$ are the $J$-adic completions.


examples-defos-lemma-smooth-completion

In Example \ref{example-rings} let $P$ be a $k$-algebra. Let $J \subset P$ be an ideal. Denote $P^\wedge$ the $J$-adic completion. If \begin{enumerate} \item $k \to P$ is of finite type, and \item $\Spec(P) \to \Spec(k)$ is smooth on the complement of $V(J)$. \end{enumerate} then the functor between deformation categories of Lemma \ref{lemma-completion} $$ \Deformationcategory_P \longrightarrow \Deformationcategory_{P^\wedge} $$ is smooth and induces an isomorphism on tangent spaces.


examples-defos-lemma-lift-equivalence-localization

Let $A' \to A$ be a surjective map of Noetherian rings with nilpotent kernel. Let $A \to B$ be a finite type flat ring map. Let $S \subset B$ be a multiplicative subset such that if $\Spec(B) \to \Spec(A)$ is not syntomic at $\mathfrak q$, then $S \cap \mathfrak q = \emptyset$. Then $B$ has a flat lift to $A'$ if and only if $S^{-1}B$ has a flat lift to $A'$.


examples-defos-lemma-first-order-localization

Let $k$ be a field. Let $B$ be a finite type $k$-algebra. Let $S \subset B$ be a multiplicative subset ideal such that if $\Spec(B) \to \Spec(k)$ is not smooth at $\mathfrak q$ then $S \cap \mathfrak q = \emptyset$. Let $N$ be a finite $B$-module. Then there is a canonical bijection $$ \text{Exal}_k(B, N) \to \text{Exal}_k(S^{-1}B, S^{-1}N) $$


examples-defos-lemma-smooth-localization

In Example \ref{example-rings} let $P$ be a $k$-algebra. Let $S \subset P$ be a multiplicative subset. If \begin{enumerate} \item $k \to P$ is of finite type, and \item $\Spec(P) \to \Spec(k)$ is smooth at all points of $V(g)$ for all $g \in S$. \end{enumerate} then the functor between deformation categories of Lemma \ref{lemma-localization} $$ \Deformationcategory_P \longrightarrow \Deformationcategory_{S^{-1}P} $$ is smooth and induces an isomorphism on tangent spaces.


examples-defos-lemma-lift-equivalence-henselization

Let $A' \to A$ be a surjective map of Noetherian rings with nilpotent kernel. Let $A \to B$ be a finite type flat ring map. Let $\mathfrak b \subset B$ be an ideal such that $\Spec(B) \to \Spec(A)$ is syntomic on the complement of $V(\mathfrak b)$. Let $(B^h, \mathfrak b^h)$ be the henselization of the pair $(B, \mathfrak b)$. Then $B$ has a flat lift to $A'$ if and only if $B^h$ has a flat lift to $A'$.


examples-defos-lemma-first-order-henselization

Let $k$ be a field. Let $B$ be a finite type $k$-algebra. Let $J \subset B$ be an ideal such that $\Spec(B) \to \Spec(k)$ is smooth on the complement of $V(J)$. Let $N$ be a finite $B$-module. Then there is a canonical bijection $$ \text{Exal}_k(B, N) \to \text{Exal}_k(B^h, N^h) $$ Here $(B^h, J^h)$ is the henselization of $(B, J)$ and $N^h = N \otimes_B B^h$.


examples-defos-lemma-smooth-henselization

In Example \ref{example-rings} let $P$ be a $k$-algebra. Let $J \subset P$ be an ideal. Denote $(P^h, J^h)$ the henselization of the pair $(P, J)$. If \begin{enumerate} \item $k \to P$ is of finite type, and \item $\Spec(P) \to \Spec(k)$ is smooth on the complement of $V(J)$, \end{enumerate} then the functor between deformation categories of Lemma \ref{lemma-henselization} $$ \Deformationcategory_P \longrightarrow \Deformationcategory_{P^h} $$ is smooth and induces an isomorphism on tangent spaces.


examples-defos-lemma-isolated

In Example \ref{example-rings} let $P$ be a $k$-algebra. Assume that $k \to P$ is of finite type and that $\Spec(P) \to \Spec(k)$ is smooth except at the maximal ideals $\mathfrak m_1, \ldots, \mathfrak m_n$ of $P$. Let $P_{\mathfrak m_i}$, $P_{\mathfrak m_i}^h$, $P_{\mathfrak m_i}^\wedge$ be the local ring, henselization, completion. Then the maps of deformation categories $$ \Deformationcategory_P \to \prod \Deformationcategory_{P_{\mathfrak m_i}} \to \prod \Deformationcategory_{P_{\mathfrak m_i}^h} \to \prod \Deformationcategory_{P_{\mathfrak m_i}^\wedge} $$ are smooth and induce isomorphisms on their finite dimensional tangent spaces.


examples-defos-lemma-lci-unobstructed

In Example \ref{example-rings} let $P$ be a local complete intersection over $k$ (Algebra, Definition \ref{algebra-definition-lci-field}). Then $\Deformationcategory_P$ is unobstructed.


examples-defos-lemma-glueing-smooth

In Situation \ref{situation-glueing} if $U_{12} \to \Spec(k)$ is smooth, then the morphism $$ \Deformationcategory_X \longrightarrow \Deformationcategory_{U_1} \times \Deformationcategory_{U_2} = \Deformationcategory_{P_1} \times \Deformationcategory_{P_2} $$ is smooth. If in addition $U_1$ is a local complete intersection over $k$, then $$ \Deformationcategory_X \longrightarrow \Deformationcategory_{U_2} = \Deformationcategory_{P_2} $$ is smooth.


examples-defos-lemma-curve-isolated

In Example \ref{example-schemes} let $X$ be a scheme over $k$. Assume \begin{enumerate} \item $X$ is separated, finite type over $k$ and $\dim(X) \leq 1$, \item $X \to \Spec(k)$ is smooth except at the closed points $p_1, \ldots, p_n \in X$. \end{enumerate} Let $\mathcal{O}_{X, p_1}$, $\mathcal{O}_{X, p_1}^h$, $\mathcal{O}_{X, p_1}^\wedge$ be the local ring, henselization, completion. Consider the maps of deformation categories $$ \Deformationcategory_X \longrightarrow \prod \Deformationcategory_{\mathcal{O}_{X, p_i}} \longrightarrow \prod \Deformationcategory_{\mathcal{O}_{X, p_i}^h} \longrightarrow \prod \Deformationcategory_{\mathcal{O}_{X, p_i}^\wedge} $$ The first arrow is smooth and the second and third arrows are smooth and induce isomorphisms on tangent spaces.


examples-defos-lemma-curve-isolated-lci

In Example \ref{example-schemes} let $X$ be a scheme over $k$. Assume \begin{enumerate} \item $X$ is separated, finite type over $k$ and $\dim(X) \leq 1$, \item $X$ is a local complete intersection over $k$, and \item $X \to \Spec(k)$ is smooth except at finitely many points. \end{enumerate} Then $\Deformationcategory_X$ is unobstructed.


examples-defos-lemma-criterion-smoothing

Let $k$ be a field. Set $S = \Spec(k[[t]])$ and $S_n = \Spec(k[t]/(t^n))$. Let $Y \to S$ be a proper, flat morphism of schemes whose special fibre $X$ is Cohen-Macaulay and equidimensional of dimension $d$. Denote $X_n = Y \times_S S_n$. If for some $n \geq 1$ the $d$the Fitting ideal of $\Omega_{X_n/S_n}$ contains $t^{n - 1}$, then the generic fibre of $Y \to S$ is smooth.


examples-defos-lemma-jouanolou-type-thing

Let $k$ be a field. Let $1 \leq c \leq n$ be integers. Let $f_1, \ldots, f_c \in k[x_1, \ldots x_n]$ be elements. Let $a_{ij}$, $0 \leq i \leq n$, $1 \leq j \leq c$ be variables. Consider $$ g_j = f_j + a_{0j} + a_{1j}x_1 + \ldots + a_{nj}x_n \in k[a_{ij}][x_1, \ldots, x_n] $$ Denote $Y \subset \mathbf{A}^{n + c(n + 1)}_k$ the closed subscheme cut out by $g_1, \ldots, g_c$. Denote $\pi : Y \to \mathbf{A}^{c(n + 1)}_k$ the projection onto the affine space with variables $a_{ij}$. Then there is a nonempty Zariski open  of $\mathbf{A}^{c(n + 1)}_k$ over which $\pi$ is smooth.


examples-defos-lemma-smoothing-affine-lci

Let $k$ be a field. Let $A$ be a global complete interesection over $k$. There exists a flat finite type ring map $k[[t]] \to B$ with $B/tB \cong A$ such that $B[1/t]$ is smooth over $k((t))$.


examples-defos-lemma-smoothing-artinian-lci

Let $k$ be a field. Let $A$ be a finite dimensional $k$-algebra which is a local complete intersection over $k$. Then there is a finite flat $k[[t]]$-algebra $B$ with $B/tB \cong A$ and $B[1/t]$ \'etale over $k((t))$.


examples-defos-lemma-smoothing-at-lci-point

Let $k$ be a field. Let $A$ be a $k$-algebra. Assume \begin{enumerate} \item $A$ is a local ring essentially of finite type over $k$, \item $A$ is a complete intersection over $k$ (Algebra, Definition \ref{algebra-definition-lci-local-ring}). \end{enumerate} Set $d = \dim(A) + \text{trdeg}_k(\kappa)$ where $\kappa$ is the residue field of $A$. Then there exists an integer $n$ and a flat, essentially of finite type ring map $k[[t]] \to B$ with $B/tB \cong A$ such that $t^n$ is in the $d$th Fitting ideal of $\Omega_{B/k[[t]]}$.


examples-defos-lemma-smoothing-proper-curve-isolated-lci

Let $X$ be a scheme over a field $k$. Assume \begin{enumerate} \item $X$ is proper over $k$, \item $X$ is a local complete intersection over $k$, \item $X$ has dimension $\leq 1$, and \item $X \to \Spec(k)$ is smooth except at finitely many points. \end{enumerate} Then there exists a flat projective morphism $Y \to \Spec(k[[t]])$ whose generic fibre is smooth and whose special fibre is isomorphic to $X$.


sets-theorem-reflection-principle

Suppose given $\phi_1(x_1, \ldots, x_n), \ldots, \phi_m(x_1, \ldots, x_n)$ a {\bf finite} collection of formulas of set theory. Let $M_0$ be a set. There exists a set $M$ such that $M_0 \subset M$ and $\forall x_1, \ldots, x_n \in M$, we have $$ \forall i = 1, \ldots, m, \ \phi_i^{M}(x_1, \ldots, x_n) \Leftrightarrow \forall i = 1, \ldots, m, \ \phi_i(x_1, \ldots, x_n). $$ In fact we may take $M = V_\alpha$ for some limit ordinal $\alpha$.


sets-lemma-axiom-regularity

Every set is an element of $V_\alpha$ for some ordinal $\alpha$.


sets-lemma-map-from-set-lifts

Suppose that $T = \colim_{\alpha < \beta} T_\alpha$ is a colimit of sets indexed by ordinals less than a given ordinal $\beta$. Suppose that $\varphi : S \to T$ is a map of sets. Then $\varphi$ lifts to a map into $T_\alpha$ for some $\alpha < \beta$ provided that $\beta$ is not a limit of ordinals indexed by $S$, in other words, if $\beta$ is an ordinal with $\text{cf}(\beta) > |S|$.


sets-lemma-bounded-size

For every cardinal $\kappa$, there exists a set $A$ such that every element of $A$ is a scheme and such that for every scheme $S$ with $\text{size}(S) \leq \kappa$, there is an element $X \in A$ such that $X \cong S$ (isomorphism of schemes).


