cohomology-lemma-K-flat-two-out-of-three-ses

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $0 \to \mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet \to \mathcal{K}_3^\bullet \to 0$ be a short exact sequence of complexes such that the terms of $\mathcal{K}_3^\bullet$ are flat $\mathcal{O}_X$-modules. If two out of three of $\mathcal{K}_i^\bullet$ are K-flat, so is the third.


cohomology-lemma-inverse-limit-complexes

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $(\mathcal{F}_n^\bullet)$ be an inverse system of complexes of $\mathcal{O}_X$-modules. Let $m \in \mathbf{Z}$. Assume there exist a set $\mathcal{B}$ of open subsets of $X$ and an integer $n_0$ such that \begin{enumerate} \item every open in $X$ has a covering whose members are elements of $\mathcal{B}$, \item for every $U \in \mathcal{B}$ \begin{enumerate} \item the systems of abelian groups $\mathcal{F}_n^{m - 2}(U)$ and $\mathcal{F}_n^{m - 1}(U)$ have vanishing $R^1\lim$ (for example these have the Mittag-Leffler condition), \item the system of abelian groups $H^{m - 1}(\mathcal{F}_n^\bullet(U))$ has vanishing $R^1\lim$ (for example it has the Mittag-Leffler condition), and \item we have $H^m(\mathcal{F}_n^\bullet(U)) = H^m(\mathcal{F}_{n_0}^\bullet(U))$ for all $n \geq n_0$. \end{enumerate} \end{enumerate} Then the maps $H^m(\mathcal{F}^\bullet) \to \lim H^m(\mathcal{F}_n^\bullet) \to H^m(\mathcal{F}_{n_0}^\bullet)$ are isomorphisms of sheaves where $\mathcal{F}^\bullet = \lim \mathcal{F}_n^\bullet$ is the termwise inverse limit.


cohomology-lemma-RHom-well-defined

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $(\mathcal{I}')^\bullet \to \mathcal{I}^\bullet$ be a quasi-isomorphism of K-injective complexes of $\mathcal{O}_X$-modules. Let $(\mathcal{L}')^\bullet \to \mathcal{L}^\bullet$ be a quasi-isomorphism of complexes of $\mathcal{O}_X$-modules. Then $$ \SheafHom^\bullet(\mathcal{L}^\bullet, (\mathcal{I}')^\bullet) \longrightarrow \SheafHom^\bullet((\mathcal{L}')^\bullet, \mathcal{I}^\bullet) $$ is a quasi-isomorphism.


cohomology-lemma-n-pseudo-module

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. \begin{enumerate} \item $\mathcal{F}$ viewed as an object of $D(\mathcal{O}_X)$ is $0$-pseudo-coherent if and only if $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module, and \item $\mathcal{F}$ viewed as an object of $D(\mathcal{O}_X)$ is $(-1)$-pseudo-coherent if and only if $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation. \end{enumerate}


cohomology-lemma-tensor-perfect

Let $(X, \mathcal{O}_X)$ be a ringed space. If $K, L$ are perfect objects of $D(\mathcal{O}_X)$, then so is $K \otimes_{\mathcal{O}_X}^\mathbf{L} L$.


cohomology-lemma-left-dual-complex

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}^\bullet$ be a complex of $\mathcal{O}_X$-modules. If $\mathcal{F}^\bullet$ has a left dual in the monoidal category of complexes of $\mathcal{O}_X$-modules (Categories, Definition \ref{categories-definition-dual}) then $\mathcal{F}^\bullet$ is a locally bounded complex whose terms are locally direct summands of finite free $\mathcal{O}_X$-modules and the left dual is as constructed in Example \ref{example-dual}.


cohomology-lemma-perfect-is-compact

Let $X$ be a ringed space. Assume that the underlying topological space of $X$ has the following properties: \begin{enumerate} \item $X$ is quasi-compact, \item there exists a basis of quasi-compact open subsets, and \item the intersection of any two quasi-compact opens is quasi-compact. \end{enumerate} Let $K$ be a perfect object of $D(\mathcal{O}_X)$. Then \begin{enumerate} \item[(a)] $K$ is a compact object of $D^+(\mathcal{O}_X)$ in the following sense: if $M = \bigoplus_{i \in I} M_i$ is bounded below, then $\Hom(K, M) = \bigoplus_{i \in I} \Hom(K, M_i)$. \item[(b)] If $X$ has finite cohomological dimension, i.e., if there exists a $d$ such that $H^i(X, \mathcal{F}) = 0$ for $i > d$, then $K$ is a compact object of $D(\mathcal{O}_X)$. \end{enumerate}


restricted-lemma-base-change-finite-type-red

Consider the property $P$ on arrows of $\textit{WAdm}^{count}$ defined in Lemma \ref{lemma-finite-type-red}. Then $P$ is stable under base change (Formal Spaces, Situation \ref{formal-spaces-situation-base-change-local-property}).


restricted-lemma-rig-closed-point-in-image

Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$ which is adic and topologically of finite type. Let $\mathfrak p \subset A$ be rig-closed. Let $\mathfrak a \subset A$ and $\mathfrak b \subset B$ be the ideals of topologically nilpotent elements. If $\varphi$ is flat, then the following are equivalent \begin{enumerate} \item the maximal ideal of $A/\mathfrak p$ is in the image of $\Spec(B/\mathfrak b) \to \Spec(A/\mathfrak a)$, \item there exists a rig-closed prime ideal $\mathfrak q \subset B$ such that $\mathfrak p = \varphi^{-1}(\mathfrak q)$. \end{enumerate} and if so then $\varphi$, $\mathfrak p$, and $\mathfrak q$ satisfy the conclusions of Lemma \ref{lemma-rig-closed-point-relative}.


restricted-lemma-rig-flat-morphisms

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is rig-flat, \item for every commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and \'etale, the morphism $U \to V$ corresponds to a rig-flat map in $\textit{WAdm}^{Noeth}$, \item there exists a covering $\{Y_j \to Y\}$ as in Formal Spaces, Definition \ref{formal-spaces-definition-formal-algebraic-space} and for each $j$ a covering $\{X_{ji} \to Y_j \times_Y X\}$ as in Formal Spaces, Definition \ref{formal-spaces-definition-formal-algebraic-space} such that each $X_{ji} \to Y_j$  corresponds to a rig-flat map in $\textit{WAdm}^{Noeth}$, and \item there exist a covering $\{X_i \to X\}$ as in Formal Spaces, Definition \ref{formal-spaces-definition-formal-algebraic-space} and for each $i$ a factorization $X_i \to Y_i \to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \to Y$ is representable by algebraic spaces and \'etale, and $X_i \to Y_i$ corresponds to a rig-flat map in $\textit{WAdm}^{Noeth}$. \end{enumerate}


restricted-lemma-base-change-rig-flat

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Z \to Y$ be morphisms of locally Noetherian formal algebraic spaces over $S$. If $f$ is rig-flat and $g$ is locally of finite type, then the base change $X \times_Y Z \to Z$ is rig-flat.


restricted-lemma-composition-rig-flat

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of locally Noetherian formal algebraic spaces over $S$. If $f$ and $g$ are rig-flat, then so is $g \circ f$.


restricted-lemma-rig-smooth-rig-flat-morphism

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. If $f$ is rig-smooth, then $f$ is rig-flat.


restricted-lemma-rig-etale-finite-type

A rig-\'etale morphism of locally Noetherian formal algebraic spaces is locally of finite type.


restricted-lemma-Noetherian-local-ring-properties

Notation and assumptions as in Lemma \ref{lemma-Noetherian-local-ring}. Let $f : X' \to \Spec(A)$ correspond to $g : Y' \to \Spec(B)$ via the equivalence. Then $f$ is quasi-compact, quasi-separated, separated, proper, finite, and add more here if and only if $g$ is so.


restricted-lemma-dominate-by-admissible-blowup

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $f : X \to S$ be an object of (\ref{equation-modification}). Then there exists a $U$-admissible blowup $S' \to S$ which dominates $X$.


more-groupoids-lemma-groupoid-on-field-translate-open

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. If $U$ is the spectrum of a field, $W \subset R$ is open, and $Z \to R$ is a morphism of schemes, then the image of the composition $Z \times_{s, U, t} W \to R \times_{s, U, t} R \to R$ is open.


more-groupoids-lemma-groupoid-characteristic-zero-smooth

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume \begin{enumerate} \item $U = \Spec(k)$ with $k$ a field, \item $s, t$ are locally of finite type, and \item the characteristic of $k$ is zero. \end{enumerate} Then $s, t : R \to U$ are smooth.


more-groupoids-lemma-reduced-group-scheme-perfect-field-characteristic-p-smooth

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume \begin{enumerate} \item $U = \Spec(k)$ with $k$ a field, \item $s, t$ are locally of finite type, \item $R$ is reduced, and \item $k$ is perfect. \end{enumerate} Then $s, t : R \to U$ are smooth.


more-groupoids-lemma-quasi-compact-groupoid-on-field-image

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U$ is the spectrum of a field and assume $R$ is quasi-compact (equivalently $s, t$ are quasi-compact). Let $Z \subset U \times_S U$ be the scheme theoretic image (see Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-image}) of $j = (t, s) : R \to U \times_S U$. Then $$ (U, Z, \text{pr}_0|_Z, \text{pr}_1|_Z, \text{pr}_{02}|_{Z \times_{\text{pr}_1, U, \text{pr}_0} Z}) $$ is a groupoid scheme over $S$.


examples-lemma-countable-coherent

Let $R$ be a countable ring. Then $R$ is coherent if and only if $R^\mathbf{N}$ is a flat $R$-module.


examples-lemma-almost-integral-when-powerseries-flat

Let $R$ be a domain with fraction field $K$. If $R[[x]]$ is flat over $R[x]$, then $R$ is normal if and only if $R$ is completely normal (Algebra, Definition \ref{algebra-definition-almost-integral}).


examples-lemma-completion-polynomial-ring-not-flat-bis

If $R$ is a valuation ring of dimension $> 1$, then $R[[x]]$ is flat over $R$ but not flat over $R[x]$.


examples-lemma-ideal-generated-by-idempotents-projective

Let $R$ be a ring. Let $I \subset R$ be an ideal generated by a countable collection of idempotents. Then $I$ is projective as an $R$-module.


examples-lemma-excellent-regular-local-rings

Let $(A, \mathfrak m, \kappa)$ be a regular local ring of characteristic $p > 0$. Suppose $[\kappa : \kappa^p] < \infty$. Then $A$ is excellent if and only if $A \to A^\wedge$ is formally \'etale.


examples-lemma-sum-is-product

Let $\mathcal{X}$ be an algebraic stack. If $\mathcal{F}_n$ is a collection of locally quasi-coherent sheaves with the flat base change property on $\mathcal{X}$, then $\oplus_n \mathcal{F}_n[n] \to \prod_n \mathcal{F}_n[n]$ is an isomorphism in $D(\mathcal{O}_\mathcal{X})$.


examples-lemma-nonadditivity-of-trace

There exists a ring $R$, a distinguished triangle $(K, L, M, \alpha, \beta, \gamma)$ in the homotopy category $K(R)$, and an endomorphism $(a, b, c)$ of this distinguished triangle, such that $K$, $L$, $M$ are perfect complexes and $\text{Tr}_K(a) + \text{Tr}_M(c) \not = \text{Tr}_L(b)$.


examples-lemma-proper-spaces-not-algebraic

The stack in groupoids $$ p'_{fp, flat, proper} : \Spacesstack'_{fp, flat, proper} \longrightarrow \Sch_{fppf} $$ whose category of sections over a scheme $S$ is the category of flat, proper, finitely presented algebraic spaces over $S$ (see Quot, Section \ref{quot-section-stack-of-spaces}) is not an algebraic stack.


examples-lemma-Z-not-quasi-compact

The scheme $\Spec(\mathbf{Z})$ is not quasi-compact in the canonical topology on the category of schemes.


bootstrap-lemma-representable-by-spaces-transformation-diagonal

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$ be representable by algebraic spaces. Then $\Delta_{F/G} : F \to F \times_G F$ is representable by algebraic spaces.


bootstrap-lemma-spaces-smooth-locally-representable

Denote the common underlying category of $\Sch_{fppf}$ and $\Sch_\etale$ by $\Sch_\alpha$ (see Topologies, Remark \ref{topologies-remark-choice-sites}). Let $S$ be an object of $\Sch_\alpha$. Let $$ F : (\Sch_\alpha/S)^{opp} \longrightarrow \textit{Sets} $$ be a presheaf with the following properties: \begin{enumerate} \item $F$ is a sheaf for the \'etale topology, \item there exists an algebraic space $U$ over $S$ and a map $U \to F$ which is representable by algebraic spaces, surjective, and smooth. \end{enumerate} Then $F$ is an algebraic space in the sense of Algebraic Spaces, Definition \ref{spaces-definition-algebraic-space}.


spaces-perfect-lemma-direct-image-coherator

Let $S$ be a scheme and let $f : X \to Y$ be a morphism of algebraic spaces over $S$. 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.


