decent-spaces-proposition-reasonable-open-dense-scheme

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is reasonable, then there exists a dense open subspace of $X$ which is a scheme.


decent-spaces-proposition-reasonable-sober

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Then the topological space $|X|$ is sober (see Topology, Definition \ref{topology-definition-generic-point}).


decent-spaces-proposition-characterize-universally-closed

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact, and $X$ is decent. Then $f$ is universally closed if and only if the existence part of the valuative criterion holds.


groupoids-lemma-restrict-relation

Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j : R \to U \times_S U$ be a pre-relation. Let $g : U' \to U$ be a morphism of schemes. Finally, set $$ R' = (U' \times_S U')\times_{U \times_S U} R \xrightarrow{j'} U' \times_S U' $$ Then $j'$ is a pre-relation on $U'$ over $S$. If $j$ is a relation, then $j'$ is a relation. If $j$ is a pre-equivalence relation, then $j'$ is a pre-equivalence relation. If $j$ is an equivalence relation, then $j'$ is an equivalence relation.


groupoids-lemma-pre-equivalence-equivalence-relation-points

Let $j : R \to U \times_S U$ be a pre-relation. Consider the relation on points of the scheme $U$ defined by the rule $$ x \sim y \Leftrightarrow \exists\ r \in R : t(r) = x, s(r) = y. $$ If $j$ is a pre-equivalence relation then this is an equivalence relation.


groupoids-lemma-etale-equivalence-relation

Let $j : R \to U \times_S U$ be a pre-relation. Assume \begin{enumerate} \item $s, t$ are unramified, \item for any algebraically closed field $k$ over $S$ the map $R(k) \to U(k) \times U(k)$ is an equivalence relation, \item there are morphisms $e : U \to R$, $i : R \to R$, $c : R \times_{s, U, t} R \to R$ such that $$ \xymatrix{ U \ar[r]_e \ar[d]_\Delta & R \ar[d]_j & R \ar[d]^j \ar[r]_i & R \ar[d]^j & R \times_{s, U, t} R \ar[d]^{j \times j} \ar[r]_c & R \ar[d]^j \\ U \times_S U \ar[r] & U \times_S U & U \times_S U \ar[r]^{flip} & U \times_S U & U \times_S U \times_S U \ar[r]^{\text{pr}_{02}} & U \times_S U } $$ are commutative. \end{enumerate} Then $j$ is an equivalence relation.


groupoids-lemma-base-change-group-scheme

Let $(G, m)$ be a group scheme over $S$. Let $S' \to S$ be a morphism of schemes. The pullback $(G_{S'}, m_{S'})$ is a group scheme over $S'$.


groupoids-lemma-closed-subgroup-scheme

Let $S$ be a scheme. Let $(G, m, e, i)$ be a group scheme over $S$. \begin{enumerate} \item A closed subscheme $H \subset G$ is a closed subgroup scheme if and only if $e : S \to G$, $m|_{H \times_S H} : H \times_S H \to G$, and $i|_H : H \to G$ factor through $H$. \item An open subscheme $H \subset G$ is an open subgroup scheme if and only if $e : S \to G$, $m|_{H \times_S H} : H \times_S H \to G$, and $i|_H : H \to G$ factor through $H$. \end{enumerate}


groupoids-lemma-group-scheme-separated

Let $S$ be a scheme. Let $G$ be a group scheme over $S$. Then $G \to S$ is separated (resp.\ quasi-separated) if and only if the identity morphism $e : S \to G$ is a closed immersion (resp.\ quasi-compact).


groupoids-lemma-group-scheme-module-differentials

\begin{reference} \cite[Proposition 3.15]{BookAV} \end{reference} Let $(G, m, e, i)$ be a group scheme over the scheme $S$. Denote $f : G \to S$ the structure morphism. Then there exist canonical isomorphisms $$ \Omega_{G/S} \cong f^*\mathcal{C}_{S/G} \cong f^*e^*\Omega_{G/S} $$ where $\mathcal{C}_{S/G}$ denotes the conormal sheaf of the immersion $e$. In particular, if $S$ is the spectrum of a field, then $\Omega_{G/S}$ is a free $\mathcal{O}_G$-module.


groupoids-lemma-group-scheme-addition-tangent-vectors

Let $S$ be a scheme. Let $G$ be a group scheme over $S$. Let $s \in S$. Then the composition $$ T_{G/S, e(s)} \oplus T_{G/S, e(s)} = T_{G \times_S G/S, (e(s), e(s))} \rightarrow T_{G/S, e(s)} $$ is addition of tangent vectors. Here the $=$ comes from Varieties, Lemma \ref{varieties-lemma-tangent-space-product} and the right arrow is induced from $m : G \times_S G \to G$ via Varieties, Lemma \ref{varieties-lemma-map-tangent-spaces}.


groupoids-lemma-group-scheme-over-field-open-multiplication

If $(G, m)$ is a group scheme over a field $k$, then the multiplication map $m : G \times_k G \to G$ is open.


groupoids-lemma-group-scheme-over-field-translate-open

If $(G, m)$ is a group scheme over a field $k$. Let $U \subset G$ open and $T \to G$ a morphism of schemes. Then the image of the composition $T \times_k U \to G \times_k G \to G$ is open.


groupoids-lemma-group-scheme-over-field-separated

Let $G$ be a group scheme over a field. Then $G$ is a separated scheme.


groupoids-lemma-group-scheme-field-geometrically-irreducible

Let $G$ be a group scheme over a field $k$. Then \begin{enumerate} \item every local ring $\mathcal{O}_{G, g}$ of $G$ has a unique minimal prime ideal, \item there is exactly one irreducible component $Z$ of $G$ passing through $e$, and \item $Z$ is geometrically irreducible over $k$. \end{enumerate}


groupoids-lemma-reduced-subgroup-scheme-perfect

Let $G$ be a group scheme over a perfect field $k$. Then the reduction $G_{red}$ of $G$ is a closed subgroup scheme of $G$.


groupoids-lemma-open-subgroup-closed-over-field

Let $k$ be a field. Let $\psi : G' \to G$ be a morphism of group schemes over $k$. If $\psi(G')$ is open in $G$, then $\psi(G')$ is closed in $G$.


groupoids-lemma-irreducible-group-scheme-over-field-qc

Let $G$ be a group scheme over a field $k$. If $G$ is irreducible, then $G$ is quasi-compact.


groupoids-lemma-connected-group-scheme-over-field-irreducible

Let $G$ be a group scheme over a field $k$. If $G$ is connected, then $G$ is irreducible.


groupoids-lemma-profinite-product-over-field

Let $k$ be a field. Let $T = \Spec(A)$ where $A$ is a directed colimit of algebras which are finite products of copies of $k$. For any scheme $X$ over $k$ we have $|T \times_k X| = |T| \times |X|$ as topological spaces.


groupoids-lemma-compact-set-in-affine

Let $k$ be an algebraically closed field. Let $G$ be a group scheme over $k$. Assume that $G$ is Jacobson and that all closed points are $k$-rational. Let $T = \Spec(A)$ where $A$ is a directed colimit of algebras which are finite products of copies of $k$. For any morphism $f : T \to G$ there exists an affine open $U \subset G$ containing $f(T)$.


groupoids-lemma-group-scheme-finite-type-field

Let $k$ be a field. Let $G$ be a locally algebraic group scheme over $k$. Then $G$ is equidimensional and $\dim(G) = \dim_g(G)$ for all $g \in G$. For any closed point $g \in G$ we have $\dim(G) = \dim(\mathcal{O}_{G, g})$.


groupoids-lemma-group-scheme-characteristic-zero-smooth

Let $k$ be a field of characteristic $0$. Let $G$ be a locally algebraic group scheme over $k$. Then the structure morphism $G \to \Spec(k)$ is smooth, i.e., $G$ is a smooth group scheme.


groupoids-lemma-reduced-group-scheme-prefect-field-characteristic-p-smooth

Let $k$ be a perfect field of characteristic $p > 0$ (see Lemma \ref{lemma-group-scheme-characteristic-zero-smooth} for the characteristic zero case). Let $G$ be a locally algebraic group scheme over $k$. If $G$ is reduced then the structure morphism $G \to \Spec(k)$ is smooth, i.e., $G$ is a smooth group scheme.


groupoids-lemma-points-in-affine

Let $k$ be an algebraically closed field. Let $G$ be a locally algebraic group scheme over $k$. Let $g_1, \ldots, g_n \in G(k)$ be $k$-rational points. Then there exists an affine open $U \subset G$ containing $g_1, \ldots, g_n$.


groupoids-lemma-algebraic-quasi-projective

Let $k$ be a field. Let $G$ be an algebraic group scheme over $k$. Then $G$ is quasi-projective over $k$.


groupoids-lemma-abelian-variety-projective

Let $k$ be a field. Let $A$ be an abelian variety over $k$. Then $A$ is projective.


groupoids-lemma-abelian-variety-change-field

Let $k$ be a field. Let $A$ be an abelian variety over $k$. For any field extension $K/k$ the base change $A_K$ is an abelian variety over $K$.


groupoids-lemma-abelian-variety-smooth

Let $k$ be a field. Let $A$ be an abelian variety over $k$. Then $A$ is smooth over $k$.


groupoids-lemma-abelian-variety-abelian

An abelian variety is an abelian group scheme, i.e., the group law is commutative.


groupoids-lemma-apply-cube

Let $k$ be a field. Let $A$ be an abelian variety over $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_A$-module. Then there is an isomorphism $$ m_{1, 2, 3}^*\mathcal{L} \otimes m_1^*\mathcal{L} \otimes m_2^*\mathcal{L} \otimes m_3^*\mathcal{L} \cong m_{1, 2}^*\mathcal{L} \otimes m_{1, 3}^*\mathcal{L} \otimes m_{2, 3}^*\mathcal{L} $$ of invertible modules on $A \times_k A \times_k A$ where $m_{i_1, \ldots, i_t} : A \times_k A \times_k A \to A$ is the morphism $(x_1, x_2, x_3) \mapsto \sum x_{i_j}$.


groupoids-lemma-pullbacks-by-n

Let $k$ be a field. Let $A$ be an abelian variety over $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_A$-module. Then $$ [n]^*\mathcal{L} \cong \mathcal{L}^{\otimes n(n + 1)/2} \otimes ([-1]^*\mathcal{L})^{\otimes n(n - 1)/2} $$ where $[n] : A \to A$ sends $x$ to $x + x + \ldots + x$ with $n$ summands and where $[-1] : A \to A$ is the inverse of $A$.


groupoids-lemma-degree-multiplication-by-d

Let $k$ be a field. Let $A$ be an abelian variety over $k$. Let $[d] : A \to A$ be the multiplication by $d$. Then $[d]$ is finite locally free of degree $d^{2\dim(A)}$.


groupoids-lemma-abelian-variety-multiplication-by-d-etale

\begin{slogan} Multiplication by an integer on an abelian variety is an etale morphism if and only if the integer is invertible in the base field. \end{slogan} Let $k$ be a field. Let $A$ be a nonzero abelian variety over $k$. Then $[d] : A \to A$ is \'etale if and only if $d$ is invertible in $k$.


groupoids-lemma-abelian-variety-multiplication-by-p

Let $k$ be a field of characteristic $p > 0$. Let $A$ be an abelian variety over $k$. The fibre of $[p] : A \to A$ over $0$ has at most $p^g$ distinct points.


groupoids-lemma-free-action

Situation as in Definition \ref{definition-free-action}, The action $a$ is free if and only if $$ G \times_S X \to X \times_S X, \quad (g, x) \mapsto (a(g, x), x) $$ is a monomorphism.


groupoids-lemma-characterize-trivial-pseudo-torsors

In the situation of Definition \ref{definition-pseudo-torsor}. \begin{enumerate} \item The scheme $X$ is a pseudo $G$-torsor if and only if for every scheme $T$ over $S$ the set $X(T)$ is either empty or the action of the group $G(T)$ on $X(T)$ is simply transitive. \item A pseudo $G$-torsor $X$ is trivial if and only if the morphism $X \to S$ has a section. \end{enumerate}


groupoids-lemma-torsor

Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$. Let $X$ be a scheme over $S$, and let $a : G \times_S X \to X$ be an action of $G$ on $X$. Let $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$. Then $X$ is a $G$-torsor in the $\tau$-topology if and only if $\underline{X}$ is a $\underline{G}$-torsor on $(\Sch/S)_\tau$.


groupoids-lemma-pullback-equivariant

Let $S$ be a scheme. Let $G$ be a group scheme over $S$. Let $f : Y \to X$ be a $G$-equivariant morphism between $S$-schemes endowed with $G$-actions. Then pullback $f^*$ given by $(\mathcal{F}, \alpha) \mapsto (f^*\mathcal{F}, (1_G \times f)^*\alpha)$ defines a functor from the category of $G$-equivariant quasi-coherent $\mathcal{O}_X$-modules  to the category of $G$-equivariant quasi-coherent $\mathcal{O}_Y$-modules.


groupoids-lemma-complete-reducibility-Gm

Let $a : \mathbf{G}_m \times X \to X$ be an action on an affine scheme. Then $X$ is the spectrum of a $\mathbf{Z}$-graded ring and the action is as in Example \ref{example-Gm-on-affine}.


groupoids-lemma-Gm-equivariant-module

Let $A$ be a graded ring. Let $X = \Spec(A)$ with action $a : \mathbf{G}_m \times X \to X$ as in Example \ref{example-Gm-on-affine}. Let $\mathcal{F}$ be a $\mathbf{G}_m$-equivariant quasi-coherent $\mathcal{O}_X$-module. Then $M = \Gamma(X, \mathcal{F})$ has a canonical grading such that it is a graded $A$-module and such that the isomorphism $\widetilde{M} \to \mathcal{F}$ (Schemes, Lemma \ref{schemes-lemma-quasi-coherent-affine}) is an isomorphism of $\mathbf{G}_m$-equivariant modules where the $\mathbf{G}_m$-equivariant structure on $\widetilde{M}$ is the one from Example \ref{example-Gm-on-affine}.


groupoids-lemma-groupoid-pre-equivalence

Given a groupoid scheme $(U, R, s, t, c)$ over $S$ the morphism $j : R \to U \times_S U$ is a pre-equivalence relation.


