etale-cohomology-lemma-brauer-inverse

Let $A$ be a finite central simple algebra over $K$. Then $$ \begin{matrix} A \otimes_K A^{opp} & \longrightarrow & \text{End}_K(A) \\ \ a \otimes a' & \longmapsto & (x \mapsto a x a') \end{matrix} $$ is an isomorphism of algebras over $K$.


etale-cohomology-lemma-annihilated-by-degree

\begin{reference} Argument taken from \cite{Saltman-torsion}. \end{reference} Let $S$ be a scheme. Let $\mathcal{A}$ be an Azumaya algebra which is locally free of rank $d^2$ over $S$. Then the class of $\mathcal{A}$ in the Brauer group of $S$ is annihilated by $d$.


etale-cohomology-lemma-constructible-constructible

Let $X$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{F}$ be a sheaf of sets, abelian groups, $\Lambda$-modules (with $\Lambda$ Noetherian) on $X_\etale$. If there exist constructible locally closed subschemes $T_i \subset X$ such that (a) $X = \bigcup T_j$ and (b) $\mathcal{F}|_{T_j}$ is constructible, then $\mathcal{F}$ is constructible.


etale-cohomology-lemma-cohomology-with-support-sheaf-on-support

Let $i : Z \to X$ be a closed immersion of schemes. Let $\mathcal{G}$ be an injective abelian sheaf on $Z_\etale$. Then $\mathcal{H}^p_Z(i_*\mathcal{G}) = 0$ for $p > 0$.


etale-cohomology-lemma-complexes-with-support-on-closed

Let $i : Z \to X$ be a closed immersion of schemes. The map $Ri_{small, *} = i_{small, *} : D(Z_\etale) \to D(X_\etale)$ induces an equivalence $D(Z_\etale) \to D_Z(X_\etale)$ with quasi-inverse $$ i_{small}^{-1}|_{D_Z(X_\etale)} = R\mathcal{H}_Z|_{D_Z(X_\etale)} $$


etale-cohomology-lemma-affine-only-closed-points

Let $S$ be an affine scheme such that (1) all points are closed, and (2) all residue fields are separably algebraically closed. Then for any abelian sheaf $\mathcal{F}$ on $S_\etale$ we have $H^i(S_\etale, \mathcal{F}) = 0$ for $i > 0$.


etale-cohomology-lemma-base-change-f-star-general-stalks

Consider a cartesian diagram of schemes $$ \xymatrix{ X \ar[d]_f & Y \ar[l]^h \ar[d]^e \\ S & T \ar[l]_g } $$ where $g : T \to S$ is quasi-compact and quasi-separated. Let $\mathcal{F}$ be an abelian sheaf on $T_\etale$. Let $q \geq 0$. The following are equivalent \begin{enumerate} \item For every geometric point $\overline{x}$ of $X$ with image $\overline{s} = f(\overline{x})$ we have $$ H^q(\Spec(\mathcal{O}^{sh}_{X, \overline{x}}) \times_S T, \mathcal{F}) = H^q(\Spec(\mathcal{O}^{sh}_{S, \overline{s}}) \times_S T, \mathcal{F}) $$ \item $f^{-1}R^qg_*\mathcal{F} \to R^qh_*e^{-1}\mathcal{F}$ is an isomorphism. \end{enumerate}


etale-cohomology-lemma-base-change-separably-closed

Let $K/k$ be an extension of separably closed fields. Let $X$ be a proper scheme over $k$. Let $\mathcal{F}$ be a torsion abelian sheaf on $X_\etale$. Then the map $H^q_\etale(X, \mathcal{F}) \to H^q_\etale(X_K, \mathcal{F}|_{X_K})$ is an isomorphism for $q \geq 0$.


etale-cohomology-lemma-kunneth

Let $k$ be a separably closed field. Let $X$ and $Y$ be finite type schemes over $k$. Let $n \geq 1$ be an integer invertible in $k$. Then for $E \in D(X_\etale, \mathbf{Z}/n\mathbf{Z})$ and $K \in D(Y_\etale, \mathbf{Z}/n\mathbf{Z})$ we have $$ R\Gamma(X \times_{\Spec(k)} Y, \text{pr}_1^{-1}E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} \text{pr}_2^{-1}K ) = R\Gamma(X, E) \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} R\Gamma(Y, K) $$


etale-cohomology-lemma-compare-cohomology-etale-fppf

For a scheme $X$ and $a_X : \Sh((\Sch/X)_{fppf}) \to \Sh(X_\etale)$ as above: \begin{enumerate} \item $H^q(X_\etale, \mathcal{F}) = H^q_{fppf}(X, a_X^{-1}\mathcal{F})$ for an abelian sheaf $\mathcal{F}$ on $X_\etale$, \item $H^q(X_\etale, K) = H^q_{fppf}(X, a_X^{-1}K)$ for $K \in D^+(X_\etale)$. \end{enumerate} Example: if $A$ is an abelian group, then $H^q_\etale(X, \underline{A}) = H^q_{fppf}(X, \underline{A})$.


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

Let $S$ be a scheme. For $\mathcal{F}$ a quasi-coherent $\mathcal{O}_S$-module on $S_\etale$ the maps $$ \pi_S^*\mathcal{F} \longrightarrow R\epsilon_{S, *}(a_S^*\mathcal{F}) \quad\text{and}\quad \mathcal{F} \longrightarrow Ra_{S, *}(a_S^*\mathcal{F}) $$ are isomorphisms with $a_S : \Sh((\Sch/S)_{fppf}) \to \Sh(S_\etale)$ as above.


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

In Lemma \ref{lemma-push-pull-ph-etale} if $f$ is proper, then we have $a_Y^{-1} \circ f_{small, *} = f_{big, ph, *} \circ a_X^{-1}$.


etale-cohomology-lemma-proper-push-pull-h-etale

In Lemma \ref{lemma-push-pull-h-etale} if $f$ is proper, then we have $a_Y^{-1} \circ f_{small, *} = f_{big, h, *} \circ a_X^{-1}$.


etale-cohomology-lemma-compare-cohomology-etale-h

For a scheme $X$ and $a_X : \Sh((\Sch/X)_h) \to \Sh(X_\etale)$ as above: \begin{enumerate} \item $H^q(X_\etale, \mathcal{F}) = H^q_h(X, a_X^{-1}\mathcal{F})$ for a torsion abelian sheaf $\mathcal{F}$ on $X_\etale$, \item $H^q(X_\etale, K) = H^q_h(X, a_X^{-1}K)$ for $K \in D^+(X_\etale)$ with torsion cohomology sheaves. \end{enumerate} Example: if $A$ is a torsion abelian group, then $H^q_\etale(X, \underline{A}) = H^q_h(X, \underline{A})$.


etale-cohomology-lemma-glue-etale-sheaf-fppf

Let $\{f_i : X_i \to X\}$ be an fppf covering of schemes. The functor $$ \Sh(X_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{f_i : X_i \to X\} $$ is an equivalence of categories.


etale-cohomology-lemma-blow-up-square-etale-cohomology

Let $X$ be a scheme and let $K \in D^+(X_\etale)$ have torsion cohomology sheaves. Let $Z \subset X$ be a closed subscheme cut out by a quasi-coherent ideal of finite type. Consider the corresponding blow up square $$ \xymatrix{ E \ar[d] \ar[r] & X' \ar[d]^b \\ Z \ar[r] & X } $$ Then there is a canonical long exact sequence $$ H^p_\etale(X, K) \to H^p_\etale(X', K|_{X'}) \oplus H^p_\etale(Z, K|_Z) \to H^p_\etale(E, K|_E) \to H^{p + 1}_\etale(X, K) $$


etale-cohomology-proposition-cohomology-restrict-small-site

Let $S$ be a scheme and $\mathcal{F}$ an abelian sheaf on $(\Sch/S)_\etale$. Then $\mathcal{F}|_{S_\etale}$ is a sheaf on $S_\etale$ and $$ H^p_\etale(S, \mathcal{F}|_{S_\etale}) = H^p_\etale(S, \mathcal{F}) $$ for all $p \geq 0$.


equiv-theorem-bondal-van-den-bergh

\begin{reference} \cite[Theorem A.1]{BvdB} \end{reference} Let $X$ be a projective scheme over a field $k$. Let $F : D_{perf}(\mathcal{O}_X)^{opp} \to \text{Vect}_k$ be a $k$-linear cohomological functor such that $$ \sum\nolimits_{n \in \mathbf{Z}} \dim_k F(E[n]) < \infty $$ for all $E \in D_{perf}(\mathcal{O}_X)$. Then $F$ is isomorphic to a functor of the form $E \mapsto \Hom_X(E, K)$ for some $K \in D^b_{\textit{Coh}}(\mathcal{O}_X)$.


equiv-theorem-countable

\begin{reference} Slight improvement of \cite{AT} \end{reference} Let $K$ be an algebraically closed field. Let $\mathbf{X}$ be a smooth proper scheme over $K$. There are at most countably many isomorphism classes of smooth proper schemes $\mathbf{Y}$ over $K$ which are derived equivalent to $\mathbf{X}$.


equiv-lemma-Serre-functor

In the situation of Definition \ref{definition-Serre-functor}. If a Serre functor exists, then it is unique up to unique isomorphism and it is an exact functor of triangulated categories.


equiv-lemma-Serre-functor-Gorenstein-proper

Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is Gorenstein. Consider the complex $\omega_X^\bullet$ of Duality for Schemes, Lemmas \ref{duality-lemma-duality-proper-over-field}. Then the functor $$ S : D_{perf}(\mathcal{O}_X) \longrightarrow D_{perf}(\mathcal{O}_X),\quad K \longmapsto S(K) = \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} K $$ is a Serre functor.


equiv-lemma-on-product-general

Let $R$ be a ring. Let $X$, $Y$ be quasi-compact and quasi-separated schemes over $R$ having the resolution property. For any finite type quasi-coherent $\mathcal{O}_{X \times_R Y}$-module $\mathcal{F}$ there exist a surjection $\mathcal{E} \boxtimes \mathcal{G} \to \mathcal{F}$ where $\mathcal{E}$ is a finite locally free $\mathcal{O}_X$-module and $\mathcal{G}$ is a finite locally free $\mathcal{O}_Y$-module.


stacks-more-morphisms-proposition-affine-smooth-lift-to-first-order

\begin{reference} Email of Matthew Emerton dated April 27, 2016. \end{reference} Let $\mathcal{X} \subset \mathcal{X}'$ be a first order thickening of algebraic stacks. Let $W$ be an affine scheme and let $W \to \mathcal{X}$ be a smooth morphism. Then there exists a cartesian diagram $$ \xymatrix{ W \ar[d] \ar[r] & W' \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{X}' } $$ with $W' \to \mathcal{X}'$ smooth and $W'$ affine.


perfect-lemma-support-quasi-coherent

Let $X$ be a scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is a retrocompact open of $X$. Let $i : T \to X$ be the inclusion. \begin{enumerate} \item For $E$ in $D_\QCoh(\mathcal{O}_X)$ we have $i_*R\mathcal{H}_T(E)$ in $D_{\QCoh, T}(\mathcal{O}_X)$. \item The functor $i_* \circ R\mathcal{H}_T : D_\QCoh(\mathcal{O}_X) \to D_{\QCoh, T}(\mathcal{O}_X)$ is right adjoint to the inclusion functor $D_{\QCoh, T}(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_X)$. \end{enumerate}


perfect-lemma-support-direct-sums

Let $X$ be a scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is a retrocompact open of $X$. Then for a family of objects $E_i$, $i \in I$ of $D_\QCoh(\mathcal{O}_X)$ we have $R\mathcal{H}_T(\bigoplus E_i) = \bigoplus R\mathcal{H}_T(E_i)$.


perfect-lemma-supported-map-global

Let $X$ be a scheme. Let $Z \to X$ be a closed immersion of finite presentation whose conormal sheaf $\mathcal{C}_{Z/X}$ is locally free of rank $c$. Then there is a canonical map $$ c : \wedge^c(\mathcal{C}_{Z/X})^\vee \otimes_{\mathcal{O}_Z} i^*\mathcal{F} \longrightarrow \mathcal{H}_Z^c(\mathcal{F}) $$ functorial in the quasi-coherent module $\mathcal{F}$.


perfect-lemma-direct-image-coherator

Let $f : X \to Y$ be a morphism of schemes. Assume $X$ and $Y$ are quasi-compact and have affine diagonal. Then, denoting $$ \Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y)) $$ the right derived functor of $f_* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)$ the diagram $$ \xymatrix{ D(\QCoh(\mathcal{O}_X)) \ar[d]_\Phi \ar[r] & D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\ D(\QCoh(\mathcal{O}_Y)) \ar[r] & D_\QCoh(\mathcal{O}_Y) } $$ is commutative.


perfect-lemma-kunneth-Ext

In the situation above, assume $a$ and $b$ are quasi-compact and quasi-separated and $X$ and $Y$ are tor independent over $S$. If $K$ is perfect, $K' \in D_\QCoh(\mathcal{O}_X)$, $M$ is perfect, and $M' \in D_\QCoh(\mathcal{O}_Y)$, then (\ref{equation-kunneth-ext}) is an isomorphism.


perfect-lemma-pullback-and-limits

Let $R$ be a ring. Let $X$ be a scheme and let $f : X \to \Spec(R)$ be proper, flat, and of finite presentation. Let $(M_n)$ be an inverse system of $R$-modules with surjective transition maps. Then the canonical map $$ \mathcal{O}_X \otimes_R (\lim M_n) \longrightarrow \lim \mathcal{O}_X \otimes_R M_n $$ induces an isomorphism from the source to $DQ_X$ applied to the target.