spaces-perfect-lemma-flat-pullback-injective-quasi-coherent

Let $S$ be a scheme. Let $f : X \to Y$ be a flat, quasi-compact, and quasi-separated morphism of algebraic spaces over $S$. If $\mathcal{J}$ is an injective object of $\QCoh(\mathcal{O}_X)$, then $f_*\mathcal{J}$ is an injective object of $\QCoh(\mathcal{O}_Y)$.


spaces-perfect-lemma-pseudo-coherent-hocolim-with-support

Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is quasi-compact. Let $K \in D(\mathcal{O}_X)$ supported on $T$. The following are equivalent \begin{enumerate} \item $K$ is pseudo-coherent, and \item $K = \text{hocolim} K_n$ where $K_n$ is perfect, supported on $T$, and $\tau_{\geq -n}K_n \to \tau_{\geq -n}K$ is an isomorphism for all $n$. \end{enumerate}


spaces-perfect-lemma-chi-locally-constant

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \in D(\mathcal{O}_X)$ be perfect. The function $$ \chi_E : |X| \longrightarrow \mathbf{Z},\quad x \longmapsto \sum (-1)^i \beta_i(x) $$ is locally constant on $X$.


dualizing-lemma-product-injectives

Let $R$ be a ring. Any product of injective $R$-modules is injective.


dualizing-lemma-projective-covers-local

Let $(R, \mathfrak m, \kappa)$ be a local ring. Any finite $R$-module has a projective cover.


dualizing-lemma-injective-hull

Let $R$ be a ring. Any $R$-module has an injective hull.


dualizing-lemma-formal-fibres-lci

Properties (A), (B), (C), (D), and (E) of More on Algebra, Section \ref{more-algebra-section-properties-formal-fibres} hold for $P(k \to R) =$``$R$ is a local complete intersection''. See Divided Power Algebra, Definition \ref{dpa-definition-lci}.


properties-lemma-catenary-dimension-function

Let $S$ be a locally Noetherian scheme. The following are equivalent: \begin{enumerate} \item $S$ is catenary, and \item locally in the Zariski topology there exists a dimension function on $S$ (see Topology, Definition \ref{topology-definition-dimension-function}). \end{enumerate}


properties-lemma-scheme-regular-iff-all-Rk

Let $X$ be a locally Noetherian scheme. Then $X$ is regular if and only if $X$ has $(R_k)$ for all $k \geq 0$.


properties-lemma-nagata-universally-Japanese

Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item $X$ is Nagata, and \item $X$ is locally Noetherian and universally Japanese. \end{enumerate}


properties-lemma-geometrically-unibranch

\begin{reference} Compare with \cite[Proposition 2.3]{Etale-coverings} \end{reference} Let $X$ be a Noetherian scheme. The following are equivalent \begin{enumerate} \item $X$ is geometrically unibranch (Definition \ref{definition-unibranch}), \item for every point $x \in X$ which is not the generic point of an irreducible component of $X$, the punctured spectrum of the strict henselization $\mathcal{O}_{X, x}^{sh}$ is connected. \end{enumerate}


properties-lemma-section-maps-backwards

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$ be a section. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent $\mathcal{O}_X$-modules. \begin{enumerate} \item If $X$ is quasi-compact and $\mathcal{F}$ is of finite type, then (\ref{equation-hom-invert-s}) is injective, and \item if $X$ is quasi-compact and quasi-separated and $\mathcal{F}$ is of finite presentation, then (\ref{equation-hom-invert-s}) is bijective. \end{enumerate}


properties-lemma-quasi-affine-invertible-nonvanishing-section

Let $X$ be a quasi-affine scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $E \subset W \subset X$ with $E$ finite and $W$ open. Then there exists an $s \in \Gamma(X, \mathcal{L})$ such that $X_s$ is affine and $E \subset X_s \subset W$.


criteria-lemma-stack-in-setoids-descent

Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be stacks in groupoids over $(\Sch/S)_{fppf}$. Suppose that $\mathcal{X} \to \mathcal{Y}$ and $\mathcal{Z} \to \mathcal{Y}$ are $1$-morphisms. If \begin{enumerate} \item $\mathcal{Y}$, $\mathcal{Z}$ are representable by algebraic spaces $Y$, $Z$ over $S$, \item the associated morphism of algebraic spaces $Y \to Z$ is surjective, flat and locally of finite presentation, and \item $\mathcal{Y} \times_\mathcal{Z} \mathcal{X}$ is a stack in setoids, \end{enumerate} then $\mathcal{X}$ is a stack in setoids.


criteria-lemma-etale-covering-restriction-of-scalars

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


criteria-lemma-etale-map-hilbert

In the situation of Lemma \ref{lemma-map-hilbert}. Assume that $G$, $H$ are representable by algebraic spaces and \'etale. Then $\mathcal{H}_d(\mathcal{X}'/\mathcal{Y}') \to \mathcal{H}_d(\mathcal{X}/\mathcal{Y})$ is representable by algebraic spaces and \'etale. If also $H$ is surjective and the induced functor $\mathcal{X}' \to \mathcal{Y}' \times_\mathcal{Y} \mathcal{X}$ is surjective, then $\mathcal{H}_d(\mathcal{X}'/\mathcal{Y}') \to \mathcal{H}_d(\mathcal{X}/\mathcal{Y})$ is surjective.


quot-theorem-polarized-algebraic

The stack $\Polarizedstack$ (Situation \ref{situation-polarized}) is algebraic. In fact, for any algebraic space $B$ the stack $B\textit{-Polarized}$ (Remark \ref{remark-polarized-base-change}) is algebraic.


quot-theorem-curves-algebraic

\begin{reference} See \cite[Proposition 3.3, page 8]{dJHS} and \cite[Appendix B by Jack Hall, Theorem B.1]{Smyth}. \end{reference} The stack $\Curvesstack$ (Situation \ref{situation-curves}) is algebraic. In fact, for any algebraic space $B$ the stack $B\text{-}\Curvesstack$ (Remark \ref{remark-curves-base-change}) is algebraic.


quot-theorem-complexes-algebraic

\begin{reference} \cite{lieblich-complexes} \end{reference} Let $S$ be a scheme. Let $f : X \to B$ be morphism of algebraic spaces over $S$. Assume that $f$ is proper, flat, and of finite presentation. Then $\Complexesstack_{X/B}$ is an algebraic stack over $S$.


quot-lemma-hom-limits

In Situation \ref{situation-hom}. If $\mathcal{F}$ is of finite presentation and $f$ is quasi-compact and quasi-separated, then $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ is limit preserving.


quot-lemma-isom-sheaf

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


quot-lemma-q-sheaf-in-X

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


quot-lemma-pic-with-section-stack

In Situation \ref{situation-pic} let $\sigma : B \to X$ be a section. Then $\Picardstack_{X/B, \sigma}$ as defined above is a stack in groupoids over $(\Sch/S)_{fppf}$.


quot-lemma-curves-fibred-in-groupoids

The category $\Curvesstack$ is fibred in groupoids over $\Sch_{fppf}$.


quot-lemma-curves-stack

The category $\Curvesstack$ is a stack in groupoids over $\Sch_{fppf}$.


quot-lemma-curves-tangent-space

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


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

Let $A$ be a henselian local ring. Let $X$ be an algebraic space over $A$ such that $f : X \to \Spec(A)$ be a proper morphism. Let $X_0 \subset X$ be the fibre of $f$ over the closed point. For any sheaf $\mathcal{F}$ on $X_\etale$ we have $\Gamma(X, \mathcal{F}) = \Gamma(X_0, \mathcal{F}|_{X_0})$.


spaces-more-cohomology-lemma-proper-pushforward-stalk

Let $S$ be a scheme. Let $f : Y \to X$ be a proper morphism of algebraic spaces over $S$. Let $\overline{x} \to X$ be a geometric point. For any sheaf $\mathcal{F}$ on $Y_\etale$ the canonical map $$ (f_*\mathcal{F})_{\overline{x}} \longrightarrow \Gamma(Y_{\overline{x}}, \mathcal{F}_{\overline{x}}) $$ is bijective.


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

Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces. Let $\overline{y} \to Y$ be a geometric point. \begin{enumerate} \item For a torsion abelian sheaf $\mathcal{F}$ on $X_\etale$ we have $(R^nf_*\mathcal{F})_{\overline{y}} = H^n_\etale(X_{\overline{y}}, \mathcal{F}_{\overline{y}})$. \item For $E \in D^+(X_\etale)$ with torsion cohomology sheaves we have $(R^nf_*E)_{\overline{y}} = H^n_\etale(X_{\overline{y}}, E_{\overline{y}})$. \end{enumerate}


spaces-more-cohomology-lemma-base-change-separably-closed

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


spaces-more-cohomology-lemma-compare-cohomology-etale-fppf

Let $S$ be a scheme and let $X$ be an algebraic space over $S$. With $a_X : \Sh((\textit{Spaces}/X)_{fppf}) \to \Sh(X_\etale)$ 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 $H^q_\etale(X, \underline{A}) = H^q_{fppf}(X, \underline{A})$.


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

In Lemma \ref{lemma-push-pull-fppf-etale} if $f$ is finite, then $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$ for $K$ in $D^+(X_\etale)$.


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

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $K \in D_\QCoh(\mathcal{O}_X)$ the maps $$ L\pi_X^*K \longrightarrow R\epsilon_{X, *}(La_X^*\mathcal{F}) \quad\text{and}\quad K \longrightarrow Ra_{X, *}(La_X^*K) $$ are isomorphisms. Here $a_X : \Sh((\textit{Spaces}/X)_{fppf}) \to \Sh(X_\etale)$ is as above.


spaces-more-cohomology-lemma-compare-cohomology-etale-ph

Let $S$ be a scheme and let $X$ be an algebraic space over $S$. With $a_X : \Sh((\textit{Spaces}/X)_{ph}) \to \Sh(X_\etale)$ as above: \begin{enumerate} \item $H^q(X_\etale, \mathcal{F}) = H^q_{ph}(X, a_X^{-1}\mathcal{F})$ for a torsion abelian sheaf $\mathcal{F}$ on $X_\etale$, \item $H^q(X_\etale, K) = H^q_{ph}(X, a_X^{-1}K)$ for $K \in D^+(X_\etale)$ with torsion cohomology sheaves \end{enumerate} Example: if $A$ is a torsion abelian group, then $H^q_\etale(X, \underline{A}) = H^q_{ph}(X, \underline{A})$.


coherent-lemma-hom-into-S2

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. \begin{enumerate} \item If $\mathcal{G}$ has property $(S_1)$, then $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ has property $(S_1)$. \item If $\mathcal{G}$ has property $(S_2)$, then $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ has property $(S_2)$. \end{enumerate}


coherent-lemma-cohomology-powers-ideal-times-sheaf

Given a morphism of schemes $f : X \to Y$, a quasi-coherent sheaf $\mathcal{F}$ on $X$, and a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_Y$. Assume $Y$ locally Noetherian, $f$ proper, and $\mathcal{F}$ coherent. Then $$ \mathcal{M} = \bigoplus\nolimits_{n \geq 0} R^pf_*(\mathcal{I}^n\mathcal{F}) $$ is a graded $\mathcal{A} = \bigoplus_{n \geq 0} \mathcal{I}^n$-module which is quasi-coherent and of finite type.


coherent-lemma-cat-module-support-proper-over-base

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$, $\mathcal{G}$ be finite type, quasi-coherent $\mathcal{O}_X$-module. \begin{enumerate} \item If the supports of $\mathcal{F}$, $\mathcal{G}$ are proper over $S$, then the same is true for $\mathcal{F} \oplus \mathcal{G}$, for any extension of $\mathcal{G}$ by $\mathcal{F}$, for $\Im(u)$ and $\Coker(u)$ given any $\mathcal{O}_X$-module map $u : \mathcal{F} \to \mathcal{G}$, and for any quasi-coherent quotient of $\mathcal{F}$ or $\mathcal{G}$. \item If $S$ is locally Noetherian, then the category of coherent $\mathcal{O}_X$-modules with support proper over $S$ is a Serre subcategory (Homology, Definition \ref{homology-definition-serre-subcategory}) of the abelian category of coherent $\mathcal{O}_X$-modules. \end{enumerate}


formal-defos-lemma-S1-small-extensions

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda$. Then $\mathcal{F}$ satisfies (S1) if the condition of (S1) is assumed to hold only when $A_2 \to A$ is a small extension.


formal-defos-lemma-miniversal-object-unique

Let $\mathcal{F}$ be a predeformation category. Let $\xi$ be a versal formal object of $\mathcal{F}$ such that (\ref{equation-bijective-orbits}) holds. Then $\xi$ is a minimal versal formal object. In particular, such $\xi$ are unique up to isomorphism.


formal-defos-lemma-infaut-vector-space

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$ satisfying (RS). Let $x_0 \in \Ob(\mathcal{F}(k))$. Then $\text{Inf}_{x_0}(\mathcal{F})$ is equal as a set to $T_{\text{id}_{x_0}} \mathit{Aut}(x_0)$, and so has a natural $k$-vector space structure such that addition agrees with composition of automorphisms.


formal-defos-lemma-map-between-smooth

Let $\mathcal{F} \to \mathcal{G} \to \mathcal{H}$ be maps of categories cofibred in groupoids over $\mathcal{C}_\Lambda$. If \begin{enumerate} \item $\mathcal{F}$, $\mathcal{G}$ are deformation categories \item the map $T\mathcal{F} \to T\mathcal{G}$ is surjective, and \item $\mathcal{F} \to \mathcal{H}$ is smooth. \end{enumerate} Then $\mathcal{F} \to \mathcal{G}$ is smooth.