groupoids-lemma-equivalence-groupoid

Given an equivalence relation $j : R \to U$ over $S$ there is a unique way to extend it to a groupoid $(U, R, s, t, c)$ over $S$.


groupoids-lemma-diagram

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. In the commutative diagram $$ \xymatrix{ & U & \\ R \ar[d]_s \ar[ru]^t & R \times_{s, U, t} R \ar[l]^-{\text{pr}_0} \ar[d]^{\text{pr}_1} \ar[r]_-c & R \ar[d]^s \ar[lu]_t \\ U & R \ar[l]_t \ar[r]^s & U } $$ the two lower squares are fibre product squares. Moreover, the triangle on top (which is really a square) is also cartesian.


groupoids-lemma-diagram-pull

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


groupoids-lemma-base-change-groupoid

Let $(U, R, s, t, c)$ be a groupoid over a scheme $S$. Let $S' \to S$ be a morphism. Then the base changes $U' = S' \times_S U$, $R' = S' \times_S R$ endowed with the base changes $s'$, $t'$, $c'$ of the morphisms $s, t, c$ form a groupoid scheme $(U', R', s', t', c')$ over $S'$ and the projections determine a morphism $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoid schemes over $S$.


groupoids-lemma-pullback

Let $S$ be a scheme. Consider a morphism $f : (U, R, s, t, c) \to (U', R', s', t', c')$ of groupoid schemes over $S$. Then pullback $f^*$ given by $$ (\mathcal{F}, \alpha) \mapsto (f^*\mathcal{F}, f^*\alpha) $$ defines a functor from the category of quasi-coherent sheaves on $(U', R', s', t', c')$ to the category of quasi-coherent sheaves on $(U, R, s, t, c)$.


groupoids-lemma-pushforward

Let $S$ be a scheme. Consider a morphism $f : (U, R, s, t, c) \to (U', R', s', t', c')$ of groupoid schemes over $S$. Assume that \begin{enumerate} \item $f : U \to U'$ is quasi-compact and quasi-separated, \item the square $$ \xymatrix{ R \ar[d]_t \ar[r]_f & R' \ar[d]^{t'} \\ U \ar[r]^f & U' } $$ is cartesian, and \item $s'$ and $t'$ are flat. \end{enumerate} Then pushforward $f_*$ given by $$ (\mathcal{F}, \alpha) \mapsto (f_*\mathcal{F}, f_*\alpha) $$ defines a functor from the category of quasi-coherent sheaves on $(U, R, s, t, c)$ to the category of quasi-coherent sheaves on $(U', R', s', t', c')$ which is right adjoint to pullback as defined in Lemma \ref{lemma-pullback}.


groupoids-lemma-colimits

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. The category of quasi-coherent modules on $(U, R, s, t, c)$ has colimits.


groupoids-lemma-abelian

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. If $s$, $t$ are flat, then the category of quasi-coherent modules on $(U, R, s, t, c)$ is abelian.


groupoids-lemma-construct-quasi-coherent

Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $s, t$ are flat, quasi-compact, and quasi-separated. For any quasi-coherent module $\mathcal{G}$ on $U$, there exists a canonical isomorphism $\alpha : t^*t_*s^*\mathcal{G} \to s^*t_*s^*\mathcal{G}$ which turns $(t_*s^*\mathcal{G}, \alpha)$ into a quasi-coherent module on $(U, R, s, t, c)$. This construction defines a functor $$ \QCoh(\mathcal{O}_U) \longrightarrow \QCoh(U, R, s, t, c) $$ which is a right adjoint to the forgetful functor $(\mathcal{F}, \beta) \mapsto \mathcal{F}$.


groupoids-lemma-push-pull

Let $f : Y \to X$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module, let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module, and let $\varphi : \mathcal{G} \to f^*\mathcal{F}$ be a module map. Assume \begin{enumerate} \item $\varphi$ is injective, \item $f$ is quasi-compact, quasi-separated, flat, and surjective, \item $X$, $Y$ are locally Noetherian, and \item $\mathcal{G}$ is a coherent $\mathcal{O}_Y$-module. \end{enumerate} Then $\mathcal{F} \cap f_*\mathcal{G}$ defined as the pullback $$ \xymatrix{ \mathcal{F} \ar[r] & f_*f^*\mathcal{F} \\ \mathcal{F} \cap f_*\mathcal{G} \ar[u] \ar[r] & f_*\mathcal{G} \ar[u] } $$ is a coherent $\mathcal{O}_X$-module.


groupoids-lemma-colimit-coherent

Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume that \begin{enumerate} \item $U$, $R$ are Noetherian, \item $s, t$ are flat, quasi-compact, and quasi-separated. \end{enumerate} Then every quasi-coherent module $(\mathcal{F}, \alpha)$ on $(U, R, s, t, c)$ is a filtered colimit of coherent modules.


groupoids-lemma-set-of-iso-classes

Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $\kappa$ be a cardinal. There exists a set $T$ and a family $(\mathcal{F}_t, \alpha_t)_{t \in T}$ of $\kappa$-generated quasi-coherent modules on $(U, R, s, t, c)$ such that every $\kappa$-generated quasi-coherent module on $(U, R, s, t, c)$ is isomorphic to one of the $(\mathcal{F}_t, \alpha_t)$.


groupoids-lemma-colimit-kappa

Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume that $s, t$ are flat. There exists a cardinal $\kappa$ such that every quasi-coherent module $(\mathcal{F}, \alpha)$ on $(U, R, s, t, c)$ is the directed colimit of its $\kappa$-generated quasi-coherent submodules.


groupoids-lemma-groupoid-from-action

Let $S$ be a scheme. Let $Y$ be a scheme over $S$. Let $(G, m)$ be a group scheme over $Y$ with identity $e_G$ and inverse $i_G$. Let $X/Y$ be a scheme over $Y$ and let $a : G \times_Y X \to X$ be an action of $G$ on $X/Y$. Then we get a groupoid scheme $(U, R, s, t, c, e, i)$ over $S$ in the following manner: \begin{enumerate} \item We set $U = X$, and $R = G \times_Y X$. \item We set $s : R \to U$ equal to $(g, x) \mapsto x$. \item We set $t : R \to U$ equal to $(g, x) \mapsto a(g, x)$. \item We set $c : R \times_{s, U, t} R \to R$ equal to $((g, x), (g', x')) \mapsto (m(g, g'), x')$. \item We set $e : U \to R$ equal to $x \mapsto (e_G(x), x)$. \item We set $i : R \to R$ equal to $(g, x) \mapsto (i_G(g), a(g, x))$. \end{enumerate}


groupoids-lemma-groupoid-stabilizer

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. The scheme $G$ defined by the cartesian square $$ \xymatrix{ G \ar[r] \ar[d] & R \ar[d]^{j = (t, s)} \\ U \ar[r]^-{\Delta} & U \times_S U } $$ is a group scheme over $U$ with composition law $m$ induced by the composition law $c$.


groupoids-lemma-groupoid-action-stabilizer

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$, and let $G/U$ be its stabilizer. Denote $R_t/U$ the scheme $R$ seen as a scheme over $U$ via the morphism $t : R \to U$. There is a canonical left action $$ a : G \times_U R_t \longrightarrow R_t $$ induced by the composition law $c$.


groupoids-lemma-groupoid-action-stabilizer-pseudo-torsor

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $G$ be the stabilizer group scheme of $R$. Let $$ G_0 = G \times_{U, \text{pr}_0} (U \times_S U) = G \times_S U $$ as a group scheme over $U \times_S U$. The action of $G$ on $R$ of Lemma \ref{lemma-groupoid-action-stabilizer} induces an action of $G_0$ on $R$ over $U \times_S U$ which turns $R$ into a pseudo $G_0$-torsor over $U \times_S U$.


groupoids-lemma-fibres-j

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $p \in U \times_S U$ be a point. Denote $R_p$ the scheme theoretic fibre of $j = (t, s) : R \to U \times_S U$. If $R_p \not = \emptyset$, then the action $$ G_{0, \kappa(p)} \times_{\kappa(p)} R_p \longrightarrow R_p $$ (see Lemma \ref{lemma-groupoid-action-stabilizer-pseudo-torsor}) which turns $R_p$ into a $G_{\kappa(p)}$-torsor over $\kappa(p)$.


groupoids-lemma-restrict-groupoid

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ be a morphism of schemes. Consider the following diagram $$ \xymatrix{ R' \ar[d] \ar[r] \ar@/_3pc/[dd]_{t'} \ar@/^1pc/[rr]^{s'}& R \times_{s, U} U' \ar[r] \ar[d] & U' \ar[d]^g \\ U' \times_{U, t} R \ar[d] \ar[r] & R \ar[r]^s \ar[d]_t & U \\ U' \ar[r]^g & U } $$ where all the squares are fibre product squares. Then there is a canonical composition law $c' : R' \times_{s', U', t'} R' \to R'$ such that $(U', R', s', t', c')$ is a groupoid scheme over $S$ and such that $U' \to U$, $R' \to R$ defines a morphism $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoid schemes over $S$. Moreover, for any scheme $T$ over $S$ the functor of groupoids $$ (U'(T), R'(T), s', t', c') \to (U(T), R(T), s, t, c) $$ is the restriction (see above) of $(U(T), R(T), s, t, c)$ via the map $U'(T) \to U(T)$.


groupoids-lemma-restrict-groupoid-relation

The notions of restricting groupoids and (pre-)equivalence relations defined in Definitions \ref{definition-restrict-groupoid} and \ref{definition-restrict-relation} agree via the constructions of Lemmas \ref{lemma-groupoid-pre-equivalence} and \ref{lemma-equivalence-groupoid}.


groupoids-lemma-restrict-stabilizer

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ be a morphism of schemes. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$. Let $G$ be the stabilizer of $(U, R, s, t, c)$ and let $G'$ be the stabilizer of $(U', R', s', t', c')$. Then $G'$ is the base change of $G$ by $g$, i.e., there is a canonical identification $G' = U' \times_{g, U} G$.


groupoids-lemma-constructing-invariant-opens

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. \begin{enumerate} \item For any subset $W \subset U$ the subset $t(s^{-1}(W))$ is set-theoretically $R$-invariant. \item If $s$ and $t$ are open, then for every open $W \subset U$ the open $t(s^{-1}(W))$ is an $R$-invariant open subscheme. \item If $s$ and $t$ are open and quasi-compact, then $U$ has an open covering consisting of $R$-invariant quasi-compact open subschemes. \end{enumerate}


groupoids-lemma-first-observation

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $s$ and $t$ quasi-compact and flat and $U$ quasi-separated. Let $W \subset U$ be quasi-compact open. Then $t(s^{-1}(W))$ is an intersection of a nonempty family of quasi-compact open subsets of $U$.


groupoids-lemma-second-observation

Assumptions and notation as in Lemma \ref{lemma-first-observation}. There exists an $R$-invariant open $V \subset U$ and a quasi-compact open $W'$ such that $W \subset V \subset W' \subset U$.


groupoids-lemma-criterion-quotient-representable

In the situation of Definition \ref{definition-quotient-sheaf}. Assume there is a scheme $M$, and a morphism $U \to M$ such that \begin{enumerate} \item the morphism $U \to M$ equalizes $s, t$, \item the morphism $U \to M$ induces a surjection of sheaves $h_U \to h_M$ in the $\tau$-topology, and \item the induced map $(t, s) : R \to U \times_M U$ induces a surjection of sheaves $h_R \to h_{U \times_M U}$ in the $\tau$-topology. \end{enumerate} In this case $M$ represents the quotient sheaf $U/R$.


groupoids-lemma-quotient-pre-equivalence

Let $\tau \in \{Zariski, \etale, fppf, smooth, syntomic\}$. Let $S$ be a scheme. Let $j : R \to U \times_S U$ be a pre-equivalence relation over $S$. Assume $U, R, S$ are objects of a $\tau$-site $\Sch_\tau$. For $T \in \Ob((\Sch/S)_\tau)$ and $a, b \in U(T)$ the following are equivalent: \begin{enumerate} \item $a$ and $b$ map to the same element of $(U/R)(T)$, and \item there exists a $\tau$-covering $\{f_i : T_i \to T\}$ of $T$ and morphisms $r_i : T_i \to R$ such that $a \circ f_i = s \circ r_i$ and $b \circ f_i = t \circ r_i$. \end{enumerate} In other words, in this case the map of $\tau$-sheaves $$ h_R \longrightarrow h_U \times_{U/R} h_U $$ is surjective.


groupoids-lemma-quotient-pre-equivalence-relation-restrict

Let $\tau \in \{Zariski, \etale, fppf, smooth, syntomic\}$. Let $S$ be a scheme. Let $j : R \to U \times_S U$ be a pre-equivalence relation over $S$ and $g : U' \to U$ a morphism of schemes over $S$. Let $j' : R' \to U' \times_S U'$ be the restriction of $j$ to $U'$. Assume  $U, U', R, S$ are objects of a $\tau$-site $\Sch_\tau$. The map of quotient sheaves $$ U'/R' \longrightarrow U/R $$ is injective. If $g$ defines a surjection $h_{U'} \to h_U$ of sheaves in the $\tau$-topology (for example if $\{g : U' \to U\}$ is a $\tau$-covering), then $U'/R' \to U/R$ is an isomorphism.


groupoids-lemma-quotient-groupoid-restrict

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


groupoids-lemma-criterion-fibre-product

Let $S$ be a scheme. Let $f : (U, R, j) \to (U', R', j')$ be a morphism between equivalence relations over $S$. Assume that $$ \xymatrix{ R \ar[d]_s \ar[r]_f & R' \ar[d]^{s'} \\ U \ar[r]^f & U' } $$ is cartesian. For any $\tau \in  \{Zariski, \etale, fppf, smooth, syntomic\}$ the diagram $$ \xymatrix{ U \ar[d] \ar[r] & U/R \ar[d]^f \\ U' \ar[r] & U'/R' } $$ is a fibre product square of $\tau$-sheaves.


groupoids-lemma-characterize-cartesian-schemes

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. The category of groupoid schemes cartesian over $(U, R, s, t, c)$ is equivalent to the category of pairs $(V, \varphi)$ where $V$ is a scheme over $U$ and $$ \varphi : V \times_{U, t} R \longrightarrow R \times_{s, U} V $$ is an isomorphism over $R$ such that $e^*\varphi = \text{id}_V$ and such that $$ c^*\varphi = \text{pr}_1^*\varphi \circ \text{pr}_0^*\varphi $$ as morphisms of schemes over $R \times_{s, U, t} R$.