sets-lemma-construct-category

With notations $\text{size}$, $Bound$ and $\Sch_\alpha$ as above. Let $S_0$ be a set of schemes. There exists a limit ordinal $\alpha$ with the following properties: \begin{enumerate} \item \label{item-inclusion} We have $S_0 \subset V_\alpha$; in other words, $S_0 \subset \Ob(\Sch_\alpha)$. \item \label{item-bounded} For any $S \in \Ob(\Sch_\alpha)$ and any scheme $T$ with $\text{size}(T) \leq Bound(\text{size}(S))$, there exists a scheme $S' \in \Ob(\Sch_\alpha)$ such that $T \cong S'$. \item \label{item-limit} For any countable\footnote{Both the set of objects and the morphism sets are countable. In fact you can prove the lemma with $\aleph_0$ replaced by any cardinal whatsoever in (3) and (4).} diagram category $\mathcal{I}$ and any functor $F : \mathcal{I} \to \Sch_\alpha$, the limit $\lim_\mathcal{I} F$ exists in $\Sch_\alpha$ if and only if it exists in $\Sch$ and moreover, in this case, the natural morphism between them is an isomorphism. \item \label{item-colimit} For any countable diagram category $\mathcal{I}$ and any functor $F : \mathcal{I} \to \Sch_\alpha$, the colimit $\colim_\mathcal{I} F$ exists in $\Sch_\alpha$ if and only if it exists in $\Sch$ and moreover, in this case, the natural morphism between them is an isomorphism. \end{enumerate}


sets-lemma-bound-affine

Let $S$ be an affine scheme. Let $R = \Gamma(S, \mathcal{O}_S)$. Then the size of $S$ is equal to $\max\{ \aleph_0, |R|\}$.


sets-lemma-bound-size

Let $S$ be a scheme. Let $S = \bigcup_{i \in I} S_i$ be an open covering. Then $\text{size}(S) \leq \max\{|I|, \sup_i\{\text{size}(S_i)\}\}$.


sets-lemma-bound-size-fibre-product

Let $f : X \to S$, $g : Y \to S$ be morphisms of schemes. Then we have $\text{size}(X \times_S Y) \leq \max\{\text{size}(X), \text{size}(Y)\}$.


sets-lemma-bound-finite-type

Let $S$ be a scheme. Let $f : X \to S$ be locally of finite type with $X$ quasi-compact. Then $\text{size}(X) \leq \text{size}(S)$.


sets-lemma-bound-monomorphism

Let $f : X \to Y$ be a monomorphism of schemes. If at least one of the following properties holds, then $\text{size}(X) \leq \text{size}(Y)$: \begin{enumerate} \item $f$ is quasi-compact, \item $f$ is locally of finite presentation, \item add more here as needed. \end{enumerate} But the bound does not hold for monomorphisms which are locally of finite type.


sets-lemma-what-is-in-it

Let $\alpha$ be an ordinal as in Lemma \ref{lemma-construct-category} above. The category $\Sch_\alpha$ satisfies the following properties: \begin{enumerate} \item If $X, Y, S \in \Ob(\Sch_\alpha)$, then for any morphisms $f : X \to S$, $g : Y \to S$ the fibre product $X \times_S Y$ in $\Sch_\alpha$ exists and is a fibre product in the category of schemes. \item Given any at most countable collection $S_1, S_2, \ldots$ of elements of $\Ob(\Sch_\alpha)$, the coproduct $\coprod_i S_i$ exists in $\Ob(\Sch_\alpha)$ and is a coproduct in the category of schemes. \item For any $S \in \Ob(\Sch_\alpha)$ and any open immersion $U \to S$, there exists a $V \in \Ob(\Sch_\alpha)$ with $V \cong U$. \item For any $S \in \Ob(\Sch_\alpha)$ and any closed immersion $T \to S$, there exists a $S' \in \Ob(\Sch_\alpha)$ with $S' \cong T$. \item For any $S \in \Ob(\Sch_\alpha)$ and any finite type morphism $T \to S$, there exists a $S' \in \Ob(\Sch_\alpha)$ with $S' \cong T$. \item Suppose $S$ is a scheme which has an open covering $S = \bigcup_{i \in I} S_i$ such that there exists a $T \in \Ob(\Sch_\alpha)$ with (a) $\text{size}(S_i) \leq \text{size}(T)^{\aleph_0}$ for all $i \in I$, and (b) $|I| \leq \text{size}(T)^{\aleph_0}$. Then $S$ is isomorphic to an object of $\Sch_\alpha$. \item For any $S \in \Ob(\Sch_\alpha)$ and any morphism $f : T \to S$ locally of finite type such that $T$ can be covered by at most $\text{size}(S)^{\aleph_0}$ open affines, there exists a $S' \in \Ob(\Sch_\alpha)$ with $S' \cong T$. For example this holds if $T$ can be covered by at most $|\mathbf{R}| = 2^{\aleph_0} = \aleph_0^{\aleph_0}$ open affines. \item For any $S \in \Ob(\Sch_\alpha)$ and any monomorphism $T \to S$ which is either locally of finite presentation or quasi-compact, there exists a $S' \in \Ob(\Sch_\alpha)$ with $S' \cong T$. \item Suppose that $T \in \Ob(\Sch_\alpha)$ is affine. Write $R = \Gamma(T, \mathcal{O}_T)$. Then any of the following schemes is isomorphic to a scheme in $\Sch_\alpha$: \begin{enumerate} \item For any ideal $I \subset R$ with completion $R^* = \lim_n R/I^n$, the scheme $\Spec(R^*)$. \item For any finite type $R$-algebra $R'$, the scheme $\Spec(R')$. \item For any localization $S^{-1}R$, the scheme $\Spec(S^{-1}R)$. \item For any prime $\mathfrak p \subset R$, the scheme $\Spec(\overline{\kappa(\mathfrak p)})$. \item For any subring $R' \subset R$, the scheme $\Spec(R')$. \item Any scheme of finite type over a ring of cardinality at most $|R|^{\aleph_0}$. \item And so on. \end{enumerate} \end{enumerate}


sets-lemma-bound-by-covering

Let $f : X \to Y$ be a morphism of schemes. Assume there exists an fpqc covering $\{g_j : Y_j \to Y\}_{j \in J}$ such that $g_j$ factors through $f$. Then $\text{size}(Y) \leq \text{size}(X)$.


sets-lemma-bound-fppf-covering

Let $\{f_i : X_i \to X\}_{i \in I}$ be an fppf covering of a scheme. There exists an fppf covering $\{W_j \to X\}_{j \in J}$ which is a refinement of $\{X_i \to X\}_{i \in I}$ such that $\text{size}(\coprod W_j) \leq \text{size}(X)$.


sets-lemma-sets-with-group-action

With notations $G$, $G\textit{-Sets}_\alpha$, $\text{size}$, and $Bound$ as above. Let $S_0$ be a set of $G$-sets. There exists a limit ordinal $\alpha$ with the following properties: \begin{enumerate} \item We have $S_0 \cup \{{}_GG\} \subset \Ob(G\textit{-Sets}_\alpha)$. \item For any $S \in \Ob(G\textit{-Sets}_\alpha)$ and any $G$-set $T$ with $\text{size}(T) \leq Bound(\text{size}(S))$, there exists a $S' \in \Ob(G\textit{-Sets}_\alpha)$ that is isomorphic to $T$. \item For any countable diagram category $\mathcal{I}$ and any functor $F : \mathcal{I} \to G\textit{-Sets}_\alpha$, the limit $\lim_\mathcal{I} F$ and colimit $\colim_\mathcal{I} F$ exist in $G\textit{-Sets}_\alpha$ and are the same as in $G\textit{-Sets}$. \end{enumerate}


sets-lemma-what-is-in-it-G-sets

Let $\alpha$ be an ordinal as in Lemma \ref{lemma-sets-with-group-action} above. The category $G\textit{-Sets}_\alpha$ satisfies the following properties: \begin{enumerate} \item The $G$-set ${}_GG$ is an object of $G\textit{-Sets}_\alpha$. \item (Co)Products, fibre products, and pushouts exist in $G\textit{-Sets}_\alpha$ and are the same as their counterparts in $G\textit{-Sets}$. \item Given an object $U$ of $G\textit{-Sets}_\alpha$, any $G$-stable subset $O \subset U$  is isomorphic to an object of $G\textit{-Sets}_\alpha$. \end{enumerate}


sets-lemma-coverings-site

With notations as above. Let $\text{Cov}_0 \subset \text{Cov}(\mathcal{C})$ be a set contained in $\text{Cov}(\mathcal{C})$. There exist a cardinal $\kappa$ and a limit ordinal $\alpha$ with the following properties: \begin{enumerate} \item We have $\text{Cov}_0 \subset \text{Cov}(\mathcal{C})_{\kappa, \alpha}$. \item The set of coverings $\text{Cov}(\mathcal{C})_{\kappa, \alpha}$ satisfies (1), (2), and (3) of Sites, Definition \ref{sites-definition-site} (see above). In other words $(\mathcal{C}, \text{Cov}(\mathcal{C})_{\kappa, \alpha})$ is a site. \item Every covering in $\text{Cov}(\mathcal{C})$ is combinatorially equivalent to a covering in $\text{Cov}(\mathcal{C})_{\kappa, \alpha}$. \end{enumerate}


sets-lemma-abelian-injectives

Suppose given a big category $\mathcal{A}$ (see Categories, Remark \ref{categories-remark-big-categories}). Assume $\mathcal{A}$ is abelian and has enough injectives. See Homology, Definitions \ref{homology-definition-abelian-category} and \ref{homology-definition-enough-injectives}. Then for any given set of objects $\{A_s\}_{s\in S}$ of $\mathcal{A}$, there is an abelian subcategory $\mathcal{A}' \subset \mathcal{A}$ with the following properties: \begin{enumerate} \item $\Ob(\mathcal{A}')$ is a set, \item $\Ob(\mathcal{A}')$ contains $A_s$ for each $s \in S$, \item $\mathcal{A}'$ has enough injectives, and \item an object of $\mathcal{A}'$ is injective if and only if it is an injective object of $\mathcal{A}$. \end{enumerate}


sets-proposition-exist-ordinals-large-cofinality

Let $\kappa$ be a cardinal. Then there exists an ordinal whose cofinality is bigger than $\kappa$.


more-etale-lemma-section-support-in-locally-closed-pre

Let $X$ be a scheme. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. Let $\varphi : U' \to U$ be a morphism of $X_\etale$. Let $Z' \subset U'$ be a closed subscheme such that $Z' \to U' \to U$ is a closed immersion with image $Z \subset U$. Then there is a canonical bijection $$ \{s \in \mathcal{F}(U) \mid \text{Supp}(s) \subset Z\} = \{s' \in \mathcal{F}(U') \mid \text{Supp}(s') \subset Z'\} $$ which is given by restriction if $\varphi^{-1}(Z) = Z'$.


more-etale-lemma-section-support-in-locally-closed

Let $X$ be a scheme. Let $Z \subset X$ be a locally closed subscheme. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. Given $U, U' \subset X$ open containing $Z$ as a closed subscheme, there is a canonical bijection $$ \{s \in \mathcal{F}(U) \mid \text{Supp}(s) \subset Z\} = \{s \in \mathcal{F}(U') \mid \text{Supp}(s) \subset Z\} $$ which is given by restriction if $U' \subset U$.


more-etale-lemma-f-shriek-separated

Let $f : X \to Y$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. The rule $$ Y_\etale \longrightarrow \textit{Ab},\quad V \longmapsto \{s \in f_*\mathcal{F}(V) = \mathcal{F}(X_V) \mid \text{Supp}(s) \subset X_V \text{ is proper over }V\} $$ is an abelian subsheaf of $f_*\mathcal{F}$.


more-etale-lemma-proper-f-shriek

Let $f : X \to Y$ be a proper morphism of schemes. Then $f_! = f_*$.


more-etale-lemma-compactify-f-shriek-separated

Let $Y$ be a scheme. Let $j : X \to \overline{X}$ be an open immersion of schemes over $Y$ with $\overline{X}$ proper over $Y$. Denote $f : X \to Y$ and $\overline{f} : \overline{X} \to Y$ the structure morphisms. For $\mathcal{F} \in \textit{Ab}(X_\etale)$ there is a canonical isomorphism (see proof) $$ f_!\mathcal{F} \longrightarrow \overline{f}_!j_!\mathcal{F} $$ As we have $\overline{f}_! = \overline{f}_*$ by Lemma \ref{lemma-proper-f-shriek} we obtain $\overline{f}_* \circ j_! = f_!$ as functors $\textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$.


more-etale-lemma-proper-compact-support

Let $X$ be a proper scheme over a field $k$. Then $H^0_c(X, \mathcal{F}) = H^0(X, \mathcal{F})$.


more-etale-lemma-compactify-compact-support

Let $k$ be a field. Let $j : X \to \overline{X}$ be an open immersion of schemes over $k$ with $\overline{X}$ proper over $k$. For $\mathcal{F} \in \textit{Ab}(X_\etale)$ there is a canonical isomorphism (see proof) $$ H^0_c(X, \mathcal{F}) \longrightarrow H^0_c(\overline{X}, j_!\mathcal{F}) = H^0(\overline{X}, j_!\mathcal{F}) $$ where we have the equality on the right by Lemma \ref{lemma-proper-compact-support}.


