cohomology-lemma-triangle-sections-with-support-sheaves

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset. Denote $j : U = X \setminus Z \to X$ the inclusion of the complement. There is a distinguished triangle $$ i_*R\mathcal{H}_Z(K) \to K \to Rj_*(K|_U) \to i_*R\mathcal{H}_Z(K)[1] $$ in $D(\mathcal{O}_X)$ functorial for $K$ in $D(\mathcal{O}_X)$.


cohomology-lemma-sections-support-in-closed-disjoint-open

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $Z \subset X$ be a closed subset. Let $j : U \to X$ be the inclusion of an open subset with $U \cap Z = \emptyset$. Then $R\mathcal{H}_Z(Rj_*K) = 0$ for all $K$ in $D(\mathcal{O}_U)$.


cohomology-lemma-sections-support-abelian-unbounded

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $Z \subset X$ be a closed subset. Let $K$ be an object of $D(\mathcal{O}_X)$ and denote $K_{ab}$ its image in $D(\underline{\mathbf{Z}}_X)$. \begin{enumerate} \item There is a canonical map $R\Gamma_Z(X, K) \to R\Gamma_Z(X, K_{ab})$ which is an isomorphism in $D(\textit{Ab})$. \item There is a canonical map $R\mathcal{H}_Z(K) \to R\mathcal{H}_Z(K_{ab})$ which is an isomorphism in $D(\underline{\mathbf{Z}}_Z)$. \end{enumerate}


cohomology-lemma-support-cup-product

With notation as in Remark \ref{remark-support-cup-product} the diagram $$ \xymatrix{ H^i(X, K) \times H^j_Z(X, M) \ar[r] \ar[d] & H^{i + j}_Z(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} M) \ar[d] \\ H^i(X, K) \times H^j(X, M) \ar[r] & H^{i + j}(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} M) } $$ commutes where the top horizontal arrow is the cup product of Remark \ref{remark-support-cup-product-global}.


cohomology-lemma-support-functorial

With notation and assumptions as in Remark \ref{remark-support-functorial} the diagram $$ \xymatrix{ H^p_Z(X, K) \ar[r] \ar[d] & H^p_{Z'}(X, Lf^*K) \ar[d] \\ H^p(X, K) \ar[r] & H^p(X', Lf^*K) } $$ commutes. Here the top horizontal arrow comes from the identifications $H^p_Z(X, K) = H^p(Z, R\mathcal{H}_Z(K))$ and $H^p_{Z'}(X', Lf^*K) = H^p(Z', R\mathcal{H}_{Z'}(K'))$, the pullback map $H^p(Z, R\mathcal{H}_Z(K)) \to H^p(Z', L(f|_{Z'})^*R\mathcal{H}_Z(K))$, and the map constructed in Remark \ref{remark-support-functorial}.


cohomology-lemma-RGamma-commutes-with-Rlim

Let $(X, \mathcal{O}_X)$ be a ringed space. For $U \subset X$ open the functor $R\Gamma(U, -)$ commutes with $R\lim$. Moreover, there are short exact sequences $$ 0 \to R^1\lim H^{m - 1}(U, K_n) \to H^m(U, R\lim K_n) \to \lim H^m(U, K_n) \to 0 $$ for any inverse system $(K_n)$ in $D(\mathcal{O}_X)$ and any $m \in \mathbf{Z}$.


cohomology-lemma-Rf-commutes-with-Rlim

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Then $Rf_*$ commutes with $R\lim$, i.e., $Rf_*$ commutes with derived limits.


cohomology-lemma-inverse-limit-is-derived-limit

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $(\mathcal{F}_n)$ be an inverse system of $\mathcal{O}_X$-modules. Let $\mathcal{B}$ be a set of opens of $X$. Assume \begin{enumerate} \item every open of $X$ has a covering whose members are elements of $\mathcal{B}$, \item $H^p(U, \mathcal{F}_n) = 0$ for $p > 0$ and $U \in \mathcal{B}$, \item the inverse system $\mathcal{F}_n(U)$ has vanishing $R^1\lim$ for $U \in \mathcal{B}$. \end{enumerate} Then $R\lim \mathcal{F}_n = \lim \mathcal{F}_n$ and we have $H^p(U, \lim \mathcal{F}_n) = 0$ for $p > 0$ and $U \in \mathcal{B}$.


cohomology-lemma-cohomology-derived-limit-injective

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $(K_n)$ be an inverse system in $D(\mathcal{O}_X)$. Let $x \in X$ and $m \in \mathbf{Z}$. Assume there exist an integer $n(x)$ and a fundamental system $\mathfrak{U}_x$ of open neighbourhoods of $x$ such that for $U \in \mathfrak{U}_x$ \begin{enumerate} \item $R^1\lim H^{m - 1}(U, K_n) = 0$, and \item $H^m(U, K_n) \to H^m(U, K_{n(x)})$ is injective for $n \geq n(x)$. \end{enumerate} Then the map on stalks $H^m(R\lim K_n)_x \to H^m(K_{n(x)})_x$ is injective.


cohomology-lemma-is-limit-per-point

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $E \in D(\mathcal{O}_X)$. Assume that for every $x \in X$ there exist a function $p(x, -) : \mathbf{Z} \to \mathbf{Z}$ and a fundamental system $\mathfrak{U}_x$ of open neighbourhoods of $x$ such that $$ H^p(U, H^{m - p}(E)) = 0 \text{ for } U \in \mathfrak{U}_x \text{ and } p > p(x, m) $$ Then the canonical map $E \to R\lim \tau_{\geq -n} E$ is an isomorphism in $D(\mathcal{O}_X)$.


cohomology-lemma-is-limit-spaltenstein

\begin{reference} \cite[Proposition 3.13]{Spaltenstein} \end{reference} Let $(X, \mathcal{O}_X)$ be a ringed space. Let $E \in D(\mathcal{O}_X)$. Assume that for every $x \in X$ there exist an integer $d_x \geq 0$ and a fundamental system $\mathfrak{U}_x$ of open neighbourhoods of $x$ such that $$ H^p(U, H^q(E)) = 0 \text{ for } U \in \mathfrak{U}_x,\ p > d_x, \text{ and }q < 0 $$ Then the canonical map $E \to R\lim \tau_{\geq -n} E$ is an isomorphism in $D(\mathcal{O}_X)$.


cohomology-lemma-is-limit

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $E \in D(\mathcal{O}_X)$. Assume there exist a function $p(-) : \mathbf{Z} \to \mathbf{Z}$ and a set $\mathcal{B}$ of opens of $X$ such that \begin{enumerate} \item every open in $X$ has a covering whose members are elements of $\mathcal{B}$, and \item $H^p(U, H^{m - p}(E)) = 0$ for $p > p(m)$ and $U \in \mathcal{B}$. \end{enumerate} Then the canonical map $E \to R\lim \tau_{\geq -n} E$ is an isomorphism in $D(\mathcal{O}_X)$.


cohomology-lemma-is-limit-dimension

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $E \in D(\mathcal{O}_X)$. Assume there exist an integer $d \geq 0$ and a basis $\mathcal{B}$ for the topology of $X$ such that $$ H^p(U, H^q(E)) = 0 \text{ for } U \in \mathcal{B},\ p > d, \text{ and }q < 0 $$ Then the canonical map $E \to R\lim \tau_{\geq -n} E$ is an isomorphism in $D(\mathcal{O}_X)$.


cohomology-lemma-cohomology-over-U-trivial

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $K$ be an object of $D(\mathcal{O}_X)$. Let $\mathcal{B}$ be a set of opens of $X$. Assume \begin{enumerate} \item every open of $X$ has a covering whose members are elements of $\mathcal{B}$, \item $H^p(U, H^q(K)) = 0$ for all $p > 0$, $q \in \mathbf{Z}$, and $U \in \mathcal{B}$. \end{enumerate} Then $H^q(U, K) = H^0(U, H^q(K))$ for $q \in \mathbf{Z}$ and $U \in \mathcal{B}$.


cohomology-lemma-derived-limit-suitable-system

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $(K_n)$ be an inverse system of objects of $D(\mathcal{O}_X)$. Let $\mathcal{B}$ be a set of opens of $X$. Assume \begin{enumerate} \item every open of $X$ has a covering whose members are elements of $\mathcal{B}$, \item for all $U \in \mathcal{B}$ and all $q \in \mathbf{Z}$ we have \begin{enumerate} \item $H^p(U, H^q(K_n)) = 0$ for $p > 0$, \item the inverse system $H^0(U, H^q(K_n))$ has vanishing $R^1\lim$. \end{enumerate} \end{enumerate} Then $H^q(R\lim K_n) = \lim H^q(K_n)$ for $q \in \mathbf{Z}$.


cohomology-lemma-K-injective

In the situation described above. Denote $\mathcal{H}^m = H^m(\mathcal{F}^\bullet)$ the $m$th cohomology sheaf. Let $\mathcal{B}$ be a set of open subsets of $X$. Let $d \in \mathbf{N}$. Assume \begin{enumerate} \item every open in $X$ has a covering whose members are elements of $\mathcal{B}$, \item for every $U \in \mathcal{B}$ we have $H^p(U, \mathcal{H}^q) = 0$ for $p > d$ and $q < 0$\footnote{It suffices if $\forall m$, $\exists p(m)$, $H^p(U. \mathcal{H}^{m - p}) = 0$ for $p > p(m)$, see Lemma \ref{lemma-is-limit}.}. \end{enumerate} Then (\ref{equation-into-candidate-K-injective}) is a quasi-isomorphism.