groupoids-lemma-cartesian-equivalent-descent-datum

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. The construction of Lemma \ref{lemma-characterize-cartesian-schemes} determines an equivalence $$ \begin{matrix} \text{category of groupoid schemes} \\ \text{cartesian over } (X, X \times_Y X, \ldots) \end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data} \\ \text{ relative to } X/Y \end{matrix} $$


groupoids-lemma-diagram-diagonal

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $G \to U$ be the stabilizer group scheme. The commutative diagram $$ \xymatrix{ R \ar[d]^{\Delta_{R/U \times_S U}} \ar[rrr]_{f \mapsto (f, s(f))} & & & R \times_{s, U} U \ar[d] \ar[r] & U \ar[d] \\ R \times_{(U \times_S U)} R \ar[rrr]^{(f, g) \mapsto (f, f^{-1} \circ g)} & & & R \times_{s, U} G \ar[r] & G } $$ the two left horizontal arrows are isomorphisms and the right square is a fibre product square.


groupoids-lemma-diagonal

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $G \to U$ be the stabilizer group scheme. \begin{enumerate} \item The following are equivalent \begin{enumerate} \item $j : R \to U \times_S U$ is separated, \item $G \to U$ is separated, and \item $e : U \to G$ is a closed immersion. \end{enumerate} \item The following are equivalent \begin{enumerate} \item $j : R \to U \times_S U$ is quasi-separated, \item $G \to U$ is quasi-separated, and \item $e : U \to G$ is quasi-compact. \end{enumerate} \end{enumerate}


groupoids-lemma-determinant-trick

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \Spec(A)$ and $R = \Spec(B)$ are affine and $s, t : R \to U$ finite locally free. Let $C$ be as in (\ref{equation-invariants}). Let $f \in A$. Then $\text{Norm}_{s^\sharp}(t^\sharp(f)) \in C$.


groupoids-lemma-finite-locally-free-disjoint-free

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $s, t : R \to U$ finite locally free. Then $$ U = \coprod\nolimits_{r \geq 1} U_r $$ is a disjoint union of $R$-invariant opens such that the restriction $R_r$ of $R$ to $U_r$ has the property that $s, t : R_r \to U_r$ are finite locally free of rank $r$.


groupoids-lemma-integral-over-invariants

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \Spec(A)$ and $R = \Spec(B)$ are affine and $s, t : R \to U$ finite locally free. Let $C \subset A$ be as in (\ref{equation-invariants}). Then $A$ is integral over $C$.


groupoids-lemma-invariants-base-change

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \Spec(A)$ and $R = \Spec(B)$ are affine and $s, t : R \to U$ finite locally free. Let $C \subset A$ be as in (\ref{equation-invariants}). Let $C \to C'$ be a ring map, and set $U' = \Spec(A \otimes_C C')$, $R' = \Spec(B \otimes_C C')$. Then \begin{enumerate} \item The maps $s, t, c$ induce maps $s', t', c'$ such that $(U', R', s', t', c')$ is a groupoid scheme. Let $C^1 \subset A'$ be the $R'$-invariant functions on $U'$. \item The canonical map $\varphi : C' \to C^1$ satisfies \begin{enumerate} \item for every $f \in C^1$ there exists an $n > 0$ and a polynomial $P \in C'[x]$ whose image in $C^1[x]$ is $(x - f)^n$, and \item for every $f \in \Ker(\varphi)$ there exists an $n > 0$ such that $f^n = 0$. \end{enumerate} \item If $C \to C'$ is flat then $\varphi$ is an isomorphism. \end{enumerate}


groupoids-lemma-points

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \Spec(A)$ and $R = \Spec(B)$ are affine and $s, t : R \to U$ finite locally free. Let $C \subset A$ be as in (\ref{equation-invariants}). Then $U \to M = \Spec(C)$ has the following properties: \begin{enumerate} \item the map on points $|U| \to |M|$ is surjective and $u_0, u_1 \in |U|$ map to the same point if and only if there exists a $r \in |R|$ with $t(r) = u_0$ and $s(r) = u_1$, in a formula $$ |M| = |U|/|R| $$ \item for any algebraically closed field $k$ we have $$ M(k) = U(k)/R(k) $$ \end{enumerate}


groupoids-lemma-etale

Let $S$ be a scheme. Let $f : (U', R', s', t') \to (U, R, s, t, c)$ be a morphism of groupoid schemes over $S$. \begin{enumerate} \item $U$, $R$, $U'$, $R'$ are affine, \item $s, t, s', t'$ are finite locally free, \item the diagrams $$ \xymatrix{ R' \ar[d]_{s'} \ar[r]_f & R \ar[d]^s \\ U' \ar[r]^f & U } \quad \quad \xymatrix{ R' \ar[d]_{t'} \ar[r]_f & R \ar[d]^t \\ U' \ar[r]^f & U } \quad \quad \xymatrix{ G' \ar[d] \ar[r]_f & G \ar[d] \\ U' \ar[r]^f & U } $$ are cartesian where $G$ and $G'$ are the stabilizer group schemes, and \item $f : U' \to U$ is \'etale. \end{enumerate} Then the map $C \to C'$ from the $R$-invariant functions on $U$ to the $R'$-invariant functions on $U'$ is \'etale and $U' = \Spec(C') \times_{\Spec(C)} U$.


groupoids-lemma-basis

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume \begin{enumerate} \item $U = \Spec(A)$, and $R = \Spec(B)$ are affine, and \item there exist elements $x_i \in A$, $i \in I$ such that $B = \bigoplus_{i \in I} s^\sharp(A)t^\sharp(x_i)$. \end{enumerate} Then $A = \bigoplus_{i\in I} Cx_i$, and $B \cong A \otimes_C A$ where $C \subset A$ is the $R$-invariant functions on $U$ as in (\ref{equation-invariants}).


groupoids-lemma-find-invariant-affine

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $s$, $t$ are finite locally free. Let $u \in U$ be a point such that $t(s^{-1}(\{u\}))$ is contained in an affine open of $U$. Then there exists an $R$-invariant affine open neighbourhood of $u$ in $U$.


groupoids-lemma-descend-along-finite

Let $X \to Y$ be a surjective finite locally free morphism. Let $V$ be a scheme over $X$ such that for all $(y, v_1, \ldots, v_d)$ where $y \in Y$ and $v_1, \ldots, v_d \in V_y$ there exists an affine open $U \subset V$ with $v_1, \ldots, v_d \in U$. Then any descent datum on $V/X/Y$ is effective.


groupoids-proposition-connected-component

Let $G$ be a group scheme over a field $k$. There exists a canonical closed subgroup scheme $G^0 \subset G$ with the following properties \begin{enumerate} \item $G^0 \to G$ is a flat closed immersion, \item $G^0 \subset G$ is the connected component of the identity, \item $G^0$ is geometrically irreducible, and \item $G^0$ is quasi-compact. \end{enumerate}


groupoids-proposition-review-abelian-varieties

\begin{reference} Wonderfully explained in \cite{AVar}. \end{reference} Let $A$ be an abelian variety over a field $k$. Then \begin{enumerate} \item $A$ is projective over $k$, \item $A$ is a commutative group scheme, \item the morphism $[n] : A \to A$ is surjective for all $n \geq 1$, \item if $k$ is algebraically closed, then $A(k)$ is a divisible abelian group, \item $A[n] = \Ker([n] : A \to A)$ is a finite group scheme of degree $n^{2\dim A}$ over $k$, \item $A[n]$ is \'etale over $k$ if and only if $n \in k^*$, \item if $n \in k^*$ and $k$ is algebraically closed, then $A(k)[n] \cong (\mathbf{Z}/n\mathbf{Z})^{\oplus 2\dim(A)}$, \item if $k$ is algebraically closed of characteristic $p > 0$, then there exists an integer $0 \leq f \leq \dim(A)$ such that $A(k)[p^m] \cong (\mathbf{Z}/p^m\mathbf{Z})^{\oplus f}$ for all $m \geq 1$. \end{enumerate}


groupoids-proposition-finite-flat-equivalence

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume \begin{enumerate} \item $U = \Spec(A)$, and $R = \Spec(B)$ are affine, \item $s, t : R \to U$ finite locally free, and \item $j = (t, s)$ is an equivalence. \end{enumerate} In this case, let $C \subset A$ be as in (\ref{equation-invariants}). Then $U \to M = \Spec(C)$ is finite locally free and $R = U \times_M U$. Moreover, $M$ represents the quotient sheaf $U/R$ in the fppf topology (see Definition \ref{definition-quotient-sheaf}).


local-cohomology-theorem-finiteness

\begin{reference} This is a special case of \cite[Satz 2]{Faltings-finiteness}. \end{reference} Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Set $Z = V(I) \subset \Spec(A)$. Let $M$ be a finite $A$-module. Set $s = s_{A, I}(M)$ as in (\ref{equation-cutoff}). Assume that \begin{enumerate} \item $A$ is universally catenary, \item the formal fibres of the local rings of $A$ are Cohen-Macaulay. \end{enumerate} Then $H^i_Z(M)$ is finite for $0 \leq i < s$ and $H^s_Z(M)$ is not finite.


local-cohomology-lemma-local-cohomology-is-local-cohomology

Let $A$ be a ring and let $I$ be a finitely generated ideal. Set $Z = V(I) \subset X = \Spec(A)$. For $K \in D(A)$ corresponding to $\widetilde{K} \in D_\QCoh(\mathcal{O}_X)$ via Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-compare-bounded} there is a functorial isomorphism $$ R\Gamma_Z(K) = R\Gamma_Z(X, \widetilde{K}) $$ where on the left we have Dualizing Complexes, Equation (\ref{dualizing-equation-local-cohomology}) and on the right we have the functor of Cohomology, Section \ref{cohomology-section-cohomology-support-bis}.


local-cohomology-lemma-local-cohomology

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Set $X = \Spec(A)$, $Z = V(I)$, $U = X \setminus Z$, and $j : U \to X$ the inclusion morphism. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_U$-module. Then \begin{enumerate} \item there exists an $A$-module $M$ such that $\mathcal{F}$ is the restriction of $\widetilde{M}$ to $U$, \item given $M$ there is an exact sequence $$ 0 \to H^0_Z(M) \to M \to H^0(U, \mathcal{F}) \to H^1_Z(M) \to 0 $$ and isomorphisms $H^p(U, \mathcal{F}) = H^{p + 1}_Z(M)$ for $p \geq 1$, \item we may take $M = H^0(U, \mathcal{F})$ in which case we have $H^0_Z(M) = H^1_Z(M) = 0$. \end{enumerate}


local-cohomology-lemma-already-torsion

Let $I, J \subset A$ be finitely generated ideals of a ring $A$. If $M$ is an $I$-power torsion module, then the canonical map $$ H^i_{V(I) \cap V(J)}(M) \to H^i_{V(J)}(M) $$ is an isomorphism for all $i$.


local-cohomology-lemma-multiplicative

Let $S \subset A$ be a multiplicative set of a ring $A$. Let $M$ be an $A$-module with $S^{-1}M = 0$. Then $\colim_{f \in S} H^0_{V(f)}(M) = M$ and $\colim_{f \in S} H^1_{V(f)}(M) = 0$.


local-cohomology-lemma-elements-come-from-bigger

Let $I \subset A$ be a finitely generated ideal of a ring $A$. Let $\mathfrak p$ be a prime ideal. Let $M$ be an $A$-module. Let $i \geq 0$ be an integer and consider the map $$ \Psi : \colim_{f \in A, f \not \in \mathfrak p} H^i_{V((I, f))}(M) \longrightarrow H^i_{V(I)}(M) $$ Then \begin{enumerate} \item $\Im(\Psi)$ is the set of elements which map to zero in $H^i_{V(I)}(M)_\mathfrak p$, \item if $H^{i - 1}_{V(I)}(M)_\mathfrak p = 0$, then $\Psi$ is injective, \item if $H^{i - 1}_{V(I)}(M)_\mathfrak p = H^i_{V(I)}(M)_\mathfrak p = 0$, then $\Psi$ is an isomorphism. \end{enumerate}


local-cohomology-lemma-isomorphism

Let $I \subset I' \subset A$ be finitely generated ideals of a Noetherian ring $A$. Let $M$ be an $A$-module. Let $i \geq 0$ be an integer. Consider the map $$ \Psi : H^i_{V(I')}(M) \to H^i_{V(I)}(M) $$ The following are true: \begin{enumerate} \item if $H^i_{\mathfrak pA_\mathfrak p}(M_\mathfrak p) = 0$ for all $\mathfrak p \in V(I) \setminus V(I')$, then $\Psi$ is surjective, \item if $H^{i - 1}_{\mathfrak pA_\mathfrak p}(M_\mathfrak p) = 0$ for all $\mathfrak p \in V(I) \setminus V(I')$, then $\Psi$ is injective, \item if $H^i_{\mathfrak pA_\mathfrak p}(M_\mathfrak p) = H^{i - 1}_{\mathfrak pA_\mathfrak p}(M_\mathfrak p) = 0$ for all $\mathfrak p \in V(I) \setminus V(I')$, then $\Psi$ is an isomorphism. \end{enumerate}


local-cohomology-lemma-depth-2-connected-punctured-spectrum

\begin{reference} \cite[Proposition 2.1]{Hartshorne-connectedness} \end{reference} \begin{slogan} Hartshorne's connectedness \end{slogan} Let $A$ be a Noetherian local ring of depth $\geq 2$. Then the punctured spectra of $A$, $A^h$, and $A^{sh}$ are connected.


local-cohomology-lemma-catenary-S2-equidimensional

\begin{reference} \cite[Corollary 5.10.9]{EGA} \end{reference} Let $A$ be a Noetherian local ring which is catenary and $(S_2)$. Then $\Spec(A)$ is equidimensional.