more-etale-lemma-stalk-f-shriek-separated

Let $f : X \to Y$ be a morphism of schemes which is separated and locally of finite type. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. Then there is a canonical isomorphism $$ (f_!\mathcal{F})_{\overline{y}} \longrightarrow H^0_c(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}}) $$ for any geometric point $\overline{y} : \Spec(k) \to Y$.


more-etale-lemma-base-change-f-shriek-separated

Consider a cartesian square $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ of schemes with $f$ separated and locally of finite type. For any abelian sheaf $\mathcal{F}$ on $X_\etale$ we have $f'_!(g')^{-1}\mathcal{F} = g^{-1}f_!\mathcal{F}$.


more-etale-lemma-f-shriek-composition

Let $f : X \to Y$ and $g : Y \to Z$ be composable morphisms of schemes which are separated and locally of finite type. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. Then $g_!f_!\mathcal{F} = (g \circ f)_!\mathcal{F}$ as subsheaves of $(g \circ f)_*\mathcal{F}$.


more-etale-lemma-colim-f-shriek-separated

Let $f : X \to Y$ be morphism of schemes which is separated and locally of finite type. Let $X = \bigcup_{i \in I} X_i$ be an open covering such that for all $i, j \in I$ there exists a $k$ with $X_i \cup X_j \subset X_k$. Denote $f_i : X_i \to Y$ the restriction of $f$. Then $$ f_!\mathcal{F} = \colim_{i \in I} f_{i, !}(\mathcal{F}|_{X_i}) $$ functorially in $\mathcal{F} \in \textit{Ab}(X_\etale)$ where the transition maps are the ones constructed in Remark \ref{remark-covariance-f-shriek-separated}.


more-etale-lemma-lqf-f-shriek-separated-colimits

Let $f : X \to Y$ be a morphism of schemes which is separated and locally quasi-finite. Then \begin{enumerate} \item for $\mathcal{F}$ in $\textit{Ab}(X_\etale)$ and a geometric point $\overline{y} : \Spec(k) \to Y$ we have $$ (f_!\mathcal{F})_{\overline{y}} = \bigoplus\nolimits_{f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}} $$ functorially in $\mathcal{F}$, and \item the functor $f_!$ is exact. \end{enumerate}


more-etale-lemma-finite-support-f-shriek-separated

Let $f : X \to Y$ be a separated and locally quasi-finite morphism of schemes. Functorially in $\mathcal{F} \in \textit{Ab}(X_\etale)$ there is a canonical isomorphism(!) $$ f_{p!}\mathcal{F} \longrightarrow f_!\mathcal{F} $$ of abelian presheaves which identifies the sheaf $f_!\mathcal{F}$ of Definition \ref{definition-f-shriek-separated} with the presheaf $f_{p!}\mathcal{F}$ constructed above.


more-etale-lemma-finite-support-stalk

Let $f : X \to Y$ be a morphism of schemes which is locally quasi-finite. Let $\overline{y} : \Spec(k) \to Y$ be a geometric point. Functorially in $\mathcal{F}$ in $\textit{Ab}(X_\etale)$ we have $$ (f_{p!}\mathcal{F})_{\overline{y}} = \bigoplus\nolimits_{f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}} $$


more-etale-lemma-finite-support-etale-shriek

Let $f = j : U \to X$ be an \'etale of schemes. Denote $j_{p!}$ the construction of \'Etale Cohomology, Equation (\ref{etale-cohomology-equation-j-p-shriek}) and denote $f_{p!}$ the construction above. Functorially in $\mathcal{F} \in \textit{Ab}(X_\etale)$ there is a canonical map $$ j_{p!}\mathcal{F} \longrightarrow f_{p!}\mathcal{F} $$ of abelian presheaves which identifies the sheaf $j_!\mathcal{F} = (j_{p!}\mathcal{F})^\#$ of \'Etale Cohomology, Definition \ref{etale-cohomology-definition-extension-zero} with $(f_{p!}\mathcal{F})^\#$.


more-etale-lemma-lqf-f-shriek-stalk

Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. Then \begin{enumerate} \item for $\mathcal{F}$ in $\textit{Ab}(X_\etale)$ and a geometric point $\overline{y} : \Spec(k) \to Y$ we have $$ (f_!\mathcal{F})_{\overline{y}} = \bigoplus\nolimits_{f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}} $$ functorially in $\mathcal{F}$, and \item the functor $f_! : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$ is exact and commutes with direct sums. \end{enumerate}


more-etale-lemma-lqf-colimit-f-shriek

Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. Let $X = \bigcup_{i \in I} X_i$ be an open covering. Then there exists an exact complex $$ \ldots \to \bigoplus\nolimits_{i_0, i_1, i_2} f_{i_0i_1i_2, !} \mathcal{F}|_{X_{i_0i_1i_2}} \to \bigoplus\nolimits_{i_0, i_1} f_{i_0i_1, !} \mathcal{F}|_{X_{i_0i_1}} \to \bigoplus\nolimits_{i_0} f_{i_0, !} \mathcal{F}|_{X_{i_0}} \to f_!\mathcal{F} \to 0 $$ functorial in $\mathcal{F} \in \textit{Ab}(X_\etale)$, see proof for details.


more-etale-lemma-lqf-base-change-f-shriek

Consider a cartesian square $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ of schemes with $f$ locally quasi-finite. For any abelian sheaf $\mathcal{F}$ on $X_\etale$ we have $f'_!(g')^{-1}\mathcal{F} = g^{-1}f_!\mathcal{F}$.


more-etale-lemma-lqf-separated-shriek-composition

Let $f' : X \to Y'$ and $g : Y' \to Y$ be composable morphisms of schemes with $f'$ and $f = g \circ f'$ locally quasi-finite and $g$ separated and locally of finite type. Then there is a canonical isomorphism of functors $g_! \circ f'_! = f_!$. This isomorphism is compatible with \begin{enumerate} \item[(a)] covariance with respect to open embeddings as in Remarks \ref{remark-covariance-f-shriek-separated} and \ref{remark-covariance-lqf-f-shriek}, \item[(b)] the base change isomorphisms of Lemmas \ref{lemma-lqf-base-change-f-shriek} and \ref{lemma-base-change-f-shriek-separated}, and \item[(c)] equal to the isomorphism of Lemma \ref{lemma-f-shriek-composition} via the identifications of Lemma \ref{lemma-finite-support-f-shriek-separated} in case $f'$ is separated. \end{enumerate}


more-etale-lemma-lqf-shriek-composition

Let $f : X \to Y$ and $g : Y \to Z$ be composable locally quasi-finite morphisms of schemes. Then there is a canonical isomorphism of functors $$ (g \circ f)_! \longrightarrow g_! \circ f_! $$ These isomorphisms satisfy the following properties: \begin{enumerate} \item If $f$ and $g$ are separated, then the isomorphism agrees with Lemma \ref{lemma-f-shriek-composition}. \item If $g$ is separated, then the isomorphism agrees with Lemma \ref{lemma-lqf-separated-shriek-composition}. \item For a geometric point $\overline{z} : \Spec(k) \to Z$ the diagram $$ \xymatrix{ ((g \circ f)_!\mathcal{F})_{\overline{z}} \ar[d] \ar[rr] & & \bigoplus\nolimits_{g(f(\overline{x})) = \overline{z}} \mathcal{F}_{\overline{x}} \ar@{=}[d] \\ (g_!f_!\mathcal{F})_{\overline{z}} \ar[r] & \bigoplus\nolimits_{g(\overline{y}) = \overline{z}} (f_!\mathcal{F})_{\overline{y}} \ar[r] & \bigoplus\nolimits_{g(f(\overline{x})) = \overline{z}} \mathcal{F}_{\overline{x}} } $$ is commutative where the horizontal arrows are given by Lemma \ref{lemma-lqf-f-shriek-stalk}. \item Let $h : Z \to T$ be a third locally quasi-finite morphism of schemes. Then the diagram $$ \xymatrix{ (h \circ g \circ f)_! \ar[r] \ar[d] & (h \circ g)_! \circ f_! \ar[d] \\ h_! \circ (g \circ f)_! \ar[r] & h_! \circ g_! \circ f_! } $$ commutes. \item Suppose that we have a diagram of schemes $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_c & X \ar[d]^f \\ Y' \ar[d]_{g'} \ar[r]_b & Y \ar[d]^g \\ Z' \ar[r]^a & Z } $$ with both squares cartesian and $f$ and $g$ locally quasi-finite. Then the diagram $$ \xymatrix{ a^{-1} \circ (g \circ f)_! \ar[d] \ar[rr] & & (g' \circ f')_! \circ c^{-1} \ar[d] \\ a^{-1} \circ g_! \circ f_! \ar[r] & g'_! \circ b^{-1} \circ f_! \ar[r] & g'_! \circ f'_! \circ c^{-1} } $$ commutes where the horizontal arrows are those of Lemma \ref{lemma-lqf-base-change-f-shriek}. \end{enumerate}


more-etale-lemma-lqf-f-upper-shriek

Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. The functor $f_! : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$ has a right adjoint $f^! : \textit{Ab}(Y_\etale) \to \textit{Ab}(X_\etale)$. Moreover, we have $f^!(\overline{y}_*A) = \prod_{f(\overline{x}) = \overline{y}} \overline{x}_*A$.


more-etale-lemma-etale-upper-shriek

Let $j : U \to X$ be an \'etale morphism. Then $j^! = j^{-1}$.


more-etale-lemma-upper-shriek-restriction

Let $f : X \to Y$ and $g : Y \to Z$ be separated and locally quasi-finite morphisms. There is a canonical isomorphism $(g \circ f)^! \to f^! \circ g^!$. Given a third locally quasi-finite morphism $h : Z \to T$ the diagram $$ \xymatrix{ (h \circ g \circ f)^! \ar[r] \ar[d] & f^! \circ (h \circ g)^! \ar[d] \\ (g \circ f)^! \circ h^! \ar[r] & f^! \circ g^! \circ h^! } $$ commutes.


more-etale-lemma-upper-shriek-restriction-etale

Let $j : U \to X$ and $j' : V \to U$ be \'etale morphisms. The isomorphism $(j \circ j')^{-1} = (j')^{-1} \circ j^{-1}$ and the isomorphism $(j \circ j')^! = (j')^! \circ j^!$ of Lemma \ref{lemma-upper-shriek-restriction} agree via the isomorphism of Lemma \ref{lemma-etale-upper-shriek}.


more-etale-lemma-shriek-proper-and-open

Consider a commutative diagram of schemes $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ with $f$ and $f'$ proper and $g$ and $g'$ separated and locally quasi-finite. For a torsion ring $\Lambda$ and $K$ in $D(X'_\etale, \Lambda)$ there is a canonical isomorphism $g_!Rf'_*K \to Rf_*(g'_!K)$ in $D(Y_\etale, \Lambda)$.


more-etale-lemma-shriek-proper-and-open-compose

Consider a commutative diagram of schemes $$ \xymatrix{ X' \ar[r]_k \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]_l \ar[d]_{g'} & Y \ar[d]^g \\ Z' \ar[r]^m & Z } $$ with $f$, $f'$, $g$ and $g'$ proper and $k$, $l$, and $m$ separated and locally quasi-finite. Then the isomorphisms of Lemma \ref{lemma-shriek-proper-and-open} for the two squares compose to give the isomorphism for the outer rectangle (see proof for a precise statement).


more-etale-lemma-shriek-proper-and-open-compose-horizontal

Consider a commutative diagram of schemes $$ \xymatrix{ X'' \ar[r]_{g'} \ar[d]_{f''} & X' \ar[r]_g \ar[d]_{f'} & X \ar[d]^f \\ Y'' \ar[r]^{h'} & Y' \ar[r]^h & Y } $$ with $f$, $f'$, and $f''$ proper and $g$, $g'$, $h$, and $h'$ separated and locally quasi-finite. Then the isomorphisms of Lemma \ref{lemma-shriek-proper-and-open} for the two squares compose to give the isomorphism for the outer rectangle (see proof for a precise statement).