cohomology-lemma-alternating-cech-complex-complex

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be a finite open covering. For a complex $\mathcal{F}^\bullet$ of $\mathcal{O}_X$-modules there is a canonical map $$ \text{Tot}(\check{\mathcal{C}}^\bullet_{alt}(\mathcal{U}, \mathcal{F}^\bullet)) \longrightarrow R\Gamma(X, \mathcal{F}^\bullet) $$ functorial in $\mathcal{F}^\bullet$ and compatible with (\ref{equation-global-sections-to-alternating-cech}).


cohomology-lemma-alternating-cech-complex-complex-ss

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be a finite open covering. Let $\mathcal{F}^\bullet$ be a complex of $\mathcal{O}_X$-modules. Let $\mathcal{B}$ be a set of open subsets of $X$. Assume \begin{enumerate} \item every open in $X$ has a covering whose members are elements of $\mathcal{B}$, \item we have $U_{i_0\ldots i_p} \in \mathcal{B}$ for all $i_0, \ldots, i_p \in I$, \item for every $U \in \mathcal{B}$ and $p > 0$ we have \begin{enumerate} \item $H^p(U, \mathcal{F}^q) = 0$, \item $H^p(U, \Coker(\mathcal{F}^{q - 1} \to \mathcal{F}^q)) = 0$, and \item $H^p(U, H^q(\mathcal{F})) = 0$. \end{enumerate} \end{enumerate} Then the map $$ \text{Tot}(\check{\mathcal{C}}^\bullet_{alt}(\mathcal{U}, \mathcal{F}^\bullet)) \longrightarrow R\Gamma(X, \mathcal{F}^\bullet) $$ of Lemma \ref{lemma-alternating-cech-complex-complex} is an isomorphism in $D(\textit{Ab})$.


cohomology-lemma-compose

Let $(X, \mathcal{O}_X)$ be a ringed space. Given complexes $\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet$ of $\mathcal{O}_X$-modules there is an isomorphism $$ \SheafHom^\bullet(\mathcal{K}^\bullet, \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{M}^\bullet)) = \SheafHom^\bullet(\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_X} \mathcal{L}^\bullet), \mathcal{M}^\bullet) $$ of complexes of $\mathcal{O}_X$-modules functorial in $\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet$.


cohomology-lemma-composition

Let $(X, \mathcal{O}_X)$ be a ringed space. Given complexes $\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet$ of $\mathcal{O}_X$-modules there is a canonical morphism $$ \text{Tot}\left( \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{M}^\bullet) \otimes_{\mathcal{O}_X} \SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{L}^\bullet) \right) \longrightarrow \SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{M}^\bullet) $$ of complexes of $\mathcal{O}_X$-modules.


cohomology-lemma-diagonal-better

Let $(X, \mathcal{O}_X)$ be a ringed space. Given complexes $\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet$ of $\mathcal{O}_X$-modules there is a canonical morphism $$ \text{Tot}\left( \mathcal{K}^\bullet \otimes_{\mathcal{O}_X} \SheafHom^\bullet(\mathcal{M}^\bullet, \mathcal{L}^\bullet) \right) \longrightarrow \SheafHom^\bullet(\mathcal{M}^\bullet, \text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_X} \mathcal{L}^\bullet)) $$ of complexes of $\mathcal{O}_X$-modules functorial in all three complexes.


cohomology-lemma-diagonal

Let $(X, \mathcal{O}_X)$ be a ringed space. Given complexes $\mathcal{K}^\bullet, \mathcal{L}^\bullet$ of $\mathcal{O}_X$-modules there is a canonical morphism $$ \mathcal{K}^\bullet \longrightarrow \SheafHom^\bullet(\mathcal{L}^\bullet, \text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_X} \mathcal{L}^\bullet)) $$ of complexes of $\mathcal{O}_X$-modules functorial in both complexes.


cohomology-lemma-evaluate-and-more

Let $(X, \mathcal{O}_X)$ be a ringed space. Given complexes $\mathcal{K}^\bullet, \mathcal{L}^\bullet, \mathcal{M}^\bullet$ of $\mathcal{O}_X$-modules there is a canonical morphism $$ \text{Tot}(\SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{M}^\bullet) \otimes_{\mathcal{O}_X} \mathcal{K}^\bullet) \longrightarrow \SheafHom^\bullet(\SheafHom^\bullet(\mathcal{K}^\bullet, \mathcal{L}^\bullet), \mathcal{M}^\bullet) $$ of complexes of $\mathcal{O}_X$-modules functorial in all three complexes.


cohomology-lemma-RHom-into-K-injective

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{I}^\bullet$ be a K-injective complex of $\mathcal{O}_X$-modules. Let $\mathcal{L}^\bullet$ be a complex of $\mathcal{O}_X$-modules. Then $$ H^0(\Gamma(U, \SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{I}^\bullet))) = \Hom_{D(\mathcal{O}_U)}(L|_U, M|_U) $$ for all $U \subset X$ open.


cohomology-lemma-RHom-from-K-flat-into-K-injective

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{I}^\bullet$ be a K-injective complex of $\mathcal{O}_X$-modules. Let $\mathcal{L}^\bullet$ be a K-flat complex of $\mathcal{O}_X$-modules. Then $\SheafHom^\bullet(\mathcal{L}^\bullet, \mathcal{I}^\bullet)$ is a K-injective complex of $\mathcal{O}_X$-modules.


cohomology-lemma-internal-hom

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $K, L, M$ be objects of $D(\mathcal{O}_X)$. With the construction as described above there is a canonical isomorphism $$ R\SheafHom(K, R\SheafHom(L, M)) = R\SheafHom(K \otimes_{\mathcal{O}_X}^\mathbf{L} L, M) $$ in $D(\mathcal{O}_X)$ functorial in $K, L, M$ which recovers (\ref{equation-internal-hom}) by taking $H^0(X, -)$.


cohomology-lemma-restriction-RHom-to-U

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $K, L$ be objects of $D(\mathcal{O}_X)$. The construction of $R\SheafHom(K, L)$ commutes with restrictions to opens, i.e., for every open $U$ we have $R\SheafHom(K|_U, L|_U) = R\SheafHom(K, L)|_U$.


cohomology-lemma-RHom-triangulated

Let $(X, \mathcal{O}_X)$ be a ringed space. The bifunctor $R\SheafHom(- , -)$ transforms distinguished triangles into distinguished triangles in both variables.


cohomology-lemma-internal-hom-composition

Let $(X, \mathcal{O}_X)$ be a ringed space. Given $K, L, M$ in $D(\mathcal{O}_X)$ there is a canonical morphism $$ R\SheafHom(L, M) \otimes_{\mathcal{O}_X}^\mathbf{L} R\SheafHom(K, L) \longrightarrow R\SheafHom(K, M) $$ in $D(\mathcal{O}_X)$ functorial in $K, L, M$.


cohomology-lemma-internal-hom-diagonal-better

Let $(X, \mathcal{O}_X)$ be a ringed space. Given $K, L, M$ in $D(\mathcal{O}_X)$ there is a canonical morphism $$ K \otimes_{\mathcal{O}_X}^\mathbf{L} R\SheafHom(M, L) \longrightarrow R\SheafHom(M, K \otimes_{\mathcal{O}_X}^\mathbf{L} L) $$ in $D(\mathcal{O}_X)$ functorial in $K, L, M$.


cohomology-lemma-dual

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $L$ be an object of $D(\mathcal{O}_X)$. Set $L^\vee = R\SheafHom(L, \mathcal{O}_X)$. For $M$ in $D(\mathcal{O}_X)$ there is a canonical map \begin{equation} \label{equation-eval} M \otimes^\mathbf{L}_{\mathcal{O}_X} L^\vee \longrightarrow R\SheafHom(L, M) \end{equation} which induces a canonical map $$ H^0(X, M \otimes^\mathbf{L}_{\mathcal{O}_X} L^\vee) \longrightarrow \Hom_{D(\mathcal{O}_X)}(L, M) $$ functorial in $M$ in $D(\mathcal{O}_X)$.


cohomology-lemma-internal-hom-evaluate

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $K, L, M$ be objects of $D(\mathcal{O}_X)$. There is a canonical morphism $$ R\SheafHom(L, M) \otimes_{\mathcal{O}_X}^\mathbf{L} K \longrightarrow R\SheafHom(R\SheafHom(K, L), M) $$ in $D(\mathcal{O}_X)$ functorial in $K, L, M$.


cohomology-lemma-glue

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $X = U \cup V$ be the union of two open subspaces of $X$. Suppose given \begin{enumerate} \item an object $A$ of $D(\mathcal{O}_U)$, \item an object $B$ of $D(\mathcal{O}_V)$, and \item an isomorphism $c : A|_{U \cap V} \to B|_{U \cap V}$. \end{enumerate} Then there exists an object $F$ of $D(\mathcal{O}_X)$ and isomorphisms $f : F|_U \to A$, $g : F|_V \to B$ such that $c = g|_{U \cap V} \circ f^{-1}|_{U \cap V}$. Moreover, given \begin{enumerate} \item an object $E$ of $D(\mathcal{O}_X)$, \item a morphism $a : A \to E|_U$ of $D(\mathcal{O}_U)$, \item a morphism $b : B \to E|_V$ of $D(\mathcal{O}_V)$,  \end{enumerate} such that $$ a|_{U \cap V}  = b|_{U \cap V} \circ c. $$ Then there exists a morphism $F \to E$ in $D(\mathcal{O}_X)$ whose restriction to $U$ is $a \circ f$ and whose restriction to $V$ is $b \circ g$.