perfect-proposition-perfect-resolution-property

Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Denote \begin{enumerate} \item $\mathcal{A}$ the additive category of finite locally free $\mathcal{O}_X$-modules, \item $K^b(\mathcal{A})$ the homotopy category of bounded complexes in $\mathcal{A}$, see Derived Categories, Section \ref{derived-section-homotopy}, and \item $D_{perf}(\mathcal{O}_X)$ the strictly full, saturated, triangulated subcategory of $D(\mathcal{O}_X)$ consisting of perfect objects. \end{enumerate} With this notation the obvious functor $$ K^b(\mathcal{A}) \longrightarrow D_{perf}(\mathcal{O}_X) $$ is an exact functor of trianglated categories which factors through an equivalence $S^{-1}K^b(\mathcal{A}) \to D_{perf}(\mathcal{O}_X)$ of triangulated categories where $S$ is the saturated multiplicative system of quasi-isomorphisms in $K^b(\mathcal{A})$.


spaces-flat-theorem-flattening-map

In Situation \ref{situation-iso} assume \begin{enumerate} \item $f$ is of finite presentation, \item $\mathcal{F}$ is of finite presentation, flat over $B$, and pure relative to $B$, and \item $u$ is surjective. \end{enumerate} Then $F_{iso}$ is representable by a closed immersion $Z \to B$. Moreover $Z \to S$ is of finite presentation if $\mathcal{G}$ is of finite presentation.


spaces-flat-lemma-open-in-fibre-where-flat

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $y \in |Y|$ and $F = f^{-1}(\{y\}) \subset |X|$. Then the set $$ \{x \in F \mid \mathcal{F} \text{ flat over }Y\text{ at }x\} $$ is open in $F$.


spaces-flat-lemma-check-along-closed-fibre

Let $S$ be a local scheme with closed point $s$. Let $f : X \to S$ be a morphism from an algebraic space $X$ to $S$ which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Assume that \begin{enumerate} \item every point of $\text{Ass}_{X/S}(\mathcal{F})$ specializes to a point of the closed fibre $X_s$\footnote{For example this holds if $f$ is finite type and $\mathcal{F}$ is pure along $X_s$, or if $f$ is proper.}, \item $\mathcal{F}$ is flat over $S$ at every point of $X_s$. \end{enumerate} Then $\mathcal{F}$ is flat over $S$.


spaces-flat-lemma-F-zero-somewhat-closed-points

In Situation \ref{situation-somewhat-closed}. Let $T \subset S$ be a subset. Let $s \in S$ be in the closure of $T$. For $t \in T$, let $u_t$ be the pullback of $u$ to $X_t$ and let $u_s$ be the pullback of $u$ to $X_s$. If $X$ is locally of finite presentation over $S$, $\mathcal{G}$ is of finite presentation\footnote{It would suffice if $X$ is locally of finite type over $S$ and $\mathcal{G}$ is finitely presented relative to $S$, but this notion hasn't yet been defined in the setting of algebraic spaces. The definition for schemes is given in More on Morphisms, Section \ref{more-morphisms-section-finite-type-finite-presentation}.}, and $u_t = 0$ for all $t \in T$, then $u_s = 0$.


spaces-flat-lemma-F-iso-closed

In Situation \ref{situation-iso}. Assume \begin{enumerate} \item $f$ is locally of finite presentation, \item $\mathcal{F}$ is locally of finite presentation and flat over $B$, \item the support of $\mathcal{F}$ is proper over $B$, and \item $u$ is surjective. \end{enumerate} Then the functor $F_{iso}$ is an algebraic space and $F_{iso} \to B$ is a closed immersion. If $\mathcal{G}$ is of finite presentation, then $F_{iso} \to B$ is of finite presentation.


spaces-flat-lemma-freebie

Let $S$ be the spectrum of a henselian local ring with closed point $s$. Let $X \to S$ be a morphism of algebraic spaces which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Let $E \subset |X_s|$ be a subset. There exists a closed subscheme $Z \subset S$ with the following property: for any morphism of pointed schemes $(T, t) \to (S, s)$ the following are equivalent \begin{enumerate} \item $\mathcal{F}_T$ is flat over $T$ at all points of $|X_t|$ which map to a point of $E \subset |X_s|$, and \item $\Spec(\mathcal{O}_{T, t}) \to S$ factors through $Z$. \end{enumerate} Moreover, if $X \to S$ is locally of finite presentation, $\mathcal{F}$ is of finite presentation, and $E \subset |X_s|$ is closed and quasi-compact, then $Z \to S$ is of finite presentation.


spaces-chow-lemma-additivity-sheaf-cycle

In Situation \ref{situation-setup} let $X/B$ be good. Let $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ be a short exact sequence of coherent $\mathcal{O}_X$-modules. Assume that the $\delta$-dimension of the supports of $\mathcal{F}$, $\mathcal{G}$, and $\mathcal{H}$ are $\leq k$. Then $[\mathcal{G}]_k = [\mathcal{F}]_k + [\mathcal{H}]_k$.


spaces-chow-lemma-finite-flat

In Situation \ref{situation-setup} let $X, Y/B$ be good. Let $f : X \to Y$ be a finite locally free morphism of degree $d$ (see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-finite-locally-free}). Then $f$ is both proper and flat of relative dimension $0$, and $$ f_*f^*\alpha = d\alpha $$ for every $\alpha \in Z_k(Y)$.


spaces-chow-lemma-vectorbundle

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $$ p : E = \underline{\Spec}(\text{Sym}^*(\mathcal{E})) \longrightarrow X $$ be the associated vector bundle over $X$. Then $p^* : \CH_k(X) \to \CH_{k + r}(E)$ is an isomorphism for all $k$.


spaces-chow-lemma-cap-commutative-chern

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{E}$ be a locally free $\mathcal{O}_X$-module of rank $r$. Then $c_j(\mathcal{L}) \in A^j(X)$ commutes with every element $c \in A^p(X)$. In particular, if $\mathcal{F}$ is a second locally free $\mathcal{O}_X$-module on $X$ of rank $s$, then $$ c_i(\mathcal{E}) \cap c_j(\mathcal{F}) \cap \alpha = c_j(\mathcal{F}) \cap c_i(\mathcal{E}) \cap \alpha $$ as elements of $\CH_{k - i - j}(X)$ for all $\alpha \in \CH_k(X)$.


spaces-chow-lemma-additivity-chern-classes

In Situation \ref{situation-setup} let $X/B$ be good. Suppose that $\mathcal{E}$ sits in an exact sequence $$ 0 \to \mathcal{E}_1 \to \mathcal{E} \to \mathcal{E}_2 \to 0 $$ of finite locally free sheaves $\mathcal{E}_i$ of rank $r_i$. The total Chern classes satisfy $$ c(\mathcal{E}) = c(\mathcal{E}_1) c(\mathcal{E}_2) $$ in $A^*(X)$.


spaces-chow-lemma-chern-classes-tensor-product

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{E}$ and $\mathcal{F}$ be a finite locally free $\mathcal{O}_X$-modules of ranks $r$ and $s$. Then we have $$ c_1(\mathcal{E} \otimes \mathcal{F}) = r c_1(\mathcal{F}) + s c_1(\mathcal{E}) $$ $$ c_2(\mathcal{E} \otimes \mathcal{F}) = r^2 c_2(\mathcal{F}) + rs c_1(\mathcal{F})c_1(\mathcal{E}) + s^2 c_2(\mathcal{E}) $$ and so on (see proof).


spaces-chow-lemma-degrees-and-numerical-intersections

Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $Z \subset X$ be a closed subspace of dimension $d$. Let $\mathcal{L}_1, \ldots, \mathcal{L}_d$ be invertible $\mathcal{O}_X$-modules. Then $$ (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z) = \deg( c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_1) \cap [Z]_d) $$ where the left hand side is defined in Spaces over Fields, Definition \ref{spaces-over-fields-definition-intersection-number}.


sdga-lemma-good-quotient

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a differential graded $\mathcal{A}$-module. There exists a homomorphism $\mathcal{P} \to \mathcal{M}$ of differential graded $\mathcal{A}$-modules with the following properties \begin{enumerate} \item $\mathcal{P} \to \mathcal{M}$ is surjective, \item $\Ker(\text{d}_\mathcal{P}) \to \Ker(\text{d}_\mathcal{M})$ is surjective, and \item $\mathcal{P}$ is good. \end{enumerate}


sdga-lemma-homotopy-colimit

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}_n$ be a system of differential graded $\mathcal{A}$-modules. Then the derived colimit $\text{hocolim} \mathcal{M}_n$ in $D(\mathcal{A}, \text{d})$ is represented by the differential graded module $\colim \mathcal{M}_n$.


sdga-proposition-homotopy-category-triangulated

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The homotopy category $K(\text{Mod}_{(\mathcal{A}, \text{d})})$ is a triangulated category where \begin{enumerate} \item the shift functors are those constructed in Section \ref{section-shift-dg}, \item the distinghuished triangles are those triangles in $K(\text{Mod}_{(\mathcal{A}, \text{d})})$ which are isomorphic as a triangle to a triangle $$ \mathcal{K} \to \mathcal{L} \to \mathcal{N} \xrightarrow{\delta} \mathcal{K}[1],\quad\quad \delta = \pi \circ \text{d}_\mathcal{L} \circ s $$ constructed from an admissible short exact sequence $0 \to \mathcal{K} \to \mathcal{L} \to \mathcal{N} \to 0$ in $\text{Mod}_{(\mathcal{A}, \text{d})}$ above. \end{enumerate}


stacks-morphisms-lemma-composition-affine

Compositions of affine morphisms of algebraic stacks are affine.


stacks-morphisms-lemma-finite-type-points-surjective-morphism

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. If $f$ is locally of finite type and surjective, then $f(\mathcal{X}_{\text{ft-pts}}) = \mathcal{Y}_{\text{ft-pts}}$.


stacks-morphisms-lemma-representable-by-spaces-quasi-finite

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ is representable by algebraic spaces. The following are equivalent \begin{enumerate} \item $f$ is locally quasi-finite (as in Properties of Stacks, Section \ref{stacks-properties-section-properties-morphisms}), and \item $f$ is locally of finite type and for every morphism $\Spec(k) \to \mathcal{Y}$ where $k$ is a field the space $|\Spec(k) \times_\mathcal{Y} \mathcal{X}|$ is discrete. \end{enumerate}


stacks-morphisms-lemma-characterize-locally-quasi-finite

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \begin{enumerate} \item $f$ is locally quasi-finite, \item $f$ is quasi-DM and for any morphism $V \to \mathcal{Y}$ with $V$ an algebraic space and any locally quasi-finite morphism $U \to \mathcal{X} \times_\mathcal{Y} V$ where $U$ is an algebraic space the morphism $U \to V$ is locally quasi-finite, \item for any morphism $V \to \mathcal{Y}$ from an algebraic space $V$ there exists a surjective, flat, locally finitely presented, and locally quasi-finite morphism $U \to \mathcal{X} \times_\mathcal{Y} V$ where $U$ is an algebraic space such that $U \to V$ is locally quasi-finite, \item there exists algebraic spaces $U$, $V$, a surjective, flat, and locally of finite presentation morphism $V \to \mathcal{Y}$, and a morphism $U \to \mathcal{X} \times_\mathcal{Y} V$ which is surjective, flat, locally of finite presentation, and locally quasi-finite such that $U \to V$ is locally quasi-finite. \end{enumerate}


stacks-morphisms-lemma-composition-quasi-finite

The composition of quasi-finite morphisms is quasi-finite.


stacks-morphisms-lemma-gerbe-with-section

Let $\pi : \mathcal{X} \to U$ be a morphism from an algebraic stack to an algebraic space and let $x : U \to \mathcal{X}$ be a section of $\pi$. Set $G = \mathit{Isom}_\mathcal{X}(x, x)$, see Definition \ref{definition-isom}. If $\mathcal{X}$ is a gerbe over $U$, then \begin{enumerate} \item there is a canonical equivalence of stacks in groupoids $$ x_{can} : [U/G] \longrightarrow \mathcal{X}. $$ where $[U/G]$ is the quotient stack for the trivial action of $G$ on $U$, \item $G \to U$ is flat and locally of finite presentation, and \item $U \to \mathcal{X}$ is surjective, flat, and locally of finite presentation. \end{enumerate}


stacks-morphisms-lemma-spectral-qcqs

Let $\mathcal{X}$ be a quasi-compact and quasi-separated algebraic stack. Then $|\mathcal{X}|$ is a spectral topological space.


stacks-morphisms-lemma-sober-qs

Let $\mathcal{X}$ be an algebraic stack whose diagonal is quasi-compact (for example if $\mathcal{X}$ is quasi-separated). Then there is an open covering $|\mathcal{X}| = \bigcup U_i$ with $U_i$ spectral. In particular $|\mathcal{X}|$ is a sober topological space.


stacks-morphisms-lemma-etale-local-quasi-DM-at-x-inertia

Let $\mathcal{X}$ be an algebraic stack. Assume $\mathcal{X}$ is quasi-DM with separated diagonal (equivalently $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is locally quasi-finite and separated). Let $x \in |\mathcal{X}|$. Assume $x$ can be represented by a quasi-compact morphism $\Spec(k) \to \mathcal{X}$. Then there exists a morphism of algebraic stacks $$ g : \mathcal{U} \longrightarrow \mathcal{X} $$ with the following properties \begin{enumerate} \item there exists a point $u \in |\mathcal{U}|$ mapping to $x$ and $g$ induces an isomorphism between the residual gerbes at $u$ and $x$, \item $\mathcal{U} \to \mathcal{X}$ is representable by algebraic spaces and \'etale, \item $\mathcal{U} = [U/R]$ where $(U, R, s, t, c)$ is a groupoid scheme with $U$, $R$ affine, and $s, t$ finite, flat, and locally of finite presentation. \end{enumerate}


stacks-morphisms-lemma-composition-etale

The composition of \'etale morphisms is \'etale.