more-etale-lemma-shriek-proper-and-open-base-change

Let $b : Y_1 \to Y$ be a morphism of schemes. Consider a commutative diagram of schemes $$ \vcenter{ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } } \quad\text{and let}\quad \vcenter{ \xymatrix{ X'_1 \ar[r]_{g'_1} \ar[d]_{f'_1} & X_1 \ar[d]^{f_1} \\ Y'_1 \ar[r]^{g_1} & Y_1 } } $$ be the base change by $b$. Assume $f$ and $f'$ proper and $g$ and $g'$ separated and locally quasi-finite. For a torsion ring $\Lambda$ and $K$ in $D(X'_\etale, \Lambda)$ there is commutative diagram $$ \xymatrix{ b^{-1}g_!Rf'_*K \ar[d] \ar[r] & g_{1, !}(b')^{-1}Rf'_*K \ar[r] & g_{1, !}Rf'_{1, *}(a')^{-1}K \ar[d] \\ b^{-1}Rf_*g'_!K \ar[r] & Rf_{1, *}a^{-1}g'_!K \ar[r] & Rf_{1, *}g'_{1, !}(a')^{-1}K } $$ in $D(Y_{1, \etale}, \Lambda)$ where $a : X_1 \to X$, $a' : X'_1 \to X'$, $b' : Y'_1 \to Y'$ are the projections, the vertical maps are the arrows of Lemma \ref{lemma-shriek-proper-and-open} and the horizontal arrows are the base change map (from \'Etale Cohomology, Section \ref{etale-cohomology-section-base-change-preliminaries}) and the base change map of Lemma \ref{lemma-base-change-f-shriek-separated}.


more-etale-lemma-shriek-lqf-and-proper

Consider a commutative diagram of schemes $$ \xymatrix{ X \ar[r]_f \ar[rd]_g & Y \ar[d]^h \\ & Z } $$ with $f$ and $g$ locally quasi-finite and $h$ proper. For any torsion ring $\Lambda$ and $K$ in $D(X_\etale, \Lambda)$ there is a canonical isomorphism $g_!K \to Rh_*(f_!K)$ in $D(Z_\etale, \Lambda)$.


more-etale-lemma-shriek-well-defined

Let $f : X \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. The functor $Rf_!$ is, up to canonical isomorphism, independent of the choice of the compactification.


more-etale-lemma-shriek-composition

Let $f : X \to Y$ and $g : Y \to Z$ be separated morphisms of finite type of quasi-compact and quasi-separated schemes. Then there is a canonical isomorphism $Rg_! \circ Rf_! \to R(g \circ f)_!$.


more-etale-lemma-derived-lower-shriek-commute-direct-sums

Let $f : X \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. The functor $Rf_!$ commutes with direct sums.


stacks-properties-lemma-check-representable-covering

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $W$ be an algebraic space and let $W \to \mathcal{Y}$ be surjective, locally of finite presentation, and flat. The following are equivalent \begin{enumerate} \item $f$ is representable by algebraic spaces, and \item $W \times_\mathcal{Y} \mathcal{X}$ is an algebraic space. \end{enumerate}


stacks-properties-lemma-property-spaces-too

Let $P$ be a property of morphisms of algebraic spaces as above. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. The following are equivalent: \begin{enumerate} \item $f$ has $P$, \item for every algebraic space $Z$ and morphism $Z \to \mathcal{Y}$ the morphism $Z \times_\mathcal{Y} \mathcal{X} \to Z$ has $P$. \end{enumerate}


stacks-properties-lemma-check-property-covering

Let $P$ be a property of morphisms of algebraic spaces as above. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Let $W$ be an algebraic space and let $W \to \mathcal{Y}$ be surjective, locally of finite presentation, and flat. Set $V = W \times_\mathcal{Y} \mathcal{X}$. Then $$ (f\text{ has }P) \Leftrightarrow (\text{the projection }V \to W\text{ has }P). $$


stacks-properties-lemma-check-property-weak-covering

Let $P$ be a property of morphisms of algebraic spaces as above. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Let $\mathcal{Z} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Set $\mathcal{W} = \mathcal{Z} \times_\mathcal{Y} \mathcal{X}$. Then $$ (f\text{ has }P) \Leftrightarrow (\text{the projection }\mathcal{W} \to \mathcal{Z}\text{ has }P). $$


stacks-properties-lemma-check-property-after-precomposing

Let $P$ be a property of morphisms of algebraic spaces as above. Let $\tau \in \{\etale, smooth, syntomic, fppf\}$. Let $\mathcal{X} \to \mathcal{Y}$ and $\mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks representable by algebraic spaces. Assume \begin{enumerate} \item $\mathcal{X} \to \mathcal{Y}$ is surjective and \'etale, smooth, syntomic, or flat and locally of finite presentation, \item the composition has $P$, and \item $P$ is local on the source in the $\tau$ topology. \end{enumerate} Then $\mathcal{Y} \to \mathcal{Z}$ has property $P$.


stacks-properties-lemma-representable-in-terms-presentations

Let $g : \mathcal{X}' \to \mathcal{X}$ be a morphism of algebraic stacks which is representable by algebraic spaces. Let $[U/R] \to \mathcal{X}$ be a presentation. Set $U' = U \times_\mathcal{X} \mathcal{X}'$, and $R' = R \times_\mathcal{X} \mathcal{X}'$. Then there exists a groupoid in algebraic spaces of the form $(U', R', s', t', c')$, a presentation $[U'/R'] \to \mathcal{X}'$, and the diagram $$ \xymatrix{ [U'/R'] \ar[d]_{[\text{pr}]} \ar[r] & \mathcal{X}' \ar[d]^g \\ [U/R] \ar[r] & \mathcal{X} } $$ is $2$-commutative where the morphism $[\text{pr}]$ comes from a morphism of groupoids $\text{pr} : (U', R', s', t', c') \to (U, R, s, t, c)$.


stacks-properties-lemma-equivalence

The notion above does indeed define an equivalence relation on morphisms from spectra of fields into the algebraic stack $\mathcal{X}$.


stacks-properties-lemma-points-cartesian

Let $$ \xymatrix{ \mathcal{Z} \times_\mathcal{Y} \mathcal{X} \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ \mathcal{Z} \ar[r] & \mathcal{Y} } $$ be a fibre product of algebraic stacks. Then the map of sets of points $$ |\mathcal{Z} \times_\mathcal{Y} \mathcal{X}| \longrightarrow |\mathcal{Z}| \times_{|\mathcal{Y}|} |\mathcal{X}| $$ is surjective.


stacks-properties-lemma-characterize-surjective

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent: \begin{enumerate} \item $|f| : |\mathcal{X}| \to |\mathcal{Y}|$ is surjective, and \item $f$ is surjective (in the sense of Section \ref{section-properties-morphisms}). \end{enumerate}


stacks-properties-lemma-points-presentation

Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{X} = [U/R]$ be a presentation of $\mathcal{X}$, see Algebraic Stacks, Definition \ref{algebraic-definition-presentation}. Then the image of $|R| \to |U| \times |U|$ is an equivalence relation and $|\mathcal{X}|$ is the quotient of $|U|$ by this equivalence relation.


stacks-properties-lemma-topology-points

There exists a unique topology on the sets of points of algebraic stacks with the following properties: \begin{enumerate} \item for every morphism of algebraic stacks $\mathcal{X} \to \mathcal{Y}$ the map $|\mathcal{X}| \to |\mathcal{Y}|$ is continuous, and \item for every morphism $U \to \mathcal{X}$ which is flat and locally of finite presentation with $U$ an algebraic space the map of topological spaces $|U| \to |\mathcal{X}|$ is continuous and open. \end{enumerate}


stacks-properties-lemma-space-locally-quasi-compact

Let $\mathcal{X}$ be an algebraic stack. Every point of $|\mathcal{X}|$ has a fundamental system of quasi-compact open neighbourhoods. In particular $|\mathcal{X}|$ is locally quasi-compact in the sense of Topology, Definition \ref{topology-definition-locally-quasi-compact}.


stacks-properties-lemma-composition-surjective

The composition of surjective morphisms is surjective.


stacks-properties-lemma-base-change-surjective

The base change of a surjective morphism is surjective.


stacks-properties-lemma-descent-surjective

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $\mathcal{Y}' \to \mathcal{Y}$ be a surjective morphism of algebraic stacks. If the base change $f' : \mathcal{Y}' \times_\mathcal{Y} \mathcal{X} \to \mathcal{Y}'$ of $f$ is surjective, then $f$ is surjective.


stacks-properties-lemma-surjective-permanence

Let $\mathcal{X} \to \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. If $\mathcal{X} \to \mathcal{Z}$ is surjective so is $\mathcal{Y} \to \mathcal{Z}$.


stacks-properties-lemma-quasi-compact-stack

Let $\mathcal{X}$ be an algebraic stack. The following are equivalent: \begin{enumerate} \item $\mathcal{X}$ is quasi-compact, \item there exists a surjective smooth morphism $U \to \mathcal{X}$ with $U$ a quasi-compact scheme, \item there exists a surjective smooth morphism $U \to \mathcal{X}$ with $U$ a quasi-compact algebraic space, and \item there exists a surjective morphism $\mathcal{U} \to \mathcal{X}$ of algebraic stacks such that $\mathcal{U}$ is quasi-compact. \end{enumerate}


stacks-properties-lemma-finite-disjoint-quasi-compact

A finite disjoint union of quasi-compact algebraic stacks is a quasi-compact algebraic stack.


stacks-properties-lemma-type-property

Let $\mathcal{P}$ be a property of schemes which is local in the smooth topology, see Descent, Definition \ref{descent-definition-property-local}. Let $\mathcal{X}$ be an algebraic stack. The following are equivalent \begin{enumerate} \item for some scheme $U$ and some surjective smooth morphism $U \to \mathcal{X}$ the scheme $U$ has property $\mathcal{P}$, \item for every scheme $U$ and every smooth morphism $U \to \mathcal{X}$ the scheme $U$ has property $\mathcal{P}$, \item for some algebraic space $U$ and some surjective smooth morphism $U \to \mathcal{X}$ the algebraic space $U$ has property $\mathcal{P}$, and \item for every algebraic space $U$ and every smooth morphism $U \to \mathcal{X}$ the algebraic space $U$ has property $\mathcal{P}$. \end{enumerate} If $\mathcal{X}$ is a scheme this is equivalent to $\mathcal{P}(U)$. If $\mathcal{X}$ is an algebraic space this is equivalent to $X$ having property $\mathcal{P}$.


stacks-properties-lemma-local-source-target-at-point

Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$ be a point of $\mathcal{X}$. Let $\mathcal{P}$ be a property of germs of schemes which is smooth local, see Descent, Definition \ref{descent-definition-local-at-point}. The following are equivalent \begin{enumerate} \item for any smooth morphism $U \to \mathcal{X}$ with $U$ a scheme and $u \in U$ with $a(u) = x$ we have $\mathcal{P}(U, u)$, \item for some smooth morphism $U \to \mathcal{X}$ with $U$ a scheme and some $u \in U$ with $a(u) = x$ we have $\mathcal{P}(U, u)$, \item for any smooth morphism $U \to \mathcal{X}$ with $U$ an algebraic space and $u \in |U|$ with $a(u) = x$ the algebraic space $U$ has property $\mathcal{P}$ at $u$, and \item for some smooth morphism $U \to \mathcal{X}$ with $U$ a an algebraic space and some $u \in |U|$ with $a(u) = x$ the algebraic space $U$ has property $\mathcal{P}$ at $u$. \end{enumerate} If $\mathcal{X}$ is representable, then this is equivalent to $\mathcal{P}(\mathcal{X}, x)$. If $\mathcal{X}$ is an algebraic space then this is equivalent to $\mathcal{X}$ having property $\mathcal{P}$ at $x$.


stacks-properties-lemma-base-change-monomorphism

Let $\mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $\mathcal{Z} \to \mathcal{Y}$ be a monomorphism. Then $\mathcal{Z} \times_\mathcal{Y} \mathcal{X} \to \mathcal{X}$ is a monomorphism.


stacks-properties-lemma-monomorphism

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent: \begin{enumerate} \item $f$ is a monomorphism, \item $f$ is fully faithful, \item the diagonal $\Delta_f : \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$ is an equivalence, and \item there exists an algebraic space $W$ and a surjective, flat morphism $W \to \mathcal{Y}$ which is locally of finite presentation such that $V = \mathcal{X} \times_\mathcal{Y} W$ is an algebraic space, and the morphism $V \to W$ is a monomorphism of algebraic spaces. \end{enumerate}


stacks-properties-lemma-monomorphism-injective-points

\begin{slogan} Monomorphisms of stacks are injective on points. \end{slogan} A monomorphism of algebraic stacks induces an injective map of sets of points.