cohomology-lemma-vanishing-and-glueing

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{B}$ be a basis for the topology on $Y$. \begin{enumerate} \item Assume $K$ is in $D(\mathcal{O}_X)$ such that for $V \in \mathcal{B}$ we have $H^i(f^{-1}(V), K) = 0$ for $i < 0$. Then $Rf_*K$ has vanishing cohomology sheaves in negative degrees, $H^i(f^{-1}(V), K) = 0$ for $i < 0$ for all opens $V \subset Y$, and the rule $V \mapsto H^0(f^{-1}V, K)$ is a sheaf on $Y$. \item Assume $K, L$ are in $D(\mathcal{O}_X)$ such that for $V \in \mathcal{B}$ we have $\Ext^i(K|_{f^{-1}V}, L|_{f^{-1}V}) = 0$ for $i < 0$. Then $\Ext^i(K|_{f^{-1}V}, L|_{f^{-1}V}) = 0$ for $i < 0$ for all opens $V \subset Y$ and the rule $V \mapsto \Hom(K|_{f^{-1}V}, L|_{f^{-1}V})$ is a sheaf on $Y$. \end{enumerate}


cohomology-lemma-uniqueness

In Situation \ref{situation-locally-given} assume \begin{enumerate} \item $X = \bigcup_{U \in \mathcal{B}} U$ and for $U, V \in \mathcal{B}$ we have $U \cap V = \bigcup_{W \in \mathcal{B}, W \subset U \cap V} W$, \item for any $U \in \mathcal{B}$ we have $\Ext^i(K_U, K_U) = 0$ for $i < 0$. \end{enumerate} If a solution $(K, \rho_U)$ exists, then it is unique up to unique isomorphism and moreover $\Ext^i(K, K) = 0$ for $i < 0$.


cohomology-lemma-solution-in-finite-case

In Situation \ref{situation-locally-given} assume \begin{enumerate} \item $X = U_1 \cup \ldots \cup U_n$ with $U_i \in \mathcal{B}$, \item for $U, V \in \mathcal{B}$ we have $U \cap V = \bigcup_{W \in \mathcal{B}, W \subset U \cap V} W$, \item for any $U \in \mathcal{B}$ we have $\Ext^i(K_U, K_U) = 0$ for $i < 0$. \end{enumerate} Then a solution exists and is unique up to unique isomorphism.


cohomology-lemma-glueing-increasing-union

Let $X$ be a ringed space. Let $E$ be a well ordered set and let $$ X = \bigcup\nolimits_{\alpha \in E} W_\alpha $$ be an open covering with $W_\alpha \subset W_{\alpha + 1}$ and $W_\alpha = \bigcup_{\beta < \alpha} W_\beta$ if $\alpha$ is not a successor. Let $K_\alpha$ be an object of $D(\mathcal{O}_{W_\alpha})$ with $\Ext^i(K_\alpha, K_\alpha) = 0$ for $i < 0$. Assume given isomorphisms $\rho_\beta^\alpha :  K_\alpha|_{W_\beta} \to K_\beta$ in $D(\mathcal{O}_{W_\beta})$ for all $\beta < \alpha$ with $\rho_\gamma^\alpha = \rho_\gamma^\beta \circ \rho^\alpha_\beta|_{W_\gamma}$ for $\gamma < \beta < \alpha$. Then there exists an object $K$ in $D(\mathcal{O}_X)$ and isomorphisms $K|_{W_\alpha} \to K_\alpha$ for $\alpha \in E$ compatible with the isomorphisms $\rho_\beta^\alpha$.


cohomology-lemma-cone

The cone on a morphism of strictly perfect complexes is strictly perfect.


cohomology-lemma-tensor

The total complex associated to the tensor product of two strictly perfect complexes is strictly perfect.


cohomology-lemma-strictly-perfect-pullback

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. If $\mathcal{F}^\bullet$ is a strictly perfect complex of $\mathcal{O}_Y$-modules, then $f^*\mathcal{F}^\bullet$ is a strictly perfect complex of $\mathcal{O}_X$-modules.


cohomology-lemma-local-lift-map

Let $(X, \mathcal{O}_X)$ be a ringed space. Given a solid diagram of $\mathcal{O}_X$-modules $$ \xymatrix{ \mathcal{E} \ar@{..>}[dr] \ar[r] & \mathcal{F} \\ & \mathcal{G} \ar[u]_p } $$ with $\mathcal{E}$ a direct summand of a finite free $\mathcal{O}_X$-module and $p$ surjective, then a dotted arrow making the diagram commute exists locally on $X$.


cohomology-lemma-local-homotopy

Let $(X, \mathcal{O}_X)$ be a ringed space. \begin{enumerate} \item Let $\alpha : \mathcal{E}^\bullet \to \mathcal{F}^\bullet$ be a morphism of complexes of $\mathcal{O}_X$-modules with $\mathcal{E}^\bullet$ strictly perfect and $\mathcal{F}^\bullet$ acyclic. Then $\alpha$ is locally on $X$ homotopic to zero. \item Let $\alpha : \mathcal{E}^\bullet \to \mathcal{F}^\bullet$ be a morphism of complexes of $\mathcal{O}_X$-modules with $\mathcal{E}^\bullet$ strictly perfect, $\mathcal{E}^i = 0$ for $i < a$, and $H^i(\mathcal{F}^\bullet) = 0$ for $i \geq a$. Then $\alpha$ is locally on $X$ homotopic to zero. \end{enumerate}


cohomology-lemma-lift-through-quasi-isomorphism

Let $(X, \mathcal{O}_X)$ be a ringed space. Given a solid diagram of complexes of $\mathcal{O}_X$-modules $$ \xymatrix{ \mathcal{E}^\bullet \ar@{..>}[dr] \ar[r]_\alpha & \mathcal{F}^\bullet \\ & \mathcal{G}^\bullet \ar[u]_f } $$ with $\mathcal{E}^\bullet$ strictly perfect, $\mathcal{E}^j = 0$ for $j < a$ and $H^j(f)$ an isomorphism for $j > a$ and surjective for $j = a$, then a dotted arrow making the diagram commute up to homotopy exists locally on $X$.


cohomology-lemma-local-actual

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{E}^\bullet$, $\mathcal{F}^\bullet$ be complexes of $\mathcal{O}_X$-modules with $\mathcal{E}^\bullet$ strictly perfect. \begin{enumerate} \item For any element $\alpha \in \Hom_{D(\mathcal{O}_X)}(\mathcal{E}^\bullet, \mathcal{F}^\bullet)$ there exists an open covering $X = \bigcup U_i$ such that $\alpha|_{U_i}$ is given by a morphism of complexes $\alpha_i : \mathcal{E}^\bullet|_{U_i} \to \mathcal{F}^\bullet|_{U_i}$. \item Given a morphism of complexes $\alpha : \mathcal{E}^\bullet \to \mathcal{F}^\bullet$ whose image in the group $\Hom_{D(\mathcal{O}_X)}(\mathcal{E}^\bullet, \mathcal{F}^\bullet)$ is zero, there exists an open covering $X = \bigcup U_i$ such that $\alpha|_{U_i}$ is homotopic to zero. \end{enumerate}


cohomology-lemma-Rhom-strictly-perfect

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{E}^\bullet$, $\mathcal{F}^\bullet$ be complexes of $\mathcal{O}_X$-modules with $\mathcal{E}^\bullet$ strictly perfect. Then the internal hom $R\SheafHom(\mathcal{E}^\bullet, \mathcal{F}^\bullet)$ is represented by the complex $\mathcal{H}^\bullet$ with terms $$ \mathcal{H}^n = \bigoplus\nolimits_{n = p + q} \SheafHom_{\mathcal{O}_X}(\mathcal{E}^{-q}, \mathcal{F}^p) $$ and differential as described in Section \ref{section-internal-hom}.


cohomology-lemma-Rhom-complex-of-direct-summands-finite-free

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{E}^\bullet$, $\mathcal{F}^\bullet$ be complexes of $\mathcal{O}_X$-modules with \begin{enumerate} \item $\mathcal{F}^n = 0$ for $n \ll 0$, \item $\mathcal{E}^n = 0$ for $n \gg 0$, and \item $\mathcal{E}^n$ isomorphic to a direct summand of a finite free $\mathcal{O}_X$-module. \end{enumerate} Then the internal hom $R\SheafHom(\mathcal{E}^\bullet, \mathcal{F}^\bullet)$ is represented by the complex $\mathcal{H}^\bullet$ with terms $$ \mathcal{H}^n = \bigoplus\nolimits_{n = p + q} \SheafHom_{\mathcal{O}_X}(\mathcal{E}^{-q}, \mathcal{F}^p) $$ and differential as described in Section \ref{section-internal-hom}.


cohomology-lemma-pseudo-coherent-independent-representative

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\mathcal{O}_X)$. \begin{enumerate} \item If there exists an open covering $X = \bigcup U_i$, strictly perfect complexes $\mathcal{E}_i^\bullet$ on $U_i$, and maps $\alpha_i : \mathcal{E}_i^\bullet \to E|_{U_i}$ in $D(\mathcal{O}_{U_i})$ with $H^j(\alpha_i)$ an isomorphism for $j > m$ and $H^m(\alpha_i)$ surjective, then $E$ is $m$-pseudo-coherent. \item If $E$ is $m$-pseudo-coherent, then any complex representing $E$ is $m$-pseudo-coherent. \end{enumerate}


cohomology-lemma-pseudo-coherent-pullback

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $E$ be an object of $D(\mathcal{O}_Y)$. If $E$ is $m$-pseudo-coherent, then $Lf^*E$ is $m$-pseudo-coherent.