stacks-morphisms-lemma-etale

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \begin{enumerate} \item $f$ is \'etale, \item $f$ is DM and for any morphism $V \to \mathcal{Y}$ where $V$ is an algebraic space and any \'etale morphism $U \to V \times_\mathcal{Y} \mathcal{X}$ where $U$ is an algebraic space, the morphism $U \to V$ is \'etale, \item there exists some surjective, locally of finite presentation, and flat morphism $W \to \mathcal{Y}$ where $W$ is an algebraic space and some surjective \'etale morphism $T \to W \times_\mathcal{Y} \mathcal{X}$ where $T$ is an algebraic space such that the morphism $T \to W$ is \'etale. \end{enumerate}


stacks-morphisms-lemma-unramified

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \begin{enumerate} \item $f$ is unramified, \item $f$ is DM and for any morphism $V \to \mathcal{Y}$ where $V$ is an algebraic space and any \'etale morphism $U \to V \times_\mathcal{Y} \mathcal{X}$ where $U$ is an algebraic space, the morphism $U \to V$ is unramified, \item there exists some surjective, locally of finite presentation, and flat morphism $W \to \mathcal{Y}$ where $W$ is an algebraic space and some surjective \'etale morphism $T \to W \times_\mathcal{Y} \mathcal{X}$ where $T$ is an algebraic space such that the morphism $T \to W$ is unramified. \end{enumerate}


stacks-morphisms-lemma-permanence-unramified

Let $\mathcal{X} \to \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. If $\mathcal{X} \to \mathcal{Z}$ is unramified and $\mathcal{Y} \to \mathcal{Z}$ is DM, then $\mathcal{X} \to \mathcal{Y}$ is unramified.


stacks-morphisms-lemma-base-change-proper

A base change of a proper morphism is proper.


stacks-morphisms-lemma-composition-uniqueness

The composition of morphisms of algebraic stacks which satisfy the uniqueness part of the valuative criterion is another morphism of algebraic stacks which satisfies the uniqueness part of the valuative criterion.


stacks-morphisms-lemma-flat-base-change-lci

A flat base change of a local complete intersection morphism is a local complete intersection morphism.


schemes-lemma-colimit-quasi-coherent

Let $X = \Spec(R)$ be an affine scheme. The direct sum of an arbitrary collection of quasi-coherent sheaves on $X$ is quasi-coherent. The same holds for colimits.


schemes-lemma-section-immersion

Let $f : X \to S$ be a morphism of schemes. Let $s : S \to X$ be a section of $f$ (in a formula $f \circ s = \text{id}_S$). Then $s$ is an immersion. If $f$ is separated then $s$ is a closed immersion. If $f$ is quasi-separated, then $s$ is quasi-compact.


schemes-lemma-curiosity

Let $f : X \to S$ be a morphism. Assume $f$ is separated and $S$ is a separated scheme. Suppose $U \subset X$ and $V \subset X$ are affine. Then $U \cap V$ is affine (and a closed subscheme of $U \times V$).


schemes-lemma-injective-points

Let $j : X \to Y$ be a morphism of schemes. If $j$ is injective on points, then $j$ is separated.


brauer-lemma-base-change

Let $A$ be a finite central simple algebra over $k$. Let $k \subset k'$ be a field extension. Then $A' = A \otimes_k k'$ is a finite central simple algebra over $k'$.


brauer-lemma-automorphism-inner

Let $A$ be a finite central simple $k$-algebra. Any automorphism of $A$ is inner. In particular, any automorphism of $\text{Mat}(n \times n, k)$ is inner.


brauer-lemma-splitting-field-degree

Consider a finite central skew field $K$ over $k$. Let $d^2 = [K : k]$. For any finite splitting field $k'$ for $K$ the degree $[k' : k]$ is divisible by $d$.


divisors-lemma-relative-assassin-affine-open

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $U \subset X$ and $V \subset S$ be affine opens with $f(U) \subset V$. Write $U = \Spec(A)$, $V = \Spec(R)$, and set $M = \Gamma(U, \mathcal{F})$. Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime. Then $$ \mathfrak p \in \text{Ass}_{A/R}(M) \Rightarrow x \in \text{Ass}_{X/S}(\mathcal{F}) $$ If all fibres $X_s$ of $f$ are locally Noetherian, then $\mathfrak p \in \text{Ass}_{A/R}(M) \Leftrightarrow x \in \text{Ass}_{X/S}(\mathcal{F})$ for all pairs $(\mathfrak p, x)$ as above.


divisors-lemma-torsion

Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The torsion sections of $\mathcal{F}$ form a quasi-coherent $\mathcal{O}_X$-submodule $\mathcal{F}_{tors} \subset \mathcal{F}$. The quotient module $\mathcal{F}/\mathcal{F}_{tors}$ is torsion free.


divisors-lemma-flat-pullback-torsion

Let $f : X \to Y$ be a flat morphism of integral schemes. Let $\mathcal{G}$ be a torsion free quasi-coherent $\mathcal{O}_Y$-module. Then $f^*\mathcal{G}$ is a torsion free $\mathcal{O}_X$-module.


divisors-lemma-reflexive-torsion-free

Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. \begin{enumerate} \item If $\mathcal{F}$ is reflexive, then $\mathcal{F}$ is torsion free. \item The map $j : \mathcal{F} \longrightarrow \mathcal{F}^{**}$ is injective if and only if $\mathcal{F}$ is torsion free \end{enumerate}


divisors-lemma-regular-section-associated-points

Let $X$ be a locally Noetherian scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$. Then $s$ is a regular section if and only if $s$ does not vanish in the associated points of $X$.


divisors-lemma-flat-relative-Cartier-divisor

Let $f : X \to S$ be a morphism of schemes. Let $D \subset X$ be a relative effective Cartier divisor. If $f$ is locally of finite presentation, then there exists an open subscheme $U \subset X$ such that $D \subset U$ and such that $f|_U : U \to S$ is flat.


divisors-lemma-michael-artin

Let $f : X \to S$ be a morphism of schemes. Let $D \subset X$ be a relative effective Cartier divisor on $X/S$. If $f$ is flat at all points of $X \setminus D$, then $f$ is flat.


divisors-lemma-koszul-regular-smooth-locally-regular

Let $i : Z \to X$ be a Koszul regular closed immersion. Then there exists a surjective smooth morphism $X' \to X$ such that the base change $i' : Z \times_X X' \to X'$ of $i$ is a regular immersion.


divisors-lemma-flat-relative-H1-regular

Let $X \to S$ be a morphism of schemes. Let $Z \to X$ be a relative $H_1$-regular immersion. Assume $X \to S$ is locally of finite presentation. Then \begin{enumerate} \item there exists an open subscheme $U \subset X$ such that $Z \subset U$ and such that $U \to S$ is flat, and \item $Z \to X$ is a regular immersion and the same remains true after any base change. \end{enumerate}


divisors-lemma-section-smooth-regular-immersion

Let $f : X \to S$ be a smooth morphism of schemes. Let $\sigma : S \to X$ be a section of $f$. Then $\sigma$ is a regular immersion.


divisors-lemma-locally-Noetherian-K

Let $X$ be a locally Noetherian scheme. \begin{enumerate} \item For any $x \in X$ we have $\mathcal{S}_x \subset \mathcal{O}_{X, x}$ is the set of nonzerodivisors, and hence $\mathcal{K}_{X, x}$ is the total quotient ring of $\mathcal{O}_{X, x}$. \item For any affine open $U \subset X$ the ring $\mathcal{K}_X(U)$ equals the total quotient ring of $\mathcal{O}_X(U)$. \end{enumerate}


divisors-lemma-reduced-normalization

Let $X$ be a scheme. Assume $X$ is reduced and any quasi-compact open $U \subset X$ has a finite number of irreducible components. Then the normalization morphism $\nu : X^\nu \to X$ is the morphism $$ \underline{\Spec}_X(\mathcal{O}') \longrightarrow X $$ where $\mathcal{O}' \subset \mathcal{K}_X$ is the integral closure of $\mathcal{O}_X$ in the sheaf of meromorphic functions.


divisors-lemma-regular-meromorphic-section-exists

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. In each of the following cases $\mathcal{L}$ has a regular meromorphic section: \begin{enumerate} \item $X$ is integral, \item $X$ is reduced and any quasi-compact open has a finite number of irreducible components, \item $X$ is locally Noetherian and has no embedded points. \end{enumerate}


divisors-lemma-reflexive-normal

Let $X$ be an integral locally Noetherian normal scheme. For $\mathcal{F}$ and $\mathcal{G}$ coherent reflexive $\mathcal{O}_X$-modules the map $$ (\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{O}_X) \otimes_{\mathcal{O}_X} \mathcal{G})^{**} \to \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) $$ is an isomorphism. The rule $\mathcal{F}, \mathcal{G} \mapsto (\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G})^{**}$ defines an abelian group law on the set of isomorphism classes of rank $1$ coherent reflexive $\mathcal{O}_X$-modules.


divisors-lemma-normal-class-group

Let $X$ be an integral locally Noetherian normal scheme. The group of rank $1$ coherent reflexive $\mathcal{O}_X$-modules is isomorphic to the Weil divisor class group $\text{Cl}(X)$ of $X$.


divisors-lemma-affine-Xs

Assumptions and notation as in Lemma \ref{lemma-structure-sheaf-Xs}. The following are equivalent \begin{enumerate} \item the inclusion morphism $j : U \to X$ is affine, and \item for every $x \in X \setminus U$ there is an $n > 0$ such that $s^n \in \mathfrak m_x \mathcal{F}^{[n]}_x$. \end{enumerate}


spaces-lemma-algebraic-space-coproduct-sheaves

Let $S \in \Ob(\Sch_{fppf})$. Let $F$ be an algebraic space over $S$. Given a set $I$ and sheaves $F_i$ on $\Ob((\Sch/S)_{fppf})$, if $F \cong \coprod_{i\in I} F_i$ as sheaves, then each $F_i$ is an algebraic space over $S$.


topology-lemma-baire-category-locally-compact

Let $X$ be a locally quasi-compact Hausdorff space. Let $U_n \subset X$, $n \geq 1$ be dense open subsets. Then $\bigcap_{n \geq 1} U_n$ is dense in $X$.


topology-lemma-noetherian-partition-refined-by-stratification

Let $X$ be a Noetherian topological space. Any finite partition of $X$ can be refined by a finite good stratification.


topology-lemma-topological-ring-colimits

The category of topological rings has colimits and colimits commute with the forgetful functor to the category of rings.


topology-lemma-topological-module-colimits

Let $R$ be a topological ring. The category of topological modules over $R$ has colimits and colimits commute with the forgetful functor to the category of modules over $R$.


hypercovering-lemma-covering-sheaf

Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $n \geq 0$ be an integer. Let $u : \mathcal{F} \to F(K_n)$ be a morphism of presheaves which becomes surjective on sheafification. Then there exists a morphism of hypercoverings $f: L \to K$ such that $F(f_n) : F(L_n) \to F(K_n)$ factors through $u$.


hypercovering-lemma-homotopy

Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K, L$ be hypercoverings of $X$. Let $a, b : K \to L$ be morphisms of hypercoverings. There exists a morphism of hypercoverings $c : K' \to K$ such that $a \circ c$ is homotopic to $b \circ c$.


hypercovering-proposition-cohomology-hypercoverings

Let $\mathcal{C}$ be a site with fibre products and products of pairs. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$. Let $i \geq 0$. Then \begin{enumerate} \item for every $\xi \in H^i(\mathcal{F})$ there exists a hypercovering $K$ such that $\xi$ is in the image of the canonical map $\check{H}^i(K, \mathcal{F}) \to H^i(\mathcal{F})$, and \item if $K, L$ are hypercoverings and $\xi_K \in \check{H}^i(K, \mathcal{F})$, $\xi_L \in \check{H}^i(L, \mathcal{F})$ are elements mapping to the same element of $H^i(\mathcal{F})$, then there exists a hypercovering $M$ and morphisms $M \to K$ and $M \to L$ such that $\xi_K$ and $\xi_L$ map to the same element of $\check{H}^i(M, \mathcal{F})$. \end{enumerate} In other words, modulo set theoretical issues, the cohomology groups of $\mathcal{F}$ on $\mathcal{C}$ are the colimit of the {\v C}ech cohomology groups of $\mathcal{F}$ over all hypercoverings.


algebraic-lemma-product-representable-transformations

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}_i, \mathcal{Y}_i$ be categories fibred in groupoids over $(\Sch/S)_{fppf}$, $i = 1, 2$. Let $f_i : \mathcal{X}_i \to \mathcal{Y}_i$, $i = 1, 2$ be representable $1$-morphisms. Then $$ f_1 \times f_2 : \mathcal{X}_1 \times \mathcal{X}_2 \longrightarrow \mathcal{Y}_1 \times \mathcal{Y}_2 $$ is a representable $1$-morphism.


algebraic-lemma-get-a-stack

\begin{reference} Lemma in an email of Matthew Emerton dated June 15, 2016 \end{reference} Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X} \to \mathcal{Z}$ and $\mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $\mathcal{X} \to \mathcal{Z}$ is representable by algebraic spaces and $\mathcal{Y}$ is a stack in groupoids, then $\mathcal{X} \times_\mathcal{Z} \mathcal{Y}$ is a stack in groupoids.