local-cohomology-lemma-cd

Let $I \subset A$ be a finitely generated ideal of a ring $A$. Set $Y = V(I) \subset X = \Spec(A)$. Let $d \geq -1$ be an integer. The following are equivalent \begin{enumerate} \item $H^i_Y(A) = 0$ for $i > d$, \item $H^i_Y(M) = 0$ for $i > d$ for every $A$-module $M$, and \item if $d = -1$, then $Y = \emptyset$, if $d = 0$, then $Y$ is open and closed in $X$, and if $d > 0$ then $H^i(X \setminus Y, \mathcal{F}) = 0$ for $i \geq d$ for every quasi-coherent $\mathcal{O}_{X \setminus Y}$-module $\mathcal{F}$. \end{enumerate}


local-cohomology-lemma-bound-cd

Let $I \subset A$ be a finitely generated ideal of a ring $A$. Then \begin{enumerate} \item $\text{cd}(A, I)$ is at most equal to the number of generators of $I$, \item $\text{cd}(A, I) \leq r$ if there exist $f_1, \ldots, f_r \in A$ such that $V(f_1, \ldots, f_r) = V(I)$, \item $\text{cd}(A, I) \leq c$ if $\Spec(A) \setminus V(I)$ can be covered by $c$ affine opens. \end{enumerate}


local-cohomology-lemma-cd-sum

Let $I, J \subset A$ be finitely generated ideals of a ring $A$. Then $\text{cd}(A, I + J) \leq \text{cd}(A, I) + \text{cd}(A, J)$.


local-cohomology-lemma-cd-change-rings

Let $A \to B$ be a ring map. Let $I \subset A$ be a finitely generated ideal. Then $\text{cd}(B, IB) \leq \text{cd}(A, I)$. If $A \to B$ is faithfully flat, then equality holds.


local-cohomology-lemma-cd-local

Let $I \subset A$ be a finitely generated ideal of a ring $A$. Then $\text{cd}(A, I) = \max \text{cd}(A_\mathfrak p, I_\mathfrak p)$.


local-cohomology-lemma-cd-dimension

Let $I \subset A$ be a finitely generated ideal of a ring $A$. If $M$ is a finite $A$-module, then $H^i_{V(I)}(M) = 0$ for $i > \dim(\text{Supp}(M))$. In particular, we have $\text{cd}(A, I) \leq \dim(A)$.


local-cohomology-lemma-cd-is-one

Let $I \subset A$ be a finitely generated ideal of a ring $A$. If $\text{cd}(A, I) = 1$ then $\Spec(A) \setminus V(I)$ is nonempty affine.


local-cohomology-lemma-cd-maximal

Let $(A, \mathfrak m)$ be a Noetherian local ring of dimension $d$. Then $H^d_\mathfrak m(A)$ is nonzero and $\text{cd}(A, \mathfrak m) = d$.


local-cohomology-lemma-cd-bound-dim-local

Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be a proper ideal. Let $\mathfrak p \subset A$ be a prime ideal such that $V(\mathfrak p) \cap V(I) = \{\mathfrak m\}$. Then $\dim(A/\mathfrak p) \leq \text{cd}(A, I)$.


local-cohomology-lemma-adjoint

Let $A$ be a Noetherian ring. Let $T \subset \Spec(A)$ be a subset stable under specialization. The functor $RH^0_T$ is the right adjoint to the functor $D(\text{Mod}_{A, T}) \to D(A)$.


local-cohomology-lemma-adjoint-ext

Let $A$ be a Noetherian ring. Let $T \subset \Spec(A)$ be a subset stable under specialization. For any object $K$ of $D(A)$ we have $$ H^i(RH^0_T(K)) = \colim_{Z \subset T\text{ closed}} H^i_Z(K) $$


local-cohomology-lemma-equal-plus

Let $A$ be a Noetherian ring. Let $T \subset \Spec(A)$ be a subset stable under specialization. The functor $D^+(\text{Mod}_{A, T}) \to D^+_T(A)$ is an equivalence.


local-cohomology-lemma-equal-full

Let $A$ be a Noetherian ring. Let $T \subset \Spec(A)$ be a subset stable under specialization. If $\dim(A) < \infty$, then functor $D(\text{Mod}_{A, T}) \to D_T(A)$ is an equivalence.


local-cohomology-lemma-torsion-change-rings

Let $A \to B$ be a flat homomorphism of Noetherian rings. Let $T \subset \Spec(A)$ be a subset stable under specialization. Let $T' \subset \Spec(B)$ be the inverse image of $T$. Then the canonical map $$ R\Gamma_T(K) \otimes_A^\mathbf{L} B \longrightarrow R\Gamma_{T'}(K \otimes_A^\mathbf{L} B) $$ is an isomorphism for $K \in D^+(A)$. If $A$ and $B$ have finite dimension, then this is true for $K \in D(A)$.


local-cohomology-lemma-local-cohomology-ss

Let $A$ be a ring and let $T, T' \subset \Spec(A)$ subsets stable under specialization. For $K \in D^+(A)$ there is a spectral sequence $$ E_2^{p, q} = H^p_T(H^p_{T'}(K)) \Rightarrow H^{p + q}_{T \cap T'}(K) $$ as in Derived Categories, Lemma \ref{derived-lemma-grothendieck-spectral-sequence}.


local-cohomology-lemma-filter-local-cohomology

Let $A$ be a Noetherian ring. Let $T \subset \Spec(A)$ be a subset stable under specialization. Let $T' \subset T$ be the set of nonminimal primes in $T$. Then $T'$ is a subset of $\Spec(A)$ stable under specialization and for every $A$-module $M$ there is an exact sequence $$ 0 \to \colim_{Z, f} H^1_f(H^{i - 1}_Z(M)) \to H^i_{T'}(M) \to H^i_T(M) \to \bigoplus\nolimits_{\mathfrak p \in T \setminus T'} H^i_{\mathfrak p A_\mathfrak p}(M_\mathfrak p) $$ where the colimit is over closed subsets $Z \subset T$ and $f \in A$ with $V(f) \cap Z \subset T'$.


local-cohomology-lemma-zero

Let $A$ be a Noetherian ring of finite dimension. Let $T \subset \Spec(A)$ be a subset stable under specialization. Let $\{M_n\}_{n \geq 0}$ be an inverse system of $A$-modules. Let $i \geq 0$ be an integer. Assume that for every $m$ there exists an integer $m'(m) \geq m$ such  that for all $\mathfrak p \in T$ the induced map $$ H^i_{\mathfrak p A_\mathfrak p}(M_{k, \mathfrak p}) \longrightarrow H^i_{\mathfrak p A_\mathfrak p}(M_{m, \mathfrak p}) $$ is zero for $k \geq m'(m)$. Let $m'' : \mathbf{N} \to \mathbf{N}$ be the $2^{\dim(T)}$-fold self-composition of $m'$. Then the map $H^i_T(M_k) \to H^i_T(M_m)$ is zero for all $k \geq m''(m)$.


local-cohomology-lemma-essential-image

Let $A$ be a Noetherian ring. Let $T \subset \Spec(A)$ be a subset stable under specialization. Let $\{M_n\}_{n \geq 0}$ be an inverse system of $A$-modules. Let $i \geq 0$ be an integer. Assume the dimension of $A$ is finite and that for every $m$ there exists an integer $m'(m) \geq m$ such that for all $\mathfrak p \in T$ we have \begin{enumerate} \item $H^{i - 1}_{\mathfrak p A_\mathfrak p}(M_{k, \mathfrak p}) \to H^{i - 1}_{\mathfrak p A_\mathfrak p}(M_{m, \mathfrak p})$ is zero for $k \geq m'(m)$, and \item $ H^i_{\mathfrak p A_\mathfrak p}(M_{k, \mathfrak p}) \to H^i_{\mathfrak p A_\mathfrak p}(M_{m, \mathfrak p})$ has image $G(\mathfrak p, m)$ independent of $k \geq m'(m)$ and moreover $G(\mathfrak p, m)$ maps injectively into $H^i_{\mathfrak p A_\mathfrak p}(M_{0, \mathfrak p})$. \end{enumerate} Then there exists an integer $m_0$ such that for every $m \geq m_0$ there exists an integer $m''(m) \geq m$ such that for $k \geq m''(m)$ the image of $H^i_T(M_k) \to H^i_T(M_m)$ maps injectively into $H^i_T(M_{m_0})$.


local-cohomology-lemma-check-finiteness-local-cohomology-by-annihilator

\begin{reference} \cite[Lemma 3]{Faltings-annulators} \end{reference} Let $A$ be a Noetherian ring. Let $T \subset \Spec(A)$ be a subset stable under specialization. Let $M$ be a finite $A$-module. Let $n \geq 0$. The following are equivalent \begin{enumerate} \item $H^i_T(M)$ is finite for $i \leq n$, \item there exists an ideal $J \subset A$ with $V(J) \subset T$ such that $J$ annihilates $H^i_T(M)$ for $i \leq n$. \end{enumerate} If $T = V(I) = Z$ for an ideal $I \subset A$, then these are also equivalent to \begin{enumerate} \item[(3)] there exists an $e \geq 0$ such that $I^e$ annihilates $H^i_Z(M)$ for $i \leq n$. \end{enumerate}


local-cohomology-lemma-check-finiteness-local-cohomology-locally

\begin{reference} This is a special case of \cite[Satz 1]{Faltings-finiteness}. \end{reference} Let $A$ be a Noetherian ring, $I \subset A$ an ideal, $M$ a finite $A$-module, and $n \geq 0$ an integer. Let $Z = V(I)$. The following are equivalent \begin{enumerate} \item the modules $H^i_Z(M)$ are finite for $i \leq n$, and \item for all $\mathfrak p \in \Spec(A)$ the modules $H^i_Z(M)_\mathfrak p$, $i \leq n$ are finite $A_\mathfrak p$-modules. \end{enumerate}


local-cohomology-lemma-annihilate-local-cohomology

Let $A$ be a ring and let $J \subset I \subset A$ be finitely generated ideals. Let $i \geq 0$ be an integer. Set $Z = V(I)$. If $H^i_Z(A)$ is annihilated by $J^n$ for some $n$, then $H^i_Z(M)$ annihilated by $J^m$ for some $m = m(M)$ for every finitely presented $A$-module $M$ such that $M_f$ is a finite locally free $A_f$-module for all $f \in I$.


local-cohomology-lemma-local-finiteness-for-finite-locally-free

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Set $Z = V(I)$. Let $n \geq 0$ be an integer. If $H^i_Z(A)$ is finite for $0 \leq i \leq n$, then the same is true for $H^i_Z(M)$, $0 \leq i \leq n$ for any finite $A$-module $M$ such that $M_f$ is a finite locally free $A_f$-module for all $f \in I$.


local-cohomology-lemma-check-finiteness-pushforward-on-associated-points

Let $X$ be a locally Noetherian scheme. Let $j : U \to X$ be the inclusion of an open subscheme with complement $Z$. For $x \in U$ let $i_x : W_x \to U$ be the integral closed subscheme with generic point $x$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module. The following are equivalent \begin{enumerate} \item for all $x \in \text{Ass}(\mathcal{F})$ the $\mathcal{O}_X$-module $j_*i_{x, *}\mathcal{O}_{W_x}$ is coherent, \item $j_*\mathcal{F}$ is coherent. \end{enumerate}


local-cohomology-lemma-finiteness-pushforwards-and-H1-local

Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Set $X = \Spec(A)$, $Z = V(I)$, $U = X \setminus Z$, and $j : U \to X$ the inclusion morphism. Let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module. Then \begin{enumerate} \item there exists a finite $A$-module $M$ such that $\mathcal{F}$ is the restriction of $\widetilde{M}$ to $U$, \item given $M$ there is an exact sequence $$ 0 \to H^0_Z(M) \to M \to H^0(U, \mathcal{F}) \to H^1_Z(M) \to 0 $$ and isomorphisms $H^p(U, \mathcal{F}) = H^{p + 1}_Z(M)$ for $p \geq 1$, \item given $M$ and $p \geq 0$ the following are equivalent \begin{enumerate} \item $R^pj_*\mathcal{F}$ is coherent, \item $H^p(U, \mathcal{F})$ is a finite $A$-module, \item $H^{p + 1}_Z(M)$ is a finite $A$-module, \end{enumerate} \item if the equivalent conditions in (3) hold for $p = 0$, we may take $M = \Gamma(U, \mathcal{F})$ in which case we have $H^0_Z(M) = H^1_Z(M) = 0$. \end{enumerate}


local-cohomology-lemma-finiteness-pushforward

Let $X$ be a locally Noetherian scheme. Let $j : U \to X$ be the inclusion of an open subscheme with complement $Z$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module. Assume \begin{enumerate} \item $X$ is Nagata, \item $X$ is universally catenary, and \item for $x \in \text{Ass}(\mathcal{F})$ and $z \in Z \cap \overline{\{x\}}$ we have $\dim(\mathcal{O}_{\overline{\{x\}}, z}) \geq 2$. \end{enumerate} Then $j_*\mathcal{F}$ is coherent.


local-cohomology-lemma-sharp-finiteness-pushforward

Let $X$ be an integral locally Noetherian scheme. Let $j : U \to X$ be the inclusion of a nonempty open subscheme with complement $Z$. Assume that for all $z \in Z$ and any associated prime $\mathfrak p$ of the completion $\mathcal{O}_{X, z}^\wedge$ we have $\dim(\mathcal{O}_{X, z}^\wedge/\mathfrak p) \geq 2$. Then $j_*\mathcal{O}_U$ is coherent.