cohomology-lemma-cone-pseudo-coherent

Let $(X, \mathcal{O}_X)$ be a ringed space and $m \in \mathbf{Z}$. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\mathcal{O}_X)$. \begin{enumerate} \item If $K$ is $(m + 1)$-pseudo-coherent and $L$ is $m$-pseudo-coherent then $M$ is $m$-pseudo-coherent. \item If $K$ and $M$ are $m$-pseudo-coherent, then $L$ is $m$-pseudo-coherent. \item If $L$ is $(m + 1)$-pseudo-coherent and $M$ is $m$-pseudo-coherent, then $K$ is $(m + 1)$-pseudo-coherent. \end{enumerate}


cohomology-lemma-tensor-pseudo-coherent

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $K, L$ be objects of $D(\mathcal{O}_X)$. \begin{enumerate} \item If $K$ is $n$-pseudo-coherent and $H^i(K) = 0$ for $i > a$ and $L$ is $m$-pseudo-coherent and $H^j(L) = 0$ for $j > b$, then $K \otimes_{\mathcal{O}_X}^\mathbf{L} L$ is $t$-pseudo-coherent with $t = \max(m + a, n + b)$. \item If $K$ and $L$ are pseudo-coherent, then $K \otimes_{\mathcal{O}_X}^\mathbf{L} L$ is pseudo-coherent. \end{enumerate}


cohomology-lemma-summands-pseudo-coherent

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $m \in \mathbf{Z}$. If $K \oplus L$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) in $D(\mathcal{O}_X)$ so are $K$ and $L$.


cohomology-lemma-complex-pseudo-coherent-modules

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $m \in \mathbf{Z}$. Let $\mathcal{F}^\bullet$ be a (locally) bounded above complex of $\mathcal{O}_X$-modules such that $\mathcal{F}^i$ is $(m - i)$-pseudo-coherent for all $i$. Then $\mathcal{F}^\bullet$ is $m$-pseudo-coherent.


cohomology-lemma-cohomology-pseudo-coherent

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_X)$. If $E$ is (locally) bounded above and $H^i(E)$ is $(m - i)$-pseudo-coherent for all $i$, then $E$ is $m$-pseudo-coherent.


cohomology-lemma-finite-cohomology

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $K$ be an object of $D(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$. \begin{enumerate} \item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$ for $i > m$, then $H^m(K)$ is a finite type $\mathcal{O}_X$-module. \item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$ for $i > m + 1$, then $H^{m + 1}(K)$ is a finitely presented $\mathcal{O}_X$-module. \end{enumerate}


cohomology-lemma-last-one-flat

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{E}^\bullet$ be a bounded above complex of flat $\mathcal{O}_X$-modules with tor-amplitude in $[a, b]$. Then $\Coker(d_{\mathcal{E}^\bullet}^{a - 1})$ is a flat $\mathcal{O}_X$-module.


cohomology-lemma-tor-amplitude

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\mathcal{O}_X)$. Let $a, b \in \mathbf{Z}$ with $a \leq b$. The following are equivalent \begin{enumerate} \item $E$ has tor-amplitude in $[a, b]$. \item $E$ is represented by a complex $\mathcal{E}^\bullet$ of flat $\mathcal{O}_X$-modules with $\mathcal{E}^i = 0$ for $i \not \in [a, b]$. \end{enumerate}


cohomology-lemma-tor-amplitude-pullback

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $E$ be an object of $D(\mathcal{O}_Y)$. If $E$ has tor amplitude in $[a, b]$, then $Lf^*E$ has tor amplitude in $[a, b]$.


cohomology-lemma-tor-amplitude-stalk

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\mathcal{O}_X)$. Let $a, b \in \mathbf{Z}$ with $a \leq b$. The following are equivalent \begin{enumerate} \item $E$ has tor-amplitude in $[a, b]$. \item for every $x \in X$ the object $E_x$ of $D(\mathcal{O}_{X, x})$ has tor-amplitude in $[a, b]$. \end{enumerate}


cohomology-lemma-cone-tor-amplitude

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\mathcal{O}_X)$. Let $a, b \in \mathbf{Z}$. \begin{enumerate} \item If $K$ has tor-amplitude in $[a + 1, b + 1]$ and $L$ has tor-amplitude in $[a, b]$ then $M$ has tor-amplitude in $[a, b]$. \item If $K$ and $M$ have tor-amplitude in $[a, b]$, then $L$ has tor-amplitude in $[a, b]$. \item If $L$ has tor-amplitude in $[a + 1, b + 1]$ and $M$ has tor-amplitude in $[a, b]$, then $K$ has tor-amplitude in $[a + 1, b + 1]$. \end{enumerate}


cohomology-lemma-tensor-tor-amplitude

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $K, L$ be objects of $D(\mathcal{O}_X)$. If $K$ has tor-amplitude in $[a, b]$ and $L$ has tor-amplitude in $[c, d]$ then $K \otimes_{\mathcal{O}_X}^\mathbf{L} L$ has tor amplitude in $[a + c, b + d]$.


cohomology-lemma-summands-tor-amplitude

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $a, b \in \mathbf{Z}$. For $K$, $L$ objects of $D(\mathcal{O}_X)$ if $K \oplus L$ has tor amplitude in $[a, b]$ so do $K$ and $L$.


cohomology-lemma-perfect-independent-representative

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\mathcal{O}_X)$. \begin{enumerate} \item If there exists an open covering $X = \bigcup U_i$ and strictly perfect complexes $\mathcal{E}_i^\bullet$ on $U_i$ such that $\mathcal{E}_i^\bullet$ represents $E|_{U_i}$ in $D(\mathcal{O}_{U_i})$, then $E$ is perfect. \item If $E$ is perfect, then any complex representing $E$ is perfect. \end{enumerate}


cohomology-lemma-perfect-on-locally-ringed

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\mathcal{O}_X)$. Assume that all stalks $\mathcal{O}_{X, x}$ are local rings. Then the following are equivalent \begin{enumerate} \item $E$ is perfect, \item there exists an open covering $X = \bigcup U_i$ such that $E|_{U_i}$ can be represented by a finite complex of finite locally free $\mathcal{O}_{U_i}$-modules, and \item there exists an open covering $X = \bigcup U_i$ such that $E|_{U_i}$ can be represented by a finite complex of finite free $\mathcal{O}_{U_i}$-modules. \end{enumerate}


cohomology-lemma-perfect-precise

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\mathcal{O}_X)$. Let $a \leq b$ be integers. If $E$ has tor amplitude in $[a, b]$ and is $(a - 1)$-pseudo-coherent, then $E$ is perfect.


cohomology-lemma-perfect

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item $E$ is perfect, and \item $E$ is pseudo-coherent and locally has finite tor dimension. \end{enumerate}


cohomology-lemma-perfect-pullback

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $E$ be an object of $D(\mathcal{O}_Y)$. If $E$ is perfect in $D(\mathcal{O}_Y)$, then $Lf^*E$ is perfect in $D(\mathcal{O}_X)$.


cohomology-lemma-two-out-of-three-perfect

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\mathcal{O}_X)$. If two out of three of $K, L, M$ are perfect then the third is also perfect.


cohomology-lemma-summands-perfect

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


cohomology-lemma-pushforward-perfect

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $j : U \to X$ be an open subspace. Let $E$ be a perfect object of $D(\mathcal{O}_U)$ whose cohomology sheaves are supported on a closed subset $T \subset U$ with $j(T)$ closed in $X$. Then $Rj_*E$ is a perfect object of $D(\mathcal{O}_X)$.


cohomology-lemma-symmetric-monoidal-cat-complexes

Let $(X, \mathcal{O}_X)$ be a ringed space. The category of complexes of $\mathcal{O}_X$-modules with tensor product defined by $\mathcal{F}^\bullet \otimes \mathcal{G}^\bullet = \text{Tot}(\mathcal{F}^\bullet \otimes_{\mathcal{O}_X} \mathcal{G}^\bullet)$ is a symmetric monoidal category (for sign rules, see More on Algebra, Section \ref{more-algebra-section-sign-rules}).


cohomology-lemma-internal-hom-evaluate-isom

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $K, L, M \in D(\mathcal{O}_X)$. If $K$ is perfect, then the map $$ R\SheafHom(L, M) \otimes_{\mathcal{O}_X}^\mathbf{L} K \longrightarrow R\SheafHom(R\SheafHom(K, L), M) $$ of Lemma \ref{lemma-internal-hom-evaluate} is an isomorphism.


cohomology-lemma-dual-perfect-complex

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $K$ be a perfect object of $D(\mathcal{O}_X)$. Then $K^\vee = R\SheafHom(K, \mathcal{O}_X)$ is a perfect object too and $(K^\vee)^\vee \cong K$. There are functorial isomorphisms $$ M \otimes^\mathbf{L}_{\mathcal{O}_X} K^\vee = R\SheafHom(K, M) $$ and $$ H^0(X, M \otimes^\mathbf{L}_{\mathcal{O}_X} K^\vee) = \Hom_{D(\mathcal{O}_X)}(K, M) $$ for $M$ in $D(\mathcal{O}_X)$.