algebraic-lemma-descent-representable-transformations-property

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\mathcal{P}$ be a property as in Definition \ref{definition-relative-representable-property}. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism representable by algebraic spaces. Let $g : \mathcal{Z} \to \mathcal{Y}$ be any $1$-morphism. Consider the fibre product diagram $$ \xymatrix{ \mathcal{Z} \times_{g, \mathcal{Y}, f} \mathcal{X} \ar[r]_-{g'} \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{Z} \ar[r]^g & \mathcal{Y} } $$ Assume that for every scheme $U$ and object $x$ of $\mathcal{Y}_U$, there exists an fppf covering $\{U_i \to U\}$ such that $x|_{U_i}$ is in the essential image of the functor $g : \mathcal{Z}_{U_i} \to \mathcal{Y}_{U_i}$. In this case, if $f'$ has $\mathcal{P}$, then $f$ has $\mathcal{P}$.


algebraic-lemma-product-representable-transformations-property

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{P}$ be a property as in Definition \ref{definition-relative-representable-property} which is stable under composition. Let $\mathcal{X}_i, \mathcal{Y}_i$ be categories fibred in groupoids over $(\Sch/S)_{fppf}$, $i = 1, 2$. Let $f_i : \mathcal{X}_i \to \mathcal{Y}_i$, $i = 1, 2$ be $1$-morphisms representable by algebraic spaces. If $f_1$ and $f_2$ have property $\mathcal{P}$ so does $ f_1 \times f_2 : \mathcal{X}_1 \times \mathcal{X}_2 \to \mathcal{Y}_1 \times \mathcal{Y}_2 $.


algebraic-lemma-product-spaces

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}$, $\mathcal{Y}$ be algebraic stacks over $S$. Then $\mathcal{X} \times_{(\Sch/S)_{fppf}} \mathcal{Y}$ is an algebraic stack, and is a product in the $2$-category of algebraic stacks over $S$.


sites-lemma-localize-pushforward

In the situation of Lemma \ref{lemma-localize-topos}, the functor $j_{\mathcal{F}, *}$ is the one associates to $\varphi : \mathcal{G} \to \mathcal{F}$ the sheaf $$ U \longmapsto \{\alpha : \mathcal{F}|_U \to \mathcal{G}|_U \text{ such that } \alpha \text{ is a right inverse to }\varphi|_U \}. $$


sites-lemma-check-morphism-sites

Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous functor of sites. Let $\{(q_i, v_i)\}_{i\in I}$ be a conservative family of points of $\mathcal{D}$. If each functor $u_i = v_i \circ u$ defines a point of $\mathcal{C}$, then $u$ defines a morphism of sites $f : \mathcal{D} \to \mathcal{C}$.


sites-lemma-characterize-empty

Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$. The following are equivalent: \begin{enumerate} \item $U$ is sheaf theoretically empty, \item $\mathcal{F}(U)$ is a singleton for each sheaf $\mathcal{F}$, \item $\emptyset^\#(U)$ is a singleton, \item $\emptyset^\#(U)$ is nonempty, and \item the empty family is a covering of $U$ in $\mathcal{C}$. \end{enumerate} Moreover, if $U$ is sheaf theoretically empty, then for any morphism $U' \to U$ of $\mathcal{C}$ the object $U'$ is sheaf theoretically empty.


sites-lemma-finer-topology

Assumption and notation as in Theorem \ref{theorem-topology-and-topos}. Then $J \subset J'$ if and only if every sheaf for the topology $J'$ is a sheaf for the topology $J$.


sites-proposition-sheaves-on-group

The functors $\mathcal{F} \mapsto \mathcal{F}({}_GG)$ and $S \mapsto \mathcal{F}_S$ define quasi-inverse equivalences between $\Sh(\mathcal{T}_G)$ and $G\textit{-Sets}$.


examples-defos-lemma-continuous-representations-hull

In Example \ref{example-continuous-representations} assume $\Gamma$ is topologically 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-graded-algebras-hull

In Example \ref{example-graded-algebras} assume $P$ is a finitely generated graded $k$-algebra. Assume $\Lambda$ is a complete local ring with residue field $k$ (the classical case). Then the functor $$ F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \Ob(\Deformationcategory_P(A))/\cong $$ of isomorphism classes of objects has a hull.


examples-defos-lemma-strict-henselization

In Example \ref{example-rings} let $P$ be a $k$-algebra. Assume $P$ is a local ring and let $P^{sh}$ be a strict henselization of $P$. There is a natural functor $$ \Deformationcategory_P \longrightarrow \Deformationcategory_{P^{sh}} $$ of deformation categories.


examples-defos-lemma-schemes-hull

In Example \ref{example-schemes} assume $X$ is a proper $k$-scheme. Assume $\Lambda$ is a complete local ring with residue field $k$ (the classical case). Then the functor $$ F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \Ob(\Deformationcategory_X(A))/\cong $$ of isomorphism classes of objects has a hull. If $\text{Der}_k(\mathcal{O}_X, \mathcal{O}_X) = 0$, then $F$ is prorepresentable.


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

Let $k$ be a field and 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 there exists a flat, separated, finite type morphism $Y \to \Spec(k[[t]])$ whose generic fibre is smooth and whose special fibre is isomorphic to $X$.


more-etale-lemma-separated-etale-shriek

Let $j : U \to X$ be a separated \'etale morphism. Let $\mathcal{F}$ be an abelian sheaf on $U_\etale$. The image of the injective map $j_!\mathcal{F} \to j_*\mathcal{F}$ of \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-shriek-into-star-separated-etale} is the subsheaf of Lemma \ref{lemma-f-shriek-separated}.


more-etale-lemma-base-change-shriek

Consider a cartesian square $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ of quasi-compact and quasi-separated schemes with $f$ separated and of finite type. Then there is a canonical isomorphism $$ g^{-1} \circ Rf_! \to Rf'_! \circ (g')^{-1} $$ Moreover, these isomorphisms are compatible with the isomorphisms of Lemma \ref{lemma-shriek-composition}.


more-etale-lemma-upper-shriek-derived

Let $f : X \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. The functor $Rf_! : D(X_\etale, \Lambda) \to D(Y_\etale, \Lambda)$ has a right adjoint $Rf^! : D(Y_\etale, \Lambda) \to D(X_\etale, \Lambda)$.


stacks-properties-lemma-composition-monomorphism

Compositions of monomorphisms of algebraic stacks are monomorphisms.


stacks-properties-lemma-residual-gerbe-regular

A reduced, locally Noetherian algebraic stack $\mathcal{Z}$ such that $|\mathcal{Z}|$ is a singleton is regular.


stacks-lemma-stack-gives-stack-groupoids

Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a stack. Let $p' : \mathcal{S}' \to \mathcal{C}$ be the category fibred in groupoids associated to $\mathcal{S}$ constructed in Categories, Lemma \ref{categories-lemma-fibred-gives-fibred-groupoids}. Then $p' : \mathcal{S}' \to \mathcal{C}$ is a stack in groupoids.


stacks-lemma-fibred-groupoids-category-pullback

With notation and assumptions as in Lemma \ref{lemma-fibred-category-pullback}. If $\mathcal{S}$ is fibred in groupoids, then $u_p\mathcal{S}$ is fibred in groupoids.


stacks-lemma-when-localization-stack

Let $\mathcal{C}$ be a site. Let $U \in \Ob(\mathcal{C})$. Then $j_U : \mathcal{C}/U \to \mathcal{C}$ is a stack over $\mathcal{C}$ if and only if $h_U$ is a sheaf.


spaces-simplicial-lemma-augmentation-spectral-sequence-modules

With notation as above for any $K$ in $D^+(\mathcal{O})$ there is a spectral sequence $(E_r, d_r)_{r \geq 0}$ in $\textit{Mod}(\mathcal{O}_\mathcal{D})$ with $$ E_1^{p, q} = R^qa_{p, *} K_p $$ converging to $R^{p + q}a_*K$. This spectral sequence is functorial in $K$.


spaces-simplicial-lemma-derived-cartesian-modules

In Situation \ref{situation-simplicial-site}. \begin{enumerate} \item An object $K$ of $D(\mathcal{C}_{total})$ is cartesian if and only if $H^q(K)$ is a cartesian abelian sheaf for all $q$. \item Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$ such that the morphisms $f_{\delta^n_j} : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}_{n - 1}), \mathcal{O}_{n - 1})$ are flat. Then an object $K$ of $D(\mathcal{O})$ is cartesian if and only if $H^q(K)$ is a cartesian $\mathcal{O}$-module for all $q$. \end{enumerate}


spaces-simplicial-lemma-has-P

Let $\mathcal{C}$ be a site. \begin{enumerate} \item For $K$ in $\text{SR}(\mathcal{C})$ the functor $j : \mathcal{C}/K \to \mathcal{C}$ is continuous, cocontinuous, and has property P of Sites, Remark \ref{sites-remark-cartesian-cocontinuous}. \item For $f : K \to L$ in $\text{SR}(\mathcal{C})$ the functor $v : \mathcal{C}/K \to \mathcal{C}/L$ (see above) is continuous, cocontinuous, and has property P of Sites, Remark \ref{sites-remark-cartesian-cocontinuous}. \end{enumerate}


spaces-simplicial-lemma-compare-cohomology-hypercovering-X-simple

Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$. Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above. Then we have a canonical isomorphism $$ R\Gamma(X, E) = R\Gamma((\mathcal{C}/U)_{total}, a^{-1}E) $$ for $E \in D(\mathcal{C}/X)$.


spaces-simplicial-lemma-hypercovering-X-simple-equivalence-bounded-modules

Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings. Let $U$ be a hypercovering of $X$ in $\mathcal{C}$. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ denote the weak Serre subcategory of cartesian $\mathcal{O}$-modules. Then the functor $La^*$ defines an equivalence $$ D^+(\mathcal{O}_X) \longrightarrow D_\mathcal{A}^+(\mathcal{O}) $$ with quasi-inverse $Ra_*$.


spaces-simplicial-lemma-bbd-unbounded-glueing

In Situation \ref{situation-locally-given}. Assume \begin{enumerate} \item $\mathcal{C}$ has equalizers and fibre products, \item there is a morphism of sites $f : \mathcal{C} \to \mathcal{D}$ given by a continuous functor $u : \mathcal{D} \to \mathcal{C}$ such that \begin{enumerate} \item $\mathcal{D}$ has equalizers and fibre products and $u$ commutes with them, \item $\mathcal{B}$ is a full subcategory of $\mathcal{D}$ and $u : \mathcal{B} \to \mathcal{C}$ is the restriction of $u$, \item every object of $\mathcal{D}$ has a covering whose members are objects of $\mathcal{B}$, \end{enumerate} \item all negative self-exts of $E_U$ in $D(\mathcal{O}_{u(U)})$ are zero, and \item there exist weak Serre subcategories $\mathcal{A}_U \subset \textit{Mod}(\mathcal{O}_U)$ for all $U \in \Ob(\mathcal{C})$ satisfying conditions (\ref{item-restriction}), (\ref{item-local}), and (\ref{item-bounded-dimension}), \item $E_U \in D_{\mathcal{A}_U}(\mathcal{O}_U)$. \end{enumerate} Then there exists a solution unique up to unique isomorphism.


spaces-simplicial-lemma-proper-hypercovering-equivalence-bounded

Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. Let $\mathcal{A} \subset \textit{Ab}(U_{Zar})$ denote the weak Serre subcategory of cartesian abelian sheaves. If $U$ is a proper hypercovering of $X$, then the functor $a^{-1}$ defines an equivalence $$ D^+(X) \longrightarrow D_\mathcal{A}^+(U_{Zar}) $$ with quasi-inverse $Ra_*$ where $a : \Sh(U_{Zar}) \to \Sh(X)$ is as in Lemma \ref{lemma-augmentation}.


spaces-simplicial-lemma-spectral-sequence-proper-hypercovering

Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. Let $\mathcal{F}$ be an abelian sheaf on $X$. Let $\mathcal{F}_n$ be the pullback to $U_n$. If $U$ is a proper hypercovering of $X$, then there exists a canonical spectral sequence $$ E_1^{p, q} = H^q(U_p, \mathcal{F}_p) $$ converging to $H^{p + q}(X, \mathcal{F})$.


spaces-simplicial-lemma-adjoint-functors-cartesian-modules

Let $f : V \to U$ be a cartesian morphism of simplicial schemes. Assume the morphisms $d^n_j : U_n \to U_{n - 1}$ are flat and the morphisms $V_n \to U_n$ are quasi-compact and quasi-separated. Then $f^*$ and $f_*$ form an adjoint pair of functors between the categories of quasi-coherent modules on $U_{Zar}$ and $V_{Zar}$.


spaces-simplicial-lemma-cartesian-modules-with-section

Let $f : X \to S$ be a morphism of schemes which has a section\footnote{In fact, it would be enough to assume that $f$ has fpqc locally on $S$ a section, since we have descent of quasi-coherent modules by Descent, Section \ref{descent-section-fpqc-descent-quasi-coherent}.}. Let $(X/S)_\bullet$ be the simplicial scheme associated to $X \to S$, see Definition \ref{definition-fibre-products-simplicial-scheme}. Then pullback defines an equivalence between the category of quasi-coherent $\mathcal{O}_S$-modules and the category of quasi-coherent modules on $((X/S)_\bullet)_{Zar}$.


spaces-simplicial-lemma-quasi-coherent-groupoid-simplicial

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $X$ be the simplicial scheme over $S$ constructed in Lemma \ref{lemma-groupoid-simplicial}. Then the category of quasi-coherent modules on $(U, R, s, t, c)$ is equivalent to the category of quasi-coherent modules on $X_{Zar}$.