stacks-properties-lemma-monomorphism-diagonal

Let $\mathcal{X} \to \mathcal{X}' \to \mathcal{Y}$ be morphisms of algebraic stacks. If $\mathcal{X} \to \mathcal{X}'$ is a monomorphism then the canonical diagram $$ \xymatrix{ \mathcal{X} \ar[r] \ar[d] & \mathcal{X} \times_\mathcal{Y} \mathcal{X} \ar[d] \\ \mathcal{X}' \ar[r] & \mathcal{X}' \times_\mathcal{Y} \mathcal{X}' } $$ is a fibre product square.


stacks-properties-lemma-base-change-immersion

Let $\mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $\mathcal{Z} \to \mathcal{Y}$ be a (closed, resp.\ open) immersion. Then $\mathcal{Z} \times_\mathcal{Y} \mathcal{X} \to \mathcal{X}$ is a (closed, resp.\ open) immersion.


stacks-properties-lemma-composition-immersion

Compositions of immersions of algebraic stacks are immersions. Similarly for closed immersions and open immersions.


stacks-properties-lemma-check-immersion-covering

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. let $W$ be an algebraic space and let $W \to \mathcal{Y}$ be a surjective, flat morphism which is locally of finite presentation. The following are equivalent: \begin{enumerate} \item $f$ is an (open, resp.\ closed) immersion, and \item $V = W \times_\mathcal{Y} \mathcal{X}$ is an algebraic space, and $V \to W$ is an (open, resp.\ closed) immersion. \end{enumerate}


stacks-properties-lemma-immersion-monomorphism

An immersion is a monomorphism.


stacks-properties-lemma-immersion-into-presentation

Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces. Let $i : \mathcal{Z} \to [U/R]$ be an immersion. Then there exists an $R$-invariant locally closed subspace $Z \subset U$ and a presentation $[Z/R_Z] \to \mathcal{Z}$ where $R_Z$ is the restriction of $R$ to $Z$ such that $$ \xymatrix{ [Z/R_Z] \ar[dr] \ar[rr] & & \mathcal{Z} \ar[ld]^i \\ & [U/R] } $$ is $2$-commutative. If $i$ is a closed (resp.\ open) immersion then $Z$ is a closed (resp.\ open) subspace of $U$.


stacks-properties-lemma-immersion-presentation

Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces. Let $\mathcal{X} = [U/R]$ be the associated algebraic stack, see Algebraic Stacks, Theorem \ref{algebraic-theorem-smooth-groupoid-gives-algebraic-stack}. Let $Z \subset U$ be an $R$-invariant locally closed subspace. Then $$ [Z/R_Z] \longrightarrow [U/R] $$ is an immersion of algebraic stacks, where $R_Z$ is the restriction of $R$ to $Z$. If $Z \subset U$ is open (resp.\ closed) then the morphism is an open (resp.\ closed) immersion of algebraic stacks.


stacks-properties-lemma-substack-image

For any immersion $i : \mathcal{Z} \to \mathcal{X}$ there exists a unique locally closed substack $\mathcal{X}' \subset \mathcal{X}$ such that $i$ factors as the composition of an equivalence $i' : \mathcal{Z} \to \mathcal{X}'$ followed by the inclusion morphism $\mathcal{X}' \to \mathcal{X}$. If $i$ is a closed (resp.\ open) immersion, then $\mathcal{X}'$ is a closed (resp.\ open) substack of $\mathcal{X}$.


stacks-properties-lemma-substacks-presentation

Let $[U/R] \to \mathcal{X}$ be a presentation of an algebraic stack. There is a canonical bijection $$ \text{locally closed substacks }\mathcal{Z}\text{ of }\mathcal{X} \longrightarrow R\text{-invariant locally closed subspaces }Z\text{ of }U $$ which sends $\mathcal{Z}$ to $U \times_\mathcal{X} \mathcal{Z}$. Moreover, a morphism of algebraic stacks $f : \mathcal{Y} \to \mathcal{X}$ factors through $\mathcal{Z}$ if and only if $\mathcal{Y} \times_\mathcal{X} U \to U$ factors through $Z$. Similarly for closed substacks and open substacks.


stacks-properties-lemma-open-substacks

Let $\mathcal{X}$ be an algebraic stack. The rule $\mathcal{U} \mapsto |\mathcal{U}|$ defines an inclusion preserving bijection between open substacks of $\mathcal{X}$ and open subsets of $|\mathcal{X}|$.


stacks-properties-lemma-union-open-substacks

Let $\mathcal X$ be an algebraic stack and $\mathcal{X}_i \subset \mathcal X$ a collection of open substacks indexed by $i \in I$. Then there exists an open substack, which we denote $\bigcup_{i\in I} \mathcal{X}_i \subset \mathcal X$, such that the $\mathcal{X}_i$ are open substacks covering it.


stacks-properties-lemma-zariski-open-cover-stack-is-space

Let $\mathcal X$ be an algebraic stack. Let $\mathcal{X}_i$, $i \in I$ be a set of open substacks of $\mathcal{X}$. Assume \begin{enumerate} \item $\mathcal{X} = \bigcup_{i \in I} \mathcal{X}_i$, and \item each $\mathcal{X}_i$ is an algebraic space. \end{enumerate} Then $\mathcal{X}$ is an algebraic space.


stacks-properties-lemma-local-source

Let $\mathcal{P}, \mathcal{Q}, \mathcal{R}$ be properties of morphisms of algebraic spaces. Assume \begin{enumerate} \item $\mathcal{P}, \mathcal{Q}, \mathcal{R}$ are fppf local on the target and stable under arbitrary base change, \item $\text{smooth} \Rightarrow \mathcal{R}$, \item for any morphism $f : X \to Y$ which has $\mathcal{Q}$ there exists a largest open subspace $W(\mathcal{P}, f) \subset X$ such that $f|_{W(\mathcal{P}, f)}$ has $\mathcal{P}$, and \item for any morphism $f : X \to Y$ which has $\mathcal{Q}$, and any morphism $Y' \to Y$ which has $\mathcal{R}$ we have $Y' \times_Y W(\mathcal{P}, f) = W(\mathcal{P}, f')$, where $f' : X_{Y'} \to Y'$ is the base change of $f$. \end{enumerate} Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Assume $f$ has $\mathcal{Q}$. Then \begin{enumerate} \item[(A)] there exists a largest open substack $\mathcal{X}' \subset \mathcal{X}$ such that $f|_{\mathcal{X}'}$ has $\mathcal{P}$, and \item[(B)] if $\mathcal{Z} \to \mathcal{Y}$ is a morphism of algebraic stacks representable by algebraic spaces which has $\mathcal{R}$ then $\mathcal{Z} \times_\mathcal{Y} \mathcal{X}'$ is the largest open substack of $\mathcal{Z} \times_\mathcal{Y} \mathcal{X}$ over which the base change $\text{id}_\mathcal{Z} \times f$ has property $\mathcal{P}$. \end{enumerate}


stacks-properties-lemma-reduced-closed-substack

Let $\mathcal{X}$ be an algebraic stack. Let $T \subset |\mathcal{X}|$ be a closed subset. There exists a unique closed substack $\mathcal{Z} \subset \mathcal{X}$ with the following properties: (a) we have $|\mathcal{Z}| = T$, and (b) $\mathcal{Z}$ is reduced.


stacks-properties-lemma-reduced-stack-determined-by-points

Let $\mathcal{X}$ be an algebraic stack. If $\mathcal{X}' \subset \mathcal{X}$ is a closed substack, $\mathcal{X}$ is reduced and $|\mathcal{X}'| = |\mathcal{X}|$, then $\mathcal{X}' = \mathcal{X}$.


stacks-properties-lemma-flat-cover-by-field

Let $\mathcal{Z}$ be an algebraic stack. Let $k$ be a field and let $\Spec(k) \to \mathcal{Z}$ be surjective and flat. Then any morphism $\Spec(k') \to \mathcal{Z}$ where $k'$ is a field is surjective and flat.


stacks-properties-lemma-unique-point

Let $\mathcal{Z}$ be an algebraic stack. The following are equivalent \begin{enumerate} \item $\mathcal{Z}$ is reduced and $|\mathcal{Z}|$ is a singleton, \item there exists a surjective flat morphism $\Spec(k) \to \mathcal{Z}$ where $k$ is a field, and \item there exists a locally of finite type, surjective, flat morphism $\Spec(k) \to \mathcal{Z}$ where $k$ is a field. \end{enumerate}


stacks-properties-lemma-unique-point-better

Let $\mathcal{Z}$ be an algebraic stack. The following are equivalent \begin{enumerate} \item $\mathcal{Z}$ is reduced, locally Noetherian, and $|\mathcal{Z}|$ is a singleton, and \item there exists a locally finitely presented, surjective, flat morphism $\Spec(k) \to \mathcal{Z}$ where $k$ is a field. \end{enumerate}


stacks-properties-lemma-monomorphism-into-point

Let $\mathcal{Z}' \to \mathcal{Z}$ be a monomorphism of algebraic stacks. Assume there exists a field $k$ and a locally finitely presented, surjective, flat morphism $\Spec(k) \to \mathcal{Z}$. Then either $\mathcal{Z}'$ is empty or $\mathcal{Z}' \to \mathcal{Z}$ is an equivalence.


stacks-properties-lemma-improve-unique-point

Let $\mathcal{Z}$ be an algebraic stack. Assume $\mathcal{Z}$ satisfies the equivalent conditions of Lemma \ref{lemma-unique-point}. Then there exists a unique strictly full subcategory $\mathcal{Z}' \subset \mathcal{Z}$ such that $\mathcal{Z}'$ is an algebraic stack which satisfies the equivalent conditions of Lemma \ref{lemma-unique-point-better}. The inclusion morphism $\mathcal{Z}' \to \mathcal{Z}$ is a monomorphism of algebraic stacks.


stacks-properties-lemma-residual-gerbe

Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$ be a point. The following are equivalent \begin{enumerate} \item there exists an algebraic stack $\mathcal{Z}$ and a monomorphism $\mathcal{Z} \to \mathcal{X}$ such that $|\mathcal{Z}|$ is a singleton and such that the image of $|\mathcal{Z}|$ in $|\mathcal{X}|$ is $x$, \item there exists a reduced algebraic stack $\mathcal{Z}$ and a monomorphism $\mathcal{Z} \to \mathcal{X}$ such that $|\mathcal{Z}|$ is a singleton and such that the image of $|\mathcal{Z}|$ in $|\mathcal{X}|$ is $x$, \item there exists an algebraic stack $\mathcal{Z}$, a monomorphism $f : \mathcal{Z} \to \mathcal{X}$, and a surjective flat morphism $z : \Spec(k) \to \mathcal{Z}$ where $k$ is a field such that $x = f(z)$. \end{enumerate} Moreover, if these conditions hold, then there exists a unique strictly full subcategory $\mathcal{Z}_x \subset \mathcal{X}$ such that $\mathcal{Z}_x$ is a reduced, locally Noetherian algebraic stack and $|\mathcal{Z}_x|$ is a singleton which maps to $x$ via the map $|\mathcal{Z}_x| \to |\mathcal{X}|$.


stacks-properties-lemma-residual-gerbe-points

Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. Assume that the residual gerbe $\mathcal{Z}_x$ of $\mathcal{X}$ exists. Let $f : \Spec(K) \to \mathcal{X}$ be a morphism where $K$ is a field in the equivalence class of $x$. Then $f$ factors through the inclusion morphism $\mathcal{Z}_x \to \mathcal{X}$.


stacks-properties-lemma-residual-gerbe-unique

Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. Let $\mathcal{Z}$ be an algebraic stack satisfying the equivalent conditions of Lemma \ref{lemma-unique-point-better} and let $\mathcal{Z} \to \mathcal{X}$ be a monomorphism such that the image of $|\mathcal{Z}| \to |\mathcal{X}|$ is $x$. Then the residual gerbe $\mathcal{Z}_x$ of $\mathcal{X}$ at $x$ exists and $\mathcal{Z} \to \mathcal{X}$ factors as $\mathcal{Z} \to \mathcal{Z}_x \to \mathcal{X}$ where the first arrow is an equivalence.