cohomology-lemma-symmetric-monoidal-derived

Let $(X, \mathcal{O}_X)$ be a ringed space. The derived category $D(\mathcal{O}_X)$ is a symmetric monoidal category with tensor product given by derived tensor product with usual associativity and commutativity constraints (for sign rules, see More on Algebra, Section \ref{more-algebra-section-sign-rules}).


cohomology-lemma-left-dual-derived

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $M$ be an object of $D(\mathcal{O}_X)$. If $M$ has a left dual in the monoidal category $D(\mathcal{O}_X)$ (Categories, Definition \ref{categories-definition-dual}) then $M$ is perfect and the left dual is as constructed in Example \ref{example-dual-derived}.


cohomology-lemma-colim-and-lim-of-duals

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $(K_n)_{n \in \mathbf{N}}$ be a system of perfect objects of $D(\mathcal{O}_X)$. Let $K = \text{hocolim} K_n$ be the derived colimit (Derived Categories, Definition \ref{derived-definition-derived-colimit}). Then for any object $E$ of $D(\mathcal{O}_X)$ we have $$ R\SheafHom(K, E) = R\lim E \otimes^\mathbf{L}_{\mathcal{O}_X} K_n^\vee $$ where $(K_n^\vee)$ is the inverse system of dual perfect complexes.


cohomology-lemma-ext-composition-is-cup

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $K$ and $E$ be objects of $D(\mathcal{O}_X)$ with $E$ perfect. The diagram $$ \xymatrix{ H^0(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} E^\vee) \times H^0(X, E) \ar[r] \ar[d] & H^0(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} E^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E) \ar[d] \\ \Hom_X(E, K) \times H^0(X, E) \ar[r] & H^0(X, K) } $$ commutes where the top horizontal arrow is the cup product, the right vertical arrow uses $\epsilon : E^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E \to \mathcal{O}_X$ (Example \ref{example-dual-derived}), the left vertical arrow uses Lemma \ref{lemma-dual-perfect-complex}, and the bottom horizontal arrow is the obvious one.


cohomology-lemma-category-summands-finite-free

Let $(X, \mathcal{O}_X)$ be a ringed space. Set $R = \Gamma(X, \mathcal{O}_X)$. The category of $\mathcal{O}_X$-modules which are summands of finite free $\mathcal{O}_X$-modules is equivalent to the category of finite projective $R$-modules.


cohomology-lemma-invertible-derived

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $M$ be an object of $D(\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item $M$ is invertible in $D(\mathcal{O}_X)$, see Categories, Definition \ref{categories-definition-invertible}, and \item there is a locally finite direct product decomposition $$ \mathcal{O}_X = \prod\nolimits_{n \in \mathbf{Z}} \mathcal{O}_n $$ and for each $n$ there is an invertible $\mathcal{O}_n$-module $\mathcal{H}^n$ (Modules, Definition \ref{modules-definition-invertible}) and $M = \bigoplus \mathcal{H}^n[-n]$ in $D(\mathcal{O}_X)$. \end{enumerate} If (1) and (2) hold, then $M$ is a perfect object of $D(\mathcal{O}_X)$. If $\mathcal{O}_{X, x}$ is a local ring for all $x \in X$ these condition are also equivalent to \begin{enumerate} \item[(3)] there exists an open covering $X = \bigcup U_i$ and for each $i$ an integer $n_i$ such that $M|_{U_i}$ is represented by an invertible $\mathcal{O}_{U_i}$-module placed in degree $n_i$. \end{enumerate}


cohomology-lemma-when-jshriek-compact

Let $X$ be a ringed space. Let $j : U \to X$ be the inclusion of an open. The $\mathcal{O}_X$-module $j_!\mathcal{O}_U$ is a compact object of $D(\mathcal{O}_X)$ if there exists an integer $d$ such that \begin{enumerate} \item $H^p(U, \mathcal{F}) = 0$ for all $p > d$, and \item the functors $\mathcal{F} \mapsto H^p(U, \mathcal{F})$ commute with direct sums. \end{enumerate}


cohomology-lemma-injective-tensor-finite-locally-free

Let $X$ be a ringed space. Let $\mathcal{I}$ be an injective $\mathcal{O}_X$-module. Let $\mathcal{E}$ be an $\mathcal{O}_X$-module. Assume $\mathcal{E}$ is finite locally free on $X$, see Modules, Definition \ref{modules-definition-locally-free}. Then $\mathcal{E} \otimes_{\mathcal{O}_X} \mathcal{I}$ is an injective $\mathcal{O}_X$-module.


cohomology-lemma-projection-formula

Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Let $\mathcal{E}$ be an $\mathcal{O}_Y$-module. Assume $\mathcal{E}$ is finite locally free on $Y$, see Modules, Definition \ref{modules-definition-locally-free}. Then there exist isomorphisms $$ \mathcal{E} \otimes_{\mathcal{O}_Y} R^qf_*\mathcal{F} \longrightarrow R^qf_*(f^*\mathcal{E} \otimes_{\mathcal{O}_X} \mathcal{F}) $$ for all $q \geq 0$. In fact there exists an isomorphism $$ \mathcal{E} \otimes_{\mathcal{O}_Y} Rf_*\mathcal{F} \longrightarrow Rf_*(f^*\mathcal{E} \otimes_{\mathcal{O}_X} \mathcal{F}) $$ in $D^{+}(Y)$ functorial in $\mathcal{F}$.


cohomology-lemma-projection-formula-perfect

Let $f : X \to Y$ be a morphism of ringed spaces. Let $E \in D(\mathcal{O}_X)$ and $K \in D(\mathcal{O}_Y)$. If $K$ is perfect, then $$ Rf_*E \otimes^\mathbf{L}_{\mathcal{O}_Y} K = Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} Lf^*K) $$ in $D(\mathcal{O}_Y)$.


cohomology-lemma-projection-formula-closed-immersion

Let $f : X \to Y$ be a morphism of ringed spaces such that $f$ is a homeomorphism onto a closed subset. Then (\ref{equation-projection-formula-map}) is an isomorphism always.


cohomology-proposition-vanishing-Noetherian

\begin{reference} \cite[Theorem 3.6.5]{Tohoku}. \end{reference} Let $X$ be a Noetherian topological space. If $\dim(X) \leq d$, then $H^p(X, \mathcal{F}) = 0$ for all $p > d$ and any abelian sheaf $\mathcal{F}$ on $X$.


cohomology-proposition-cohomological-dimension-spectral

\begin{reference} Part (1) is the main theorem of \cite{Scheiderer}. \end{reference} Let $X$ be a spectral space of Krull dimension $d$. Let $\mathcal{F}$ be an abelian sheaf on $X$. \begin{enumerate} \item $H^q(X, \mathcal{F}) = 0$ for $q > d$, \item $H^d(X, \mathcal{F}) \to H^d(U, \mathcal{F})$ is surjective for every quasi-compact open $U \subset X$, \item $H^q_Z(X, \mathcal{F}) = 0$ for $q > d$ and any constructible closed subset $Z \subset X$. \end{enumerate}


stacks-introduction-lemma-key-fact

The functor $\Sch^{opp} \to \textit{Sets}$, $T \mapsto \{(a, a', \alpha)\text{ as above}\}$ is representable by a scheme $S \times_{\mathcal{M}_{1, 1}} S'$.


stacks-introduction-lemma-Weierstrass-smooth-cover

The morphism $W \xrightarrow{(E_W, f_W, 0_W)} \mathcal{M}_{1, 1}$ is smooth and surjective.


restricted-theorem-dilatations-general

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $T \subset |X|$ be a closed subset. Let $U \subset X$ be the open subspace with $|U| = |X| \setminus T$. The completion functor (\ref{equation-completion-functor}) $$ \left\{ \begin{matrix} \text{morphisms of algebraic spaces}\\ f : X' \to X\text{ which are locally}\\ \text{of finite type and such that}\\ f^{-1}U \to U\text{ is an isomorphism} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{morphisms }g : W \to X_{/T}\\ \text{of formal algebraic spaces}\\ \text{with }W\text{ locally Noetherian}\\ \text{and }g\text{ rig-\'etale} \end{matrix} \right\} $$ sending $f : X' \to X$ to $f_{/T} : X'_{/T'} \to X_{/T}$ is an equivalence.


restricted-theorem-dilatations

\begin{reference} \cite[Theorem 3.2]{ArtinII} \end{reference} Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $T \subset |X|$ be a closed subset. Let $\mathfrak X = X_{/T}$ be the formal completion of $X$ along $T$. Let $$ \mathfrak f : \mathfrak X' \to \mathfrak X $$ be a formal modification (Definition \ref{definition-formal-modification}). Then there exists a unique proper morphism $f : X' \to X$ which is an isomorphism over the complement of $T$ in $X$ whose completion $f_{/T}$ recovers $\mathfrak f$.


restricted-lemma-topologically-finite-type

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. The functor $$ \mathcal{C} \longrightarrow \mathcal{C}',\quad (B_n) \longmapsto B = \lim B_n $$ is a quasi-inverse to (\ref{equation-from-complete-to-systems}). The completions $A[x_1, \ldots, x_r]^\wedge$ are in $\mathcal{C}'$ and any object of $\mathcal{C}'$ is of the form $$ B = A[x_1, \ldots, x_r]^\wedge / J $$ for some ideal $J \subset A[x_1, \ldots, x_r]^\wedge$.