spaces-simplicial-lemma-spectral-sequence-fppf-hypercovering

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. Let $\mathcal{F}_n$ be the pullback to $U_{n, \etale}$. If $U$ is an fppf hypercovering of $X$, then there exists a canonical spectral sequence $$ E_1^{p, q} = H^q_\etale(U_p, \mathcal{F}_p) $$ converging to $H^{p + q}_\etale(X, \mathcal{F})$.


spaces-simplicial-lemma-spectral-sequence-fppf-hypercovering-modules

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. Let $\mathcal{F}$ be quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{F}_n$ be the pullback to $U_{n, \etale}$. If $U$ is an fppf hypercovering of $X$, then there exists a canonical spectral sequence $$ E_1^{p, q} = H^q_\etale(U_p, \mathcal{F}_p) $$ converging to $H^{p + q}_\etale(X, \mathcal{F})$.


spaces-simplicial-lemma-ph-hypercovering-equivalence-bounded

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. Let $\mathcal{A} \subset \textit{Ab}(U_\etale)$ denote the weak Serre subcategory of cartesian abelian sheaves. If $U$ is a proper hypercovering of $X$, then the functor $a^{-1}$ defines an equivalence $$ D^+(X_\etale) \longrightarrow D_\mathcal{A}^+(U_\etale) $$ with quasi-inverse $Ra_*$. Here $a : \Sh(U_\etale) \to \Sh(X_\etale)$ is as in Section \ref{section-simplicial-algebraic-spaces}.


spaces-simplicial-lemma-spectral-sequence-ph-hypercovering

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. Let $\mathcal{F}_n$ be the pullback to $U_{n, \etale}$. If $U$ is a ph hypercovering of $X$, then there exists a canonical spectral sequence $$ E_1^{p, q} = H^q_\etale(U_p, \mathcal{F}_p) $$ converging to $H^{p + q}_\etale(X, \mathcal{F})$.


examples-stacks-lemma-stack-of-quasi-coherent-sheaves

The functor $p : \QCohstack \to (\Sch/S)_{fppf}$ satisfies conditions (1), (2) and (3) of Stacks, Definition \ref{stacks-definition-stack}.


examples-stacks-lemma-stack-ft-spaces

There exists a subcategory $\Spacesstack_{ft, small} \subset \Spacesstack_{ft}$ with the following properties: \begin{enumerate} \item the inclusion functor $\Spacesstack_{ft, small} \to \Spacesstack_{ft}$ is fully faithful and essentially surjective, and \item the functor $p_{ft, small} : \Spacesstack_{ft, small} \to (\Sch/S)_{fppf}$ turns $\Spacesstack_{ft, small}$ into a stack over $(\Sch/S)_{fppf}$. \end{enumerate}


examples-stacks-lemma-variant-torsors-sheaf-stack-in-groupoids

Up to a replacement as in Stacks, Remark \ref{stacks-remark-stack-make-small} the functor $$ p : \mathcal{G}/\mathcal{B}\textit{-Torsors} \longrightarrow (\Sch/S)_{fppf} $$ defines a stack in groupoids over $(\Sch/S)_{fppf}$.


examples-stacks-lemma-group-quotient-stack-in-groupoids

Up to a replacement as in Stacks, Remark \ref{stacks-remark-stack-make-small} the functor $$ p : [[X/G]] \longrightarrow (\Sch/S)_{fppf} $$ defines a stack in groupoids over $(\Sch/S)_{fppf}$.


examples-stacks-lemma-classifying-stacks

\begin{slogan} The classifying stack of a group scheme or group algebraic space. \end{slogan} Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Then the stacks in groupoids $$ [B/G],\quad [[B/G]],\quad G\textit{-Torsors},\quad \mathcal{G}/\mathcal{B}\textit{-Torsors} $$ are all canonically equivalent. If $G \to B$ is flat and locally of finite presentation, then these are also equivalent to $G\textit{-Principal}$.


examples-stacks-lemma-hilbert-d-stack

The category $\mathcal{H}_d(\mathcal{X}/\mathcal{Y})$ endowed with the functor $p$ above defines a stack in groupoids over $(\Sch/S)_{fppf}$.


models-lemma-picard-rank-1

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. The Picard group of $T$ is a finitely generated abelian group of rank $1$.


models-lemma-genus-reduction-smaller

In Situation \ref{situation-regular-model} assume $X$ is a minimal model, $\gcd(m_1, \ldots, m_n) = 1$, and $H^0((X_k)_{red}, \mathcal{O}) = k$. Then the map $$ H^1(X_k, \mathcal{O}_{X_k}) \to H^1((X_k)_{red}, \mathcal{O}_{(X_k)_{red}}) $$ is surjective and has a nontrivial kernel as soon as $(X_k)_{red} \not = X_k$.


models-lemma-equality-genus-reduction-bigger-than

If equality holds in Lemma \ref{lemma-genus-reduction-bigger-than} then \begin{enumerate} \item the unique irreducible component of $X_k$ containing $x$ is a smooth projective geometrically irreducible curve over $k$, \item if $C \subset X_k$ is another irreducible component, then $\kappa = H^0(C, \mathcal{O}_C)$ is a finite separable extension of $k$, $C$ has a $\kappa$-rational point, and $C$ is smooth over $\kappa$ \end{enumerate}


spaces-groupoids-lemma-diagram-pull

Let $B \to S$ be as in Section \ref{section-notation}. Let $(U, R, s, t, c, e, i)$ be a groupoid in algebraic spaces over $B$. The diagram \begin{equation} \label{equation-pull} \xymatrix{ R \times_{t, U, t} R \ar@<1ex>[r]^-{\text{pr}_1} \ar@<-1ex>[r]_-{\text{pr}_0} \ar[d]_{\text{pr}_0 \times c \circ (i, 1)} & R \ar[r]^t \ar[d]^{\text{id}_R} & U \ar[d]^{\text{id}_U} \\ R \times_{s, U, t} R \ar@<1ex>[r]^-c \ar@<-1ex>[r]_-{\text{pr}_0} \ar[d]_{\text{pr}_1} & R \ar[r]^t \ar[d]^s & U \\ R \ar@<1ex>[r]^s \ar@<-1ex>[r]_t & U } \end{equation} is commutative. The two top rows are isomorphic via the vertical maps given. The two lower left squares are cartesian.


spaces-groupoids-lemma-colimits

Let $B \to S$ be as in Section \ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The category of quasi-coherent modules on $(U, R, s, t, c)$ has colimits.


spaces-groupoids-lemma-action-groupoid-modules

Let $B \to S$ as in Section \ref{section-notation}. Let $(G, m)$ be a group algebraic space over $B$. Let $X$ be an algebraic space over $B$ and let $a : G \times_B X \to X$ be an action of $G$ on $X$ over $B$. Let $(U, R, s, t, c)$ be the groupoid in algebraic spaces constructed in Lemma \ref{lemma-groupoid-from-action}. The rule $(\mathcal{F}, \alpha) \mapsto (\mathcal{F}, \alpha)$ defines an equivalence of categories between $G$-equivariant $\mathcal{O}_X$-modules and the category of quasi-coherent modules on $(U, R, s, t, c)$.


spaces-groupoids-lemma-quotient-groupoid-restrict

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $g : U' \to U$ a morphism of algebraic spaces over $B$. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ to $U'$. The map of quotient sheaves $$ U'/R' \longrightarrow U/R $$ is injective. If the composition $$ \xymatrix{ U' \times_{g, U, t} R \ar[r]_-{\text{pr}_1} \ar@/^3ex/[rr]^h & R \ar[r]_s & U } $$ is a surjection of fppf sheaves then the map is bijective. This holds for example if $\{h : U' \times_{g, U, t} R \to U\}$ is an $fppf$-covering, or if $U' \to U$ is a surjection of sheaves, or if $\{g : U' \to U\}$ is a covering in the fppf topology.


spaces-descent-lemma-finite-type-descends

Let $X$ be an algebraic space over a scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that each $f_i^*\mathcal{F}$ is a finite type $\mathcal{O}_{X_i}$-module. Then $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module.


spaces-descent-lemma-flat-descends

Let $X$ be an algebraic space over a scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that each $f_i^*\mathcal{F}$ is a flat $\mathcal{O}_{X_i}$-module. Then $\mathcal{F}$ is a flat $\mathcal{O}_X$-module.


spaces-descent-lemma-syntomic-permanence

Let $S$ be a scheme. Let $$ \xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & B } $$ be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that \begin{enumerate} \item $f$ is surjective, flat, and locally of finite presentation, \item $p$ is syntomic. \end{enumerate} Then both $q$ and $f$ are syntomic.


spaces-descent-lemma-descending-property-universal-homeomorphism

The property $\mathcal{P}(f) =$``$f$ is a universal homeomorphism'' is fpqc local on the base.


spaces-descent-lemma-descending-property-finite-locally-free

The property $\mathcal{P}(f) =$``$f$ is finite locally free'' is fpqc local on the base.


spaces-descent-lemma-open-fppf-local-source

The property $\mathcal{P}(f)=$``$f$ is open'' is fppf local on the source.


decent-spaces-lemma-birational-dominant

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which are decent and have finitely many irreducible components. If $f$ is birational then $f$ is dominant.


decent-spaces-lemma-birational-generic-fibres

Let $S$ be a scheme. Let $f : X \to Y$ be a birational morphism of algebraic spaces over $S$ which are decent and have finitely many irreducible components. If $y \in Y$ is the generic point of an irreducible component, then the base change $X \times_Y \Spec(\mathcal{O}_{Y, y}) \to \Spec(\mathcal{O}_{Y, y})$ is an isomorphism.


decent-spaces-lemma-normalization-normal

Let $S$ be a scheme. Let $f : X \to Y$ be an integral birational morphism of decent algebraic spaces over $S$ which have finitely many irreducible components. Then there exists a factorization $Y^\nu \to X \to Y$ and $Y^\nu \to X$ is the normalization of $X$.


decent-spaces-lemma-decent-Jacobson

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Then $X$ is Jacobson if and only if $|X|$ is Jacobson.


decent-spaces-lemma-universally-catenary

Let $S$ be a scheme. Let $X$ be a decent, locally Noetherian, and universally catenary algebraic space over $S$. Then any decent algebraic space locally of finite type over $X$ is universally catenary.


groupoids-lemma-algebraic-center

Let $k$ be a field. Let $G$ be a locally algebraic group scheme over $k$. Then the center of $G$ is a closed subgroup scheme of $G$.


groupoids-lemma-colimit-finite-type

Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume that \begin{enumerate} \item $U$, $R$ are affine, \item there exist $e_i \in \mathcal{O}_R(R)$ such that every element $g \in \mathcal{O}_R(R)$ can be uniquely written as $\sum s^*(f_i)e_i$ for some $f_i \in \mathcal{O}_U(U)$. \end{enumerate} Then every quasi-coherent module $(\mathcal{F}, \alpha)$ on $(U, R, s, t, c)$ is a filtered colimit of finite type quasi-coherent modules.


groupoids-lemma-descend-along-finite-quasi-projective

Let $X \to Y$ be a surjective finite locally free morphism. Let $V$ be a scheme over $X$ such that one of the following holds \begin{enumerate} \item $V \to X$ is projective, \item $V \to X$ is quasi-projective, \item there exists an ample invertible sheaf on $V$, \item there exists an $X$-ample invertible sheaf on $V$, \item there exists an $X$-very ample invertible sheaf on $V$. \end{enumerate} Then any descent datum on $V/X/Y$ is effective.


local-cohomology-lemma-kollar-finiteness-H1-local

Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Set $Z = V(I)$. Let $M$ be a finite $A$-module. The following are equivalent \begin{enumerate} \item $H^1_Z(M)$ is a finite $A$-module, and \item for all $\mathfrak p \in \text{Ass}(M)$, $\mathfrak p \not \in Z$ and all $\mathfrak q \in V(\mathfrak p + I)$ the completion of $(A/\mathfrak p)_\mathfrak q$ does not have associated primes of dimension $1$. \end{enumerate}


local-cohomology-lemma-ideal-depth-function

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $\mathfrak p \in V(I)$ be a prime ideal. Assume $e = \text{depth}_{IA_\mathfrak p}(M_\mathfrak p) < \infty$. Then there exists a nonempty open $U \subset V(\mathfrak p)$ such that $\text{depth}_{IA_\mathfrak q}(M_\mathfrak q) \geq e$ for all $\mathfrak q \in U$.


local-cohomology-lemma-finite-nr-points-next-S

Let $X$ be a Noetherian scheme with dualizing complex $\omega_X^\bullet$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Let $k \geq 0$ be an integer. Assume $\mathcal{F}$ is $(S_k)$. Then there is a finite number of points $x \in X$ such that $$ \text{depth}(\mathcal{F}_x) = k \quad\text{and}\quad \dim(\text{Supp}(\mathcal{F}_x)) > k $$


local-cohomology-lemma-make-S2-along-Z

Let $X$ be a Noetherian scheme. Let $Z \subset X$ be a closed subscheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Assume $X$ is universally catenary and the formal fibres of local rings have $(S_1)$. Then there exists a unique map $\mathcal{F} \to \mathcal{F}''$ of coherent $\mathcal{O}_X$-modules such that \begin{enumerate} \item $\mathcal{F}_x \to \mathcal{F}''_x$ is an isomorphism for $x \in X \setminus Z$, \item $\mathcal{F}_x \to \mathcal{F}''_x$ is surjective and $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}''_x) = 1$ for $x \in Z$ such that there exists an immediate specialization $x' \leadsto x$ with $x' \not \in Z$ and $x' \in \text{Ass}(\mathcal{F})$, \item $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}''_x) \geq 2$ for the remaining $x \in Z$. \end{enumerate} If $f : Y \to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian, then $f^*\mathcal{F} \to f^*\mathcal{F}''$ satisfies the same properties with respect to $f^{-1}(Z) \subset Y$.