formal-defos-lemma-homotopy

Being formally homotopic is an equivalence relation on sets of morphisms in $\widehat{\mathcal{C}}_\Lambda$.


adequate-lemma-direct-image-parasitic-adequate

Let $f : T \to S$ be a quasi-compact and quasi-separated morphism of schemes. For any parasitic adequate $\mathcal{O}_T$-module on $(\Sch/T)_\tau$ the pushforward $f_*\mathcal{F}$ and the higher direct images $R^if_*\mathcal{F}$ are parasitic adequate $\mathcal{O}_S$-modules on $(\Sch/S)_\tau$.


adequate-lemma-describe-Dplus-adequate

Let $U = \Spec(A)$ be an affine scheme. The bounded below derived category $D^+(\textit{Adeq}(\mathcal{O}))$ is the localization of $K^+(\QCoh(\mathcal{O}_U))$ at the multiplicative subset of universal quasi-isomorphisms.


adequate-lemma-big-ext

Let $U = \Spec(A)$ be an affine scheme. Let $M$, $N$ be $A$-modules. For all $i$ we have a canonical isomorphism $$ \Ext^i_{\textit{Mod}(\mathcal{O})}(M^a, N^a) = \text{Pext}^i_A(M, N) $$ functorial in $M$ and $N$.


spaces-topologies-lemma-composition-etale

Let $S$ be a scheme. Given morphisms $f : X \to Y$, $g : Y \to Z$ in $(\textit{Spaces}/S)_\etale$ we have $g_{big} \circ f_{big} = (g \circ f)_{big}$ and $g_{small} \circ f_{small} = (g \circ f)_{small}$.


spaces-topologies-lemma-composition-fppf

Let $S$ be a scheme. Given morphisms $f : X \to Y$, $g : Y \to Z$ of algebraic spaces over $S$ we have $g_{big} \circ f_{big} = (g \circ f)_{big}$.


spaces-topologies-lemma-ph

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item If $X' \to X$ is an isomorphism then $\{X' \to X\}$ is a ph covering of $X$. \item If $\{X_i \to X\}_{i\in I}$ is a ph covering and for each $i$ we have a ph covering $\{X_{ij} \to X_i\}_{j\in J_i}$, then $\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is a ph covering. \item If $\{X_i \to X\}_{i\in I}$ is a ph covering and $X' \to X$ is a morphism of algebraic spaces then $\{X' \times_X X_i \to X'\}_{i\in I}$ is a ph covering. \end{enumerate}


spaces-topologies-lemma-composition-ph


spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc

Any fppf covering is an fpqc covering, and a fortiori, any syntomic, smooth, \'etale or Zariski covering is an fpqc covering.


proetale-theorem-proper-base-change

Let $f : X \to Y$ be a proper morphism of schemes. Let $g : Y' \to Y$ be a morphism of schemes giving rise to the base change diagram $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal such that $\Lambda/I$ is torsion. Let $K$ be an object of $D(X_\proetale)$ such that \begin{enumerate} \item $K$ is derived complete, and \item $K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}$ is bounded below with cohomology sheaves coming from $X_\etale$, \item $\Lambda/I^n$ is a perfect $\Lambda$-module\footnote{This assumption can be removed if $K$ is a constructible complex, see \cite{BS}.}. \end{enumerate} Then the base change map $$ Lg_{comp}^*Rf_*K \longrightarrow Rf'_*L(g')^*_{comp}K $$ is an isomorphism.


proetale-lemma-verify-site-proetale

Let $S$ be a scheme. Let $\Sch_\proetale$ be a big pro-\'etale site containing $S$. Both $S_\proetale$ and $(\textit{Aff}/S)_\proetale$ are sites.


proetale-lemma-composition-proetale

Given schemes $X$, $Y$, $Y$ in $\Sch_\proetale$ and morphisms $f : X \to Y$, $g : Y \to Z$ we have $g_{big} \circ f_{big} = (g \circ f)_{big}$ and $g_{small} \circ f_{small} = (g \circ f)_{small}$.


proetale-lemma-enough-compact-proetale

Let $S$ be a scheme. Let $\Lambda$ be a ring. \begin{enumerate} \item $D(S_\proetale)$ is compactly generated, \item $D(S_\proetale, \Lambda)$ is compactly generated, \item $D(S_\proetale, \mathcal{A})$ is compactly generated for any sheaf of rings $\mathcal{A}$ on $S_\proetale$, \item $D((\Sch/S)_\proetale)$ is compactly generated, \item $D((\Sch/S)_\proetale, \Lambda)$ is compactly generated, and \item $D((\Sch/S)_\proetale, \mathcal{A})$ is compactly generated for any sheaf of rings $\mathcal{A}$ on $(\Sch/S)_\proetale$, \end{enumerate}


proetale-lemma-describe-pullback

Let $S$ be a scheme. Consider the morphism $$ \pi_S : (\Sch/S)_\proetale \longrightarrow S_\proetale $$ of Lemma \ref{lemma-at-the-bottom}. Let $\mathcal{F}$ be a sheaf on $S_\proetale$. Then $\pi_S^{-1}\mathcal{F}$ is given by the rule $$ (\pi_S^{-1}\mathcal{F})(T) = \Gamma(T_\proetale, f_{small}^{-1}\mathcal{F}) $$ where $f : T \to S$. Moreover, $\pi_S^{-1}\mathcal{F}$ satisfies the sheaf condition with respect to fpqc coverings.


proetale-lemma-compare-higher-direct-image

Let $f : T \to S$ be a morphism of schemes. For $K$ in $D((\Sch/T)_\proetale)$ we have $$ (Rf_{big, *}K)|_{S_\proetale} = Rf_{small, *}(K|_{T_\proetale}) $$ in $D(S_\proetale)$. More generally, let $S' \in \Ob((\Sch/S)_\proetale)$ with structure morphism $g : S' \to S$. Consider the fibre product $$ \xymatrix{ T' \ar[r]_{g'} \ar[d]_{f'} & T \ar[d]^f \\ S' \ar[r]^g & S } $$ Then for $K$ in $D((\Sch/T)_\proetale)$ we have $$ i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K) $$ in $D(S'_\proetale)$ and $$ g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K) $$ in $D((\Sch/S')_\proetale)$.


proetale-lemma-cofinal-etale

Let $S$ be a scheme and let $\overline{s} : \Spec(k) \to S$ be a geometric point. The category of pro-\'etale neighbourhoods of $\overline{s}$ is cofiltered.


proetale-lemma-compare-cohomology

Let $X$ be a scheme. \begin{enumerate} \item For an abelian sheaf $\mathcal{F}$ on $X_\etale$ we have $$ H^i(X_\etale, \mathcal{F}) = H^i(X_\proetale, \epsilon^{-1}\mathcal{F}) $$ for all $i$. \item For $K \in D^+(X_\etale)$ we have $$ R\Gamma(X_\etale, K) = R\Gamma(X_\proetale, \epsilon^{-1}K) $$ \end{enumerate}


proetale-lemma-more-general-point

Let $k$ be a field. Let $G = \text{Gal}(k^{sep}/k)$ be its absolute Galois group. Further,  \begin{enumerate} \item let $M$ be a profinite abelian group with a continuous $G$-action, or \item let $\Lambda$ be a Noetherian ring and $I \subset \Lambda$ an ideal an let $M$ be an $I$-adically complete $\Lambda$-module with continuous $G$-action. \end{enumerate} Then there is a canonical sheaf $\underline{M}^\wedge$ on $\Spec(k)_\proetale$ associated to $M$ such that $$ H^i(\Spec(k), \underline{M}^\wedge) = H^i_{cont}(G, M) $$ as abelian groups or $\Lambda$-modules.


proetale-lemma-open-immersion

Let $j : U \to X$ be an open immersion of schemes. Then $\text{id} \cong j^{-1}j_!$ and $j^{-1}j_* \cong \text{id}$ and the functors $j_!$ and $j_*$ are fully faithful.


proetale-lemma-Noetherian-adic-constructible

Let $X$ be a Noetherian scheme. Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal. Let $\mathcal{F}$ be a constructible $\Lambda$-sheaf on $X_\proetale$. Then $\mathcal{F}$ is adic constructible.


proetale-lemma-proetale-cohomology-independent-partial-universe

Let $S$ be a scheme. Let $S_\proetale \subset S_\proetale'$ be two small pro-\'etale sites of $S$ as constructed in Definition \ref{definition-big-small-proetale}. Then the inclusion functor satisfies the assumptions of  Sites, Lemma \ref{sites-lemma-bigger-site}. Hence there exist morphisms of topoi $$ \xymatrix{ \Sh(S_\proetale) \ar[r]^g & \Sh(S_\proetale') \ar[r]^f & \Sh(S_\proetale) } $$ whose composition is isomorphic to the identity and with $f_* = g^{-1}$. Moreover, \begin{enumerate} \item for $\mathcal{F}' \in \textit{Ab}(S_\proetale')$ we have $H^p(S_\proetale', \mathcal{F}') = H^p(S_\proetale, g^{-1}\mathcal{F}')$, \item for $\mathcal{F} \in \textit{Ab}(S_\proetale)$ we have $$ H^p(S_\proetale, \mathcal{F}) = H^p(S_\proetale', g_*\mathcal{F}) = H^p(S_\proetale', f^{-1}\mathcal{F}). $$ \end{enumerate}


proetale-lemma-cohomology-enlarge-partial-universe

Let $S$ be a scheme. Let $(\Sch/S)_\proetale$ and $(\Sch'/S)_\proetale$ be two big pro-\'etale sites of $S$ as in Definition \ref{definition-big-small-proetale}. Assume that the first is contained in the second. In this case \begin{enumerate} \item for any abelian sheaf $\mathcal{F}'$ defined on $(\Sch'/S)_\proetale$ and any object $U$ of $(\Sch/S)_\proetale$ we have $$ H^p(U, \mathcal{F}'|_{(\Sch/S)_\proetale}) = H^p(U, \mathcal{F}') $$ In words: the cohomology of $\mathcal{F}'$ over $U$ computed in the bigger site agrees with the cohomology of $\mathcal{F}'$ restricted to the smaller site over $U$. \item for any abelian sheaf $\mathcal{F}$ on $(\Sch/S)_\proetale$ there is an abelian sheaf $\mathcal{F}'$ on $(\Sch/S)_\proetale'$ whose restriction to $(\Sch/S)_\proetale$ is isomorphic to $\mathcal{F}$. \end{enumerate}


proetale-proposition-Noetherian-adic-constructible

Let $X$ be a Noetherian scheme. Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal. Let $K$ be an object of $D_{cons}(X, \Lambda)$. Then $K$ is adic constructible (Definition \ref{definition-adic-constructible}).


formal-spaces-lemma-completion-adic-star

Let $R$ be a topological ring. Let $M$ be a topological $R$-module. Let $I \subset R$ be a finitely generated ideal. Assume $M$ has an open submodule whose topology is $I$-adic. Then $M^\wedge$ has an open submodule whose topology is $I$-adic and we have $M^\wedge/I^n M^\wedge = M/I^nM$ for all $n \geq 1$.


formal-spaces-lemma-weakly-admissible-henselian

Let $A$ be a weakly admissible topological ring. Let $I \subset A$ be a weak ideal of definition. Then $(A, I)$ is a henselian pair.


formal-spaces-lemma-taut-weakly-admissible

Let $\varphi : A \to B$ be a continuous map of weakly admissible topological rings. The following are equivalent \begin{enumerate} \item $\varphi$ is taut, \item for every weak ideal of definition $I \subset A$ the closure of $\varphi(I)B$ is a weak ideal of definition of $B$ and these form a fundamental system of weak ideals of definition of $B$. \end{enumerate}


formal-spaces-lemma-characterize-affine-formal-algebraic-space

Let $S$ be a scheme. Let $X$ be a sheaf on $(\Sch/S)_{fppf}$. Then $X$ is an affine formal algebraic space if and only if the following hold \begin{enumerate} \item any morphism $U \to X$ where $U$ is an affine scheme over $S$ factors through a morphism $T \to X$ which is representable and a thickening with $T$ an affine scheme over $S$, and \item a set theoretic condition as in Remark \ref{remark-set-theoretic}. \end{enumerate}


formal-spaces-lemma-completion-countably-indexed

\begin{reference} Email by Ofer Gabber of September 11, 2014. \end{reference} Let $S$ be a scheme. Let $X = \Spec(A)$ be an affine scheme over $S$. Let $T \subset X$ be a closed subscheme. \begin{enumerate} \item If the formal completion $X_{/T}$ is countably indexed and there exist countably many $f_1, f_2, f_3, \ldots \in A$ such that $T = V(f_1, f_2, f_3, \ldots)$, then $X_{/T}$ is adic*. \item The conclusion of (1) is wrong if we omit the assumption that $T$ can be cut out by countably many functions in $X$. \end{enumerate}


formal-spaces-lemma-base-change-representable

A base change of a morphism representable by algebraic spaces is representable by algebraic spaces. The same holds for representable (by schemes).