local-cohomology-lemma-finiteness-pushforward-general

Let $X$ be a locally Noetherian scheme. Let $j : U \to X$ be the inclusion of an open subscheme with complement $Z$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module. Assume \begin{enumerate} \item $X$ is universally catenary, \item for every $z \in Z$ the formal fibres of $\mathcal{O}_{X, z}$ are $(S_1)$. \end{enumerate} In this situation the following are equivalent \begin{enumerate} \item[(a)] for $x \in \text{Ass}(\mathcal{F})$ and $z \in Z \cap \overline{\{x\}}$ we have $\dim(\mathcal{O}_{\overline{\{x\}}, z}) \geq 2$, and \item[(b)] $j_*\mathcal{F}$ is coherent. \end{enumerate}


local-cohomology-lemma-depth-function

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


local-cohomology-lemma-sitting-in-degrees

Let $(A, \mathfrak m)$ be a Noetherian local ring with normalized dualizing complex $\omega_A^\bullet$. Let $M$ be a finite $A$-module. Set $E^i = \text{Ext}_A^{-i}(M, \omega_A^\bullet)$. Then \begin{enumerate} \item $E^i$ is a finite $A$-module nonzero only for $0 \leq i \leq \dim(\text{Supp}(M))$, \item $\dim(\text{Supp}(E^i)) \leq i$, \item $\text{depth}(M)$ is the smallest integer $\delta \geq 0$ such that $E^\delta \not = 0$, \item $\mathfrak p \in \text{Supp}(E^0 \oplus \ldots \oplus E^i) \Leftrightarrow \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim(A/\mathfrak p) \leq i$, \item the annihilator of $E^i$ is equal to the annihilator of $H^i_\mathfrak m(M)$. \end{enumerate}


local-cohomology-lemma-kill-local-cohomology-at-prime

Let $I$ be an ideal of a Noetherian ring $A$. Let $M$ be a finite $A$-module, let $\mathfrak p \subset A$ be a prime ideal, and let $s \geq 0$ be an integer. Assume \begin{enumerate} \item $A$ has a dualizing complex, \item $\mathfrak p \not \in V(I)$, and \item for all primes $\mathfrak p' \subset \mathfrak p$ and $\mathfrak q \in V(I)$ with $\mathfrak p' \subset \mathfrak q$ we have $$ \text{depth}_{A_{\mathfrak p'}}(M_{\mathfrak p'}) + \dim((A/\mathfrak p')_\mathfrak q) > s $$ \end{enumerate} Then there exists an $f \in A$, $f \not \in \mathfrak p$ which annihilates $H^i_{V(I)}(M)$ for $i \leq s$.


local-cohomology-lemma-cutoff

Let $A \to B$ be a finite homomorphism of Noetherian rings. Let $I \subset A$ be an ideal and set $J = IB$. Let $M$ be a finite $B$-module. If $A$ is universally catenary, then $s_{B, J}(M) = s_{A, I}(M)$.


local-cohomology-lemma-change-completion

Let $A$ be a Noetherian ring which has a dualizing complex. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $A', M'$ be the $I$-adic completions of $A, M$. Let $\mathfrak p' \subset \mathfrak q'$ be prime ideals of $A'$ with $\mathfrak q' \in V(IA')$ lying over $\mathfrak p \subset \mathfrak q$ in $A$. Then $$ \text{depth}_{A_{\mathfrak p'}}(M'_{\mathfrak p'}) \geq \text{depth}_{A_\mathfrak p}(M_\mathfrak p) $$ and $$ \text{depth}_{A_{\mathfrak p'}}(M'_{\mathfrak p'}) + \dim((A'/\mathfrak p')_{\mathfrak q'}) = \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) $$


local-cohomology-lemma-cutoff-completion

Let $A$ be a universally catenary Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Then $$ s_{A, I}(M) \geq s_{A^\wedge, I^\wedge}(M^\wedge) $$ If the formal fibres of $A$ are $(S_n)$, then $\min(n + 1, s_{A, I}(M)) \leq s_{A^\wedge, I^\wedge}(M^\wedge)$.


local-cohomology-lemma-local-annihilator

\begin{reference} This is a special case of \cite[Satz 1]{Faltings-annulators}. \end{reference} Let $A$ be a Gorenstein Noetherian local ring. Let $I \subset A$ be an ideal and set $Z = V(I) \subset \Spec(A)$. Let $M$ be a finite $A$-module. Let $s = s_{A, I}(M)$ as in (\ref{equation-cutoff}). Then $H^i_Z(M)$ is finite for $i < s$, but $H^s_Z(M)$ is not finite.


local-cohomology-lemma-finiteness-Rjstar

Let $X$ be a locally Noetherian scheme. Let $j : U \to X$ be the inclusion of an open subscheme with complement $Z$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module. Let $n \geq 0$ be an integer. Assume \begin{enumerate} \item $X$ is universally catenary, \item for every $z \in Z$ the formal fibres of $\mathcal{O}_{X, z}$ are $(S_n)$. \end{enumerate} In this situation the following are equivalent \begin{enumerate} \item[(a)] for $x \in \text{Supp}(\mathcal{F})$ and $z \in Z \cap \overline{\{x\}}$ we have $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) + \dim(\mathcal{O}_{\overline{\{x\}}, z}) > n$, \item[(b)] $R^pj_*\mathcal{F}$ is coherent for $0 \leq p < n$. \end{enumerate}


local-cohomology-lemma-finiteness-for-finite-locally-free

Let $X$ be a locally Noetherian scheme. Let $j : U \to X$ be the inclusion of an open subscheme with complement $Z$. Let $n \geq 0$ be an integer. If $R^pj_*\mathcal{O}_U$ is coherent for $0 \leq p < n$, then the same is true for $R^pj_*\mathcal{F}$, $0 \leq p < n$ for any finite locally free $\mathcal{O}_U$-module $\mathcal{F}$.


local-cohomology-lemma-annihilate-Hp

\begin{reference} \cite[Lemma 1.9]{Bhatt-local} \end{reference} Let $A$ be a ring and let $J \subset I \subset A$ be finitely generated ideals. Let $p \geq 0$ be an integer. Set $U = \Spec(A) \setminus V(I)$. If $H^p(U, \mathcal{O}_U)$ is annihilated by $J^n$ for some $n$, then $H^p(U, \mathcal{F})$ annihilated by $J^m$ for some $m = m(\mathcal{F})$ for every finite locally free $\mathcal{O}_U$-module $\mathcal{F}$.


local-cohomology-lemma-check-finiteness-local-cohomology-by-annihilator-complex

Let $A$ be a Noetherian ring. Let $T \subset \Spec(A)$ be a subset stable under specialization. Let $K$ be an object of $D_{\textit{Coh}}^+(A)$. Let $n \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item $H^i_T(K)$ is finite for $i \leq n$, \item there exists an ideal $J \subset A$ with $V(J) \subset T$ such that $J$ annihilates $H^i_T(K)$ for $i \leq n$. \end{enumerate} If $T = V(I) = Z$ for an ideal $I \subset A$, then these are also equivalent to \begin{enumerate} \item[(3)] there exists an $e \geq 0$ such that $I^e$ annihilates $H^i_Z(K)$ for $i \leq n$. \end{enumerate}


local-cohomology-lemma-get-depth-1-along-Z

Let $X$ be a Noetherian scheme. Let $T \subset X$ be a subset stable under specialization. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then there is a unique map $\mathcal{F} \to \mathcal{F}'$ of coherent $\mathcal{O}_X$-modules such that \begin{enumerate} \item $\mathcal{F} \to \mathcal{F}'$ is surjective, \item $\mathcal{F}_x \to \mathcal{F}'_x$ is an isomorphism for $x \not \in T$, \item $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}'_x) \geq 1$ for $x \in T$. \end{enumerate} If $f : Y \to X$ is a flat morphism with $Y$ Noetherian, then $f^*\mathcal{F} \to f^*\mathcal{F}'$ is the corresponding quotient for $f^{-1}(T) \subset Y$ and $f^*\mathcal{F}$.


local-cohomology-lemma-get-depth-2-along-Z

Let $j : U \to X$ be an open immersion of Noetherian schemes. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Assume $\mathcal{F}' = j_*(\mathcal{F}|_U)$ is coherent. Then $\mathcal{F} \to \mathcal{F}'$ is the unique map of coherent $\mathcal{O}_X$-modules such that \begin{enumerate} \item $\mathcal{F}|_U \to \mathcal{F}'|_U$ is an isomorphism, \item $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}'_x) \geq 2$ for $x \in X$, $x \not \in U$. \end{enumerate} If $f : Y \to X$ is a flat morphism with $Y$ Noetherian, then $f^*\mathcal{F} \to f^*\mathcal{F}'$ is the corresponding map for $f^{-1}(U) \subset Y$.


local-cohomology-lemma-make-S2-along-T-simple

Let $X$ be a Noetherian scheme which locally has a dualizing complex. Let $T' \subset X$ be a subset stable under specialization. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Assume that if $x \leadsto x'$ is an immediate specialization of points in $X$ with $x' \in T'$ and $x \not \in T'$, then $\text{depth}(\mathcal{F}_x) \geq 1$. Then there exists a unique map $\mathcal{F} \to \mathcal{F}''$ of coherent $\mathcal{O}_X$-modules such that \begin{enumerate} \item $\mathcal{F}_x \to \mathcal{F}''_x$ is an isomorphism for $x \not \in T'$, \item $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}''_x) \geq 2$ for $x \in T'$. \end{enumerate} If $f : Y \to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian, then $f^*\mathcal{F} \to f^*\mathcal{F}''$ satisfies the same properties with respect to $f^{-1}(T') \subset Y$.


local-cohomology-lemma-make-S2-along-T

Let $X$ be a Noetherian scheme which locally has a dualizing complex. Let $T' \subset T \subset X$ be subsets stable under specialization such that if $x \leadsto x'$ is an immediate specialization of points in $X$ and $x' \in T'$, then $x \in T$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then there exists a unique map $\mathcal{F} \to \mathcal{F}''$ of coherent $\mathcal{O}_X$-modules such that \begin{enumerate} \item $\mathcal{F}_x \to \mathcal{F}''_x$ is an isomorphism for $x \not \in T$, \item $\mathcal{F}_x \to \mathcal{F}''_x$ is surjective and $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}''_x) \geq 1$ for $x \in T$, $x \not \in T'$, and \item $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}''_x) \geq 2$ for $x \in T'$. \end{enumerate} If $f : Y \to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian, then $f^*\mathcal{F} \to f^*\mathcal{F}''$ satisfies the same properties with respect to $f^{-1}(T') \subset f^{-1}(T) \subset Y$.


local-cohomology-lemma-cd-top-vanishing

Let $A$ be a Noetherian ring of dimension $d$. Let $I \subset I' \subset A$ be ideals. If $I'$ is contained in the Jacobson radical of $A$ and $\text{cd}(A, I') < d$, then $\text{cd}(A, I) < d$.


local-cohomology-lemma-cd-top-vanishing-some-module

Let $A$ be a Noetherian ring of dimension $d$. Let $I \subset A$ be an ideal. If $H^d_{V(I)}(M) = 0$ for some finite $A$-module whose support contains all the irreducible components of dimension $d$, then $\text{cd}(A, I) < d$.


local-cohomology-lemma-top-coh-divisible

Let $A$ be a Noetherian local ring of dimension $d$. Let $f \in A$ be an element which is not contained in any minimal prime of dimension $d$. Then $f : H^d_{V(I)}(M) \to H^d_{V(I)}(M)$ is surjective for any finite $A$-module $M$ and any ideal $I \subset A$.


local-cohomology-lemma-cd-bound-dualizing

Let $A$ be a Noetherian local ring with normalized dualizing complex $\omega_A^\bullet$. Let $I \subset A$ be an ideal. If $H^0_{V(I)}(\omega_A^\bullet) = 0$, then $\text{cd}(A, I) < \dim(A)$.


local-cohomology-lemma-inverse-system-symbolic-powers

Let $(A, \mathfrak m)$ be a complete Noetherian local domain. Let $\mathfrak p \subset A$ be a prime ideal of dimension $1$. For every $n \geq 1$ there is an $m \geq n$ such that $\mathfrak p^{(m)} \subset \mathfrak p^n$.


local-cohomology-lemma-affine-complement

Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Assume $A$ is excellent, normal, and $\dim V(I) \geq 1$. Then $\text{cd}(A, I) < \dim(A)$. In particular, if $\dim(A) = 2$, then $\Spec(A) \setminus V(I)$ is affine.


local-cohomology-lemma-1

\begin{reference} See \cite{Lech-inequalities} and \cite[Lemma 1 page 299]{MatCA}. \end{reference} Let $A$ be a ring. If $f_1, \ldots, f_{r - 1}, f_rg_r$ are independent, then $f_1, \ldots, f_r$ are independent.


local-cohomology-lemma-2

\begin{reference} See \cite{Lech-inequalities} and \cite[Lemma 2 page 300]{MatCA}. \end{reference} Let $A$ be a ring. If $f_1, \ldots, f_{r - 1}, f_rg_r$ are independent and if the $A$-module $A/(f_1, \ldots, f_{r - 1}, f_rg_r)$ has finite length, then \begin{align*} & \text{length}_A(A/(f_1, \ldots, f_{r - 1}, f_rg_r)) \\ & = \text{length}_A(A/(f_1, \ldots, f_{r - 1}, f_r)) + \text{length}_A(A/(f_1, \ldots, f_{r - 1}, g_r)) \end{align*}


local-cohomology-lemma-3

\begin{reference} See \cite{Lech-inequalities} and \cite[Lemma 3 page 300]{MatCA}. \end{reference} Let $(A, \mathfrak m)$ be a local ring. If $\mathfrak m = (x_1, \ldots, x_r)$ and $x_1^{e_1}, \ldots, x_r^{e_r}$ are independent for some $e_i > 0$, then $\text{length}_A(A/(x_1^{e_1}, \ldots, x_r^{e_r})) = e_1\ldots e_r$.


local-cohomology-lemma-flat-extension-independent

Let $\varphi : A \to B$ be a flat ring map. If $f_1, \ldots, f_r \in A$ are independent, then $\varphi(f_1), \ldots, \varphi(f_r) \in B$ are independent.


local-cohomology-lemma-frobenius-flat-regular

\begin{reference} \cite{Kunz-flat} \end{reference} Let $p$ be a prime number. Let $A$ be a Noetherian ring with $p = 0$. The following are equivalent \begin{enumerate} \item $A$ is regular, and \item $F : A \to A$, $a \mapsto a^p$ is flat. \end{enumerate}


local-cohomology-lemma-structure-torsion-D-module-regular

\begin{reference} Special case of \cite[Theorem 2.4]{Lyubeznik} \end{reference} Let $k$ be a field of characteristic $0$. Let $d \geq 1$. Let $A = k[[x_1, \ldots, x_d]]$ with maximal ideal $\mathfrak m$. Let $M$ be an $\mathfrak m$-power torsion $A$-module endowed with additive operators $D_1, \ldots, D_d$ satisfying the leibniz rule $$ D_i(fz) = \partial_i(f) z + f D_i(z) $$ for $f \in A$ and $z \in M$. Here $\partial_i$ is differentiation with respect to $x_i$. Then $M$ is isomorphic to a direct sum of copies of the injective hull $E$ of $k$.