restricted-lemma-topologically-finite-type-Noetherian

\begin{reference} \cite[Proposition 7.5.5]{EGA1} \end{reference} Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Then \begin{enumerate} \item every object of the category $\mathcal{C}'$ (\ref{equation-C-prime}) is Noetherian, \item if $B \in \Ob(\mathcal{C}')$ and $J \subset B$ is an ideal, then $B/J$ is an object of $\mathcal{C}'$, \item for a finite type $A$-algebra $C$ the $I$-adic completion $C^\wedge$ is in $\mathcal{C}'$, \item in particular the completion $A[x_1, \ldots, x_r]^\wedge$ is in $\mathcal{C}'$. \end{enumerate}


restricted-lemma-NL-up-to-homotopy

Let $A$ be a Noetherian ring and let $I \subset A$ be a ideal. Let $B$ be an object of (\ref{equation-C-prime}). The naive cotangent complex $\NL_{B/A}^\wedge$ is well defined in $K(B)$.


restricted-lemma-NL-is-completion

Let $A$ be a Noetherian ring and let $I \subset A$ be a ideal. Let $A \to B$ be a finite type ring map. Choose a presentation $\alpha : A[x_1, \ldots, x_n] \to B$. Then $\NL_{B^\wedge/A}^\wedge = \lim \NL(\alpha) \otimes_B B^\wedge$ as complexes and $\NL_{B^\wedge/A}^\wedge = \NL_{B/A} \otimes_B^\mathbf{L} B^\wedge$ in $D(B^\wedge)$.


restricted-lemma-NL-is-limit

Let $A$ be a Noetherian ring and let $I \subset A$ be a ideal. Let $B$ be an object of (\ref{equation-C-prime}). Then \begin{enumerate} \item the pro-objects $\{\NL_{B/A}^\wedge \otimes_B B/I^nB\}$ and $\{\NL_{B_n/A_n}\}$ of $D(B)$ are strictly isomorphic (see proof for elucidation), \item $\NL_{B/A}^\wedge = R\lim \NL_{B_n/A_n}$ in $D(B)$. \end{enumerate} Here $B_n$ and $A_n$ are as in Section \ref{section-two-categories}.


restricted-lemma-NL-base-change

Let $(A_1, I_1) \to (A_2, I_2)$ be as in Remark \ref{remark-base-change} with $A_1$ and $A_2$ Noetherian. Let $B_1$ be in (\ref{equation-C-prime}) for $(A_1, I_1)$. Let $B_2$ be the base change of $B_1$. Then there is a canonical map $$ \NL_{B_1/A_1} \otimes_{B_2} B_1 \to \NL_{B_2/A_2} $$ which induces and isomorphism on $H^0$ and a surjection on $H^{-1}$.


restricted-lemma-exact-sequence-NL

Let $A$ be a Noetherian ring and let $I \subset A$ be a ideal. Let $B \to C$ be morphism of (\ref{equation-C-prime}). Then there is an exact sequence $$ \xymatrix{ C \otimes_B H^0(\NL_{B/A}^\wedge) \ar[r] & H^0(\NL_{C/A}^\wedge) \ar[r] & H^0(\NL_{C/B}^\wedge) \ar[r] & 0 \\ H^{-1}(\NL_{B/A}^\wedge \otimes_B C) \ar[r] & H^{-1}(\NL_{C/A}^\wedge) \ar[r] & H^{-1}(\NL_{C/B}^\wedge) \ar[llu] } $$ See proof for elucidation.


restricted-lemma-transitive-lci-at-end

With assumptions as in Lemma \ref{lemma-exact-sequence-NL} assume that $B/I^nB \to C/I^nC$ is a local complete intersection homomorphism for all $n$. Then $H^{-1}(\NL_{B/A}^\wedge \otimes_B C) \to H^{-1}(\NL_{C/A}^\wedge)$ is injective.


restricted-lemma-equivalent-with-artin-smooth

Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Let $B$ be an object of (\ref{equation-C-prime}). Write $B = A[x_1, \ldots, x_r]^\wedge/J$ (Lemma \ref{lemma-topologically-finite-type-Noetherian}) and let $\NL_{B/A}^\wedge = (J/J^2 \to \bigoplus B\text{d}x_i)$ be its naive cotangent complex (\ref{equation-NL}). The following are equivalent \begin{enumerate} \item $B$ is rig-smooth over $(A, I)$, \item the object $\NL_{B/A}^\wedge$ of $D(B)$ satisfies the equivalent conditions (1) -- (4) of More on Algebra, Lemma \ref{more-algebra-lemma-ext-1-annihilated} with respect to the ideal $IB$, \item there exists a $c \geq 0$ such that for all $a \in I^c$ there is a map $h : \bigoplus B\text{d}x_i \to J/J^2$ such that $a : J/J^2 \to J/J^2$ is equal to $h \circ \text{d}$, \item there exist $b_1, \ldots, b_s \in B$ such that $V(b_1, \ldots, b_s) \subset V(IB)$ and such that for every $l = 1, \ldots, s$ there exist $m \geq 0$, $f_1, \ldots, f_m \in J$, and subset $T \subset \{1, \ldots, n\}$ with $|T| = m$ such that \begin{enumerate} \item $\det_{i \in T, j \leq m}(\partial f_j/ \partial x_i)$ divides $b_l$ in $B$, and \item $b_l J \subset (f_1, \ldots, f_m) + J^2$. \end{enumerate} \end{enumerate}


restricted-lemma-rig-smooth

Let $A$ be a Noetherian ring and let $I$ be an ideal. Let $B$ be a finite type $A$-algebra. \begin{enumerate} \item If $\Spec(B) \to \Spec(A)$ is smooth over $\Spec(A) \setminus V(I)$, then $B^\wedge$ is rig-smooth over $(A, I)$. \item If $B^\wedge$ is rig-smooth over $(A, I)$, then there exists $g \in 1 + IB$ such that $\Spec(B_g)$ is smooth over $\Spec(A) \setminus V(I)$. \end{enumerate}


restricted-lemma-zero-ext-1-after-modding-out

Let $(A_1, I_1) \to (A_2, I_2)$ be as in Remark \ref{remark-base-change} with $A_1$ and $A_2$ Noetherian. Let $B_1$ be in (\ref{equation-C-prime}) for $(A_1, I_1)$. Let $B_2$ be the base change of $B_1$. Let $f_1 \in B_1$ with image $f_2 \in B_2$. If $\Ext^1_{B_1}(\NL_{B_1/A_1}^\wedge, N_1)$ is annihilated by $f_1$ for every $B_1$-module $N_1$, then $\Ext^1_{B_2}(\NL_{B_2/A_2}^\wedge, N_2)$ is annihilated by $f_2$ for every $B_2$-module $N_2$.


restricted-lemma-base-change-rig-smooth-homomorphism

Let $A_1 \to A_2$ be a map of Noetherian rings. Let $I_i \subset A_i$ be an ideal such that $V(I_1A_2) = V(I_2)$. Let $B_1$ be in (\ref{equation-C-prime}) for $(A_1, I_1)$. Let $B_2$ be the base change of $B_1$ as in Remark \ref{remark-base-change}. If $B_1$ is rig-smooth over $(A_1, I_1)$, then $B_2$ is rig-smooth over $(A_2, I_2)$.


restricted-lemma-get-morphism-general-better

Assume given the following data \begin{enumerate} \item an integer $c \geq 0$, \item an ideal $I$ of a Noetherian ring $A$, \item $B$ in (\ref{equation-C-prime}) for $(A, I)$ such that $I^c$ annihilates $\Ext^1_B(\NL_{B/A}^\wedge, N)$ for any $B$-module $N$, \item a Noetherian $I$-adically complete $A$-algebra $C$; denote $d = d(\text{Gr}_I(C))$ and $q_0 = q(\text{Gr}_I(C))$ the integers found in Local Cohomology, Section \ref{local-cohomology-section-uniform}, \item an integer $n \geq \max(q_0 + (d + 1)c, 2(d + 1)c + 1)$, and \item an $A$-algebra homomorphism $\psi_n : B \to C/I^nC$. \end{enumerate} Then there exists a map $\varphi : B \to C$ of $A$-algebras such that $\psi_n \bmod I^{n - (d + 1)c} = \varphi \bmod I^{n - (d + 1)c}$.


restricted-lemma-get-morphism-nonzerodivisor

Let $I = (a)$ be a principal ideal of a Noetherian ring $A$. Let $B$ be an object of (\ref{equation-C-prime}). Assume given an integer $c \geq 0$ such that $\Ext^1_B(\NL_{B/A}^\wedge, N)$ is annihilated by $a^c$ for all $B$-modules $N$. Let $C$ be an $I$-adically complete $A$-algebra such that $a$ is a nonzerodivisor on $C$. Let $n > 2c$. For any $A$-algebra map $\psi_n : B \to C/a^nC$ there exists an $A$-algebra map $\varphi : B \to C$ such that $\psi_n \bmod a^{n - c}C = \varphi \bmod a^{n - c}C$.


restricted-lemma-get-morphism-principal

Let $I = (a)$ be a principal ideal of a Noetherian ring $A$. Let $B$ be an object of (\ref{equation-C-prime}). Assume given an integer $c \geq 0$ such that $\Ext^1_B(\NL_{B/A}^\wedge, N)$ is annihilated by $a^c$ for all $B$-modules $N$. Let $C$ be an $I$-adically complete $A$-algebra. Assume given an integer $d \geq 0$ such that $C[a^\infty] \cap a^dC = 0$. Let $n > \max(2c, c + d)$. For any $A$-algebra map $\psi_n : B \to C/a^nC$ there exists an $A$-algebra map $\varphi : B \to C$ such that $\psi_n \bmod a^{n - c} = \varphi \bmod a^{n - c}$.