formal-spaces-lemma-representable-by-algebraic-spaces

Let $S$ be a scheme. Let $Y$ be a formal algebraic space over $S$. Let $f : X \to Y$ be a map of sheaves on $(\Sch/S)_{fppf}$ which is representable by algebraic spaces. Then $X$ is a formal algebraic space.


formal-spaces-lemma-characterize-finite-type

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is of finite type, \item $f$ is representable by algebraic spaces and is of finite type in the sense of Bootstrap, Definition \ref{bootstrap-definition-property-transformation}. \end{enumerate}


formal-spaces-lemma-permanence-finite-type

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of formal algebraic spaces over $S$. If $g \circ f : X \to Z$ is locally of finite type, then $f : X \to Y$ is locally of finite type.


formal-spaces-lemma-composition-separated

All of the separation axioms listed in Definition \ref{definition-separated-morphism} are stable under composition of morphisms.


formal-spaces-lemma-structure-sheaf

Every formal algebraic space has a structure sheaf.


formal-spaces-lemma-higher-vanishing-structure-sheaf

If $X$ is a countably indexed affine formal algebraic space, then we have $H^n(X_\etale, \mathcal{O}_X) = 0$ for $n > 0$.


pione-theorem-global

Let $Y$ be an excellent regular scheme over a field. Let $f : X \to Y$ be a finite type morphism of schemes with $X$ normal. Let $V \subset X$ be the maximal open subscheme where $f$ is \'etale. Then the inclusion morphism $V \to X$ is affine.


pione-lemma-normal-pione-quotient-inertia

Let $X$ be a normal integral scheme with function field $K$. With notation as above, the following three subgroups of $\text{Gal}(K^{sep}/K) = \pi_1(\Spec(K))$ are equal \begin{enumerate} \item the kernel of the surjection $\text{Gal}(K^{sep}/K) \longrightarrow \pi_1(X)$, \item the smallest normal closed subgroup containing $I_y$ for all $y \in X^{sep}$, and \item the smallest normal closed subgroup containing $\text{Gal}(K^{sep}/K_x^{sh})$ for all $x \in  X$. \end{enumerate}


pione-lemma-unramified

Let $X$ be an integral normal scheme with function field $K$. Let $L/K$ be a finite extension. Let $Y \to X$ be the normalization of $X$ in $L$. The following are equivalent \begin{enumerate} \item $X$ is unramified in $L$ as defined in Section \ref{section-normal}, \item $Y \to X$ is an unramified morphism of schemes, \item $Y \to X$ is an \'etale morphism of schemes, \item $Y \to X$ is a finite \'etale morphism of schemes, \item for $x \in X$ the projection $Y \times_X \Spec(\mathcal{O}_{X, x}) \to \Spec(\mathcal{O}_{X, x})$ is unramified, \item same as in (5) but with $\mathcal{O}_{X, x}^h$, \item same as in (5) but with $\mathcal{O}_{X, x}^{sh}$, \item for $x \in X$ the scheme theoretic fibre $Y_x$ is \'etale over $x$ of degree $\geq [L : K]$. \end{enumerate} If $L/K$ is Galois with Galois group $G$, then these are also equivalent to \begin{enumerate} \item[(9)] for $y \in Y$ the group $I_y = \{g \in G \mid g(y) = y\text{ and } g \bmod \mathfrak m_y = \text{id}_{\kappa(y)}\}$ is trivial. \end{enumerate}


pione-lemma-specialization-map-composition

Let $f : X \to S$ be a proper morphism with geometrically connected fibres. Let $s'' \leadsto s' \leadsto s$ be specializations of points of $S$. A composition of specialization maps $\pi_1(X_{\overline{s}''}) \to \pi_1(X_{\overline{s}'}) \to \pi_1(X_{\overline{s}})$ is a specialization map $\pi_1(X_{\overline{s}''}) \to \pi_1(X_{\overline{s}})$.


pione-lemma-specialization-map-valuation-ring

Let $f : X \to S$ be a proper morphism with geometrically connected fibres. Let $s' \leadsto s$ be a specialization of points of $S$ and let $sp : \pi_1(X_{\overline{s}'}) \to \pi_1(X_{\overline{s}})$ be a specialization map. Then there exists a strictly henselian valuation ring $R$ over $S$ with algebraically closed fraction field such that $sp$ is isomorphic to $sp_R$ defined above.


pione-lemma-restriction-fully-faithful-general-special

Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme. Let $Y_n \subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$. Let $\mathcal{V}$ be the set of open subschemes $V \subset X$ containing $Y$ ordered by reverse inclusion. Assume one of the following holds \begin{enumerate} \item $X$ is quasi-affine and $$ \colim_\mathcal{V} \Gamma(V, \mathcal{O}_V) \longrightarrow \lim \Gamma(Y_n, \mathcal{O}_{Y_n}) $$ is an isomorphism, or \item $X$ has an ample invertible module $\mathcal{L}$ and $$ \colim_\mathcal{V} \Gamma(V, \mathcal{L}^{\otimes m}) \longrightarrow \lim \Gamma(Y_n, \mathcal{L}^{\otimes m}|_{Y_n}) $$ is an isomorphism for all $m \gg 0$, or \item for every $V \in \mathcal{V}$ and every finite locally free $\mathcal{O}_V$-module $\mathcal{E}$ the map $$ \colim_{V' \geq V} \Gamma(V', \mathcal{E}|_{V'}) \longrightarrow \lim \Gamma(Y_n, \mathcal{E}|_{Y_n}) $$ is an isomorphism. \end{enumerate} Then the functor $$ \colim_\mathcal{V} \textit{F\'Et}_V \to \textit{F\'Et}_Y $$ is fully faithful.


pione-lemma-pushout-along-closed-immersion-and-integral

In More on Morphisms, Situation \ref{more-morphisms-situation-pushout-along-closed-immersion-and-integral}, for example if $Z \to Y$ and $Z \to X$ are closed immersions of schemes, there is an equivalence of categories $$ \textit{F\'Et}_{Y \amalg_Z X} \longrightarrow \textit{F\'Et}_Y \times_{\textit{F\'Et}_Z} \textit{F\'Et}_X $$


pione-lemma-lift-simple

In Situation \ref{situation-local-lefschetz}. Let $V$ be finite \'etale over $U$. Assume \begin{enumerate} \item[(a)] $A$ has a dualizing complex, \item[(b)] the pair $(A, (f))$ is henselian, \item[(c)] one of the following is true \begin{enumerate} \item[(i)] $A_f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\geq 3$, or \item[(ii)] for every prime $\mathfrak p \subset A$, $f \not \in \mathfrak p$ we have $\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 2$. \end{enumerate} \item[(d)] $V_0 = V \times_U U_0$ is equal to $Y_0 \times_{X_0} U_0$ for some $Y_0 \to X_0$ finite \'etale. \end{enumerate} Then $V = Y \times_X U$ for some $Y \to X$ finite \'etale.


pione-lemma-faithful-general

In Situation \ref{situation-local-lefschetz}. Let $U' \subset U$ be open and contain $U_0$. Assume for $\mathfrak p \subset A$ minimal with $\mathfrak p \in U'$, $\mathfrak p \not \in U_0$ we have $\dim(A/\mathfrak p) \geq 2$. Then $$ \textit{F\'Et}_{U'} \longrightarrow \textit{F\'Et}_{U_0},\quad V' \longmapsto V_0 = V' \times_{U'} U_0 $$ is a faithful functor. Moreover, there exists a $U'$ satisfying the assumption and any smaller open $U'' \subset U'$ containing $U_0$ also satisfies this assumption. In particular, the restriction functor $$ \colim_{U_0 \subset U' \subset U\text{ open}} \textit{F\'Et}_{U'} \longrightarrow \textit{F\'Et}_{U_0} $$ is faithful.


pione-lemma-fully-faithful-general-better

In Situation \ref{situation-local-lefschetz} assume \begin{enumerate} \item $A$ has a dualizing complex and is $f$-adically complete, \item every irreducible component of $X$ not contained in $X_0$ has dimension $\geq 3$. \end{enumerate} Then the restriction functor $$ \colim_{U_0 \subset U' \subset U\text{ open}} \textit{F\'Et}_{U'} \longrightarrow \textit{F\'Et}_{U_0} $$ is fully faithful.


pione-lemma-fully-faithful-general

In Situation \ref{situation-local-lefschetz} assume \begin{enumerate} \item $A$ is $f$-adically complete, \item $f$ is a nonzerodivisor. \item $H^1_\mathfrak m(A/fA)$ is a finite $A$-module. \end{enumerate} Then the restriction functor $$ \colim_{U_0 \subset U' \subset U\text{ open}} \textit{F\'Et}_{U'} \longrightarrow \textit{F\'Et}_{U_0} $$ is fully faithful.


stacks-cohomology-lemma-general-pushforward-fppf

Let $\mathcal{M}$ be a rule which associates to every algebraic stack $\mathcal{X}$ a subcategory $\mathcal{M}_\mathcal{X}$ of $\textit{Mod}(\mathcal{O}_\mathcal{X})$ such that \begin{enumerate} \item $\mathcal{M}_\mathcal{X}$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_\mathcal{X})$ for all algebraic stacks $\mathcal{X}$, \item for a smooth morphism of algebraic stacks $f : \mathcal{Y} \to \mathcal{X}$ the functor $f^*$ maps $\mathcal{M}_\mathcal{X}$ into $\mathcal{M}_\mathcal{Y}$, \item if $f_i : \mathcal{X}_i \to \mathcal{X}$ is a family of smooth morphisms of algebraic stacks with $|\mathcal{X}| = \bigcup |f_i|(|\mathcal{X}_i|)$, then an object $\mathcal{F}$ of $\textit{Mod}(\mathcal{O}_\mathcal{X})$ is in $\mathcal{M}_\mathcal{X}$ if and only if $f_i^*\mathcal{F}$ is in $\mathcal{M}_{\mathcal{X}_i}$ for all $i$, and \item if $f : \mathcal{Y} \to \mathcal{X}$ is a morphism of algebraic stacks and $\mathcal{X}$ and $\mathcal{Y}$ are representable by affine schemes, then $R^if_*$ maps $\mathcal{M}_\mathcal{Y}$ into $\mathcal{M}_\mathcal{X}$. \end{enumerate} Then for any quasi-compact and quasi-separated morphism  $f : \mathcal{Y} \to \mathcal{X}$ of algebraic stacks $R^if_*$ maps $\mathcal{M}_\mathcal{Y}$ into $\mathcal{M}_\mathcal{X}$. (Higher direct images computed in fppf topology.)


stacks-cohomology-lemma-quasi-coherent-weak-serre

Let $\mathcal{X}$ be an algebraic stack. \begin{enumerate} \item $\QCoh(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$. \item $\QCoh(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$. \end{enumerate}


stacks-cohomology-proposition-smooth-covering-compute-cohomology

Let $f : \mathcal{U} \to \mathcal{X}$ be a morphism of algebraic stacks. Assume $f$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_\mathcal{X}$-module. Then there is a spectral sequence $$ E_2^{p, q} = H^q(\mathcal{U}_p, f_p^*\mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}, \mathcal{F}) $$ where $f_p$ is the morphism $\mathcal{U} \times_\mathcal{X} \ldots \times_\mathcal{X} \mathcal{U} \to \mathcal{X}$ ($p + 1$ factors).


sites-cohomology-lemma-compute-cohomology-on-sheaf-sets

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be a presheaf of sets on $\mathcal{C}$. Let $\mathcal{F}$ be an $\mathcal{O}$-module and denote $\mathcal{F}_{ab}$ the underlying sheaf of abelian groups. Then $H^p(K, \mathcal{F}) = H^p(K, \mathcal{F}_{ab})$.


sites-cohomology-lemma-cup-compatible-with-naive

In the situation above the following diagram commutes $$ \xymatrix{ f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} f_*\mathcal{M}^\bullet \ar[r] \ar[d] & Rf_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} Rf_*\mathcal{M}^\bullet \ar[d]^{\text{Remark \ref{remark-cup-product}}} \\ \text{Tot}( f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} f_*\mathcal{M}^\bullet) \ar[d]_{\text{naive cup product}} & Rf_*(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} \mathcal{M}^\bullet) \ar[d] \\ f_*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet) \ar[r] & Rf_*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} \mathcal{M}^\bullet) } $$


sites-cohomology-lemma-internal-hom-diagonal

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given $K, L$ in $D(\mathcal{O})$ there is a canonical morphism $$ K \longrightarrow R\SheafHom(L, K \otimes_\mathcal{O}^\mathbf{L} L) $$ in $D(\mathcal{O})$ functorial in both $K$ and $L$.