local-cohomology-lemma-structure-torsion-Frobenius-regular

\begin{reference} Follows from \cite[Corollary 3.6]{Huneke-Sharp} with a little bit of work. Also follows directly from \cite[Theorem 1.4]{Lyubeznik2}. \end{reference} Let $p$ be a prime number. Let $(A, \mathfrak m, k)$ be a regular local ring with $p = 0$. Denote $F : A \to A$, $a \mapsto a^p$ be the Frobenius endomorphism. Let $M$ be a $\mathfrak m$-power torsion module such that $M \otimes_{A, F} A \cong M$. Then $M$ is isomorphic to a direct sum of copies of the injective hull $E$ of $k$.


local-cohomology-lemma-derivation

Let $A$ be a ring. Let $I \subset A$ be a finitely generated ideal. Set $Z = V(I)$. For each derivation $\theta : A \to A$ there exists a canonical additive operator $D$ on the local cohomology modules $H^i_Z(A)$ satisfying the Leibniz rule with respect to $\theta$.


local-cohomology-lemma-etale-derivation

Let $A$ be a ring. Let $V \to \Spec(A)$ be quasi-compact, quasi-separated, and \'etale. For each derivation $\theta : A \to A$ there exists a canonical additive operator $D$ on $H^i(V, \mathcal{O}_V)$ satisfying the Leibniz rule with respect to $\theta$.


local-cohomology-lemma-map-tor-1-zero

Let $R$ be a ring. Let $M \to M'$ be a map of $R$-modules with $M$ of finite presentation such that $\text{Tor}_1^R(M, N) \to \text{Tor}_1^R(M', N)$ is zero for all $R$-modules $N$. Then $M \to M'$ factors through a free $R$-module.


local-cohomology-lemma-characterize-vanishing-tor-ext-above-e

Let $R$ be a ring. Let $\alpha : M \to M'$ be a map of $R$-modules. Let $P_\bullet \to M$ and $P'_\bullet \to M'$ be resolutions by projective $R$-modules. Let $e \geq 0$ be an integer. Consider the following conditions \begin{enumerate} \item We can find a map of complexes $a_\bullet : P_\bullet \to P'_\bullet$ inducing $\alpha$ on cohomology with $a_i = 0$ for $i > e$. \item We can find a map of complexes $a_\bullet : P_\bullet \to P'_\bullet$ inducing $\alpha$ on cohomology with $a_{e + 1} = 0$. \item The map $\Ext^i_R(M', N) \to \Ext^i_R(M, N)$ is zero for all $R$-modules $N$ and $i > e$. \item The map $\Ext^{e + 1}_R(M', N) \to \Ext^{e + 1}_R(M, N)$ is zero for all $R$-modules $N$. \item Let $N = \Im(P'_{e + 1} \to P'_e)$ and denote $\xi \in \Ext^{e + 1}_R(M', N)$ the canonical element (see proof). Then $\xi$ maps to zero in $\Ext^{e + 1}_R(M, N)$. \item The map $\text{Tor}_i^R(M, N) \to \text{Tor}_i^R(M', N)$ is zero for all $R$-modules $N$ and $i > e$. \item The map $\text{Tor}_{e + 1}^R(M, N) \to \text{Tor}_{e + 1}^R(M', N)$ is zero for all $R$-modules $N$. \end{enumerate} Then we always have the implications $$ (1) \Leftrightarrow (2) \Leftrightarrow (3) \Leftrightarrow (4) \Leftrightarrow (5) \Rightarrow (6) \Leftrightarrow (7) $$ If $M$ is $(-e - 1)$-pseudo-coherent (for example if $R$ is Noetherian and $M$ is a finite $R$-module), then all conditions are equivalent.


local-cohomology-lemma-cd-sequence-Koszul

Let $I$ be an ideal of a Noetherian ring $A$. For all $n \geq 1$ there exists an $m > n$ such that the map $A/I^m \to A/I^n$ satisfies the equivalent conditions of Lemma \ref{lemma-characterize-vanishing-tor-ext-above-e} with $e = \text{cd}(A, I)$.


local-cohomology-lemma-maps-zero-fixed-torsion

Let $I$ be an ideal of a Noetherian ring $A$. For every $m \geq 0$ and $i > 0$ there exist a $c = c(A, I, m, i) \geq 0$ such that for every $A$-module $M$ annihilated by $I^m$ the map $$ \text{Tor}^A_i(M, A/I^n) \to \text{Tor}^A_i(M, A/I^{n - c}) $$ is zero for all $n \geq c$.


local-cohomology-lemma-annihilates-affine

Let $I = (a_1, \ldots, a_t)$ be an ideal of a Noetherian ring $A$. Set $a = a_1$ and denote $B = A[\frac{I}{a}]$ the affine blowup algebra. There exists a $c > 0$ such that $\text{Tor}_i^A(B, M)$ is annihilated by $I^c$ for all $A$-modules $M$ and $i \geq t$.


local-cohomology-lemma-compute-tor-Iq

In the situation above, for $q \geq q(A, I)$ and any $A$-module $M$ we have $$ R\Gamma(X, Lp^*\widetilde{M}(q)) \cong M \otimes_A^\mathbf{L} I^q $$ in $D(A)$.


local-cohomology-lemma-annihilates

In the situation above, let $t$ be an upper bound on the number of generators for $I$. There exists an integer $c = c(A, I) \geq 0$ such that for any $A$-module $M$ the cohomology sheaves $H^j(Lp^*\widetilde{M})$ are annihilated by $I^c$ for $j \leq -t$.


local-cohomology-lemma-annihilates-tors

In the situation above, let $t$ be an upper bound on the number of generators for $I$. There exists an integer $c = c(A, I) \geq 0$ such that for any $A$-module $M$ the tor modules $\text{Tor}_i^A(M, A/I^q)$ are annihilated by $I^c$ for $i > t$ and all $q \geq 0$.


local-cohomology-lemma-tor-maps-vanish

Let $I$ be an ideal of a Noetherian ring $A$. Let $t \geq 0$ be an upper bound on the number of generators of $I$. There exist $N, c \geq 0$ such that the maps $$ \text{Tor}_{t + 1}^A(M, A/I^n) \to \text{Tor}_{t + 1}^A(M, A/I^{n - c}) $$ are zero for any $A$-module $M$ and all $n \geq N$.


local-cohomology-lemma-bound-q-and-d

With $q_0 = q(S)$ and $d = d(S)$ as above, we have \begin{enumerate} \item for $n \geq 1$, $q \geq q_0$, and $i > 0$ we have $H^i(X, \mathcal{O}_{Y_n}(q)) = 0$, \item for $n \geq 1$ and $q \geq q_0$ we have $H^0(X, \mathcal{O}_{Y_n}(q)) = I^q/I^{q + n}$, \item for $q \geq q_0$ and $i > 0$ we have $H^i(X, \mathcal{O}_X(q)) = 0$, \item for $q \geq q_0$ we have $H^0(X, \mathcal{O}_X(q)) = I^q$. \end{enumerate}


local-cohomology-lemma-almost-exactness

Let $0 \to K \to L \to M \to 0$ be a short exact sequence of $A$-modules such that $K$ and $L$ are annihilated by $I^n$ and $M$ is an $(A, n, c)$-module. Then the kernel of $p^*K \to p^*L$ is scheme theoretically supported on $Y_c$.


local-cohomology-lemma-annihilated

Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $\mathcal{F}$ is scheme theoretically supported on $Y_c$ if and only if the canonical map $\mathcal{F} \to \mathcal{F}(c)$ is zero.


local-cohomology-lemma-vanishing-coh-almost-projective

With $q_0 = q(S)$ and $d = d(S)$ as above, suppose we have integers $n \geq c \geq 0$, an $(A, n, c)$-module $M$, an index $i \in \{0, 1, \ldots, d\}$, and an integer $q$. Then we distinguish the following cases \begin{enumerate} \item In the case $i = d \geq 1$ and $q \geq q_0$ we have $H^d(X, p^*M(q)) = 0$. \item In the case $i = d - 1 \geq 1$ and $q \geq q_0$ we have $H^{d - 1}(X, p^*M(q)) = 0$. \item In the case $d - 1 > i > 0$ and $q \geq q_0 + (d - 1 - i)c$ the map $H^i(X, p^*M(q)) \to H^i(X, p^*M(q - (d - 1 - i)c))$ is zero. \item In the case $i = 0$, $d \in \{0, 1\}$, and $q \geq q_0$, there is a surjection $$ I^qM \longrightarrow H^0(X, p^*M(q)) $$ \item In the case $i = 0$, $d > 1$, and $q \geq q_0 + (d - 1)c$ the map $$ H^0(X, p^*M(q)) \to H^0(X, p^*M(q - (d - 1)c)) $$ has image contained in the image of the canonical map $I^{q - (d - 1)c}M \to H^0(X, p^*M(q - (d - 1)c))$. \end{enumerate}


local-cohomology-lemma-factor-hom

With $q_0 = q(S)$ and $d = d(S)$ as above, let $M$ be an $(A, n, c)$-module and let $\varphi : M \to I^n/I^{2n}$ be an $A$-linear map. Assume $n \geq \max(q_0 + (1 + d)c, (2 + d)c)$ and if $d = 0$ assume $n \geq q_0 + 2c$. Then the composition $$ M \xrightarrow{\varphi} I^n/I^{2n} \to I^{n - (1 + d)c}/I^{2n - (1 + d)c} $$ is of the form $\sum a_i \psi_i$ with $a_i \in I^c$ and $\psi_i : M \to I^{n - (2 + d)c}/I^{2n - (2 + d)c}$.


local-cohomology-lemma-bound-two-term-complex

With $d = d(S)$ and $q_0 = q(S)$ as above. Then \begin{enumerate} \item for integers $n \geq c \geq 0$ with $n \geq \max(q_0 + (1 + d)c, (2 + d)c)$, \item for $K$ of $D(A/I^n)$ with $H^i(K) = 0$ for $i \not = -1, 0$ and $H^i(K)$ finite for $i = -1, 0$ such that $\Ext^1_{A/I^c}(K, N)$ is annihilated by $I^c$ for all finite $A/I^c$-modules $N$ \end{enumerate} the map $$ \Ext^1_{A/I^n}(K, I^n/I^{2n}) \longrightarrow \Ext^1_{A/I^n}(K, I^{n - (1 + d)c}/I^{2n - 2(1 + d)c}) $$ is zero.


local-cohomology-proposition-kollar

\begin{reference} See \cite{k-coherent} and see \cite[IV, Proposition 7.2.2]{EGA} for a special case. \end{reference} \begin{slogan} Weak analogue of Hartogs' Theorem: On Noetherian schemes, the restriction of a coherent sheaf to an open set with complement of codimension 2 in the sheaf's support, is coherent. \end{slogan} Let $j : U \to X$ be an open immersion of locally Noetherian schemes with complement $Z$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module. The following are equivalent \begin{enumerate} \item $j_*\mathcal{F}$ is coherent, \item for $x \in \text{Ass}(\mathcal{F})$ and $z \in Z \cap \overline{\{x\}}$ and any associated prime $\mathfrak p$ of the completion $\mathcal{O}_{\overline{\{x\}}, z}^\wedge$ we have $\dim(\mathcal{O}_{\overline{\{x\}}, z}^\wedge/\mathfrak p) \geq 2$. \end{enumerate}


local-cohomology-proposition-annihilator

\begin{reference} \cite{Faltings-annulators}. \end{reference} Let $A$ be a Noetherian ring which has a dualizing complex. Let $T \subset T' \subset \Spec(A)$ be subsets stable under specialization. Let $s \geq 0$ an integer. Let $M$ be a finite $A$-module. The following are equivalent \begin{enumerate} \item there exists an ideal $J \subset A$ with $V(J) \subset T'$ such that $J$ annihilates $H^i_T(M)$ for $i \leq s$, and \item for all $\mathfrak p \not \in T'$, $\mathfrak q \in T$ with $\mathfrak p \subset \mathfrak q$ we have $$ \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > s $$ \end{enumerate}


local-cohomology-proposition-finiteness

\begin{reference} \cite{Faltings-annulators}. \end{reference} Let $A$ be a Noetherian ring which has a dualizing complex. Let $T \subset \Spec(A)$ be a subset stable under specialization. Let $s \geq 0$ an integer. Let $M$ be a finite $A$-module. The following are equivalent \begin{enumerate} \item $H^i_T(M)$ is a finite $A$-module for $i \leq s$, and \item for all $\mathfrak p \not \in T$, $\mathfrak q \in T$ with $\mathfrak p \subset \mathfrak q$ we have $$ \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > s $$ \end{enumerate}


local-cohomology-proposition-annihilator-complex

Let $A$ be a Noetherian ring which has a dualizing complex. Let $T \subset T' \subset \Spec(A)$ be subsets stable under specialization. Let $s \in \mathbf{Z}$. Let $K$ be an object of $D_{\textit{Coh}}^+(A)$. The following are equivalent \begin{enumerate} \item there exists an ideal $J \subset A$ with $V(J) \subset T'$ such that $J$ annihilates $H^i_T(K)$ for $i \leq s$, and \item for all $\mathfrak p \not \in T'$, $\mathfrak q \in T$ with $\mathfrak p \subset \mathfrak q$ we have $$ \text{depth}_{A_\mathfrak p}(K_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > s $$ \end{enumerate}


local-cohomology-proposition-Hartshorne-Lichtenbaum-vanishing

\begin{reference} \cite[Theorem 3.1]{CD} \end{reference} Let $A$ be a Noetherian local ring with completion $A^\wedge$. Let $I \subset A$ be an ideal such that $$ \dim V(IA^\wedge + \mathfrak p) \geq 1 $$ for every minimal prime $\mathfrak p \subset A^\wedge$ of dimension $\dim(A)$. Then $\text{cd}(A, I) < \dim(A)$.


local-cohomology-proposition-uniform-artin-rees

Let $I$ be an ideal of a Noetherian ring $A$. Let $t \geq 0$ be an upper bound on the number of generators of $I$. There exist $N, c \geq 0$ such that for $n \geq N$ the maps $$ A/I^n \to A/I^{n - c} $$ satisfy the equivalent conditions of Lemma \ref{lemma-characterize-vanishing-tor-ext-above-e} with $e = t$.