stacks-properties-lemma-residual-gerbe-functorial

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $x \in |\mathcal{X}|$ with image $y \in |\mathcal{Y}|$. If the residual gerbes $\mathcal{Z}_x \subset \mathcal{X}$ and $\mathcal{Z}_y \subset \mathcal{Y}$ of $x$ and $y$ exist, then $f$ induces a commutative diagram $$ \xymatrix{ \mathcal{X} \ar[d]_f & \mathcal{Z}_x \ar[l] \ar[d] \\ \mathcal{Y} & \mathcal{Z}_y \ar[l] } $$


stacks-properties-lemma-residual-gerbe-isomorphic

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $x \in |\mathcal{X}|$ with image $y \in |\mathcal{Y}|$. Assume the residual gerbes $\mathcal{Z}_x \subset \mathcal{X}$ and $\mathcal{Z}_y \subset \mathcal{Y}$ of $x$ and $y$ exist and that there exists a morphism $\Spec(k) \to \mathcal{X}$ in the equivalence class of $x$ such that $$ \Spec(k) \times_\mathcal{X} \Spec(k) \longrightarrow \Spec(k) \times_\mathcal{Y} \Spec(k) $$ is an isomorphism. Then $\mathcal{Z}_x \to \mathcal{Z}_y$ is an isomorphism.


stacks-properties-lemma-dimension-at-point-well-defined

Let $\mathcal{X}$ be a locally Noetherian algebraic stack over a scheme $S$. Let $x \in |\mathcal{X}|$ be a point of $\mathcal{X}$. Let $[U/R] \to \mathcal{X}$ be a presentation (Algebraic Stacks, Definition \ref{algebraic-definition-presentation}) where $U$ is a scheme. Let $u \in U$ be a point that maps to $x$. Let $e : U \to R$ be the ``identity'' map and let $s : R \to U$ be the ``source'' map, which is a smooth morphism of algebraic spaces. Let $R_u$ be the fiber of $s : R \to U$ over $u$. The element $$ \dim_x(\mathcal{X}) = \dim_u(U) - \dim_{e(u)}(R_u) \in \mathbf{Z} \cup \infty $$ is independent of the choice of presentation and the point $u$ over $x$.


stacks-properties-lemma-UR-quasi-compact-above-x

Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$ be a point. The following are equivalent \begin{enumerate} \item some morphism $\Spec(k) \to \mathcal{X}$ in the equivalence class of $x$ is quasi-compact, and \item any morphism $\Spec(k) \to \mathcal{X}$ in the equivalence class of $x$ is quasi-compact. \end{enumerate}


stacks-lemma-painful

This actually does give a presheaf.


stacks-lemma-presheaf-mor-map-fibred-categories

Let $F : \mathcal{S}_1 \to \mathcal{S}_2$ be a $1$-morphism of fibred categories over the category $\mathcal{C}$. Let $U \in \Ob(\mathcal{C})$ and $x, y\in \Ob((\mathcal{S}_1)_U)$. Then $F$ defines a canonical morphism of presheaves $$ \mathit{Mor}_{\mathcal{S}_1}(x, y) \longrightarrow \mathit{Mor}_{\mathcal{S}_2}(F(x), F(y)) $$ on $\mathcal{C}/U$.


stacks-lemma-isom-as-2-fibre-product

Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category, see Categories, Section \ref{categories-section-fibred-categories}. Let $U \in \Ob(\mathcal{C})$ and let $x, y \in \Ob(\mathcal{S}_U)$. Denote $x, y : \mathcal{C}/U \to \mathcal{S}$ also the corresponding $1$-morphisms, see Categories, Lemma \ref{categories-lemma-yoneda-2category}. Then \begin{enumerate} \item the $2$-fibre product $\mathcal{S} \times_{\mathcal{S} \times \mathcal{S}, (x, y)} \mathcal{C}/U$ is fibred in setoids over $\mathcal{C}/U$, and \item $\mathit{Isom}(x, y)$ is the presheaf of sets corresponding to this category fibred in setoids, see Categories, Lemma \ref{categories-lemma-2-category-fibred-setoids}. \end{enumerate}


stacks-lemma-pullback

(Pullback of descent data.) Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Make a choice pullbacks as in Categories, Definition \ref{categories-definition-pullback-functor-fibred-category}. Let $\mathcal{U} = \{f_i : U_i \to U\}_{i \in I}$, and $\mathcal{V} = \{V_j \to V\}_{j \in J}$ be a families of morphisms of $\mathcal{C}$ with fixed target. Assume all the fibre products $U_i \times_U U_{i'}$, $U_i \times_U U_{i'} \times_U U_{i''}$, $V_j \times_V V_{j'}$, and $V_j \times_V V_{j'} \times_V V_{j''}$ exist. Let $\alpha : I \to J$, $h : U \to V$ and $g_i : U_i \to V_{\alpha(i)}$ be a morphism of families of maps with fixed target, see Sites, Definition \ref{sites-definition-morphism-coverings}. \begin{enumerate} \item Let $(Y_j, \varphi_{jj'})$ be a descent datum relative to the family $\{V_j \to V\}$. The system $$ \left( g_i^*Y_{\alpha(i)}, (g_i \times g_{i'})^*\varphi_{\alpha(i)\alpha(i')} \right) $$ is a descent datum relative to $\mathcal{U}$. \item This construction defines a functor between descent data relative to $\mathcal{V}$ and descent data relative to $\mathcal{U}$. \item Given a second $\alpha' : I \to J$, $h' : U \to V$ and $g'_i : U_i \to V_{\alpha'(i)}$ morphism of families of maps with fixed target, then if $h = h'$ the two resulting functors between descent data are canonically isomorphic. \end{enumerate}


stacks-lemma-trivial-cocycle

In the situation of Definition \ref{definition-effective-descent-datum} part (2) the maps $can_{ij} : \text{pr}_0^*f_i^*X \to \text{pr}_1^*f_j^*X$ are equal to $(\alpha_{\text{pr}_1, f_j})_X \circ (\alpha_{\text{pr}_0, f_i})_X^{-1}$ where $\alpha_{\cdot, \cdot}$ is as in Categories, Lemma \ref{categories-lemma-fibred} and where we use the equality $f_i \circ \text{pr}_0 = f_j \circ \text{pr}_1$ as maps $U_i \times_U U_j \to U$.


stacks-lemma-compare-descent-condition

Let $\mathcal{C}$ be a category. Let $\mathcal{V} = \{V_j \to U\}_{j \in J} \to \mathcal{U} = \{U_i \to U\}_{i \in I}$ be a morphism of families of maps with fixed target of $\mathcal{C}$ given by $\text{id} : U \to U$, $\alpha : J \to I$ and $f_j : V_j \to U_{\alpha(j)}$. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. If \begin{enumerate} \item for $0 \leq p \leq 3$ and $0 \leq q \leq 3$ with $p + q \geq 2$ and $i_1, \ldots, i_p \in I$ and $j_1, \ldots, j_q \in J$ the fibre products $U_{i_1} \times_U \ldots \times_U U_{i_p} \times_U V_{j_1} \times_U \ldots \times_U V_{j_q}$ exist, \item the functor $\mathcal{S}_U \to DD(\mathcal{V})$ is an equivalence, \item for every $i \in I$ the functor $\mathcal{S}_{U_i} \to DD(\mathcal{V}_i)$ is fully faithful, and \item for every $i, i' \in I$ the functor $\mathcal{S}_{U_i \times_U U_{i'}} \to DD(\mathcal{V}_{ii'})$ is faithful. \end{enumerate} Here $\mathcal{V}_i = \{U_i \times_U V_j \to U_i\}_{j \in J}$ and $\mathcal{V}_{ii'} = \{U_i \times_U U_{i'} \times_U V_j \to U_i \times_U U_{i'}\}_{j \in J}$. Then $\mathcal{S}_U \to DD(\mathcal{U})$ is an equivalence.


stacks-lemma-stack-equivalences

Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category over $\mathcal{C}$. The following are equivalent \begin{enumerate} \item $\mathcal{S}$ is a stack over $\mathcal{C}$, and \item for any covering $\mathcal{U} = \{f_i : U_i \to U\}_{i \in I}$ of the site $\mathcal{C}$ the functor $$ \mathcal{S}_U \longrightarrow DD(\mathcal{U}) $$ which associates to an object its canonical descent datum is an equivalence. \end{enumerate}


stacks-lemma-substack

Let $p : \mathcal{S} \to \mathcal{C}$ be a stack over the site $\mathcal{C}$. Let $\mathcal{S}'$ be a subcategory of $\mathcal{S}$. Assume \begin{enumerate} \item if $\varphi : y \to x$ is a strongly cartesian morphism of $\mathcal{S}$ and $x$ is an object of $\mathcal{S}'$, then $y$ is isomorphic to an object of $\mathcal{S}'$, \item $\mathcal{S}'$ is a full subcategory of $\mathcal{S}$, and \item if $\{f_i : U_i \to U\}$ is a covering of $\mathcal{C}$, and $x$ an object of $\mathcal{S}$ over $U$ such that $f_i^*x$ is isomorphic to an object of $\mathcal{S}'$ for each $i$, then $x$ is isomorphic to an object of $\mathcal{S}'$. \end{enumerate} Then $\mathcal{S}' \to \mathcal{C}$ is a stack.


stacks-lemma-stack-equivalent

Let $\mathcal{C}$ be a site. Let $\mathcal{S}_1$, $\mathcal{S}_2$ be categories over $\mathcal{C}$. Suppose that $\mathcal{S}_1$ and $\mathcal{S}_2$ are equivalent as categories over $\mathcal{C}$. Then $\mathcal{S}_1$ is a stack over $\mathcal{C}$ if and only if $\mathcal{S}_2$ is a stack over $\mathcal{C}$.


stacks-lemma-2-product-stacks

Let $\mathcal{C}$ be a site. The $(2, 1)$-category of stacks over $\mathcal{C}$ has 2-fibre products, and they are described as in Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}.


stacks-lemma-characterize-ff

Let $\mathcal{C}$ be a site. Let $\mathcal{S}_1$, $\mathcal{S}_2$ be stacks over $\mathcal{C}$. Let $F : \mathcal{S}_1 \to \mathcal{S}_2$ be a $1$-morphism. Then the following are equivalent \begin{enumerate} \item $F$ is fully faithful, \item for every $U \in \Ob(\mathcal{C})$ and for every $x, y \in \Ob(\mathcal{S}_{1, U})$ the map $$ F : \mathit{Mor}_{\mathcal{S}_1}(x, y) \longrightarrow \mathit{Mor}_{\mathcal{S}_2}(F(x), F(y)) $$ is an isomorphism of sheaves on $\mathcal{C}/U$. \end{enumerate}


stacks-lemma-characterize-essentially-surjective-when-ff

Let $\mathcal{C}$ be a site. Let $\mathcal{S}_1$, $\mathcal{S}_2$ be stacks over $\mathcal{C}$. Let $F : \mathcal{S}_1 \to \mathcal{S}_2$ be a $1$-morphism which is fully faithful. Then the following are equivalent \begin{enumerate} \item $F$ is an equivalence, \item for every $U \in \Ob(\mathcal{C})$ and for every $x \in \Ob(\mathcal{S}_{2, U})$ there exists a covering $\{f_i : U_i \to U\}$ such that $f_i^*x$ is in the essential image of the functor $F : \mathcal{S}_{1, U_i} \to \mathcal{S}_{2, U_i}$. \end{enumerate}


stacks-lemma-stack-in-groupoids-stack

Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a category over $\mathcal{C}$. The following are equivalent \begin{enumerate} \item $\mathcal{S}$ is a stack in groupoids over $\mathcal{C}$, \item $\mathcal{S}$ is a stack over $\mathcal{C}$ and all fibre categories are groupoids, and \item $\mathcal{S}$ is fibred in groupoids over $\mathcal{C}$ and is a stack over $\mathcal{C}$. \end{enumerate}


stacks-lemma-stack-in-groupoids-equivalent

Let $\mathcal{C}$ be a site. Let $\mathcal{S}_1$, $\mathcal{S}_2$ be categories over $\mathcal{C}$. Suppose that $\mathcal{S}_1$ and $\mathcal{S}_2$ are equivalent as categories over $\mathcal{C}$. Then $\mathcal{S}_1$ is a stack in groupoids over $\mathcal{C}$ if and only if $\mathcal{S}_2$ is a stack in groupoids over $\mathcal{C}$.