sites-cohomology-lemma-special-square-cocontinuous

Assume given a commutative diagram $$ \xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{(g', (g')^\sharp)} \ar[d]_{(f', (f')^\sharp)} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^{(g, g^\sharp)} & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) } $$ of ringed topoi. Assume \begin{enumerate} \item $f$, $f'$, $g$, and $g'$ correspond to cocontinuous functors $u$, $u'$, $v$, and $v'$ as in Sites, Lemma \ref{sites-lemma-cocontinuous-morphism-topoi}, \item $v \circ u' = u \circ v'$, \item $v$ and $v'$ are continuous as well as cocontinuous, \item for any object $V'$ of $\mathcal{D}'$ the functor ${}^{u'}_{V'}\mathcal{I} \to {}^{\ \ \ u}_{v(V')}\mathcal{I}$ given by $v$ is cofinal, \item $g^{-1}\mathcal{O}_{\mathcal{D}} = \mathcal{O}_{\mathcal{D}'}$ and $(g')^{-1}\mathcal{O}_{\mathcal{C}} = \mathcal{O}_{\mathcal{C}'}$, and \item $g'_! : \textit{Ab}(\mathcal{C}') \to \textit{Ab}(\mathcal{C})$ is exact\footnote{Holds if fibre products and equalizers exist in $\mathcal{C}'$ and $v'$ commutes with them, see Modules on Sites, Lemma \ref{sites-modules-lemma-exactness-lower-shriek}.}. \end{enumerate} Then we have $Rf'_* \circ (g')^* = g^* \circ Rf_*$ as functors $D(\mathcal{O}_\mathcal{C}) \to D(\mathcal{O}_{\mathcal{D}'})$.


sites-cohomology-lemma-special-square-continuous

Consider a commutative diagram $$ \xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'} \ar[r]_{(g', (g')^\sharp)} \ar[d]_{(f', (f')^\sharp)} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^{(g, g^\sharp)} & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) } $$ of ringed topoi and suppose we have functors $$ \xymatrix{ \mathcal{C}' \ar[r]_{v'} & \mathcal{C} \\ \mathcal{D}' \ar[r]^v \ar[u]^{u'} & \mathcal{D} \ar[u]_u } $$ such that (with notation as in Sites, Sections \ref{sites-section-morphism-sites} and \ref{sites-section-cocontinuous-morphism-topoi}) we have \begin{enumerate} \item $u$ and $u'$ are continuous and give rise to the morphisms $f$ and $f'$, \item $v$ and $v'$ are cocontinuous giving rise to the morphisms $g$ and $g'$, \item $u \circ v = v' \circ u'$, \item $v$ and $v'$ are continuous as well as cocontinuous, and \item $g^{-1}\mathcal{O}_{\mathcal{D}} = \mathcal{O}_{\mathcal{D}'}$ and $(g')^{-1}\mathcal{O}_{\mathcal{C}} = \mathcal{O}_{\mathcal{C}'}$. \end{enumerate} Then $Rf'_* \circ (g')^* = g^* \circ Rf_*$ as functors $D^+(\mathcal{O}_\mathcal{C}) \to D^+(\mathcal{O}_{\mathcal{D}'})$. If in addition \begin{enumerate} \item[(6)] $g'_! : \textit{Ab}(\mathcal{C}') \to \textit{Ab}(\mathcal{C})$ is exact\footnote{Holds if fibre products and equalizers exist in $\mathcal{C}'$ and $v'$ commutes with them, see Modules on Sites, Lemma \ref{sites-modules-lemma-exactness-lower-shriek}.}, \end{enumerate} then $Rf'_* \circ (g')^* = g^* \circ Rf_*$ as functors $D(\mathcal{O}_\mathcal{C}) \to D(\mathcal{O}_{\mathcal{D}'})$.


sites-cohomology-lemma-summands-perfect

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. If $K \oplus L$ is a perfect object of $D(\mathcal{O})$, then so are $K$ and $L$.


sites-cohomology-lemma-invertible-derived

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $M$ be an object of $D(\mathcal{O})$. The following are equivalent \begin{enumerate} \item $M$ is invertible in $D(\mathcal{O})$, see Categories, Definition \ref{categories-definition-invertible}, and \item there is a locally finite\footnote{This means that for every object $U$ of $\mathcal{C}$ there is a covering $\{U_i \to U\}$ such that for every $i$ the sheaf $\mathcal{O}_n|_{U_i}$ is nonzero for only a finite number of $n$.} direct product decomposition $$ \mathcal{O} = \prod\nolimits_{n \in \mathbf{Z}} \mathcal{O}_n $$ and for each $n$ there is an invertible $\mathcal{O}_n$-module $\mathcal{H}^n$ (Modules on Sites, Definition \ref{sites-modules-definition-invertible-sheaf}) and $M = \bigoplus \mathcal{H}^n[-n]$ in $D(\mathcal{O})$. \end{enumerate} If (1) and (2) hold, then $M$ is a perfect object of $D(\mathcal{O})$. If $(\mathcal{C}, \mathcal{O})$ is a locally ringed site these condition are also equivalent to \begin{enumerate} \item[(3)] for every object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}$ and for each $i$ an integer $n_i$ such that $M|_{U_i}$ is represented by an invertible $\mathcal{O}_{U_i}$-module placed in degree $n_i$. \end{enumerate}


sites-cohomology-lemma-compact-objects-if-enough-qc

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Assume every object of $\mathcal{C}$ has a covering by quasi-compact objects. Then every compact object of $D(\mathcal{O})$ is a direct summand in $D(\mathcal{O})$ of a finite complex whose terms are finite direct sums of $\mathcal{O}$-modules of the form $j_!\mathcal{O}_U$ where $U$ is a quasi-compact object of $\mathcal{C}$.


fields-lemma-algebraic-elements

Let $E/k$ be a field extension. Then the elements of $E$ algebraic over $k$ form a subextension of $E/k$.


fields-lemma-algebraic-closures-isomorphic

Any two algebraic closures of a field are isomorphic.


spaces-limits-lemma-descend-section

Notation and assumptions as in Situation \ref{situation-descent}. Suppose that $\mathcal{F}_0$ is a quasi-coherent sheaf on $X_0$. Set $\mathcal{F}_i = f_{0i}^*\mathcal{F}_0$ for $i \geq 0$ and set $\mathcal{F} = f_0^*\mathcal{F}_0$. Then $$ \Gamma(X, \mathcal{F}) = \colim_{i \geq 0} \Gamma(X_i, \mathcal{F}_i) $$


spaces-limits-lemma-eventually-affine

Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $X = \lim X_i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume \begin{enumerate} \item $Y$ quasi-compact and quasi-separated, \item $X_i$ quasi-compact and quasi-separated, \item $X \to Y$ affine. \end{enumerate} Then $X_i \to Y$ is affine for $i$ large enough.


spaces-limits-lemma-eventually-finite

Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $X = \lim X_i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume \begin{enumerate} \item $Y$ quasi-compact and quasi-separated, \item $X_i$ quasi-compact and quasi-separated, \item the transition morphisms $X_{i'} \to X_i$ are finite, \item $X_i \to Y$ locally of finite type \item $X \to Y$ integral. \end{enumerate} Then $X_i \to Y$ is finite for $i$ large enough.


spaces-limits-lemma-descend-affine

Notation and assumptions as in Situation \ref{situation-descent-property}. If $f$ is affine, then $f_i$ is affine for some $i \geq 0$.


spaces-limits-lemma-descend-monomorphism

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is a monomorphism, \item $f_0$ is locally of finite type, \end{enumerate} then $f_i$ is a monomorphism for some $i \geq 0$.


spaces-limits-lemma-push-sections-annihilated-by-ideal

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $\mathcal{I} \subset \mathcal{O}_Y$ be a quasi-coherent sheaf of ideals of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf of sections annihilated by $f^{-1}\mathcal{I}\mathcal{O}_X$. Then $f_*\mathcal{F}' \subset f_*\mathcal{F}$ is the subsheaf of sections annihilated by $\mathcal{I}$.


spaces-limits-lemma-push-sections-supported-on-closed-subset

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $T \subset |Y|$ be a closed subset. Assume $|Y| \setminus T$ corresponds to an open subspace $V \subset Y$ such that $V \to Y$ is quasi-compact. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf of sections supported on $|f|^{-1}T$. Then $f_*\mathcal{F}' \subset f_*\mathcal{F}$ is the subsheaf of sections supported on $T$.


spaces-limits-lemma-minimal-closed-subspace

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. If $X$ is not a scheme, then there exists a closed subspace $Z \subset X$ such that $Z$ is not a scheme, but every proper closed subspace $Z' \subset Z$ is a scheme.


spaces-limits-lemma-minimal-nonscheme

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Assume that every proper closed subspace $Z \subset X$ is a scheme, but $X$ is not a scheme. Then $X$ is reduced and irreducible.


spaces-limits-lemma-glueing-near-point

Let $S$ be a scheme. Let $U \subset S$ be a retrocompact open. Let $s \in S$ be a point in the complement of $U$. With $V = \Spec(\mathcal{O}_{S, s}) \cap U$ there is an equivalence of categories $$ \colim_{s \in U' \supset U\text{ open}} FP_{U'} \longrightarrow FP_U \times_{FP_V} FP_{\Spec(\mathcal{O}_{S, s})} $$ where $FP_T$  is the category of algebraic spaces of finite presentation over $T$.


spaces-limits-lemma-modifications

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$ be a closed point such that $U = X \setminus \{x\} \to X$ is quasi-compact. With $V = \Spec(\mathcal{O}_{X, x}^h) \setminus \{\mathfrak m_x^h\}$ the base change functor $$ \left\{ \begin{matrix} f : Y \to X\text{ of finite presentation} \\ f^{-1}(U) \to U\text{ is an isomorphism} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} g : Y \to \Spec(\mathcal{O}_{X, x}^h)\text{ of finite presentation} \\ g^{-1}(V) \to V\text{ is an isomorphism} \end{matrix} \right\} $$ is an equivalence of categories.


spaces-limits-lemma-reach-point-closure-Noetherian

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ finite type and $Y$ locally Noetherian. Let $y \in |Y|$ be a point in the closure of the image of $|f|$. Then there exists a commutative diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d]^f \\ \Spec(A) \ar[r] & Y } $$ where $A$ is a discrete valuation ring and $K$ is its field of fractions mapping the closed point of $\Spec(A)$ to $y$. Moreover, we can assume that the point $x \in |X|$ corresponding to $\Spec(K) \to X$ is a codimension $0$ point\footnote{See discussion in Properties of Spaces, Section \ref{spaces-properties-section-generic-points}.} and that $K$ is the residue field of a point on a scheme \'etale over $X$.


stacks-geometry-lemma-branches-multiplicity

Let $\mathcal{X}$ be an algebraic stack locally of finite type over a locally Noetherian scheme $S$. Let $x_0 : \Spec(k) \to \mathcal{X}$ is a morphism where $k$ is a field of finite type over $S$ with image $s \in S$. If $\mathcal{O}_{S, s}$ is a G-ring, then the map of Lemma \ref{lemma-branches} preserves multiplicities.


stacks-geometry-lemma-dimension-for-stacks

If $\mathcal{X}$ is a locally Noetherian algebraic stack and $x \in |\mathcal{X}|$. Let $U \to \mathcal{X}$ be a smooth morphism from an algebraic space to $\mathcal{X}$, let $u$ be any point of $|U|$ mapping to $x$. Then we have $$ \dim_x(\mathcal{X}) =  \dim_u(U) - \dim_{u}(U_x) $$ where the relative dimension $\dim_u(U_x)$ is defined by Definition \ref{definition-relative-dimension} and the dimension of $\mathcal{X}$ at $x$ is as in Properties of Stacks, Definition \ref{stacks-properties-definition-dimension-at-point}.


stacks-geometry-lemma-dimension-via-components

If $\mathcal{X}$ is a locally Noetherian algebraic stack, and if $x \in |\mathcal{X}|$, then $\dim_x(\mathcal{X}) = \sup_{\mathcal{T}} \{ \dim_x(\mathcal{T}) \} $, where $\mathcal{T}$ runs over all the irreducible components of $|\mathcal{X}|$ passing through $x$ (endowed with their induced reduced structure).


stacks-geometry-lemma-dimension-formula

Suppose that $\mathcal{X}$ is an algebraic stack, locally of finite type over a locally Noetherian scheme $S$. Let $x_0 : \Spec(k) \to \mathcal{X}$ be a morphism where $k$ is a field of finite type over $S$. Represent $\mathcal{F}_{\mathcal{X}, k, x_0}$ as in Remark \ref{remark-groupoid-defo} by a cogroupoid $(A, B, s, t, c)$ of Noetherian complete local $S$-algebras with residue field $k$. Then $$ \text{the dimension of the local ring of }\mathcal{X}\text{ at }x_0 = 2\dim A - \dim B $$


spaces-morphisms-lemma-match-separated

Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. \begin{enumerate} \item The morphism $f$ is locally separated. \item The morphism $f$ is (quasi-)separated in the sense of Definition \ref{definition-separated} above if and only if $f$ is (quasi-)separated in the sense of Section \ref{section-representable}. \end{enumerate} In particular, if $f : X \to Y$ is a morphism of schemes over $S$, then $f$ is (quasi-)separated in the sense of Definition \ref{definition-separated} if and only if $f$ is (quasi-)separated as a morphism of schemes.


spaces-morphisms-lemma-monomorphism-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is a monomorphism, \item for every scheme $Z$ and morphism $Z \to Y$ the base change $Z \times_Y X \to Z$ of $f$ is a monomorphism, \item for every affine scheme $Z$ and every morphism $Z \to Y$ the base change $Z \times_Y X \to Z$ of $f$ is a monomorphism, \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that the base change $V \times_Y X \to V$ is a monomorphism, and \item there exists a Zariski covering $Y = \bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \to Y_i$ is a monomorphism. \end{enumerate}


spaces-morphisms-lemma-scheme-theoretically-dense-quasi-compact

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U \subset X$ be an open subspace. If $U \to X$ is quasi-compact, then $U$ is scheme theoretically dense in $X$ if and only if the scheme theoretic closure of $U$ in $X$ is $X$.