more-algebra-theorem-regular-fs

Let $k$ be a field. Let $(A, \mathfrak m, K)$ be a Noetherian local $k$-algebra. If the characteristic of $k$ is zero then the following are equivalent \begin{enumerate} \item $A$ is a regular local ring, and \item $k \to A$ is formally smooth in the $\mathfrak m$-adic topology. \end{enumerate} If the characteristic of $k$ is $p > 0$ then the following are equivalent \begin{enumerate} \item $A$ is geometrically regular over $k$, \item $k \to A$ is formally smooth in the $\mathfrak m$-adic topology. \item for all $k \subset k' \subset k^{1/p}$ finite over $k$ the ring $A \otimes_k k'$ is regular, \item $A$ is regular and the canonical map $H_1(L_{K/k}) \to \mathfrak m/\mathfrak m^2$ is injective, and \item $A$ is regular and the map $\Omega_{k/\mathbf{F}_p} \otimes_k K \to \Omega_{A/\mathbf{F}_p} \otimes_A K$ is injective. \end{enumerate}


more-algebra-theorem-formal-glueing

Let $R$ be a ring, and let $f \in R$. Let $\varphi : R \to S$ be a flat ring map inducing an isomorphism $R/fR \to S/fS$. Then the functor $$ \text{Mod}_R \longrightarrow \text{Mod}_S \times_{\text{Mod}_{S_f}} \text{Mod}_{R_f}, \quad M \longmapsto (M \otimes_R S, M_f, \text{can}) $$ is an equivalence.


more-algebra-theorem-BL-glueing

\begin{reference} Slight generalization of the main theorem of \cite{Beauville-Laszlo}. \end{reference} Let $(R \to R',f)$ be a glueing pair. The functor $\text{Can} : \text{Mod}_R \longrightarrow \text{Glue}(R \to R', f)$ determines an equivalence of the category of $R$-modules glueable for $(R \to R', f)$ and the category $\text{Glue}(R \to R', f)$ of glueing data.


more-algebra-theorem-olivier

Let $A \to B$ be a local homomorphism of local rings. If $A$ is strictly henselian and $A \to B$ is weakly \'etale, then $A = B$.


more-algebra-theorem-epp

Let $A \subset B$ be an extension of discrete valuation rings with fraction fields $K \subset L$. If the characteristic of $\kappa_A$ is $p > 0$, assume that every element of $$ \bigcap\nolimits_{n \geq 1} \kappa_B^{p^n} $$ is separable algebraic over $\kappa_A$. Then there exists a finite extension $K \subset K_1$ which is a weak solution for $A \to B$ as defined in Definition \ref{definition-solution}.


more-algebra-lemma-exact-category-stably-free

Let $R$ be a ring. Let $0 \to P' \to P \to P'' \to 0$ be a short exact sequence of finite projective $R$-modules. If $2$ out of $3$ of these modules are stably free, then so is the third.


more-algebra-lemma-lift-stably-free

Let $R$ be a ring. Let $I \subset R$ be an ideal. Assume that every element of $1 + I$ is a unit (in other words $I$ is contained in the Jacobson radical of $R$). For every finite stably free $R/I$-module $E$ there exists a finite stably free $R$-module $M$ such that $M/IM \cong E$.


more-algebra-lemma-lift-projective

Let $R$ be a ring. Let $I \subset R$ be an ideal. Assume that every element of $1 + I$ is a unit (in other words $I$ is contained in the Jacobson radical of $R$). Let $M$ be a finite flat $R$-module such that $M/IM$ is a projective $R/I$-module. Then $M$ is a finite projective $R$-module.


more-algebra-lemma-isomorphic-finite-projective-lifts

Let $R$ be a ring. Let $I \subset R$ be an ideal. Assume that every element of $1 + I$ is a unit (in other words $I$ is contained in the Jacobson radical of $R$). If $P$ and $P'$ are finite projective $R$-modules, then \begin{enumerate} \item if $\varphi : P \to P'$ is an $R$-module map inducing an isomorphism $\overline{\varphi} : P/IP \to P'/IP'$, then $\varphi$ is an isomorphism, \item if $P/IP \cong P'/IP'$, then $P \cong P'$. \end{enumerate}


more-algebra-lemma-approximate-complex

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal contained in the Jacobson radical of $A$. Let $$ S : L \xrightarrow{f} M \xrightarrow{g} N \quad\text{and}\quad S' : L \xrightarrow{f'} M \xrightarrow{g'} N $$ be two complexes of finite $A$-modules as shown. Assume that \begin{enumerate} \item $c$ works in the Artin-Rees lemma for $f$ and $g$, \item the complex $S$ is exact, and \item $f' = f \bmod I^{c + 1}M$ and $g' = g \bmod I^{c + 1}N$. \end{enumerate} Then $c$ works in the Artin-Rees lemma for $g'$ and the complex $S'$ is exact.


more-algebra-lemma-approximate-complex-graded

Assumptions as in Lemma \ref{lemma-approximate-complex}. Let $Q = \Coker(g)$ and $Q' = \Coker(g')$. Then $\text{Gr}_I(Q) \cong \text{Gr}_I(Q')$ as graded $\text{Gr}_I(A)$-modules.


more-algebra-lemma-works-flat-extension

Let $A \to B$ be a flat map of Noetherian rings. Let $I \subset A$ be an ideal. Let $f : M \to N$ be a homomorphism of finite $A$-modules. Assume that $c$ works for $f$ in the Artin-Rees lemma. Then $c$ works for $f \otimes 1 : M \otimes_A B \to N \otimes_A B$ in the Artin-Rees lemma for the ideal $IB$.


more-algebra-lemma-fibre-product-finite-type

Let $R$ be a ring. Let $A \to B$ and $C \to B$ be $R$-algebra maps. Assume \begin{enumerate} \item $R$ is Noetherian, \item $A$, $B$, $C$ are of finite type over $R$, \item $A \to B$ is surjective, and \item $B$ is finite over $C$. \end{enumerate} Then $A \times_B C$ is of finite type over $R$.


more-algebra-lemma-formal-consequence

Let $R$ be a Noetherian ring. Let $I$ be a finite set. Suppose given a cartesian diagram $$ \xymatrix{ \prod B_i & \prod A_i \ar[l]^{\prod \varphi_i} \\ Q \ar[u]^{\prod \psi_i} & P \ar[u] \ar[l] } $$ with $\psi_i$ and $\varphi_i$ surjective, and $Q$, $A_i$, $B_i$ of finite type over $R$. Then $P$ is of finite type over $R$.


more-algebra-lemma-diagram-localize

Suppose given a cartesian diagram of rings $$ \xymatrix{ R & R' \ar[l]^t \\ B \ar[u]_s & B'\ar[u] \ar[l] } $$ i.e., $B' = B \times_R R'$. If $h \in B'$ corresponds to $g \in B$ and $f \in R'$ such that $s(g) = t(f)$, then the diagram $$ \xymatrix{ R_{s(g)} = R_{t(f)} & (R')_f \ar[l]^-t \\ B_g \ar[u]_s & (B')_h \ar[u] \ar[l] } $$ is cartesian too.


more-algebra-lemma-modules

Given a commutative diagram of rings $$ \xymatrix{ R & R' \ar[l] \\ B \ar[u] & B' \ar[u] \ar[l] } $$ the functor (\ref{equation-modules}) has a right adjoint, namely the functor $$ F : (N, M', \varphi) \longmapsto N \times_\varphi M' $$ (see proof for elucidation).


more-algebra-lemma-points-of-fibre-product

In Situation \ref{situation-module-over-fibre-product} we have $$ \Spec(B') = \Spec(B) \amalg_{\Spec(A)} \Spec(A') $$ as topological spaces.


more-algebra-lemma-fibre-product-integral

In Situation \ref{situation-module-over-fibre-product} if $B \to A$ is integral, then $B' \to A'$ is integral.


more-algebra-lemma-module-over-fibre-product

In Situation \ref{situation-module-over-fibre-product} the functor (\ref{equation-functor}) has a right adjoint, namely the functor $$ F : (N, M', \varphi) \longmapsto N \times_{\varphi, M} M' $$ where $M = M'/IM'$. Moreover, the composition of $F$ with (\ref{equation-functor}) is the identity functor on $\text{Mod}_B \times_{\text{Mod}_A} \text{Mod}_{A'}$. In other words, setting $N' = N \times_{\varphi, M} M'$ we have $N' \otimes_{B'} B = N$ and $N' \otimes_{B'} A' = M'$.


more-algebra-lemma-module-over-fibre-product-bis

In the situation of Lemma \ref{lemma-module-over-fibre-product} for a $B'$-module $L'$ the adjunction map $$ L' \longrightarrow  (L' \otimes_{B'} B) \times_{(L' \otimes_{B'} A)} (L' \otimes_{B'} A') $$ is surjective but in general not injective.


more-algebra-lemma-surjection-module-over-fibre-product

In Situation \ref{situation-module-over-fibre-product} let $(N_1, M'_1, \varphi_1) \to (N_2, M'_2, \varphi_2)$ be a morphism of $\text{Mod}_B \times_{\text{Mod}_A} \text{Mod}_{A'}$ with $N_1 \to N_2$ and $M'_1 \to M'_2$ surjective. Then $$ N_1 \times_{\varphi_1, M_1} M'_1 \to N_2 \times_{\varphi_2, M_2} M'_2 $$ where $M_1 = M'_1/IM'_1$ and $M_2 = M'_2/IM'_2$ is surjective.


more-algebra-lemma-finite-module-over-fibre-product

Let $A, A', B, B', I, M, M', N, \varphi$ be as in Lemma \ref{lemma-module-over-fibre-product}. If $N$ finite over $B$ and $M'$ finite over $A'$, then $N' = N \times_{\varphi, M} M'$ is finite over $B'$.


more-algebra-lemma-flat-module-over-fibre-product

With $A, A', B, B', I$ as in Situation \ref{situation-module-over-fibre-product}. \begin{enumerate} \item Let $(N, M', \varphi)$ be an object of $\text{Mod}_B \times_{\text{Mod}_A} \text{Mod}_{A'}$. If $M'$ is flat over $A'$ and $N$ is flat over $B$, then $N' = N \times_{\varphi, M} M'$ is flat over $B'$. \item If $L'$ is a flat $B'$-module, then $L' = (L \otimes_{B'} B) \times_{(L \otimes_{B'} A)} (L \otimes_{B'} A')$. \item The category of flat $B'$-modules is equivalent to the full subcategory of $\text{Mod}_B \times_{\text{Mod}_A} \text{Mod}_{A'}$ consisting of triples $(N, M', \varphi)$ with $N$ flat over $B$ and $M'$ flat over $A'$. \end{enumerate}


more-algebra-lemma-finitely-presented-module-over-fibre-product

Let $A, A', B, B', I$ be as in Situation \ref{situation-module-over-fibre-product}. The category of finite projective $B'$-modules is equivalent to the full subcategory of $\text{Mod}_B \times_{\text{Mod}_A} \text{Mod}_{A'}$ consisting of triples $(N, M', \varphi)$ with $N$ finite projective over $B$ and $M'$ finite projective over $A'$.


more-algebra-lemma-relative-module-over-fibre-product

In Situation \ref{situation-relative-module-over-fibre-product} the functor (\ref{equation-relative-functor}) has a right adjoint, namely the functor $$ F : (N, M', \varphi) \longmapsto N \times_{\varphi, M} M' $$ where $M = M'/IM'$. Moreover, the composition of $F$ with (\ref{equation-relative-functor}) is the identity functor on $\text{Mod}_D \times_{\text{Mod}_C} \text{Mod}_{C'}$. In other words, setting $N' = N \times_{\varphi, M} M'$ we have $N' \otimes_{D'} D = N$ and $N' \otimes_{D'} C' = M'$.


more-algebra-lemma-relative-surjection-ideals

In Situation \ref{situation-relative-module-over-fibre-product} the map $JD' \to IC'$ is surjective where $J = \Ker(B' \to B)$.


more-algebra-lemma-relative-finite-module-over-fibre-product

Let $A, A', B, B', C, C', D, D', I, M', M, N, \varphi$ be as in Lemma \ref{lemma-relative-module-over-fibre-product}. If $N$ finite over $D$ and $M'$ finite over $C'$, then $N' = N \times_{\varphi, M} M'$ is finite over $D'$.


more-algebra-lemma-relative-flat-module-over-fibre-product

With $A, A', B, B', C, C', D, D', I$ as in Situation \ref{situation-relative-module-over-fibre-product}. \begin{enumerate} \item Let $(N, M', \varphi)$ be an object of $\text{Mod}_D \times_{\text{Mod}_C} \text{Mod}_{C'}$. If $M'$ is flat over $A'$ and $N$ is flat over $B$, then $N' = N \times_{\varphi, M} M'$ is flat over $B'$. \item If $L'$ is a $D'$-module flat over $B'$, then $L' = (L \otimes_{D'} D) \times_{(L \otimes_{D'} C)} (L \otimes_{D'} C')$. \item The category of $D'$-modules flat over $B'$ is equivalent to the categories of objects $(N, M', \varphi)$ of $\text{Mod}_D \times_{\text{Mod}_C} \text{Mod}_{C'}$ with $N$ flat over $B$ and $M'$ flat over $A'$. \end{enumerate}


more-algebra-lemma-relative-finitely-presented-module-over-fibre-product

Let $A, A', B, B', C, C', D, D', I, M', M, N, \varphi$ be as in Lemma \ref{lemma-relative-module-over-fibre-product}. If \begin{enumerate} \item $N$ is finitely presented over $D$ and flat over $B$, \item $M'$ finitely presented over $C'$ and flat over $A'$, and \item the ring map $B' \to D'$ factors as $B' \to D'' \to D'$ with $B' \to D''$ flat and $D'' \to D'$ of finite presentation, \end{enumerate} then $N' = N \times_M M'$ is finitely presented over $D'$.