stacks-lemma-2-product-stacks-in-groupoids

Let $\mathcal{C}$ be a category. The $2$-category of stacks in groupoids over $\mathcal{C}$ has 2-fibre products, and they are described as in Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}.


stacks-lemma-when-stack-in-sets

Let $\mathcal{C}$ be a site. Under the equivalence $$ \left\{ \begin{matrix} \text{the category of presheaves}\\ \text{of sets over }\mathcal{C} \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{the category of categories}\\ \text{fibred in sets over }\mathcal{C} \end{matrix} \right\} $$ of Categories, Lemma \ref{categories-lemma-2-category-fibred-sets} the stacks in sets correspond precisely to the sheaves.


stacks-lemma-stack-in-setoids-characterize

Let $\mathcal{C}$ be a site. Let $\mathcal{S}$ be a category fibred in setoids over $\mathcal{C}$. Then $\mathcal{S}$ is a stack in setoids if and only if the unique equivalent category $\mathcal{S}'$ fibred in sets (see Categories, Lemma \ref{categories-lemma-setoid-fibres}) is a stack in sets. In other words, if and only if the presheaf $$ U \longmapsto \Ob(\mathcal{S}_U)/\!\!\cong $$ is a sheaf.


stacks-lemma-2-product-stacks-in-setoids

Let $\mathcal{C}$ be a site. The $2$-category of stacks in setoids over $\mathcal{C}$ has 2-fibre products, and they are described as in Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}.


stacks-lemma-2-fibre-product-gives-stack-in-setoids

Let $\mathcal{C}$ be a site. Let $\mathcal{S}, \mathcal{T}$ be stacks in groupoids over $\mathcal{C}$ and let $\mathcal{R}$ be a stack in setoids over $\mathcal{C}$. Let $f : \mathcal{T} \to \mathcal{S}$ and $g : \mathcal{R} \to \mathcal{S}$ be $1$-morphisms. If $f$ is faithful, then the $2$-fibre product $$ \mathcal{T} \times_{f, \mathcal{S}, g} \mathcal{R} $$ is a stack in setoids over $\mathcal{C}$.


stacks-lemma-2-fibre-product-stacks-in-setoids-over-stack-in-groupoids

Let $\mathcal{C}$ be a site. Let $\mathcal{S}$ be a stack in groupoids over $\mathcal{C}$ and let $\mathcal{S}_i$, $i = 1, 2$ be stacks in setoids over $\mathcal{C}$. Let $f_i : \mathcal{S}_i \to \mathcal{S}$ be $1$-morphisms. Then the $2$-fibre product $$ \mathcal{S}_1 \times_{f_1, \mathcal{S}, f_2} \mathcal{S}_2 $$ is a stack in setoids over $\mathcal{C}$.


stacks-lemma-faithful-descent

Let $\mathcal{C}$ be a site. Let $$ \xymatrix{ \mathcal{T}_2 \ar[r] \ar[d]_{G'} & \mathcal{T}_1 \ar[d]^G \\ \mathcal{S}_2 \ar[r]^F & \mathcal{S}_1 } $$ be a $2$-cartesian diagram of stacks in groupoids over $\mathcal{C}$. Assume \begin{enumerate} \item for every $U \in \Ob(\mathcal{C})$ and $x \in \Ob((\mathcal{S}_1)_U)$ there exists a covering $\{U_i \to U\}$ such that $x|_{U_i}$ is in the essential image of $F : (\mathcal{S}_2)_{U_i} \to (\mathcal{S}_1)_{U_i}$, and \item $G'$ is faithful, \end{enumerate} then $G$ is faithful.


stacks-lemma-stack-in-setoids-descent

Let $\mathcal{C}$ be a site. Let $$ \xymatrix{ \mathcal{T}_2 \ar[r] \ar[d] & \mathcal{T}_1 \ar[d]^G \\ \mathcal{S}_2 \ar[r]^F & \mathcal{S}_1 } $$ be a $2$-cartesian diagram of stacks in groupoids over $\mathcal{C}$. If \begin{enumerate} \item $F : \mathcal{S}_2 \to \mathcal{S}_1$ is fully faithful, \item for every $U \in \Ob(\mathcal{C})$ and $x \in \Ob((\mathcal{S}_1)_U)$ there exists a covering $\{U_i \to U\}$ such that $x|_{U_i}$ is in the essential image of $F : (\mathcal{S}_2)_{U_i} \to (\mathcal{S}_1)_{U_i}$, and \item $\mathcal{T}_2$ is a stack in setoids. \end{enumerate} then $\mathcal{T}_1$ is a stack in setoids.


stacks-lemma-relative-sheaf-over-stack-is-stack

Let $\mathcal{C}$ be a site. Let $F : \mathcal{S} \to \mathcal{T}$ be a $1$-morphism of categories fibred in groupoids over $\mathcal{C}$. Assume that \begin{enumerate} \item $\mathcal{T}$ is a stack in groupoids over $\mathcal{C}$, \item for every $U \in \Ob(\mathcal{C})$ the functor $\mathcal{S}_U \to \mathcal{T}_U$ of fibre categories is faithful, \item for each $U$ and each $y \in \Ob(\mathcal{T}_U)$ the presheaf $$ (h : V \to U) \longmapsto \{(x, f) \mid x \in \Ob(\mathcal{S}_V), f : F(x) \to f^*y\text{ over }V\}/\cong $$ is a sheaf on $\mathcal{C}/U$. \end{enumerate} Then $\mathcal{S}$ is a stack in groupoids over $\mathcal{C}$.


stacks-lemma-characterize-stack-in-setoids

Let $\mathcal{C}$ be a site. If $\mathcal{S}$ is a stack in groupoids, then the canonical $1$-morphism $\mathcal{I}_\mathcal{S} \to \mathcal{S}$ is an equivalence if and only if $\mathcal{S}$ is a stack in setoids.


stacks-lemma-stackify

Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category over $\mathcal{C}$. There exists a stack $p' : \mathcal{S}' \to \mathcal{C}$ and a $1$-morphism $G : \mathcal{S} \to \mathcal{S}'$ of fibred categories over $\mathcal{C}$ (see Categories, Definition \ref{categories-definition-fibred-categories-over-C}) such that \begin{enumerate} \item for every $U \in \Ob(\mathcal{C})$, and any $x, y \in \Ob(\mathcal{S}_U)$ the map $$ \mathit{Mor}(x, y) \longrightarrow \mathit{Mor}(G(x), G(y)) $$ induced by $G$ identifies the right hand side with the sheafification of the left hand side, and \item for every $U \in \Ob(\mathcal{C})$, and any $x' \in \Ob(\mathcal{S}'_U)$ there exists a covering $\{U_i \to U\}_{i \in I}$ such that for every $i \in I$ the object $x'|_{U_i}$ is in the essential image of the functor $G : \mathcal{S}_{U_i} \to \mathcal{S}'_{U_i}$. \end{enumerate} Moreover the stack $\mathcal{S}'$ is determined up to unique $2$-isomorphism by these conditions.


stacks-lemma-stackify-universal-property

Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category over $\mathcal{C}$. Let $p' : \mathcal{S}' \to \mathcal{C}$ and $G : \mathcal{S} \to \mathcal{S}'$ the stack and $1$-morphism constructed in Lemma \ref{lemma-stackify}. This construction has the following universal property: Given a stack $q : \mathcal{X} \to \mathcal{C}$ and a $1$-morphism $F : \mathcal{S} \to \mathcal{X}$ of fibred categories over $\mathcal{C}$ there exists a $1$-morphism $H : \mathcal{S}' \to \mathcal{X}$ such that the diagram $$ \xymatrix{ \mathcal{S} \ar[rr]_F \ar[rd]_G & & \mathcal{X} \\ & \mathcal{S}' \ar[ru]_H } $$ is $2$-commutative.


stacks-lemma-stackify-universal-property-more

Notation and assumptions as in Lemma \ref{lemma-stackify-universal-property}. There is a canonical equivalence of categories $$ \Mor_{\textit{Fib}/\mathcal{C}}(\mathcal{S}, \mathcal{X}) = \Mor_{\textit{Stacks}/\mathcal{C}}(\mathcal{S}', \mathcal{X}) $$ given by the constructions in the proof of the aforementioned lemma.


stacks-lemma-stackification-fibre-product-fibred-categories

Let $\mathcal{C}$ be a site. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Z} \to \mathcal{Y}$ be morphisms of fibred categories over $\mathcal{C}$. In this case the stackification of the $2$-fibre product is the $2$-fibre product of the stackifications.


stacks-lemma-stackification-inertia

Let $\mathcal{C}$ be a site. Let $\mathcal{X}$ be a fibred category over $\mathcal{C}$. The stackification of the inertia fibred category $\mathcal{I}_\mathcal{X}$ is inertia of the stackification of $\mathcal{X}$.


stacks-lemma-stackify-groupoids

Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids over $\mathcal{C}$. There exists a stack in groupoids $p' : \mathcal{S}' \to \mathcal{C}$ and a $1$-morphism $G : \mathcal{S} \to \mathcal{S}'$ of categories fibred in groupoids over $\mathcal{C}$ (see Categories, Definition \ref{categories-definition-categories-fibred-in-groupoids-over-C}) such that \begin{enumerate} \item for every $U \in \Ob(\mathcal{C})$, and any $x, y \in \Ob(\mathcal{S}_U)$ the map $$ \mathit{Mor}(x, y) \longrightarrow \mathit{Mor}(G(x), G(y)) $$ induced by $G$ identifies the right hand side with the sheafification of the left hand side, and \item for every $U \in \Ob(\mathcal{C})$, and any $x' \in \Ob(\mathcal{S}'_U)$ there exists a covering $\{U_i \to U\}_{i \in I}$ such that for every $i \in I$ the object $x'|_{U_i}$ is in the essential image of the functor $G : \mathcal{S}_{U_i} \to \mathcal{S}'_{U_i}$. \end{enumerate} Moreover the stack in groupoids $\mathcal{S}'$ is determined up to unique $2$-isomorphism by these conditions.


stacks-lemma-stackify-groupoids-universal-property

Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids over $\mathcal{C}$. Let $p' : \mathcal{S}' \to \mathcal{C}$ and $G : \mathcal{S} \to \mathcal{S}'$ the stack in groupoids and $1$-morphism constructed in Lemma \ref{lemma-stackify-groupoids}. This construction has the following universal property: Given a stack in groupoids $q : \mathcal{X} \to \mathcal{C}$ and a $1$-morphism $F : \mathcal{S} \to \mathcal{X}$ of categories over $\mathcal{C}$ there exists a $1$-morphism $H : \mathcal{S}' \to \mathcal{X}$ such that the diagram $$ \xymatrix{ \mathcal{S} \ar[rr]_F \ar[rd]_G & & \mathcal{X} \\ & \mathcal{S}' \ar[ru]_H } $$ is $2$-commutative.


stacks-lemma-stackification-fibre-product-categories-fibred-in-groupoids

Let $\mathcal{C}$ be a site. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Z} \to \mathcal{Y}$ be morphisms of categories fibred in groupoids over $\mathcal{C}$. In this case the stackification of the $2$-fibre product is the $2$-fibre product of the stackifications.


stacks-lemma-topology-inherited

Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Let $\text{Cov}(\mathcal{S})$ be the set of families $\{x_i \to x\}_{i \in I}$ of morphisms in $\mathcal{S}$ with fixed target such that (a) each $x_i \to x$ is strongly cartesian, and (b) $\{p(x_i) \to p(x)\}_{i \in I}$ is a covering of $\mathcal{C}$. Then $(\mathcal{S}, \text{Cov}(\mathcal{S}))$ is a site.


stacks-lemma-topology-inherited-functorial

Let $\mathcal{C}$ be a site. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of fibred categories over $\mathcal{C}$. Then $F$ is a continuous and cocontinuous functor between the structure of sites inherited from $\mathcal{C}$. Hence $F$ induces a morphism of topoi $f : \Sh(\mathcal{X}) \to \Sh(\mathcal{Y})$ with $f_* = {}_sF = {}_pF$ and $f^{-1} = F^s = F^p$. In particular $f^{-1}(\mathcal{G})(x) = \mathcal{G}(F(x))$ for a sheaf $\mathcal{G}$ on $\mathcal{Y}$ and object $x$ of $\mathcal{X}$.