spaces-morphisms-lemma-affine-representable

Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is affine (in the sense of Section \ref{section-representable}) if and only if for all affine schemes $Z$ and morphisms $Z \to Y$ the scheme $X \times_Y Z$ is affine.


spaces-morphisms-lemma-quasi-affine-representable

Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is quasi-affine (in the sense of Section \ref{section-representable}) if and only if for all affine schemes $Z$ and morphisms $Z \to Y$ the scheme $X \times_Y Z$ is quasi-affine.


spaces-morphisms-lemma-ubiquity-nagata

The following types of algebraic spaces are Nagata. \begin{enumerate} \item Any algebraic space locally of finite type over a Nagata scheme. \item Any algebraic space locally of finite type over a field. \item Any algebraic space locally of finite type over a Noetherian complete local ring. \item Any algebraic space locally of finite type over $\mathbf{Z}$. \item Any algebraic space locally of finite type over a Dedekind ring of characteristic zero. \item And so on. \end{enumerate}


spaces-morphisms-lemma-inverse-image-constructible

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $E \subset |Y|$ be a subset. If $E$ is \'etale locally constructible in $Y$, then $f^{-1}(E)$ is \'etale locally constructible in $X$.


spaces-morphisms-lemma-fpqc-quotient-topology

Let $S$ be a scheme. Let $f : X \to Y$ be a flat, quasi-compact, surjective morphism of algebraic spaces over $S$. A subset $T \subset |Y|$ is open (resp.\ closed) if and only $f^{-1}(|T|)$ is open (resp.\ closed) in $|X|$. In other words $f$ is submersive, and in fact universally submersive.


spaces-morphisms-lemma-pf-flat-module-open

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume $f$ locally finite presentation, $\mathcal{F}$ of finite type, $X = \text{Supp}(\mathcal{F})$, and $\mathcal{F}$ flat over $Y$. Then $f$ is universally open.


spaces-morphisms-lemma-composition-syntomic

The composition of syntomic morphisms is syntomic.


spaces-morphisms-lemma-integral-representable

Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is integral, resp.\ finite (in the sense of Section \ref{section-representable}), if and only if for all affine schemes $Z$ and morphisms $Z \to Y$ the scheme $X \times_Y Z$ is affine and integral, resp.\ finite, over $Z$.


spaces-morphisms-lemma-finite-locally-free-representable

Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is finite locally free (in the sense of Section \ref{section-representable}) if and only if $f$ is affine and the sheaf $f_*\mathcal{O}_X$ is a finite locally free $\mathcal{O}_Y$-module.


spaces-morphisms-lemma-rational-map-from-reduced-to-separated

Let $S$ be a scheme. Let $X$ and $Y$ be algebraic spaces over $S$. Assume $X$ is reduced and $Y$ is separated over $S$. Let $\varphi$ be a rational map from $X$ to $Y$ with domain of definition $U \subset X$. Then there exists a unique morphism $f : U \to Y$ of algebraic spaces representing $\varphi$.


spaces-morphisms-lemma-nagata-normalization-finite

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that \begin{enumerate} \item $Y$ is Nagata, \item $f$ is quasi-separated of finite type, \item $X$ is reduced. \end{enumerate} Then the normalization $\nu : Y' \to Y$ of $Y$ in $X$ is finite.


spaces-morphisms-lemma-normalization-reduced

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying the equivalent conditions of Lemma \ref{lemma-prepare-normalization}. The normalization morphism $\nu$ factors through the reduction $X_{red}$ and $X^\nu \to X_{red}$ is the normalization of $X_{red}$.


spaces-morphisms-lemma-universal-homeomorphism-representable

Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is a universal homeomorphism (in the sense of Section \ref{section-representable}) if and only if for every morphism of algebraic spaces $Z \to Y$ the base change map $Z \times_Y X \to Z$ induces a homeomorphism $|Z \times_Y X| \to |Z|$.


weil-lemma-motives

The category $M_k$ whose objects are motives over $k$ and morphisms are morphisms of motives over $k$ is a $\mathbf{Q}$-linear category. There is a contravariant functor $$ h : \{\text{smooth projective schemes over }k\} \longrightarrow M_k $$ defined by $h(X) = (X, 1, 0)$ and $h(f) = [\Gamma_f]$.


weil-lemma-Karoubian

The category $M_k$ is Karoubian.


weil-lemma-square-diagonal

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X$ be a nonempty smooth projective scheme over $k$ which is equidimensional of dimension $d$. We have $$ \sum\nolimits_i (-1)^i\dim_F H^i(X) = \deg(\Delta \cdot \Delta) = \deg(c_d(\mathcal{T}_{X/k})) $$


weil-proposition-weil-cohomology-theory

Let $k$ be a field. Let $F$ be a field of characteristic $0$. There is a $1$-to-$1$ correspondence between the following \begin{enumerate} \item data (D0), (D1), (D2), and (D3) satisfying (A), (B), and(C), and \item $\mathbf{Q}$-linear symmetric monoidal functors $$ G : M_k \longrightarrow \text{graded }F\text{-vector spaces} $$ such that $G(\mathbf{1}(1))$ is nonzero only in degree $-2$. \end{enumerate}


morphisms-lemma-composition-closed-immersion

A composition of closed immersions is a closed immersion.


morphisms-lemma-scheme-theoretically-dense-quasi-compact

Let $X$ be a scheme. Let $U \subset X$ be an open subscheme. If the inclusion morphism $U \to X$ is quasi-compact, then $U$ is scheme theoretically dense in $X$ if and only if the scheme theoretic closure of $U$ in $X$ is $X$.


morphisms-lemma-affine-diagonal

Let $X$ be a scheme such that for every point $x \in X$ there exists an invertible $\mathcal{O}_X$-module $\mathcal{L}$ and a global section $s \in \Gamma(X, \mathcal{L})$ such that $x \in X_s$ and $X_s$ is affine. Then the diagonal of $X$ is an affine morphism.


morphisms-lemma-catenary-check-irreducible

Let $S$ be a locally Noetherian scheme. Then $S$ is universally catenary if and only if the irreducible components of $S$ are universally catenary.


morphisms-lemma-descent-quasi-compact

Let $f : X \to Y$ be a morphism of schemes. Let $g : Y' \to Y$ be open and surjective such that the base change $f' : X' \to Y'$ is quasi-compact. Then $f$ is quasi-compact.


morphisms-lemma-finite-flat-is-finite-locally-free

Let $X$ be a scheme. The following are equivalent \begin{enumerate} \item every finite flat quasi-coherent $\mathcal{O}_X$-module is finite locally free, and \item every closed subset $Z \subset X$ which is closed under generalizations is open. \end{enumerate}


morphisms-lemma-universal-homeo-iso-if-invert-p

Let $p$ be a prime number. Let $A \to B$ be a ring map which induces an isomorphism $A[1/p] \to B[1/p]$ (for example if $p$ is nilpotent in $A$). The following are equivalent \begin{enumerate} \item $\Spec(B) \to \Spec(A)$ is a universal homeomorphism, and \item the kernel of $A \to B$ is a locally nilpotent ideal and for every $b \in B$ there exists a $p$-power $q$ with $qb$ and $b^q$ in the image of $A \to B$. \end{enumerate}


morphisms-lemma-birational-dominant

Let $f : X \to Y$ be a morphism of schemes having finitely many irreducible components. If $f$ is birational then $f$ is dominant.


chow-lemma-well-defined-tame-symbol

The formula (\ref{equation-tame-symbol}) determines a well defined element of $\kappa(\mathfrak m)^*$. In other words, the right hand side does not depend on the choice of the local factorizations or the choice of $B$.


chow-lemma-coherent-sheaf-cross-p1

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $\mathcal{F}$ be a coherent sheaf on $X \times \mathbf{P}^1$. Let $i_0, i_\infty : X \to X \times \mathbf{P}^1$ be the closed immersion such that $i_t(x) = (x, t)$. Denote $\mathcal{F}_0 = i_0^*\mathcal{F}$ and $\mathcal{F}_\infty = i_\infty^*\mathcal{F}$. Assume \begin{enumerate} \item $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k + 1$, \item $\dim_\delta(\text{Supp}(\mathcal{F}_0)) \leq k$, $\dim_\delta(\text{Supp}(\mathcal{F}_\infty)) \leq k$, and \item for any embedded associated point $\xi$ of $\mathcal{F}$ either $\xi \not \in (X \times \mathbf{P}^1)_0 \cup (X \times \mathbf{P}^1)_\infty$ or $\delta(\xi) < k$. \end{enumerate} Then $[\mathcal{F}_0]_k \sim_{rat} [\mathcal{F}_\infty]_k$ as $k$-cycles on $X$.


chow-lemma-Serre-subcategories

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. The categories $\textit{Coh}_{\leq k}(X)$ are Serre subcategories of the abelian category $\textit{Coh}(X)$.


chow-lemma-factors-through-rational-equivalence

\begin{reference} Very weak form of \cite[Theorem 17.1]{F} \end{reference} Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$. Let $p \in \mathbf{Z}$. Suppose given a rule which assigns to every locally of finite type morphism $Y' \to Y$ and every $k$ a map $$ c \cap - : Z_k(Y') \longrightarrow \CH_{k - p}(X') $$ where $Y' = X' \times_X Y$, satisfying condition (3) of Definition \ref{definition-bivariant-class} whenever $\mathcal{L}'|_{D'} \cong \mathcal{O}_{D'}$. Then $c \cap -$ factors through rational equivalence.


chow-lemma-bivariant-weaker

\begin{reference} Weak form of \cite[Theorem 17.1]{F} \end{reference} Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$. Let $p \in \mathbf{Z}$. Suppose given a rule which assigns to every locally of finite type morphism $Y' \to Y$ and every $k$ a map $$ c \cap - : \CH_k(Y') \longrightarrow \CH_{k - p}(X') $$ where $Y' = X' \times_X Y$, satisfying conditions (1), (2) of Definition \ref{definition-bivariant-class} and condition (3) whenever $\mathcal{L}'|_{D'} \cong \mathcal{O}_{D'}$. Then $c \cap -$ is a bivariant class.


chow-lemma-first-chern-class

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $n = \dim_\delta(X)$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. The first Chern class of $\mathcal{L}$ on $X$ of Definition \ref{definition-chern-classes} is equal to the Weil divisor associated to $\mathcal{L}$ by Definition \ref{definition-divisor-invertible-sheaf}.


chow-lemma-spell-out-degree-zero-cycle

Let $k$ be a field. Let $X$ be proper over $k$. Let $\alpha = \sum n_i[Z_i]$ be in $Z_0(X)$. Then $$ \deg(\alpha) = \sum n_i\deg(Z_i) $$ where $\deg(Z_i)$ is the degree of $Z_i \to \Spec(k)$, i.e., $\deg(Z_i) = \dim_k \Gamma(Z_i, \mathcal{O}_{Z_i})$.


chow-lemma-pre-derived-chern-class

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_X)$ be an object such that there exists a finite complex $\mathcal{E}^\bullet$ of finite locally free $\mathcal{O}_X$-modules representing $E$. Then $c(\mathcal{E}^\bullet) \in A^*(X)$, $ch(\mathcal{E}^\bullet) \in A^*(X) \otimes \mathbf{Q}$, and $P_p(\mathcal{E}^\bullet) \in A^p(X)$ are independent of the choice of the complex.


chow-lemma-chern-classes-perfect-dual

In Situation \ref{situation-setup} let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_X)$ be a perfect object whose Chern classes are defined. Then $c_i(E^\vee) = (-1)^i c_i(E)$, $P_i(E^\vee) = (-1)^iP_i(E)$, and $ch_i(E^\vee) = (-1)^ich_i(E)$ in $A^i(X)$.


chow-lemma-silly-tensor-invertible

In Lemma \ref{lemma-silly} assume $E_2$ has constant rank $0$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then $$ c'_i(E_2 \otimes \mathcal{L}) = \sum\nolimits_{j = 0}^i \binom{- i + j}{j} c'_{i - j}(E_2) c_1(\mathcal{L})^j $$


chow-lemma-independent-loc-chern

The localized class constructed above is independent of choices.


chow-lemma-loc-chern-classes-commute

In the situation of Definition \ref{definition-localized-chern} assume $P_p(Z \to X, E)$, resp.\ $c_p(Z \to X, E)$ is defined. Let $Y \to X$ be locally of finite type and $c \in A^*(Y \to X)$. Then $$ P_p(Z \to X, E) \circ c = c \circ P_p(Z \to X, E), $$ respectively $$ c_p(Z \to X, E) \circ c = c \circ c_p(Z \to X, E) $$ in $A^*(Y \times_X Z \to X)$.


chow-lemma-adams-and-chern

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. If $\psi^2$ is as in Lemma \ref{lemma-second-adams-operator} and $c$ and $ch$ are as in Remarks \ref{remark-chern-classes-K} and \ref{remark-chern-character-K} then we have $c_i(\psi^2(\alpha)) = 2^i c_i(\alpha)$ and $ch_i(\psi^2(\alpha)) = 2^i ch_i(\alpha)$ for all $\alpha \in K_0(\textit{Vect}(X))$.