restricted-lemma-close-enough

Let $I$ be an ideal of a Noetherian ring $A$. Let $r \geq 0$ and write $P = A[x_1, \ldots, x_r]$ the $I$-adic completion. Consider a resolution $$ P^{\oplus t} \xrightarrow{K} P^{\oplus m} \xrightarrow{g_1, \ldots, g_m} P \to B \to 0 $$ of a quotient of $P$. Assume $B$ is rig-smooth over $(A, I)$. Then there exists an integer $n$ such that for any complex $$ P^{\oplus t} \xrightarrow{K'} P^{\oplus m} \xrightarrow{g'_1, \ldots, g'_m} P $$ with $g_i - g'_i \in I^nP$ and $K - K' \in I^n\text{Mat}(m \times t, P)$ there exists an isomorphism $B \to B'$ of $A$-algebras where $B' = P/(g'_1, \ldots, g'_m)$.


restricted-lemma-algebraize-easy

Let $I$ be an ideal of a Noetherian ring $A$. Let $C^h$ be the henselization of a finite type $A$-algebra $C$ with respect to the ideal $IC$. Let $J \subset C^h$ be an ideal. Then there exists a finite type $A$-algebra $B$ such that $B^\wedge \cong (C^h/J)^\wedge$.


restricted-lemma-presentation-rig-smooth

Let $A$ be a ring. Let $f_1, \ldots, f_m \in A[x_1, \ldots, x_n]$ and set $B = A[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$. Assume $m \leq n$ and set $g = \det_{1 \leq i, j \leq m}(\partial f_j/\partial x_i)$. Then \begin{enumerate} \item $g$ annihilates $\Ext^1_B(\NL_{B/A}, N)$ for every $B$-module $N$, \item if $n = m$, then multiplication by $g$ on $\NL_{B/A}$ is $0$ in $D(B)$. \end{enumerate}


restricted-lemma-approximate-presentation-rig-smooth

Let $I$ be an ideal of a Noetherian ring $A$. Let $B$ be an object of (\ref{equation-C-prime}). Let $B = A[x_1, \ldots, x_r]^\wedge/J$ be a presentation. Assume there exists an element $b \in B$, $0 \leq m \leq r$, and $f_1, \ldots, f_m \in J$ such that \begin{enumerate} \item $V(b) \subset V(IB)$ in $\Spec(B)$, \item the image of $\Delta = \det_{1 \leq i, j \leq m}(\partial f_j/\partial x_i)$ in $B$ divides $b$, and \item $b J \subset (f_1, \ldots, f_m) + J^2$. \end{enumerate} Then there exists a finite type $A$-algebra $C$ and an $A$-algebra isomorphism $B \cong C^\wedge$.


restricted-lemma-equivalent-with-artin

Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Let $B$ be an object of (\ref{equation-C-prime}). Write $B = A[x_1, \ldots, x_r]^\wedge/J$ (Lemma \ref{lemma-topologically-finite-type-Noetherian}) and let $\NL_{B/A}^\wedge = (J/J^2 \to \bigoplus B\text{d}x_i)$ be its naive cotangent complex (\ref{equation-NL}). The following are equivalent \begin{enumerate} \item $B$ is rig-\'etale over $(A, I)$, \item \label{item-zero-on-NL} there exists a $c \geq 0$ such that for all $a \in I^c$ multiplication by $a$ on $\NL_{B/A}^\wedge$ is zero in $D(B)$, \item \label{item-zero-on-cohomology-NL} there exits a $c \geq 0$ such that $H^i(\NL_{B/A}^\wedge)$, $i = -1, 0$ is annihilated by $I^c$, \item \label{item-zero-on-cohomology-NL-truncations} there exists a $c \geq 0$ such that $H^i(\NL_{B_n/A_n})$, $i = -1, 0$ is annihilated by $I^c$ for all $n \geq 1$ where $A_n = A/I^n$ and $B_n = B/I^nB$, \item \label{item-condition-artin-pre-pre} for every $a \in I$ there exists a $c \geq 0$ such that \begin{enumerate} \item $a^c$ annihilates $H^0(\NL_{B/A}^\wedge)$, and \item there exist $f_1, \ldots, f_r \in J$ such that $a^c J \subset (f_1, \ldots, f_r) + J^2$. \end{enumerate} \item \label{item-condition-artin-pre} for every $a \in I$ there exist $f_1, \ldots, f_r \in J$ and $c \geq 0$ such that \begin{enumerate} \item $\det_{1 \leq i, j \leq r}(\partial f_j/\partial x_i)$ divides $a^c$ in $B$, and \item $a^c J \subset (f_1, \ldots, f_r) + J^2$. \end{enumerate} \item \label{item-condition-artin} choosing generaters $f_1, \ldots, f_t$ for $J$ we have \begin{enumerate} \item the Jacobian ideal of $B$ over $A$, namely the ideal in $B$ generated by the $r \times r$ minors of the matrx $(\partial f_j/\partial x_i)_{1 \leq i \leq r, 1 \leq j \leq t}$, contains the ideal $I^cB$ for some $c$, and \item the Cramer ideal of $B$ over $A$, namely the ideal in $B$ generated by the image in $B$ of the $r$th Fitting ideal of $J$ as an $A[x_1, \ldots, x_r]^\wedge$-module, contains $I^cB$ for some $c$. \end{enumerate} \end{enumerate}


restricted-lemma-rig-etale-rig-smooth

Let $A$ be a Noetherian ring and let $I$ be an ideal. Let $B$ be an object of (\ref{equation-C-prime}). If $B$ is rig-\'etale over $(A, I)$, then $B$ is rig-smooth over $(A, I)$.


restricted-lemma-rig-etale

Let $A$ be a Noetherian ring and let $I$ be an ideal. Let $B$ be a finite type $A$-algebra. \begin{enumerate} \item If $\Spec(B) \to \Spec(A)$ is \'etale over $\Spec(A) \setminus V(I)$, then $B^\wedge$ satisfies the equivalent conditions of Lemma \ref{lemma-equivalent-with-artin}. \item If $B^\wedge$ satisfies the equivalent conditions of Lemma \ref{lemma-equivalent-with-artin}, then there exists $g \in 1 + IB$ such that $\Spec(B_g)$ is \'etale over $\Spec(A) \setminus V(I)$. \end{enumerate}


restricted-lemma-zero-after-modding-out

Let $(A_1, I_1) \to (A_2, I_2)$ be as in Remark \ref{remark-base-change} with $A_1$ and $A_2$ Noetherian. Let $B_1$ be in (\ref{equation-C-prime}) for $(A_1, I_1)$. Let $B_2$ be the base change of $B_1$. If multiplication by $f_1 \in B_1$ on $\NL^\wedge_{B_1/A_1}$ is zero in $D(B_1)$, then multiplication by the image $f_2 \in B_2$ on $\NL^\wedge_{B_2/A_2}$ is zero in $D(B_2)$.


restricted-lemma-base-change-rig-etale-homomorphism

Let $A_1 \to A_2$ be a map of Noetherian rings. Let $I_i \subset A_i$ be an ideal such that $V(I_1A_2) = V(I_2)$. Let $B_1$ be in (\ref{equation-C-prime}) for $(A_1, I_1)$. Let $B_2$ be the base change of $B_1$ as in Remark \ref{remark-base-change}. If $B_1$ is rig-\'etale over $(A_1, I_1)$, then $B_2$ is rig-\'etale over $(A_2, I_2)$.


restricted-lemma-fully-faithful-etale-over-complement

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $B$ be a finite type $A$-algebra such that $\Spec(B) \to \Spec(A)$ is \'etale over $\Spec(A) \setminus V(I)$. Let $C$ be a Noetherian $A$-algebra. Then any $A$-algebra map $B^\wedge \to C^\wedge$ of $I$-adic completions comes from a unique $A$-algebra map $$ B \longrightarrow C^h $$ where $C^h$ is the henselization of the pair $(C, IC)$ as in More on Algebra, Lemma \ref{more-algebra-lemma-henselization}. Moreover, any $A$-algebra homomorphism $B \to C^h$ factors through some \'etale $C$-algebra $C'$ such that $C/IC \to C'/IC'$ is an isomorphism.


restricted-lemma-lift-approximation

Let $A$ be a Noetherian ring and $I \subset A$ an ideal. Let $J \subset A$ be a nilpotent ideal. Consider a commutative diagram $$ \xymatrix{ C \ar[r] & C_0 \ar@{=}[r] & C/JC \\ & B_0 \ar[u] \\ A \ar[r] \ar[uu] & A_0 \ar[u] \ar@{=}[r] & A/J } $$ whose vertical arrows are of finite type such that \begin{enumerate} \item $\Spec(C) \to \Spec(A)$ is \'etale over $\Spec(A) \setminus V(I)$, \item $\Spec(B_0) \to \Spec(A_0)$ is \'etale over $\Spec(A_0) \setminus V(IA_0)$, and \item $B_0 \to C_0$ is \'etale and induces an isomorphism $B_0/IB_0 = C_0/IC_0$. \end{enumerate} Then we can fill in the diagram above to a commutative diagram $$ \xymatrix{ C \ar[r] & C/JC \\ B \ar[u] \ar[r] & B_0 \ar[u] \\ A \ar[r] \ar[u] & A/J \ar[u] } $$ with $A \to B$ of finite type, $B/JB = B_0$, $B \to C$ \'etale, and $\Spec(B) \to \Spec(A)$ \'etale over $\Spec(A) \setminus V(I)$.