stacks-lemma-localizing

Let $\mathcal{C}$ be a site. Let $p : \mathcal{X} \to \mathcal{C}$ be a category fibred in groupoids. Let $x \in \Ob(\mathcal{X})$ lying over $U = p(x)$. The functor $p$ induces an equivalence of sites $\mathcal{X}/x \to \mathcal{C}/U$ where $\mathcal{X}$ is endowed with the topology inherited from $\mathcal{C}$.


stacks-lemma-stack-in-groupoids-over-stack-in-groupoids

Let $\mathcal{C}$ be a site. Let $p : \mathcal{X} \to \mathcal{C}$ and $q : \mathcal{Y} \to \mathcal{C}$ be stacks in groupoids. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories over $\mathcal{C}$. If $F$ turns $\mathcal{X}$ into a category fibred in groupoids over $\mathcal{Y}$, then $\mathcal{X}$ is a stack in groupoids over $\mathcal{Y}$ (with topology inherited from $\mathcal{C}$).


stacks-lemma-stack-over-stack

Let $\mathcal{C}$ be a site. Let $p : \mathcal{X} \to \mathcal{C}$ be a stack. Endow $\mathcal{X}$ with the topology inherited from $\mathcal{C}$ and let $q : \mathcal{Y} \to \mathcal{X}$ be a stack. Then $\mathcal{Y}$ is a stack over $\mathcal{C}$. If $p$ and $q$ define stacks in groupoids, then $\mathcal{Y}$ is a stack in groupoids over $\mathcal{C}$.


stacks-lemma-gerbe-equivalent

Let $\mathcal{C}$ be a site. Let $\mathcal{S}_1$, $\mathcal{S}_2$ be categories over $\mathcal{C}$. Suppose that $\mathcal{S}_1$ and $\mathcal{S}_2$ are equivalent as categories over $\mathcal{C}$. Then $\mathcal{S}_1$ is a gerbe over $\mathcal{C}$ if and only if $\mathcal{S}_2$ is a gerbe over $\mathcal{C}$.


stacks-lemma-when-gerbe

Let $\mathcal{C}$ be a site. Let $p : \mathcal{X} \to \mathcal{C}$ and $q : \mathcal{Y} \to \mathcal{C}$ be stacks in groupoids. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories over $\mathcal{C}$. The following are equivalent \begin{enumerate} \item For some (equivalently any) factorization $F = F' \circ a$ where $a : \mathcal{X} \to \mathcal{X}'$ is an equivalence of categories over $\mathcal{C}$ and $F'$ is fibred in groupoids, the map $F' : \mathcal{X}' \to \mathcal{Y}$ is a gerbe (with the topology on $\mathcal{Y}$ inherited from $\mathcal{C}$). \item The following two conditions are satisfied \begin{enumerate} \item for $y \in \Ob(\mathcal{Y})$ lying over $U \in \Ob(\mathcal{C})$ there exists a covering $\{U_i \to U\}$ in $\mathcal{C}$ and objects $x_i$ of $\mathcal{X}$ over $U_i$ such that $F(x_i) \cong y|_{U_i}$ in $\mathcal{Y}_{U_i}$, and \item for $U \in \Ob(\mathcal{C})$, $x, x' \in \Ob(\mathcal{X}_U)$, and $b : F(x) \to F(x')$ in $\mathcal{Y}_U$ there exists a covering $\{U_i \to U\}$ in $\mathcal{C}$ and morphisms $a_i : x|_{U_i} \to x'|_{U_i}$ in $\mathcal{X}_{U_i}$ with $F(a_i) = b|_{U_i}$. \end{enumerate} \end{enumerate}


stacks-lemma-base-change-gerbe

Let $\mathcal{C}$ be a site. Let $$ \xymatrix{ \mathcal{X}' \ar[r]_{G'} \ar[d]_{F'} & \mathcal{X} \ar[d]^F \\ \mathcal{Y}' \ar[r]^G & \mathcal{Y} } $$ be a $2$-fibre product of stacks in groupoids over $\mathcal{C}$. If $\mathcal{X}$ is a gerbe over $\mathcal{Y}$, then $\mathcal{X}'$ is a gerbe over $\mathcal{Y}'$.


stacks-lemma-composition-gerbe

Let $\mathcal{C}$ be a site. Let $F : \mathcal{X} \to \mathcal{Y}$ and $G : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of stacks in groupoids over $\mathcal{C}$. If $\mathcal{X}$ is a gerbe over $\mathcal{Y}$ and $\mathcal{Y}$ is a gerbe over $\mathcal{Z}$, then $\mathcal{X}$ is a gerbe over $\mathcal{Z}$.


stacks-lemma-gerbe-descent

Let $\mathcal{C}$ be a site. Let $$ \xymatrix{ \mathcal{X}' \ar[r]_{G'} \ar[d]_{F'} & \mathcal{X} \ar[d]^F \\ \mathcal{Y}' \ar[r]^G & \mathcal{Y} } $$ be a $2$-cartesian diagram of stacks in groupoids over $\mathcal{C}$. If for every $U \in \Ob(\mathcal{C})$ and $x \in \Ob(\mathcal{Y}_U)$ there exists a covering $\{U_i \to U\}$ such that $x|_{U_i}$ is in the essential image of $G : \mathcal{Y}'_{U_i} \to \mathcal{Y}_{U_i}$ and $\mathcal{X}'$ is a gerbe over $\mathcal{Y}'$, then $\mathcal{X}$ is a gerbe over $\mathcal{Y}$.


stacks-lemma-gerbe-abelian-auts

Let $p : \mathcal{S} \to \mathcal{C}$ be a gerbe over a site $\mathcal{C}$. Assume that for all $U \in \Ob(\mathcal{C})$ and $x \in \Ob(\mathcal{S}_U)$ the sheaf of groups $\mathit{Aut}(x) = \mathit{Isom}(x, x)$ on $\mathcal{C}/U$ is abelian. Then there exist \begin{enumerate} \item a sheaf $\mathcal{G}$ of abelian groups on $\mathcal{C}$, \item for every $U \in \Ob(\mathcal{C})$ and every $x \in \Ob(\mathcal{S}_U)$ an isomorphism $\mathcal{G}|_U \to \mathit{Aut}(x)$ \end{enumerate} such that for every $U$ and every morphism $\varphi : x \to y$ in $\mathcal{S}_U$ the diagram $$ \xymatrix{ \mathcal{G}|_U \ar[d] \ar@{=}[rr] & & \mathcal{G}|_U \ar[d] \\ \mathit{Aut}(x) \ar[rr]^{\alpha \mapsto \varphi \circ \alpha \circ \varphi^{-1}} & & \mathit{Aut}(y) } $$ is commutative.


stacks-lemma-fibred-category-pushforward

In the situation above, if $\mathcal{S}$ is a fibred category over $\mathcal{D}$ then $u^p\mathcal{S}$ is a fibred category over $\mathcal{C}$.


stacks-lemma-stack-pushforward

Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous functor of sites. Let $p : \mathcal{S} \to \mathcal{D}$ be a stack over $\mathcal{D}$. Then $u^p\mathcal{S}$ is a stack over $\mathcal{C}$.


stacks-lemma-right-multiplicative-system

In the situation above assume \begin{enumerate} \item $p : \mathcal{S} \to \mathcal{C}$ is a fibred category, \item $\mathcal{C}$ has nonempty finite limits, and \item $u : \mathcal{C} \to \mathcal{D}$ commutes with nonempty finite limits. \end{enumerate} Consider the set $R \subset \text{Arrows}(u_{pp}\mathcal{S})$ of morphisms of the form $$ (a, \text{id}_V, \alpha) : (U', \phi' : V \to u(U'), x') \longrightarrow (U, \phi : V \to u(U), x) $$ with $\alpha$ strongly cartesian. Then $R$ is a right multiplicative system.


stacks-lemma-fibred-category-pullback

With notation and assumptions as in Lemma \ref{lemma-right-multiplicative-system}. Set $u_p\mathcal{S} = R^{-1}u_{pp}\mathcal{S}$, see Categories, Section \ref{categories-section-localization}. Then $u_p\mathcal{S}$ is a fibred category over $\mathcal{D}$.


stacks-lemma-adjointness-pullback-pushforward

Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Let $p : \mathcal{S} \to \mathcal{C}$ and $q : \mathcal{T} \to \mathcal{D}$ be categories over $\mathcal{C}$ and $\mathcal{D}$. Assume that \begin{enumerate} \item $p : \mathcal{S} \to \mathcal{C}$ is a fibred category, \item $q : \mathcal{T} \to \mathcal{D}$ is a fibred category, \item $\mathcal{C}$ has nonempty finite limits, and \item $u : \mathcal{C} \to \mathcal{D}$ commutes with nonempty finite limits. \end{enumerate} Then we have a canonical equivalence of categories $$ \Mor_{\textit{Fib}/\mathcal{C}}(\mathcal{S}, u^p\mathcal{T}) = \Mor_{\textit{Fib}/\mathcal{D}}(u_p\mathcal{S}, \mathcal{T}) $$ of morphism categories.


stacks-lemma-adjointness-pullback-pushforward-stacks

Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by a continuous functor $u : \mathcal{C} \to \mathcal{D}$ satisfying the hypotheses and conclusions of Sites, Proposition \ref{sites-proposition-get-morphism}. Let $p : \mathcal{S} \to \mathcal{C}$ and $q : \mathcal{T} \to \mathcal{D}$ be stacks. Then we have a canonical equivalence of categories $$ \Mor_{\textit{Stacks}/\mathcal{C}}(\mathcal{S}, f_*\mathcal{T}) = \Mor_{\textit{Stacks}/\mathcal{D}}(f^{-1}\mathcal{S}, \mathcal{T}) $$ of morphism categories.


stacks-lemma-technical-up

Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by a continuous functor $u : \mathcal{C} \to \mathcal{D}$ satisfying the hypotheses and conclusions of Sites, Proposition \ref{sites-proposition-get-morphism}. Let $\mathcal{S} \to \mathcal{C}$ be a fibred category, and let $\mathcal{S} \to \mathcal{S}'$ be the stackification of $\mathcal{S}$. Then $f^{-1}\mathcal{S}'$ is the stackification of $u_p\mathcal{S}$.


stacks-lemma-bigger-site

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor satisfying the assumptions of Sites, Lemma \ref{sites-lemma-bigger-site}. Let $f : \mathcal{D} \to \mathcal{C}$ be the corresponding morphism of sites. Then \begin{enumerate} \item for every stack $p : \mathcal{S} \to \mathcal{C}$ the canonical functor $\mathcal{S} \to f_*f^{-1}\mathcal{S}$ is an equivalence of stacks, \item given stacks $\mathcal{S}, \mathcal{S}'$ over $\mathcal{C}$ the construction $f^{-1}$ induces an equivalence $$ \Mor_{\textit{Stacks}/\mathcal{C}}(\mathcal{S}, \mathcal{S}') \longrightarrow \Mor_{\textit{Stacks}/\mathcal{D}}(f^{-1}\mathcal{S}, f^{-1}\mathcal{S}') $$ of morphism categories. \end{enumerate}


stacks-lemma-localize-stacks

Assume that $\mathcal{C}$ is a site, and $U$ is an object of $\mathcal{C}$ whose associated representable presheaf is a sheaf. Constructions A and B above define mutually inverse (!) functors of $2$-categories $$ \left\{ \begin{matrix} 2\text{-category of}\\ \text{stacks over }\mathcal{C}/U \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} 2\text{-category of pairs }(\mathcal{T}, p) \text{ consisting} \\ \text{of a stack }\mathcal{T}\text{ over }\mathcal{C}\text{ and a morphism} \\ p : \mathcal{T} \to \mathcal{C}/U\text{ of stacks over }\mathcal{C} \end{matrix} \right\} $$


spaces-simplicial-lemma-simplicial-site

Let $X$ be a simplicial space. Then $X_{Zar}$ as defined above is a site.


spaces-simplicial-lemma-describe-sheaves-simplicial-site

Let $X$ be a simplicial space. There is an equivalence of categories between \begin{enumerate} \item $\Sh(X_{Zar})$, and \item category of systems $(\mathcal{F}_n, \mathcal{F}(\varphi))$ described above. \end{enumerate}


spaces-simplicial-lemma-simplicial-space-site-functorial

Let $f : Y \to X$ be a morphism of simplicial spaces. Then the functor $u : X_{Zar} \to Y_{Zar}$ which associates to the open $U \subset X_n$ the open $f_n^{-1}(U) \subset Y_n$ defines a morphism of sites $f_{Zar} : Y_{Zar} \to X_{Zar}$.