chow-lemma-projection-formula-regular

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$ such that both $X$ and $Y$ are quasi-compact, regular, have affine diagonal, and finite dimension. Then $f$ is a local complete intersection morphism. Assume moreover the gysin map is defined for $f$ and that $f$ is proper. Then $$ f_*(\alpha \cdot f^!\beta) = f_*\alpha \cdot \beta $$ in $\CH^*(Y) \otimes \mathbf{Q}$ where the intersection product is as in Section \ref{section-intersection-regular}.


chow-lemma-exterior-product-well-defined

The map $\times : \CH_n(X) \otimes_{\mathbf{Z}} \CH_m(Y) \to \CH_{n + m}(X \times_k Y)$ is well defined.


chow-lemma-intersection-regular-smooth

Let $k$ be a field. Let $X$ be a smooth scheme over $k$ which is quasi-compact and has affine diagonal. Then the intersection product on $\CH^*(X)$ constructed in this section agrees after tensoring with $\mathbf{Q}$ with the intersection product constructed in Section \ref{section-intersection-regular}.


chow-lemma-associative-dim-1

The product defined above is associative. More precisely, with $(S, \delta)$ as above, let $X$ be smooth over $S$, let $Y, Z, W$ be schemes locally of finite type over $X$, let $\alpha \in \CH_*(Y)$, $\beta \in \CH_*(Z)$, $\gamma \in \CH_*(W)$. Then $(\alpha \cdot \beta) \cdot \gamma = \alpha \cdot (\beta \cdot \gamma)$ in $\CH_*(Y \times_X Z \times_X W)$.


chow-lemma-symbol-defined

Let $A$ be a Noetherian local ring. Let $a, b \in A$. \begin{enumerate} \item If $M$ is a finite $A$-module of dimension $1$ such that $a, b$ are nonzerodivisors on $M$, then $\text{length}_A(M/abM) < \infty$ and $(M/abM, a, b)$ is a $(2, 1)$-periodic exact complex. \item If $a, b$ are nonzerodivisors and $\dim(A) = 1$ then $\text{length}_A(A/(ab)) < \infty$ and $(A/(ab), a, b)$ is a $(2, 1)$-periodic exact complex. \end{enumerate} In particular, in these cases $\det_\kappa(M/abM, a, b) \in \kappa^*$, resp.\ $\det_\kappa(A/(ab), a, b) \in \kappa^*$ are defined.


chow-lemma-symbol-is-usual-tame-symbol

Let $A$ be a discrete valuation ring with fraction field $K$. For nonzero $x, y \in K$ we have $$ d_A(x, y) = (-1)^{\text{ord}_A(x)\text{ord}_A(y)} \frac{x^{\text{ord}_A(y)}}{y^{\text{ord}_A(x)}} \bmod \mathfrak m_A, $$ in other words the symbol is equal to the usual tame symbol.


chow-lemma-symbol-is-steinberg

Let $A$ be a Noetherian local domain of dimension $1$ with fraction field $K$. For $x \in K \setminus \{0, 1\}$ we have $$ d_A(x, 1 -x) = 1 $$


chow-lemma-secondary-ramification

\begin{reference} When $A$ is an excellent ring this is \cite[Proposition 1]{Kato-Milnor-K}. \end{reference} Let $A$ be a $2$-dimensional Noetherian local domain with fraction field $K$. Let $f, g \in K^*$. Let $\mathfrak q_1, \ldots, \mathfrak q_t$ be the height $1$ primes $\mathfrak q$ of $A$ such that either $f$ or $g$ is not an element of $A^*_{\mathfrak q}$. Then we have $$ \sum\nolimits_{i = 1, \ldots, t} \text{ord}_{A/\mathfrak q_i}(d_{A_{\mathfrak q_i}}(f, g)) = 0 $$ We can also write this as $$ \sum\nolimits_{\text{height}(\mathfrak q) = 1} \text{ord}_{A/\mathfrak q}(d_{A_{\mathfrak q}}(f, g)) = 0 $$ since at any height one prime $\mathfrak q$ of $A$ where $f, g \in A^*_{\mathfrak q}$ we have $d_{A_{\mathfrak q}}(f, g) = 1$ by Lemma \ref{lemma-symbol-when-one-is-a-unit}.


chow-lemma-coherent-sheaf-cap-c1

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Assume $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k + 1$. Then the element $$ [\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}] - [\mathcal{F}] \in K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X)) $$ lies in the subgroup $B_k(X)$ of Lemma \ref{lemma-cycles-rational-equivalence-K-group} and maps to the element $c_1(\mathcal{L}) \cap [\mathcal{F}]_{k + 1}$ via the map $B_k(X) \to \CH_k(X)$.


flat-theorem-existence

In Situation \ref{situation-existence} there exists a finitely presented $\mathcal{O}_X$-module $\mathcal{F}$, flat over $A$, with support proper over $A$, such that $\mathcal{F}_n = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{O}_{X_n}$ for all $n$ compatibly with the maps $\varphi_n$.


flat-theorem-existence-derived

In Situation \ref{situation-existence-derived} there exists a pseudo-coherent $K$ in $D(\mathcal{O}_X)$ such that $K_n = K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_{X_n}$ for all $n$ compatibly with the maps $\varphi_n$.


flat-lemma-finite-type-flat-at-point-local-X

Let $f : X \to S$ be a morphism which is locally of finite type. Let $x \in X$ with image $s \in S$. If $f$ is flat at $x$ over $S$, then $\mathcal{O}_{X, x}$ is essentially of finite presentation over $\mathcal{O}_{S, s}$.


flat-lemma-finite-presentation-flat-along-fibre-X

Let $f : X \to S$ be of finite presentation. Let $s \in S$. If $X$ is flat over $S$ at all points of $X_s$, then there exists an elementary \'etale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram of schemes $$ \xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ S & S' \ar[l] } $$ with $g$ \'etale, $X_s \subset g(X')$, such that $X'$, $S'$ are affine, and such that $\Gamma(X', \mathcal{O}_{X'})$ is a projective $\Gamma(S', \mathcal{O}_{S'})$-module.


flat-lemma-weak-bourbaki

Let $R \to S$ be a ring map which is essentially of finite type. Let $N$ be a localization of a finite $S$-module flat over $R$. Let $M$ be an $R$-module. Then $$ \text{WeakAss}_S(M \otimes_R N) = \bigcup\nolimits_{\mathfrak p \in \text{WeakAss}_R(M)} \text{Ass}_{S \otimes_R \kappa(\mathfrak p)}(N \otimes_R \kappa(\mathfrak p)) $$


flat-lemma-quasi-finite-pure

Let $f : X \to S$ be a separated, finite type morphism of schemes. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. Assume that $\text{Supp}(\mathcal{F}_s)$ is finite for every $s \in S$. Then the following are equivalent \begin{enumerate} \item $\mathcal{F}$ is pure relative to $S$, \item the scheme theoretic support of $\mathcal{F}$ is finite over $S$, and \item $\mathcal{F}$ is universally pure relative to $S$. \end{enumerate} In particular, given a quasi-finite separated morphism $X \to S$ we see that $X$ is pure relative to $S$ if and only if $X \to S$ is finite.


flat-lemma-flat-geometrically-integral-fibres-pure

Let $f : X \to S$ be a finite type, flat morphism of schemes with geometrically integral fibres. Then $X$ is universally pure over $S$.


flat-lemma-flat

In Situation \ref{situation-flat}. \begin{enumerate} \item The functor $F_{flat}$ satisfies the sheaf property for the fpqc topology. \item If $f$ is quasi-compact and locally of finite presentation and $\mathcal{F}$ is of finite presentation, then the functor $F_{flat}$ is limit preserving. \end{enumerate}


flat-lemma-flattening-stratification-artinian

Let $S$ be the spectrum of an Artinian ring. For any scheme $X$ over $S$, and any quasi-coherent $\mathcal{O}_X$-module there exists a universal flattening. In fact the universal flattening is given by a closed immersion $S' \to S$, and hence is a flattening stratification for $\mathcal{F}$ as well.


flat-lemma-Noetherian-finite-type-injective-into-flat-mod-m

If in Situation \ref{situation-mod-injective} the ring $A$ is Noetherian then the lemma holds.


flat-lemma-valuation-ring-finite-type-injective-into-flat-mod-m

If in Situation \ref{situation-mod-injective} the ring $A$ is a valuation ring then the lemma holds.


flat-lemma-get-section-after-blowup

Let $S$ be a scheme. Let $U \subset W \subset S$ be open subschemes. Let $f : X \to W$ be a morphism and let $s : U \to X$ be a morphism such that $f \circ s = \text{id}_U$. Assume \begin{enumerate} \item $f$ is proper, \item $S$ is quasi-compact and quasi-separated, and \item $U$ and $W$ are quasi-compact. \end{enumerate} Then there exists a $U$-admissible blowup $b : S' \to S$ and a morphism $s' : b^{-1}(W) \to X$ extending $s$ with $f \circ s' = b|_{b^{-1}(W)}$.


flat-lemma-invert-right-multiplicative-system

Let $S$ be a quasi-compact and quasi-separated scheme. The functor $(X, \overline{X}) \mapsto X$ defines an equivalence from the category of compactifications localized (Categories, Lemma \ref{categories-lemma-right-localization}) at the right multiplicative system of Lemma \ref{lemma-right-multiplicative-system} to the category of compactifyable schemes over $S$.


flat-lemma-refine-by-h

Let $T$ be a scheme. Let $\{f_i : T_i \to T\}_{i \in I}$ be a family of morphisms such that $f_i$ is locally of finite presentation for all $i$. The following are equivalent \begin{enumerate} \item $\{T_i \to T\}_{i \in I}$ is an h covering, \item there is an h covering which refines $\{T_i \to T\}_{i \in I}$, and \item $\{\coprod_{i \in I} T_i \to T\}$ is an h covering. \end{enumerate}


flat-lemma-verify-site-h

Let $S$ be a scheme. Let $\Sch_h$ be a big h site containing $S$. Then $(\textit{Aff}/S)_h$ is a site.


flat-lemma-affine-big-site-h

Let $S$ be a scheme. Let $\Sch_h$ be a big h site containing $S$. The functor $(\textit{Aff}/S)_h \to (\Sch/S)_h$ is cocontinuous and induces an equivalence of topoi from $\Sh((\textit{Aff}/S)_h)$ to $\Sh((\Sch/S)_h)$.


flat-lemma-composition-h

Given schemes $X$, $Y$, $Y$ in $(\Sch/S)_h$ and morphisms $f : X \to Y$, $g : Y \to Z$ we have $g_{big} \circ f_{big} = (g \circ f)_{big}$.


flat-lemma-blow-up-square-ph

Let $\mathcal{F}$ be a sheaf on a site $(\Sch/S)_{ph}$, see Topologies, Definition \ref{topologies-definition-big-small-ph}. Then for any blow up square (\ref{equation-blow-up-square}) in the category $(\Sch/S)_{ph}$ the diagram $$ \xymatrix{ \mathcal{F}(E) & \mathcal{F}(X') \ar[l] \\ \mathcal{F}(Z) \ar[u] & \mathcal{F}(X) \ar[u] \ar[l] } $$ is cartesian in the category of sets.


flat-lemma-thickening-ph

Let $\mathcal{F}$ be a sheaf on a site $(\Sch/S)_{ph}$ as in Topologies, Definition \ref{topologies-definition-big-small-ph}. Let $X \to X'$ be a morphism of $(\Sch/S)_{ph}$ which is a thickening. Then $\mathcal{F}(X') \to \mathcal{F}(X)$ is bijective.


flat-lemma-h-sheaf-lim-F

Let $p$ be a prime number. Let $S$ be a scheme over $\mathbf{F}_p$. Let $(\Sch/S)_h$ be a site as in Definition \ref{definition-big-small-h}. The rule $$ \mathcal{F}(X) = \lim_F \Gamma(X, \mathcal{O}_X) $$ defines a sheaf on $(\Sch/S)_h$.


flat-lemma-blowup-map-pd1

Let $X$ be a scheme. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a homorphism of perfect $\mathcal{O}_X$-modules of tor dimension $\leq 1$. Let $U \subset X$ be a scheme theoretically dense open such that $\mathcal{F}|_U = 0$ and $\mathcal{G}|_U = 0$. Then there is a $U$-admissible blowup $b : X' \to X$ such that the kernel, image, and cokernel of $b^*\varphi$ are perfect $\mathcal{O}_{X'}$-modules of tor dimension $\leq 1$.


flat-proposition-finite-type-injective-into-flat-mod-m

Let $A \to B$ be a local ring homomorphism of local rings which is essentially of finite type. Let $M$ be a flat $A$-module, $N$ a finite $B$-module and $u : N \to M$ an $A$-module map such that $\overline{u} : N/\mathfrak m_AN \to M/\mathfrak m_AM$ is injective. Then $u$ is $A$-universally injective, $N$ is of finite presentation over $B$, and $N$ is flat over $A$.