local-cohomology-lemma-frobenius

Let $p$ be a prime number. Let $A$ be a ring with $p = 0$. Denote $F : A \to A$, $a \mapsto a^p$ the Frobenius endomorphism. Let $I \subset A$ be a finitely generated ideal. Set $Z = V(I)$. There exists an isomorphism $R\Gamma_Z(A) \otimes_{A, F}^\mathbf{L} A \cong R\Gamma_Z(A)$.


local-cohomology-lemma-etale-frobenius

Let $p$ be a prime number. Let $A$ be a ring with $p = 0$. Denote $F : A \to A$, $a \mapsto a^p$ the Frobenius endomorphism. If $V \to \Spec(A)$ is quasi-compact, quasi-separated, and \'etale, then there exists an isomorphism $R\Gamma(V, \mathcal{O}_V) \otimes_{A, F}^\mathbf{L} A \cong R\Gamma(V, \mathcal{O}_V)$.


local-cohomology-proposition-finiteness-complex

Let $A$ be a Noetherian ring which has a dualizing complex. Let $T \subset \Spec(A)$ be a subset stable under specialization. Let $s \in \mathbf{Z}$. Let $K \in D_{\textit{Coh}}^+(A)$. The following are equivalent \begin{enumerate} \item $H^i_T(K)$ is a finite $A$-module for $i \leq s$, and \item for all $\mathfrak p \not \in T$, $\mathfrak q \in T$ with $\mathfrak p \subset \mathfrak q$ we have $$ \text{depth}_{A_\mathfrak p}(K_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > s $$ \end{enumerate}


more-algebra-lemma-lift-factorization

Let $A$ be a ring, let $I \subset A$ be an ideal. Let $f \in A[x]$ be a monic polynomial. Let $\overline{f} = \overline{g} \overline{h}$ be a factorization of $f$ in $A/I[x]$ and assume that  $\overline{g}$, $\overline{h}$ generate the unit ideal in $A/I[x]$. Then there exists an \'etale ring map $A \to A'$ which induces an isomorphism $A/I \to A'/IA'$ and a factorization $f = g' h'$ in $A'[x]$ lifting the given factorization over $A/I$.


more-algebra-lemma-henselization-local-ring

\begin{slogan} Compatibility henselization of pairs and of local rings. \end{slogan} The functor of Lemma \ref{lemma-henselization} associates to a local ring $(A, \mathfrak m)$ its henselization.


more-algebra-lemma-flat-descent-flat-at-primes-over

Let $R \to S$ be a ring map. Let $I \subset R$ be an ideal. Let $M$ be an $S$-module. Let $R \to R'$ be a ring map and $IR' \subset I' \subset R'$ an ideal such that \begin{enumerate} \item the map $V(I') \to V(I)$ induced by $\Spec(R') \to \Spec(R)$ is surjective, and \item $R'_{\mathfrak p'}$ is flat over $R$ for all primes $\mathfrak p' \in V(I')$. \end{enumerate} If (\ref{equation-flat-at-primes-over}) holds for $(R' \to S \otimes_R R', I', M \otimes_R R')$, then (\ref{equation-flat-at-primes-over}) holds for $(R \to S, I, M)$.


more-algebra-lemma-base-change-flat-at-primes

In Situation \ref{situation-flattening-general} let $R' \to R''$ be an $R$-algebra map. Let $I' \subset R'$ and $I'R'' \subset I'' \subset R''$ be ideals. If (\ref{equation-flat-at-primes}) holds for $(R', I')$, then (\ref{equation-flat-at-primes}) holds for $(R'', I'')$.


more-algebra-lemma-flat-module-powers-variant

In Situation \ref{situation-flattening-general}. Let $I \subset R$ be an ideal. Assume \begin{enumerate} \item $R$ is a Noetherian ring, \item $S$ is a Noetherian ring, \item $M$ is a finite $S$-module, and \item for each $n \geq 1$ and any prime $\mathfrak q \in V(J + IS)$ the module $(M/I^n M)_{\mathfrak q}$ is flat over $R/I^n$. \end{enumerate} Then (\ref{equation-flat-at-primes}) holds for $(R, I)$, i.e., for every prime $\mathfrak q \in V(J + IS)$ the localization $M_{\mathfrak q}$ is flat over $R$.


more-algebra-lemma-descent-flatness-injective-integral

Let $R \to S$ be an injective integral ring map. Let $M$ be a finitely presented module over $R[x_1, \ldots, x_n]$. If $M \otimes_R S$ is flat over $S$, then $M$ is flat over $R$.


more-algebra-lemma-regular-field-extension

Let $k \subset K$ be a field extension. Then $k \to K$ is a regular ring map if and only if $K$ is a separable field extension of $k$.


more-algebra-lemma-completion-normal

Let $A$ be a Noetherian local ring. If $A^\wedge$ is normal, then so is $A$.


more-algebra-lemma-check-G-ring

Let $R$ be a Noetherian ring. Then $R$ is a G-ring if and only if for every finite free ring map $R \to S$ the formal fibres of $S$ are regular rings.


more-algebra-lemma-henselian-local-limit-G-rings

Let $(A, \mathfrak m)$ be a henselian local ring. Then $A$ is a filtered colimit of a system of henselian local G-rings with local transition maps.


more-algebra-lemma-formal-fibres-Rk

Fix $n \geq 0$. Properties (A), (B), (C), (D), and (E) hold for $P(k \to R) =$``$R \otimes_k k'$ has $(R_n)$ for all finite extensions $k'/k$''.


more-algebra-lemma-JM-injective

Let $R$ be a ring. For every $R$-module $M$ the $R$-module $J(M)$ is injective.


more-algebra-lemma-colim-and-lim-of-duals

\begin{slogan} Trivial duality for systems of perfect objects. \end{slogan} Let $A$ be a ring. Let $(K_n)_{n \in \mathbf{N}}$ be a system of perfect objects of $D(A)$. Let $K = \text{hocolim} K_n$ be the derived colimit (Derived Categories, Definition \ref{derived-definition-derived-colimit}). Then for any object $E$ of $D(A)$ we have $$ R\Hom_A(K, E) = R\lim E \otimes^\mathbf{L}_A K_n^\vee $$ where $(K_n^\vee)$ is the inverse system of dual perfect complexes.


more-algebra-lemma-isolate-a-cohomology-group

Let $R$ be a ring. Let $\mathfrak p \subset R$ be a prime ideal. Let $K^\bullet$ be a pseudo-coherent complex of $R$-modules. Assume that for some $i \in \mathbf{Z}$ the maps $$ H^i(K^\bullet) \otimes_R \kappa(\mathfrak p) \longrightarrow H^i(K^\bullet \otimes_R^{\mathbf{L}} \kappa(\mathfrak p)) \quad\text{and}\quad H^{i - 1}(K^\bullet) \otimes_R \kappa(\mathfrak p) \longrightarrow H^{i - 1}(K^\bullet \otimes_R^{\mathbf{L}} \kappa(\mathfrak p)) $$ are surjective. Then there exists an $f \in R$, $f \not \in \mathfrak p$ such that \begin{enumerate} \item $\tau_{\geq i + 1}(K^\bullet \otimes_R R_f)$ is a perfect object of $D(R_f)$ with tor amplitude in $[i + 1, \infty]$, \item $H^i(K^\bullet)_f$ is a finite free $R_f$-module, and \item there is a canonical direct sum decomposition $$ K^\bullet \otimes_R R_f \cong \tau_{\leq i - 1}(K^\bullet \otimes_R R_f) \oplus H^i(K^\bullet)_f[-i] \oplus \tau_{\geq i + 1}(K^\bullet \otimes_R R_f) $$ in $D(R_f)$. \end{enumerate}


more-algebra-lemma-split-using-ext-zero

Let $R$ be a ring. Let $K \in D^-(R)$. Let $a \in \mathbf{Z}$. Assume $\Ext^{-a}_R(K, M) = 0$ for any $R$-module $M$. Then there is a unique direct sum decomposition $K \cong \tau_{\leq a - 1}K \oplus \tau_{\geq a + 1}K$ and $\tau_{\geq a + 1}K$ has projective-amplitude in $[a + 1, b]$ for some $b$.


more-algebra-lemma-complex-relative-pseudo-coherent-modules

Let $R \to A$ be a finite type ring map. Let $m \in \mathbf{Z}$. Let $K^\bullet$ be a bounded above complex of $A$-modules such that $K^i$ is $(m - i)$-pseudo-coherent relative to $R$ for all $i$. Then $K^\bullet$ is $m$-pseudo-coherent relative to $R$. In particular, if $K^\bullet$ is a bounded above complex of $A$-modules pseudo-coherent relative to $R$, then $K^\bullet$ is pseudo-coherent relative to $R$.


more-algebra-lemma-cohomology-relative-pseudo-coherent

Let $R \to A$ be a finite type ring map. Let $m \in \mathbf{Z}$. Let $K^\bullet \in D^{-}(A)$ such that $H^i(K^\bullet)$ is $(m - i)$-pseudo-coherent (resp.\ pseudo-coherent) relative to $R$ for all $i$. Then $K^\bullet$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) relative to $R$.


more-algebra-lemma-map-from-hocolim

Let $\mathcal{D}$ be a triangulated category. Let $(K_n)$ be a system of objects of $\mathcal{D}$. Let $K$ be a derived colimit of the system $(K_n)$. Then for every $L$ in $\mathcal{D}$ we have a short exact sequence $$ 0 \to R^1\lim \Hom_\mathcal{D}(K_n, L[-1]) \to \Hom_\mathcal{D}(K, L) \to \lim \Hom_\mathcal{D}(K_n, L) \to 0 $$


more-algebra-lemma-cover-spec

Let $R$ be a ring, let $f \in R$, and let $R \to R'$ be a ring map which induces an isomorphism $R/fR \to R'/fR'$. The map $\Spec(R') \amalg \Spec(R_f) \to \Spec(R)$ is surjective. For example, the map $\Spec(R^\wedge) \amalg \Spec(R_f) \to \Spec(R)$ is surjective.


more-algebra-lemma-derived-complete-zero-bis

\begin{slogan} Derived Nakayama \end{slogan} \begin{reference} A related result is \cite[Proposition 6.5]{Dwyer-Greenlees}. The derived Nakayama lemma can for example be found in Bhatt's 3rd lecture on Prismatic cohomology at Columbia University in Fall 2018 as Section 2 property (2). Leonid Positselski proposed a proof in \url{https://mathoverflow.net/a/331501}. However, we follow the proof suggested by Anonymous in the comments. \end{reference} Let $I$ be a finitely generated ideal of a ring $A$. Let $K$ be a derived complete object of $D(A)$. If $K \otimes_A^\mathbf{L} A/I = 0$, then $K = 0$.


more-algebra-lemma-derived-completion-spectral-sequence

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Let $K^\bullet$ be a filtered complex of $A$-modules. There exists a canonical spectral sequence $(E_r, \text{d}_r)_{r \geq 1}$ of bigraded derived complete $A$-modules with $d_r$ of bidegree $(r, -r + 1)$ and with $$ E_1^{p, q} = H^{p + q}((\text{gr}^pK^\bullet)^\wedge) $$ If the filtration on each $K^n$ is finite, then the spectral sequence is bounded and converges to $H^*((K^\bullet)^\wedge)$.


more-algebra-lemma-restriction-derived-complete-equivalence

Let $A \to B$ be a ring map. Let $I \subset A$ be a finitely generated ideal. If $A \to B$ is flat and $A/I \cong B/IB$, then the restriction functor $D(B) \to D(A)$ induces an equivalence $D_{comp}(B, IB) \to D_{comp}(A, I)$.


more-algebra-lemma-when-does-it-work

Let $A$ be a ring and $f \in A$. Set $I = (f)$. In this situation we have the naive derived completion $K \mapsto K' = R\lim (K \otimes_A^\mathbf{L} A/f^nA)$ and the derived completion $$ K \mapsto K^\wedge = R\lim (K \otimes_A^\mathbf{L} (A \xrightarrow{f^n} A)) $$ of Lemma \ref{lemma-derived-completion-koszul}. The natural transformation of functors $K^\wedge \to K'$ is an isomorphism if and only if the $f$-power torsion of $A$ is bounded.


more-algebra-lemma-torsion-and-derived-complete

Let $I$ be a finitely generated ideal in a ring $A$. Let $M$ be a derived complete $A$-module. If $M$ is an $I$-power torsion module, then $I^nM = 0$ for some $n$.


more-algebra-lemma-derived-complete-finite

Let $I$ be an ideal of a Noetherian ring $A$. Let $M$ be a derived complete $A$-module. If $M/IM$ is a finite $A/I$-module, then $M = \lim M/I^nM$ and $M$ is a finite $A^\wedge$-module.


more-algebra-lemma-derived-completion-tensor-finite

Let $I$ be an ideal in a Noetherian ring $A$. Let ${}^\wedge$ denote derived completion with respect to $I$. Let $K \in D^-(A)$. \begin{enumerate} \item If $M$ is a finite $A$-module, then $(K \otimes_A^\mathbf{L} M)^\wedge = K^\wedge \otimes_A^\mathbf{L} M$. \item If $L \in D(A)$ is pseudo-coherent, then $(K \otimes_A^\mathbf{L} L)^\wedge = K^\wedge \otimes_A^\mathbf{L} L$. \end{enumerate}


more-algebra-lemma-Rlim-gives-complete

Let $A$ be a ring and $I \subset A$ an ideal. Suppose given $K_n \in D(A/I^n)$ and maps $K_{n + 1} \to K_n$ in $D(A/I^{n + 1})$. If \begin{enumerate} \item $A$ is Noetherian, \item $K_1$ is bounded above, and \item the maps induce isomorphisms $K_{n + 1} \otimes_{A/I^{n + 1}}^\mathbf{L} A/I^n \to K_n$, \end{enumerate} then $K = R\lim K_n$ is a derived complete object of $D^-(A)$ and $K \otimes_A^\mathbf{L} A/I^n \to K_n$ is an isomorphism for all $n$.