more-algebra-lemma-properties-algebras-over-fibre-product

Let $A, A', B, B', I$ be as in Situation \ref{situation-module-over-fibre-product}. Let $(D, C', \varphi)$ be a system consisting of an $B$-algebra $D$, a $A'$-algebra $C'$ and an isomorphism $D \otimes_B A \to C'/IC' = C$. Set $D' = D \times_C C'$ (as in Lemma \ref{lemma-module-over-fibre-product}). Then \begin{enumerate} \item $B' \to D'$ is finite type if and only if $B \to D$ and $A' \to C'$ are finite type, \item $B' \to D'$ is flat if and only if $B \to D$ and $A' \to C'$ are flat, \item $B' \to D'$ is flat and of finite presentation if and only if $B \to D$ and $A' \to C'$ are flat and of finite presentation, \item $B' \to D'$ is smooth if and only if $B \to D$ and $A' \to C'$ are smooth, \item $B' \to D'$ is \'etale if and only if $B \to D$ and $A' \to C'$ are \'etale. \end{enumerate} Moreover, if $D'$ is a flat $B'$-algebra, then $D' \to (D' \otimes_{B'} B) \times_{(D' \otimes_{B'} A)} (D' \otimes_{B'} A')$ is an isomorphism. In this way the category of flat $B'$-algebras is equivalent to the categories of systems $(D, C', \varphi)$ as above with $D$ flat over $B$ and $C'$ flat over $A'$.


more-algebra-lemma-ideals-generated-by-minors

Let $R$ be a ring. Let $A$ be an $n \times m$ matrix with coefficients in $R$. Let $I_r(A)$ be the ideal generated by the $r \times r$-minors of $A$ with the convention that $I_0(A) = R$ and $I_r(A) = 0$ if $r > \min(n, m)$. Then \begin{enumerate} \item $I_0(A) \supset I_1(A) \supset I_2(A) \supset \ldots$, \item if $B$ is an $(n + n') \times m$ matrix, and $A$ is the first $n$ rows of $B$, then $I_{r + n'}(B) \subset I_r(A)$, \item if $C$ is an $n \times n$ matrix then $I_r(CA) \subset I_r(A)$. \item If $A$ is a block matrix $$ \left( \begin{matrix} A_1 & 0 \\ 0 & A_2  \end{matrix} \right) $$ then $I_r(A) = \sum_{r_1 + r_2 = r} I_{r_1}(A_1) I_{r_2}(A_2)$. \item Add more here. \end{enumerate}


more-algebra-lemma-fitting-ideal

Let $R$ be a ring. Let $M$ be a finite $R$-module. Choose a presentation $$ \bigoplus\nolimits_{j \in J} R \longrightarrow R^{\oplus n} \longrightarrow M \longrightarrow 0. $$ of $M$. Let $A = (a_{ij})_{i = 1, \ldots, n, j \in J}$ be the matrix of the map $\bigoplus_{j \in J} R \to R^{\oplus n}$. The ideal $\text{Fit}_k(M)$ generated by the $(n - k) \times (n - k)$ minors of $A$ is independent of the choice of the presentation.


more-algebra-lemma-fitting-ideal-basics

Let $R$ be a ring. Let $M$ be a finite $R$-module. \begin{enumerate} \item If $M$ can be generated by $n$ elements, then $\text{Fit}_n(M) = R$. \item Given a second finite $R$-module $M'$ we have $$ \text{Fit}_l(M \oplus M') = \sum\nolimits_{k + k' = l} \text{Fit}_k(M)\text{Fit}_{k'}(M') $$ \item If $R \to R'$ is a ring map, then $\text{Fit}_k(M \otimes_R R')$ is the ideal of $R'$ generated by the image of $\text{Fit}_k(M)$. \item If $M$ is of finite presentation, then $\text{Fit}_k(M)$ is a finitely generated ideal. \item If $M \to M'$ is a surjection, then $\text{Fit}_k(M) \subset \text{Fit}_k(M')$. \item We have $\text{Fit}_0(M) \subset \text{Ann}_R(M)$. \item We have $V(\text{Fit}_0(M)) = \text{Supp}(M)$. \item Add more here. \end{enumerate}


more-algebra-lemma-fitting-ideal-generate-locally

Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $k \geq 0$. Let $\mathfrak p \subset R$ be a prime ideal. The following are equivalent \begin{enumerate} \item $\text{Fit}_k(M) \not \subset \mathfrak p$, \item $\dim_{\kappa(\mathfrak p)} M \otimes_R \kappa(\mathfrak p) \leq k$, \item $M_\mathfrak p$ can be generated by $k$ elements over $R_\mathfrak p$, and \item $M_f$ can be generated by $k$ elements over $R_f$ for some $f \in R$, $f \not \in \mathfrak p$. \end{enumerate}


more-algebra-lemma-fitting-ideal-finite-locally-free

Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $r \geq 0$. The following are equivalent \begin{enumerate} \item $M$ is finite locally free of rank $r$ (Algebra, Definition \ref{algebra-definition-locally-free}), \item $\text{Fit}_{r - 1}(M) = 0$ and $\text{Fit}_r(M) = R$, and \item $\text{Fit}_k(M) = 0$ for $k < r$ and $\text{Fit}_k(M) = R$ for $k \geq r$. \end{enumerate}


more-algebra-lemma-principal-fitting-ideal

Let $R$ be a local ring. Let $M$ be a finite $R$-module. Let $k \geq 0$. Assume that $\text{Fit}_k(M) = (f)$ for some $f \in R$. Let $M'$ be the quotient of $M$ by $\{x \in M \mid fx = 0\}$. Then $M'$ can be generated by $k$ elements.


more-algebra-lemma-fitting-ideals-and-pd1

Let $R$ be a ring. Let $M$ be a finitely presented $R$-module. Let $k \geq 0$. Assume that $\text{Fit}_k(M) = (f)$ for some nonzerodivisor $f \in R$ and $\text{Fit}_{k - 1}(M) = 0$. Then $M$ has projective dimension $\leq 1$.


more-algebra-lemma-lift-invertible-element

Let $A$ be a ring, let $I \subset A$ be an ideal, let $\overline{u} \in A/I$ be an invertible element. There exists an \'etale ring map $A \to A'$ which induces an isomorphism $A/I \to A'/IA'$ and an invertible element $u' \in A'$ lifting $\overline{u}$.


more-algebra-lemma-lift-idempotent

Let $A$ be a ring, let $I \subset A$ be an ideal, let $\overline{e} \in A/I$ be an idempotent. There exists an \'etale ring map $A \to A'$ which induces an isomorphism $A/I \to A'/IA'$ and an idempotent $e' \in A'$ lifting $\overline{e}$.


more-algebra-lemma-lift-open-covering

Let $A$ be a ring, let $I \subset A$ be an ideal. Let $\Spec(A/I) = \coprod_{j \in J} \overline{U}_j$ be a finite disjoint open covering. Then there exists an \'etale ring map $A \to A'$ which induces an isomorphism $A/I \to A'/IA'$ and a finite disjoint open covering $\Spec(A') = \coprod_{j \in J} U'_j$ lifting the given covering.


more-algebra-lemma-localize-upstairs

Let $A \to B$ be a ring map and $J \subset B$ an ideal. If $A \to B$ is \'etale at every prime of $V(J)$, then there exists a $g \in B$ mapping to an invertible element of $B/J$ such that $A' = B_g$ is \'etale over $A$.


more-algebra-lemma-lift-factorization-monic

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


more-algebra-lemma-lift-factorization-easy

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


more-algebra-lemma-separate-image-closed-from-closed

Let $R \to S$ be a ring map. Let $I \subset R$ be an ideal of $R$ and let $J \subset S$ be an ideal of $S$. If the closure of the image of $V(J)$ in $\Spec(R)$ is disjoint from $V(I)$, then there exists an element $f \in R$ which maps to $1$ in $R/I$ and to an element of $J$ in $S$.


more-algebra-lemma-helper-integral

Let $I$ be an ideal of a ring $A$. Let $A \to B$ be an integral ring map. Let $b \in B$ map to an idempotent in $B/IB$. Then there exists a monic $f \in A[x]$ with $f(b) = 0$ and $f \bmod I = x^d(x - 1)^d$ for some $d \geq 1$.


more-algebra-lemma-lift-idempotent-upstairs

Let $A$ be a ring, let $I \subset A$ be an ideal. Let $A \to B$ be an integral ring map. Let $\overline{e} \in B/IB$ be an idempotent. Then there exists an \'etale ring map $A \to A'$ which induces an isomorphism $A/I \to A'/IA'$ and an idempotent $e' \in B \otimes_A A'$ lifting $\overline{e}$.


more-algebra-lemma-lift-projective-module

Let $A$ be a ring, let $I \subset A$ be an ideal. Let $\overline{P}$ be a finite projective $A/I$-module. Then there exists an \'etale ring map $A \to A'$ which induces an isomorphism $A/I \to A'/IA'$ and a finite projective $A'$-module $P'$ lifting $\overline{P}$.


more-algebra-lemma-cotangent-complex-symmetric-algebra

Let $A$ be a ring. Let $0 \to K \to A^{\oplus m} \to M \to 0$ be a sequence of $A$-modules. Consider the $A$-algebra $C = \text{Sym}^*_A(M)$ with its presentation $\alpha : A[y_1, \ldots, y_m] \to C$ coming from the surjection $A^{\oplus m} \to M$. Then $$ \NL(\alpha) = (K \otimes_A C \to \bigoplus\nolimits_{j = 1, \ldots, m} C \text{d}y_j) $$ (see Algebra, Section \ref{algebra-section-netherlander}) in particular $\Omega_{C/A} = M \otimes_A C$.


more-algebra-lemma-symmetric-algebra-smooth

Let $A$ be a ring. Let $M$ be an $A$-module. Then $C = \text{Sym}_A^*(M)$ is smooth over $A$ if and only if $M$ is a finite projective $A$-module.


more-algebra-lemma-idempotents-determined-modulo-radical

Let $(A, I)$ be a Zariski pair. Then the map from idempotents of $A$ to idempotents of $A/I$ is injective.


more-algebra-lemma-check-isomorphism-zariski

Let $(A, I)$ be a Zariski pair. Let $A \to B$ be a flat, integral, finitely presented ring map such that $A/I \to B/IB$ is an isomorphism. Then $A \to B$ is an isomorphism.


more-algebra-lemma-helper-finite

Let $(A, I)$ be a Zariski pair. Let $A \to B$ be a finite ring map. Assume \begin{enumerate} \item $B/IB = B_1 \times B_2$ is a product of $A/I$-algebras \item $A/I \to B_1/IB_1$ is surjective, \item $b \in B$ maps to $(1, 0)$ in the product. \end{enumerate} Then there exists a monic $f \in A[x]$ with $f(b) = 0$ and $f \bmod I = (x - 1)x^d$ for some $d \geq 1$.


more-algebra-lemma-noetherian-zariski-jacobson-complement

Let $(A, I)$ be a Zariski pair with $A$ Noetherian. Let $f \in I$. Then $A_f$ is a Jacobson ring.


more-algebra-lemma-locally-nilpotent-henselian

Let $(A, I)$ be a pair with $I$ locally nilpotent. Then the functor $B \mapsto B/IB$ induces an equivalence between the category of \'etale algebras over $A$ and the category of \'etale algebras over $A/I$. Moreover, the pair is henselian.


more-algebra-lemma-limit-henselian

Let $A = \lim A_n$ where $(A_n)$ is an inverse system of rings whose transition maps are surjective and have locally nilpotent kernels. Then $(A, I_n)$ is a henselian pair, where $I_n = \Ker(A \to A_n)$.


more-algebra-lemma-complete-henselian

Let $(A, I)$ be a pair. If $A$ is $I$-adically complete, then the pair is henselian.


more-algebra-lemma-helper-finite-type

Let $(A, I)$ be a pair. Let $A \to B$ be a finite type ring map such that $B/IB = C_1 \times C_2$ with $A/I \to C_1$ finite. Let $B'$ be the integral closure of $A$ in $B$. Then we can write $B'/IB' = C_1 \times C'_2$ such that the map $B'/IB' \to B/IB$ preserves product decompositions and there exists a $g \in B'$ mapping to $(1, 0)$ in $C_1 \times C'_2$ with $B'_g \to B_g$ an isomorphism.


more-algebra-lemma-characterize-henselian-pair

\begin{reference} \cite[Chapter XI]{Henselian} and \cite[Proposition 1]{Gabber-henselian} \end{reference} Let $(A, I)$ be a pair. The following are equivalent \begin{enumerate} \item $(A, I)$ is a henselian pair, \item given an \'etale ring map $A \to A'$ and an $A$-algebra map $\sigma : A' \to A/I$, there exists an $A$-algebra map $A' \to A$ lifting $\sigma$, \item for any finite $A$-algebra $B$ the map $B \to B/IB$ induces a bijection on idempotents, \item for any integral $A$-algebra $B$ the map $B \to B/IB$ induces a bijection on idempotents, and \item (Gabber) $I$ is contained in the Jacobson radical of $A$ and every monic polynomial $f(T) \in A[T]$ of the form $$ f(T) = T^n(T - 1) + a_n T^n + \ldots + a_1 T + a_0 $$ with $a_n, \ldots, a_0 \in I$ and $n \ge 1$ has a root $\alpha \in 1 + I$. \end{enumerate} Moreover, in part (5) the root is unique.


more-algebra-lemma-change-ideal-henselian-pair

Let $A$ be a ring. Let $I, J \subset A$ be ideals with $V(I) = V(J)$. Then $(A, I)$ is henselian if and only if $(A, J)$ is henselian.


more-algebra-lemma-integral-over-henselian-pair

Let $(A, I)$ be a henselian pair and let $A \to B$ be an integral ring map. Then $(B, IB)$ is a henselian pair.


more-algebra-lemma-henselian-henselian-pair

Let $I \subset J \subset A$ be ideals of a ring $A$. The following are equivalent \begin{enumerate} \item $(A, I)$ and $(A/I, J/I)$ are henselian pairs, and \item $(A, J)$ is an henselian pair. \end{enumerate}


more-algebra-lemma-sum-henselian

Let $A$ be a ring and let $(A, I)$ and $(A, I')$ be henselian pairs. Then $(A, I + I')$ is an henselian pair.


more-algebra-lemma-product-henselian-pairs

Let $J$ be a set and let $\{ (A_j, I_j)\}_{j \in J}$ be a collection of pairs. Then $(\prod_{j \in J} A_j, \prod_{j\in J} I_j)$ is Henselian if and only if so is each $(A_j, I_j)$.


more-algebra-lemma-filtered-colimits-henselian

The property of being Henselian is preserved under filtered colimits of pairs. More precisely, let $J$ be a directed set and let $(A_j, I_j)$ be a system of henselian pairs over $J$. Then $A = \colim A_j$ equipped with the ideal $I = \colim I_j$ is a henselian pair $(A, I)$.