curves-lemma-extend-to-completion

Let $k$ be a field. Let $X \to Y$ be a morphism of varieties with $Y$ proper and $X$ a curve. There exists a factorization $X \to \overline{X} \to Y$ where $X \to \overline{X}$ is an open immersion and $\overline{X}$ is a projective curve.


curves-lemma-nonsingular-model-smooth

Let $k$ be a field. Let $X$ be a curve and let $Y$ be the nonsingular projective model of $X$. If $k$ is perfect, then $Y$ is a smooth projective curve.


curves-lemma-equation-plane-curve

Let $Z \subset \mathbf{P}^2_k$ be a closed subscheme which is equidimensional of dimension $1$ and has no embedded points (equivalently $Z$ is Cohen-Macaulay). Then the ideal $I(Z) \subset k[T_0, T_1, T_2]$ corresponding to $Z$ is principal.


curves-lemma-genus-zero-singular

Let $X$ be a proper curve over a field $k$ with $H^0(X, \mathcal{O}_X) = k$. Assume $X$ is singular and has genus $0$. Then there exists a diagram $$ \xymatrix{ x' \ar[d] \ar[r] & X' \ar[d]^\nu \ar[r] & \Spec(k') \ar[d] \\ x \ar[r] & X \ar[r] & \Spec(k) } $$ where \begin{enumerate} \item $k'/k$ is a nontrivial finite extension, \item $X' \cong \mathbf{P}^1_{k'}$, \item $x'$ is a $k'$-rational point of $X'$, \item $x$ is a $k$-rational point of $X$, \item $X' \setminus \{x'\} \to X \setminus \{x\}$ is an isomorphism, \item $0 \to \mathcal{O}_X \to \nu_*\mathcal{O}_{X'} \to k'/k \to 0$ is a short exact sequence where $k'/k = \kappa(x')/\kappa(x)$ indicates the skyscraper sheaf on the point $x$. \end{enumerate}


curves-lemma-rhe

Notation and assumptions as in Lemma \ref{lemma-rh}. For a closed point $x \in X$ let $d_x$ be the multiplicity of $x$ in $R$. Then $$ 2g_X - 2 = (2g_Y - 2) \deg(f) + \sum\nolimits d_x [\kappa(x) : k] $$ Moreover, we have the following results \begin{enumerate} \item $d_x = \text{length}_{\mathcal{O}_{X, x}}(\Omega_{X/Y, x})$, \item $d_x \geq e_x - 1$ where $e_x$ is the ramification index of $\mathcal{O}_{X, x}$ over $\mathcal{O}_{Y, y}$, \item $d_x = e_x - 1$ if and only if $\mathcal{O}_{X, x}$ is tamely ramified over $\mathcal{O}_{Y, y}$. \end{enumerate}


curves-lemma-dominated-by-smooth

Let $k$ be a field. Let $f : X \to Y$ be a surjective morphism of curves over $k$. If $X$ is smooth over $k$ and $Y$ is normal, then $Y$ is smooth over $k$.


curves-lemma-purely-inseparable-smooth-genus

Let $k$ be a field of characteristic $p > 0$. Let $f : X \to Y$ be a nonconstant morphism of proper nonsingular curves over $k$. Assume \begin{enumerate} \item $X$ is smooth, \item $H^0(X, \mathcal{O}_X) = k$, \item $k(X)/k(Y)$ is purely inseparable. \end{enumerate} Then $Y$ is smooth, $H^0(Y, \mathcal{O}_Y) = k$, and the genus of $Y$ is equal to the genus of $X$.


curves-lemma-ramification-to-algebraic-closure

Let $X$ be a smooth curve over a field $k$. Let $\overline{x} \in X_{\overline{k}}$ be a closed point with image $x \in X$. The ramification index of $\mathcal{O}_{X, x} \subset \mathcal{O}_{X_{\overline{k}}, \overline{x}}$ is the inseparable degree of $\kappa(x)/k$.


curves-lemma-complete-local-ring-pushout

In the situation above, let $Z = \Spec(k')$ where $k'$ is a field and $Z' = \Spec(k'_1 \times \ldots \times k'_n)$ with $k'_i/k'$ finite extensions of fields. Let $x \in X$ be the image of $Z \to X$ and $x'_i \in X'$ the image of $\Spec(k'_i) \to X'$. Then we have a fibre product diagram $$ \xymatrix{ \prod\nolimits_{i = 1, \ldots, n} k'_i & \prod\nolimits_{i = 1, \ldots, n} \mathcal{O}_{X', x'_i}^\wedge \ar[l] \\ k' \ar[u] & \mathcal{O}_{X, x}^\wedge \ar[u] \ar[l] } $$ where the horizontal arrows are given by the maps to the residue fields.


curves-lemma-multicross-gorenstein-is-nodal

Let $k$ be an algebraically closed field. Let $X$ be a reduced algebraic $1$-dimensional $k$-scheme. Let $x \in X$ be a multicross singularity (Definition \ref{definition-multicross}). If $X$ is Gorenstein, then $x$ is a node.


curves-lemma-bound-torsion-simple

Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $\ell$ be a prime number invertible in $k$. Then $$ \dim_{\mathbf{F}_\ell} \Pic(X)[\ell] \leq \dim_k H^1(X, \mathcal{O}_X) + g_{geom}(X/k) $$ where $g_{geom}(X/k)$ is as defined above.


curves-lemma-degree-more-than-2g-1-and-Z

In Situation \ref{situation-Cohen-Macaulay-curve} assume $X$ is integral and has genus $g$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $Z \subset X$ be a nonempty $0$-dimensional closed subscheme. If $\deg(\mathcal{L}) \geq 2g - 1 + \deg(Z)$, then $\mathcal{L}$ is globally generated and $H^0(X, \mathcal{L}) \to H^0(X, \mathcal{L}|_Z)$ is surjective.


etale-cohomology-theorem-cech-derived

On $\textit{PAb}(\mathcal{C})$ the functors $\check{H}^p(\mathcal{U}, -)$ are the right derived functors of $\check{H}^0(\mathcal{U}, -)$.


etale-cohomology-theorem-standard-etale

A ring map $A \to B$ is \'etale at a prime $\mathfrak q$ if and only if there exists $g \in B$, $g \not \in \mathfrak q$ such that $B_g$ is standard \'etale over $A$.


etale-cohomology-theorem-hensel

Complete local rings are henselian.


etale-cohomology-theorem-fundamental-exact-sequence

There is a short exact sequence of \'etale sheaves on $X$ $$ 0 \longrightarrow \mathbf{G}_{m, X} \longrightarrow j_* \mathbf{G}_{m, \eta} \longrightarrow \bigoplus\nolimits_{x \in X_0} {i_x}_* \underline{\mathbf{Z}} \longrightarrow 0. $$


etale-cohomology-lemma-yoneda

\begin{slogan} Morphisms between objects are in bijection with natural transformations between the functors they represent. \end{slogan} Let $\mathcal{C}$ be a category, and $X, Y \in \Ob(\mathcal{C})$. There is a natural bijection $$ \begin{matrix} \Mor_\mathcal{C}(X, Y) & \longrightarrow & \Mor_{\textit{PSh}(\mathcal{C})} (h_X, h_Y) \\ \psi & \longmapsto & h_\psi = \psi \circ - : h_X \to h_Y. \end{matrix} $$


etale-cohomology-lemma-algebra-descent

If $A \to B$ is faithfully flat, then the complex $(B/A)_\bullet$ is exact in positive degrees, and $H^0((B/A)_\bullet) = A$.


etale-cohomology-lemma-cech-presheaves

The functor $\check{\mathcal{C}}^\bullet(\mathcal{U}, -)$ is exact on the category $\textit{PAb}(\mathcal{C})$.


etale-cohomology-lemma-yoneda-presheaf

For any presheaf $\mathcal{F}$ on a category $\mathcal{C}$ there is a functorial isomorphism $$ \Hom_{\textit{PSh}(\mathcal{C})}(h_U, \mathcal{F}) = \mathcal{F}(U). $$


etale-cohomology-lemma-cech-complex-describe

The {\v C}ech complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$ can be described explicitly as follows \begin{eqnarray*} \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) & = & \left( \prod_{i_0 \in I} \Hom_{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_{U_{i_0}}, \mathcal{F}) \to \prod_{i_0, i_1 \in I} \Hom_{\textit{PAb}(\mathcal{C})}( \mathbf{Z}_{U_{i_0} \times_U U_{i_1}}, \mathcal{F}) \to \ldots \right) \\ & = & \Hom_{\textit{PAb}(\mathcal{C})}\left( \left( \bigoplus_{i_0 \in I} \mathbf{Z}_{U_{i_0}} \leftarrow \bigoplus_{i_0, i_1 \in I} \mathbf{Z}_{U_{i_0} \times_U U_{i_1}} \leftarrow \ldots \right), \mathcal{F}\right) \end{eqnarray*}


etale-cohomology-lemma-exact

The complex of abelian presheaves \begin{align*} \mathbf{Z}_\mathcal{U}^\bullet \quad : \quad \bigoplus_{i_0 \in I} \mathbf{Z}_{U_{i_0}} \leftarrow \bigoplus_{i_0, i_1 \in I} \mathbf{Z}_{U_{i_0} \times_U U_{i_1}} \leftarrow \bigoplus_{i_0, i_1, i_2 \in I} \mathbf{Z}_{U_{i_0} \times_U U_{i_1} \times_U U_{i_2}} \leftarrow \ldots \end{align*} is exact in all degrees except $0$ in $\textit{PAb}(\mathcal{C})$.


etale-cohomology-lemma-tau-affine

Let $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$. Any $\tau$-covering of an affine scheme can be refined by a standard $\tau$-covering.


etale-cohomology-lemma-etale-topos-independent-partial-universe

Let $S$ be a scheme. The \'etale topos of $S$ is independent (up to canonical equivalence) of the construction of the small \'etale site in Definition \ref{definition-tau-site}.


etale-cohomology-lemma-etale-fpqc

Any \'etale covering is an fpqc covering.


etale-cohomology-lemma-support-section-closed

Let $S$ be a scheme. Let $\mathcal{F}$ be an abelian sheaf on $S_\etale$. Let $U \in \Ob(S_\etale)$ and $\sigma \in \mathcal{F}(U)$. \begin{enumerate} \item The support of $\sigma$ is closed in $U$. \item The support of $\sigma + \sigma'$ is contained in the union of the supports of $\sigma, \sigma' \in \mathcal{F}(U)$. \item If $\varphi : \mathcal{F} \to \mathcal{G}$ is a map of abelian sheaves on $S_\etale$, then the support of $\varphi(\sigma)$ is contained in the support of $\sigma \in \mathcal{F}(U)$. \item The support of $\mathcal{F}$ is the union of the images of the supports of all local sections of $\mathcal{F}$. \item If $\mathcal{F} \to \mathcal{G}$ is surjective then the support of $\mathcal{G}$ is a subset of the support of $\mathcal{F}$. \item If $\mathcal{F} \to \mathcal{G}$ is injective then the support of $\mathcal{F}$ is a subset of the support of $\mathcal{G}$. \end{enumerate}


etale-cohomology-lemma-describe-henselization

Let $S$ be a scheme. Let $s \in S$. Then we have $$ \mathcal{O}_{S, s}^h = \colim_{(U, u)} \mathcal{O}(U) $$ where the colimit is over the filtered category of \'etale neighbourhoods $(U, u)$ of $(S, s)$ such that $\kappa(s) = \kappa(u)$.


etale-cohomology-lemma-sections-upstairs-submersive

Let $S$ be a scheme. Let $f : X \to S$ be a morphism such that \begin{enumerate} \item $f$ is submersive, and \item the geometric fibres of $f$ are connected. \end{enumerate} Let $\mathcal{F}$ be a sheaf on $S_\etale$. Then $\Gamma(S, \mathcal{F}) = \Gamma(X, f^{-1}_{small}\mathcal{F})$.


etale-cohomology-lemma-integral-universally-injective

Let $f : X \to Y$ be a morphism of schemes. Assume that $f$ is universally injective and integral (for example a closed immersion). Then \begin{enumerate} \item $f_{small, *} : \Sh(X_\etale) \to \Sh(Y_\etale)$ reflects injections and surjections, \item $f_{small, *} : \Sh(X_\etale) \to \Sh(Y_\etale)$ commutes with pushouts and coequalizers (and more generally finite connected colimits), \item $f_{small, *}$ transforms surjections into surjections (on sheaves of sets and on abelian sheaves), \item the map $f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}$ is surjective for any sheaf (of sets or of abelian groups) $\mathcal{F}$ on $X_\etale$, \item the functor $f_{small, *}$ is faithful (on sheaves of sets and on abelian sheaves), \item $f_{small, *} : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$ is exact, and \item the functor $Y_\etale \to X_\etale$, $V \mapsto X \times_Y V$ is almost cocontinuous. \end{enumerate}


etale-cohomology-lemma-modules-abelian

Let $G$ be a topological group. Let $R$ be a ring. For every $i \geq 0$ the diagram $$ \xymatrix{ \text{Mod}_{R, G} \ar[rr]_{H^i(G, -)} \ar[d] & & \text{Mod}_R \ar[d] \\ \text{Mod}_G \ar[rr]^{H^i(G, -)} & & \textit{Ab} } $$ whose vertical arrows are the forgetful functors is commutative.