more-algebra-lemma-completion-disconnected

Let $(A, \mathfrak m)$ be a Noetherian local ring. The punctured spectrum of $A^\wedge$ is disconnected if and only if the punctured spectrum of $A^h$ is disconnected.


more-algebra-lemma-geometrically-normal-fibres-universally-catenary

\begin{reference} \cite[Corollary 2.3]{Heinzer-Rotthaus-Wiegand} \end{reference} Let $(A, \mathfrak m)$ be a Noetherian local ring with geometrically normal formal fibres. Then \begin{enumerate} \item $A^h$ is universally catenary, and \item if $A$ is unibranch (for example normal), then $A$ is universally catenary. \end{enumerate}


more-algebra-lemma-composition-unramified

Let $A$ be a discrete valuation ring with fraction field $K$. Let $M/L/K$ be finite separable extensions. Let $B$ be the integral closure of $A$ in $L$. If $L/K$ is unramified with respect to $A$ and $M/L$ is unramified with respect to $B_\mathfrak m$ for every maximal ideal $\mathfrak m$ of $B$, then $M/K$ is unramified with respect to $A$.


more-algebra-lemma-unimodular-vector

Let $R$ be a B\'ezout domain. Let $n \geq 1$ and $f_1, \ldots, f_n \in R$ generate the unit ideal. There exists an invertible $n \times n$ matrix in $R$ whose first row is $f_1 \ldots f_n$.


more-algebra-lemma-symmetric-monoidal-derived

Let $R$ be a ring. The derived category $D(R)$ of $R$ is a symmetric monoidal category with tensor product given by derived tensor product and associativity and commutativity constraints as in Section \ref{section-sign-rules}.


more-algebra-proposition-regular-strong-generator

Let $R$ be a Noetherian ring. The following are equivalent \begin{enumerate} \item $R$ is regular of finite dimension, \item $D_{perf}(R)$ has a strong generator, and \item $R$ is a strong generator for $D_{perf}(R)$. \end{enumerate}


more-algebra-proposition-propdimd

\begin{reference} \cite[Lemma 3.14]{Artin-Lipman} has this result without the assumption that the ring is catenary \end{reference} Let $R$ be a catenary Noetherian local normal domain. Let $J \subset R$ be a radical ideal. Then there exists a nonzero element $f \in J$ such that $R/fR$ is reduced.


etale-theorem-flatness-grothendieck

Let $A$, $B$ be Noetherian local rings. Let $f : A \to B$ be a local homomorphism. If $M$ is a finite $B$-module that is flat as an $A$-module, and $t \in \mathfrak m_B$ is an element such that multiplication by $t$ is injective on $M/\mathfrak m_AM$, then $M/tM$ is also $A$-flat.


etale-theorem-flat-open

Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be a morphism which is locally of finite type. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. The set of points in $X$ where $\mathcal{F}$ is flat over $Y$ is an open set. In particular the set of points where $f$ is flat is open in $X$.


etale-theorem-structure-etale

Let $f : A \to B$ be an \'etale homomorphism of local rings. Then there exist $f, g \in A[t]$ such that \begin{enumerate} \item $B' = A[t]_g/(f)$ is standard \'etale -- see (a) and (b) above, and \item $B$ is isomorphic to a localization of $B'$ at a prime. \end{enumerate}


etale-theorem-geometric-structure

Let $\varphi : X \to Y$ be a morphism of schemes. Let $x \in X$. Let $V \subset Y$ be an affine open neighbourhood of $\varphi(x)$. If $\varphi$ is \'etale at $x$, then there exist exists an affine open $U \subset X$ with $x \in U$ and $\varphi(U) \subset V$ such that we have the following diagram $$ \xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-j & \Spec(R[t]_{f'}/(f)) \ar[d] \\ Y & V \ar[l] \ar@{=}[r] & \Spec(R) } $$ where $j$ is an open immersion, and $f \in R[t]$ is monic.


etale-lemma-smooth-pullback-normal-crossings

\begin{slogan} Pullback of a normal crossings divisor by a smooth morphism is a normal crossings divisor. \end{slogan} Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be a normal crossings divisor. If $f : Y \to X$ is a smooth morphism of schemes, then the pullback $f^*D$ is a normal crossings divisor on $Y$.


etale-proposition-etale-normal

Let $A$, $B$ be Noetherian local rings. Let $f : A \to B$ be an \'etale homomorphism of local rings. Then $A$ is a normal domain if and only if $B$ is so.


crystalline-theorem-cohomology-F-crystal

In Situation \ref{situation-F-crystal} let $(\mathcal{E}, F_\mathcal{E})$ be a nondegenerate $F$-crystal. Assume $A$ is a $p$-adically complete Noetherian ring and that $X \to S_0$ is proper smooth. Then the canonical map $$ F_\mathcal{E} \circ (F_X)_{\text{cris}}^* : R\Gamma(\text{Cris}(X/S), \mathcal{E}) \otimes^\mathbf{L}_{A, \sigma} A \longrightarrow R\Gamma(\text{Cris}(X/S), \mathcal{E}) $$ becomes an isomorphism after inverting $p$.


crystalline-lemma-divided-power-envelop-quotient

Let $(A, I, \gamma)$ be a divided power ring. Let $\varphi : B' \to B$ be a surjection of $A$-algebras with kernel $K$. Let $IB \subset J \subset B$ be an ideal. Let $J' \subset B'$ be the inverse image of $J$. Write $D_{B', \gamma}(J') = (D', \bar J', \bar\gamma)$. Then $D_{B, \gamma}(J) = (D'/K', \bar J'/K', \bar\gamma)$ where $K'$ is the ideal generated by the elements $\bar\gamma_n(k)$ for $n \geq 1$ and $k \in K$.


crystalline-lemma-differentials-completion

Let $A \to B$ be a ring map and let $(J, \delta)$ be a divided power structure on $B$. Let $p$ be a prime number. Assume that $A$ is a $\mathbf{Z}_{(p)}$-algebra and that $p$ is nilpotent in $B/J$. Then we have $$ \lim_e \Omega_{B_e/A, \bar\delta} = \lim_e \Omega_{B/A, \delta}/p^e\Omega_{B/A, \delta} = \lim_e \Omega_{B^\wedge/A, \delta^\wedge}/p^e \Omega_{B^\wedge/A, \delta^\wedge} $$ see proof for notation and explanation.


crystalline-lemma-divided-power-thickening-base-change-flat

In Situation \ref{situation-global}. Let $$ \xymatrix{ (U_3, T_3, \delta_3) \ar[d] \ar[r] & (U_2, T_2, \delta_2) \ar[d] \\ (U_1, T_1, \delta_1) \ar[r] & (U, T, \delta) } $$ be a fibre square in the category of divided power thickenings of $X$ relative to $(S, \mathcal{I}, \gamma)$. If $T_2 \to T$ is flat and $U_2 = T_2 \times_T U$, then $T_3 = T_1 \times_T T_2$ (as schemes).


spaces-pushouts-lemma-glue-etale-sheaf-fppf

Let $S$ be a scheme. Let $\{f_i : X_i \to X\}$ be an fppf covering of algebraic spaces over $S$. The functor $$ \Sh(X_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{f_i : X_i \to X\} $$ is an equivalence of categories.


spaces-pushouts-lemma-pushout-along-closed-immersion-and-integral

In More on Morphisms, Situation \ref{more-morphisms-situation-pushout-along-closed-immersion-and-integral} let $Y \amalg_Z X$ be the pushout in the category of schemes (More on Morphisms, Proposition \ref{more-morphisms-proposition-pushout-along-closed-immersion-and-integral}). Then $Y \amalg_Z X$ is also a pushout in the category of algebraic spaces over $S$.


spaces-pushouts-lemma-fully-faithful-on-separated

In Situation \ref{situation-formal-glueing} the functor (\ref{equation-formal-glueing-spaces}) is fully faithful on algebraic spaces separated over $X$. More precisely, it induces a bijection $$ \Mor_X(X'_1, X'_2) \longrightarrow \Mor_{\textit{Spaces}(Y \to X, Z)}(F(X'_1), F(X'_2)) $$ whenever $X'_2 \to X$ is separated.


spaces-pushouts-lemma-glueing-ff

Let $(R \to R', f)$ be a glueing pair, see above. The functor (\ref{equation-beauville-laszlo-glueing-spaces}) is fully faithful on the full subcategory of algebraic spaces $Y/X$ which are (a) glueable for $(R \to R', f)$ and (b) have affine diagonal $Y \to Y \times_X Y$.


spaces-pushouts-lemma-glueing-quasi-affines

Let $(R \to R', f)$ be a glueing pair, see above. Any object $(V, V', Y')$ of $\textit{Spaces}(U \leftarrow U' \to X')$ with $V$, $V'$, $Y'$ quasi-affine is isomorphic to the image under the functor (\ref{equation-beauville-laszlo-glueing-spaces}) of a separated algebraic space $Y$ over $X$.


varieties-lemma-birational-varieties

Let $X$ and $Y$ be varieties over a field $k$. The following are equivalent \begin{enumerate} \item $X$ and $Y$ are birational varieties, \item the function fields $k(X)$ and $k(Y)$ are isomorphic, \item there exist nonempty opens of $X$ and $Y$ which are isomorphic as varieties, \item there exists an open $U \subset X$ and a birational morphism $U \to Y$ of varieties. \end{enumerate}


varieties-lemma-Noetherian-geometrically-reduced-at-point

Let $k$ be a field. Let $X$ be a scheme over $k$. Let $x \in X$. Assume $X$ locally Noetherian and geometrically reduced at $x$. Then there exists an open neighbourhood $U \subset X$ of $x$ which is geometrically reduced over $k$.


varieties-lemma-galois-action-connected-components-continuous

Let $k$ be a field, with separable algebraic closure $\overline{k}$. Let $X$ be a scheme over $k$. Assume \begin{enumerate} \item $X$ is quasi-compact, and \item the connected components of $X_{\overline{k}}$ are open. \end{enumerate} Then \begin{enumerate} \item[(a)] $\pi_0(X_{\overline{k}})$ is finite, and \item[(b)] the action of $\text{Gal}(\overline{k}/k)$ on $\pi_0(X_{\overline{k}})$ is continuous. \end{enumerate} Moreover, assumptions (1) and (2) are satisfied when $X$ is of finite type over $k$.


varieties-lemma-irreducible-dense-rational-points-geometrically-irreducible

Let $k$ be a field. Let $X$ be a scheme over $k$. If $X$ is irreducible and has a dense set of $k$-rational points, then $X$ is geometrically irreducible.


varieties-lemma-projective-after-field-extension

Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_K$ is projective over $K$ for some field extension $k \subset K$, then $X$ is projective over $k$.


varieties-lemma-tangent-space

The set of dotted arrows making (\ref{equation-tangent-space}) commute has a canonical $\kappa(x)$-vector space structure.


varieties-lemma-noether-normalization

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ with image $s \in S$. Let $V \subset S$ be an affine open neighbourhood of $s$. If $f$ is locally of finite type and $\dim_x(X_s) = d$, then there exists an affine open $U \subset X$ with $x \in U$ and $f(U) \subset V$ and a factorization $$ U \xrightarrow{\pi} \mathbf{A}^d_V \to V $$ of $f|_U : U \to V$ such that $\pi$ is quasi-finite.


varieties-lemma-immersion-into-affine

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ with image $s \in S$. Let $V \subset S$ be an affine open neighbourhood of $s$. If $f$ is locally of finite type and $$ r = \dim_{\kappa(x)} \Omega_{X/S, x} \otimes_{\mathcal{O}_{X, x}} \kappa(x) = \dim_{\kappa(x)} \Omega_{X_s/s, x} \otimes_{\mathcal{O}_{X_s, x}} \kappa(x) = \dim_{\kappa(x)} T_{X/S, x} $$ then there exist \begin{enumerate} \item an affine open $U \subset X$ with $x \in U$ and $f(U) \subset V$ and a factorization $$ U \xrightarrow{j} \mathbf{A}^{r + 1}_V \to V $$ of $f|_U$ such that $j$ is an immersion, or \item an affine open $U \subset X$ with $x \in U$ and $f(U) \subset V$ and a factorization $$ U \xrightarrow{j} D \to V $$ of $f|_U$ such that $j$ is a closed immersion and $D \to V$ is smooth of relative dimension $r$. \end{enumerate}


varieties-lemma-generated-by-dim-plus-1-sections

Let $k$ be an infinite field. Let $X$ be an algebraic $k$-scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $V \to \Gamma(X, \mathcal{L})$ be a linear map of $k$-vector spaces whose image generates $\mathcal{L}$. Then there exists a subspace $W \subset V$ with $\dim_k(W) \leq \dim(X) + 1$ which generates $\mathcal{L}$.


varieties-lemma-units-variety

Let $k$ be a field. Let $X$ be a variety over $k$. The group $\mathcal{O}(X)^*/k^*$ is a finitely generated abelian group provided at least one of the following conditions holds: \begin{enumerate} \item $k$ is integrally closed in $\Gamma(X, \mathcal{O}_X)$, \item $k$ is algebraically closed in $k(X)$, \item $X$ is geometrically integral over $k$, or \item $k$ is the ``intersection'' of the field extensions $k \subset \kappa(x)$ where $x$ runs over the closed points of $x$. \end{enumerate}


varieties-lemma-hilbert-polynomial-H0

Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^n_k$ with Hilbert polynomial $P \in \mathbf{Q}[t]$. Then $$ P(d) = \dim_k H^0(\mathbf{P}^n_k, \mathcal{F}(d)) $$ for all $d \gg 0$.


varieties-lemma-glue-valuation-ring

In Situation \ref{situation-glue} assume that $B$ is a valuation ring. Then for every unit $u$ of $A$ either $u \in R$ or $u^{-1} \in R$.


varieties-lemma-localization-semi-local

Let $A$ be a ring. Let $\mathfrak p_1, \ldots, \mathfrak p_r$ be a finite set of a primes of $A$. Let $S = A \setminus \bigcup \mathfrak p_i$. Then $S$ is a multiplicative system and $S^{-1}A$ is a semi-local ring whose maximal ideals correspond to the maximal elements of the set $\{\mathfrak p_i\}$.


varieties-lemma-complement-codim-1-closed-points

Let $X$ be a scheme. Let $U \subset X$ be an open. Assume \begin{enumerate} \item $U$ is a retrocompact open of $X$, \item $X \setminus U$ is discrete, and \item for $x \in X \setminus U$ the local ring $\mathcal{O}_{X, x}$ is Noetherian of dimension $\leq 1$. \end{enumerate} Then (1) there exists an invertible $\mathcal{O}_X$-module $\mathcal{L}$ and a section $s$ such that $U = X_s$ and (2) the map $\Pic(X) \to \Pic(U)$ is surjective.


varieties-lemma-surjection-on-pic-quasi-finite

Let $f : X \to Y$ be a morphism of schemes. Assume $Y$ is Noetherian of dimension $\leq 1$, $f$ is finite, and there exists a dense open $V \subset Y$ such that $f^{-1}(V) \to V$ is a closed immersion. Then every invertible $\mathcal{O}_X$-module is the pullback of an invertible $\mathcal{O}_Y$-module.


varieties-lemma-dim-1-nonproper-affine

Let $X$ be a separated scheme of finite type over $k$. If $\dim(X) \leq 1$ and no irreducible component of $X$ is proper of dimension $1$, then $X$ is affine.


cotangent-lemma-mod-regular-sequence

Let $A \to B$ be a surjective ring map whose kernel $I$ is generated by a Koszul-regular sequence (for example a regular sequence). Then $L_{B/A}$ is quasi-isomorphic to $I/I^2[1]$.


cotangent-lemma-find-obstruction

In the situation above we have \begin{enumerate} \item There is a canonical element $\xi \in \Ext^2_B(L_{B/A}, N)$ whose vanishing is a sufficient and necessary condition for the existence of a solution to (\ref{equation-to-solve}). \item If there exists a solution, then the set of isomorphism classes of solutions is principal homogeneous under $\Ext^1_B(L_{B/A}, N)$. \item Given a solution $B'$, the set of automorphisms of $B'$ fitting into (\ref{equation-to-solve}) is canonically isomorphic to $\Ext^0_B(L_{B/A}, N)$. \end{enumerate}


cotangent-lemma-find-obstruction-ringed-spaces

In the situation above we have \begin{enumerate} \item There is a canonical element $\xi \in \Ext^2_{\mathcal{O}_X}(L_{X/S}, \mathcal{G})$ whose vanishing is a sufficient and necessary condition for the existence of a solution to (\ref{equation-to-solve-ringed-spaces}). \item If there exists a solution, then the set of isomorphism classes of solutions is principal homogeneous under $\Ext^1_{\mathcal{O}_X}(L_{X/S}, \mathcal{G})$. \item Given a solution $X'$, the set of automorphisms of $X'$ fitting into (\ref{equation-to-solve-ringed-spaces}) is canonically isomorphic to $\Ext^0_{\mathcal{O}_X}(L_{X/S}, \mathcal{G})$. \end{enumerate}


cotangent-lemma-cotangent-morphism-schemes

In the situation above there is a canonical isomorphism $$ L_{X/\Lambda} =  L\pi_!(Li^*\Omega_{\mathcal{O}/\underline{\Lambda}}) = L\pi_!(i^*\Omega_{\mathcal{O}/\underline{\Lambda}}) = L\pi_!(\Omega_{\mathcal{O}/\underline{\Lambda}} \otimes_\mathcal{O} \underline{\mathcal{O}}_X) $$ in $D(\mathcal{O}_X)$.


cotangent-proposition-cotangent-complex-local-complete-intersection

Let $A \to B$ be a local complete intersection map. Then $L_{B/A}$ is a perfect complex with tor amplitude in $[-1, 0]$.


spaces-cohomology-lemma-local-isomorphism

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be a homomorphism of $\mathcal{O}_X$-modules. Let $\overline{x}$ be a geometric point of $X$ lying over $x \in |X|$. \begin{enumerate} \item If $\mathcal{F}_{\overline{x}} = 0$ then there exists an open neighbourhood $X' \subset X$ of $x$ such that $\mathcal{F}|_{X'} = 0$. \item If $\varphi_{\overline{x}} : \mathcal{G}_{\overline{x}} \to \mathcal{F}_{\overline{x}}$ is injective, then there exists an open neighbourhood $X' \subset X$ of $x$ such that $\varphi|_{X'}$ is injective. \item If $\varphi_{\overline{x}} : \mathcal{G}_{\overline{x}} \to \mathcal{F}_{\overline{x}}$ is surjective, then there exists an open neighbourhood $X' \subset X$ of $x$ such that $\varphi|_{X'}$ is surjective. \item If $\varphi_{\overline{x}} : \mathcal{G}_{\overline{x}} \to \mathcal{F}_{\overline{x}}$ is bijective, then there exists an open neighbourhood $X' \subset X$ of $x$ such that $\varphi|_{X'}$ is an isomorphism. \end{enumerate}


artin-lemma-predeformation-category

The functor $p : \mathcal{F} \to \mathcal{C}_\Lambda$ defined above is a predeformation category.


artin-lemma-versal-implies-smooth

Let $S$ be a locally Noetherian scheme. Let $f : U \to V$ be a morphism of schemes locally of finite type over $S$. Let $u_0 \in U$ be a finite type point. The following are equivalent \begin{enumerate} \item $f$ is smooth at $u_0$, \item $f$ viewed as an object of $(\Sch/V)_{fppf}$ over $U$ is versal at $u_0$. \end{enumerate}


artin-lemma-algebraic-stack-RS-star

Let $\mathcal{X}$ be an algebraic stack over a base $S$. Then $\mathcal{X}$ satisfies (RS*).


artin-proposition-second-diagonal-representable

Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Assume that \begin{enumerate} \item $\Delta_\Delta : \mathcal{X} \to \mathcal{X} \times_{\mathcal{X} \times \mathcal{X}} \mathcal{X}$ is representable by algebraic spaces, \item $\mathcal{X}$ satisfies axioms [-1], [0], [1], [2], [3], [4], and [5] (see Section \ref{section-axioms}), \item $\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$. \end{enumerate} Then $\mathcal{X}$ is an algebraic stack.


obsolete-lemma-p-ring-map

Let $\varphi : R \to S$ be a ring map. If \begin{enumerate} \item for any $x \in S$ there exists $n > 0$ such that $x^n$ is in the image of $\varphi$, and \item for any $x \in \Ker(\varphi)$ there exists $n > 0$ such that $x^n = 0$, \end{enumerate} then $\varphi$ induces a homeomorphism on spectra. Given a prime number $p$ such that \begin{enumerate} \item[(a)] $S$ is generated as an $R$-algebra by elements $x$ such that there exists an $n > 0$ with $x^{p^n} \in \varphi(R)$ and $p^nx \in \varphi(R)$, and \item[(b)] the kernel of $\varphi$ is generated by nilpotent elements, \end{enumerate} then (1) and (2) hold, and for any ring map $R \to R'$ the ring map $R' \to R' \otimes_R S$ also satisfies (a), (b), (1), and (2) and in particular induces a homeomorphism on spectra.


obsolete-lemma-spec-localization-first

Let $S$ be a multiplicative set of $A$. Then the map $$ f: \Spec(S^{-1}A)\longrightarrow \Spec(A) $$ induced by the canonical ring map $A \to S^{-1}A$ is a homeomorphism onto its image and $\Im(f) = \{ \mathfrak p \in \Spec(A) : \mathfrak p\cap S = \emptyset \}$.


obsolete-lemma-faithfully-flat-injective

If $R \to S$ is a faithfully flat ring map then for every $R$-module $M$ the map $M \to S \otimes_R M$, $x \mapsto 1 \otimes x$ is injective.


obsolete-lemma-algebraize-local-cohomology-bis-bis

Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. Assume \begin{enumerate} \item[(a)] $A$ has a dualizing complex, \item[(b)] $\text{cd}(A, I) \leq d$, \item[(c)] if $\mathfrak p \not \in V(I)$ then $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) > s$ or $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim(A/\mathfrak p) > d + s$. \end{enumerate} Then the assumptions of Algebraic and Formal Geometry, Lemma \ref{algebraization-lemma-algebraize-local-cohomology-bis} hold for $A, I, \mathfrak m, M$ and $H^i_\mathfrak m(M) \to \lim H^i_\mathfrak m(M/I^nM)$ is an isomorphism for $i \leq s$ and these modules are annihilated by a power of $I$.


obsolete-lemma-combine-one

In Algebraic and Formal Geometry, Situation \ref{algebraization-situation-bootstrap} we have $H^s_\mathfrak a(M) = \lim H^s_\mathfrak a(M/I^nM)$.


obsolete-lemma-formally-smooth-smooth

Let $R$ be a ring. Let $S$ be a $R$-algebra. If $S$ is of finite presentation and formally smooth over $R$ then $S$ is smooth over $R$.


obsolete-lemma-trivialities-cohomological-descent-abelian

In Simplicial Spaces, Situation \ref{spaces-simplicial-situation-simplicial-site} let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Simplicial Spaces, Remark \ref{spaces-simplicial-remark-augmentation-site}. Suppose given strictly full weak Serre subcategories $$ \mathcal{A} \subset \textit{Ab}(\mathcal{D}),\quad \mathcal{A}_n \subset \textit{Ab}(\mathcal{C}_n) $$ Then \begin{enumerate} \item[(1)] the collection of abelian sheaves $\mathcal{F}$ on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_n$ is in $\mathcal{A}_n$ for all $n$ is a strictly full weak Serre subcategory $\mathcal{A}_{total} \subset \textit{Ab}(\mathcal{C}_{total})$. \end{enumerate} If $a_n^{-1}$ sends $\mathcal{A}$ into $\mathcal{A}_n$ for all $n$, then \begin{enumerate} \item[(2)] $a^{-1}$ sends $\mathcal{A}$ into $\mathcal{A}_{total}$ and \item[(3)] $a^{-1}$ sends $D_\mathcal{A}(\mathcal{D})$ into $D_{\mathcal{A}_{total}}(\mathcal{C}_{total})$. \end{enumerate} If $R^qa_{n, *}$ sends $\mathcal{A}_n$ into $\mathcal{A}$ for all $n, q$, then \begin{enumerate} \item[(4)] $R^qa_*$ sends $\mathcal{A}_{total}$ into $\mathcal{A}$ for all $q$, and \item[(5)] $Ra_*$ sends $D_{\mathcal{A}_{total}}^+(\mathcal{C}_{total})$ into $D_\mathcal{A}^+(\mathcal{D})$. \end{enumerate}


obsolete-lemma-glue-f-upper-shriek

Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. There exists a unique functor $f^! : \textit{Ab}(Y_\etale) \to \textit{Ab}(X_\etale)$ such that \begin{enumerate} \item for any open $j : U \to X$ with $f \circ j$ separated there is a canonical isomorphism $j^! \circ f^! = (f \circ j)^!$, and \item these isomorphisms for $U \subset U' \subset X$ are compatible with the isomorphisms in More \'Etale Cohomology, Lemma \ref{more-etale-lemma-upper-shriek-restriction}. \end{enumerate}


obsolete-lemma-lqf-f-upper-shriek-stalk

Let $f : X \to Y$ be a morphism of schemes which is locally quasi-finite. For an abelian group $A$ and a geometric point $\overline{y} : \Spec(k) \to Y$ we have $f^!(\overline{y}_*A) = \prod\nolimits_{f(\overline{x}) = \overline{y}} \overline{x}_*A$.


obsolete-lemma-P-not-preserved-base-change

Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras. Let $N$ be a differential graded $(A, B)$-bimodule with property (P). Let $M$ be a differential graded $A$-module with property (P). Then $Q = M \otimes_A N$ is a differential graded $B$-module which represents $M \otimes_A^\mathbf{L} N$ in $D(B)$ and which has a filtration $$ 0 = F_{-1}Q \subset F_0Q \subset F_1Q \subset \ldots \subset Q $$ by differential graded submodules such that $Q = \bigcup F_pQ$, the inclusions $F_iQ \to F_{i + 1}Q$ are admissible monomorphisms, the quotients $F_{i + 1}Q/F_iQ$ are isomorphic as differential graded $B$-modules to a direct sum of $(A \otimes_R B)[k]$.


obsolete-lemma-stein-projective

Let $(R, \mathfrak m, \kappa)$ be a local ring. Let $X \subset \mathbf{P}^n_R$ be a closed subscheme. Assume that $R = \Gamma(X, \mathcal{O}_X)$. Then the special fibre $X_k$ is geometrically connected.