restricted-lemma-approximate-principal

\begin{reference} The rig-\'etale case of \cite[III Theorem 7]{Elkik} \end{reference} Let $A$ be a Noetherian ring and $I = (a)$ a principal ideal. Let $B$ be an object of (\ref{equation-C-prime}) which is rig-\'etale over $(A, I)$. Then there exists a finite type $A$-algebra $C$ and an isomorphism $B \cong C^\wedge$.


restricted-lemma-approximate

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $B$ be an object of (\ref{equation-C-prime}) which is rig-\'etale over $(A, I)$. Then there exists a finite type $A$-algebra $C$ and an isomorphism $B \cong C^\wedge$.


restricted-lemma-approximate-by-etale-over-complement

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $B$ be an $I$-adically complete $A$-algebra with $A/I \to B/IB$ of finite type. The equivalent conditions of Lemma \ref{lemma-equivalent-with-artin} are also equivalent to \begin{enumerate} \item[(8)] \label{item-algebraize} there exists a finite type $A$-algebra $C$ such that $\Spec(C) \to \Spec(A)$ is \'etale over $\Spec(A) \setminus V(I)$ and such that $B \cong C^\wedge$. \end{enumerate}


restricted-lemma-finite-type

Let $A$ and $B$ be adic topological rings which have a finitely generated ideal of definition. Let $\varphi : A \to B$ be a continuous ring homomorphism. The following are equivalent: \begin{enumerate} \item $\varphi$ is adic and $B$ is topologically of finite type over $A$, \item $\varphi$ is taut and $B$ is topologically of finite type over $A$, \item there exists an ideal of definition $I \subset A$ such that the topology on $B$ is the $I$-adic topology and there exist an ideal of definition $I' \subset A$ such that $A/I' \to B/I'B$ is of finite type, \item for all ideals of definition $I \subset A$ the topology on $B$ is the $I$-adic topology and $A/I \to B/IB$ is of finite type, \item there exists an ideal of definition $I \subset A$ such that the topology on $B$ is the $I$-adic topology and $B$ is in the category (\ref{equation-C-prime}), \item for all ideals of definition $I \subset A$ the topology on $B$ is the $I$-adic topology and $B$ is in the category (\ref{equation-C-prime}), \item $B$ as a topological $A$-algebra is the quotient of $A\{x_1, \ldots, x_r\}$ by a closed ideal, \item $B$ as a topological $A$-algebra is the quotient of $A[x_1, \ldots, x_r]^\wedge$ by a closed ideal where $A[x_1, \ldots, x_r]^\wedge$ is the completion of $A[x_1, \ldots, x_r]$ with respect to some ideal of definition of $A$, and \item add more here. \end{enumerate} Moreover, these equivalent conditions define a local property of morphisms of $\text{WAdm}^{adic*}$ as defined in Formal Spaces, Remark \ref{formal-spaces-remark-variant-adic-star}.


restricted-lemma-composition-finite-type

Consider the property $P$ on arrows of $\textit{WAdm}^{adic*}$ defined in Lemma \ref{lemma-finite-type}. Then $P$ is stable under composition as defined in Formal Spaces, Remark \ref{formal-spaces-remark-composition-variant-adic-star}.


restricted-lemma-finite-type-morphisms

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


restricted-lemma-finite-type-red

For an arrow $\varphi : A \to B$ in $\text{WAdm}^{count}$ consider the property $P(\varphi)=$``the induced ring homomorphism $A/\mathfrak a \to B/\mathfrak b$ is of finite type'' where $\mathfrak a \subset A$ and $\mathfrak b \subset B$ are the ideals of topologically nilpotent elements. Then $P$ is a local property as defined in Formal Spaces, Situation \ref{formal-spaces-situation-local-property}.


restricted-lemma-composition-finite-type-red

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


restricted-lemma-finite-type-finite-type-red

Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{count}$. If $\varphi$ is taut and topologically of finite type, then $\varphi$ satisfies the condition defined in Lemma \ref{lemma-finite-type-red}.


restricted-lemma-Noetherian-finite-type-red

Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$ satisfying the condition defined in Lemma \ref{lemma-finite-type-red}. Then $A \to B$ is topologically of finite type.


restricted-lemma-finite-type-red-morphisms

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


restricted-lemma-flat-axioms

The property $P(\varphi)=$``$\varphi$ is flat'' on arrows of $\textit{WAdm}^{Noeth}$ is a local property as defined in Formal Spaces, Remark \ref{formal-spaces-remark-variant-Noetherian}.


restricted-lemma-base-change-flat-continuous

Denote $P$ the property of arrows of $\textit{WAdm}^{Noeth}$ defined in Lemma \ref{lemma-flat-axioms}. Denote $Q$ the property defined in Lemma \ref{lemma-finite-type-red} viewed as a property of arrows of $\textit{WAdm}^{Noeth}$. Denote $R$ the property defined in Lemma \ref{lemma-finite-type} viewed as a property of arrows of $\textit{WAdm}^{Noeth}$. Then \begin{enumerate} \item $P$ is stable under base change by $Q$ (Formal Spaces, Remark \ref{formal-spaces-remark-base-change-variant-variant-Noetherian}), and \item $P + R$ is stable under base change (Formal Spaces, Remark \ref{formal-spaces-remark-base-change-variant-Noetherian}). \end{enumerate}


restricted-lemma-composition-flat-continuous

Denote $P$ the property of arrows of $\textit{WAdm}^{Noeth}$ defined in Lemma \ref{lemma-flat-axioms}. Then $P$ is stable under composition (Formal Spaces, Remark \ref{formal-spaces-remark-composition-variant-Noetherian}).


restricted-lemma-flat-morphisms

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


restricted-lemma-representable-flat

Let $S$ be a scheme. Let $f : X \to Y$ be a morphisms of locally Noetherian formal algebraic spaces over $S$. If $f$ is representable by algebraic spaces and flat in the sense of Bootstrap, Definition \ref{bootstrap-definition-property-transformation}, then $f$ is flat in the sense of Definition \ref{definition-flat}.


restricted-lemma-rig-point

Let $A$ be a Noetherian adic topological ring. Let $\mathfrak q \subset A$ be a prime ideal. The following are equivalent \begin{enumerate} \item for some ideal of definition $I \subset A$ we have $I \not \subset \mathfrak q$ and $\mathfrak q$ is maximal with respect to this property, \item for some ideal of definition $I \subset A$ the prime $\mathfrak q$ defines a closed point of $\Spec(A) \setminus V(I)$, \item for any ideal of definition $I \subset A$ we have $I \not \subset \mathfrak q$ and $\mathfrak q$ is maximal with respect to this property, \item for any ideal of definition $I \subset A$ the prime $\mathfrak q$ defines a closed point of $\Spec(A) \setminus V(I)$, \item $\dim(A/\mathfrak q) = 1$ and for some ideal of definition $I \subset A$ we have $I \not \subset \mathfrak q$, \item $\dim(A/\mathfrak q) = 1$ and for any ideal of definition $I \subset A$ we have $I \not \subset \mathfrak q$, \item $\dim(A/\mathfrak q) = 1$ and the induced topology on $A/\mathfrak q$ is nontrivial, \item $A/\mathfrak q$ is a $1$-dimensional Noetherian complete local domain whose maximal ideal is the radical of the image of any ideal of definition of $A$, and \item add more here. \end{enumerate}


restricted-lemma-rig-closed-point-relative-residue-field

Let $\varphi : A \to B$ in $\textit{WAdm}^{Noeth}$. Denote $\mathfrak a \subset A$ and $\mathfrak b \subset B$ the ideals of topologically nilpotent elements. Assume $A/\mathfrak a \to B/\mathfrak b$ is of finite type. Let $\mathfrak q \subset B$ be rig-closed. The residue field $\kappa$ of the local ring $B/\mathfrak q$ is a finite type $A/\mathfrak a$-algebra.


restricted-lemma-rig-closed-point-relative

Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$ which is adic and topologically of finite type. Let $\mathfrak q \subset B$ be rig-closed. Let $\mathfrak p = \varphi^{-1}(\mathfrak q) \subset A$. Let $\mathfrak a \subset A$ be the ideal of topologically nilpotent elements. The following are equivalent \begin{enumerate} \item the residue field $\kappa$ of $B/\mathfrak q$ is finite over $A/\mathfrak a$, \item $\mathfrak p \subset A$ is rig-closed, \item $A/\mathfrak p \subset B/\mathfrak q$ is a finite extension of rings. \end{enumerate}


restricted-lemma-rig-closed-jacobson

Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$ which is adic and topologically of finite type. Let $\mathfrak q \subset B$ be rig-closed. If $A/I$ is Jacobson for some ideal of definition $I \subset A$, then $\mathfrak p = \varphi^{-1}(\mathfrak q) \subset A$ is rig-closed.


restricted-lemma-rig-closed-point-in-localization

Let $A$ be an adic Noetherian topological ring. Let $\mathfrak p \subset A$ be a prime ideal. Let $f \in A$ be an element mapping to a unit in $A/\mathfrak p$. Then $$ \mathfrak p A_{\{f\}} = \mathfrak p(A_f)^\wedge = \mathfrak p \otimes_A (A_f)^\wedge = (\mathfrak p_f)^\wedge $$ is a prime ideal with quotient $$ A/\mathfrak p = (A/\mathfrak p) \otimes_A (A_f)^\wedge = (A_f)^\wedge / \mathfrak p (A_f)^\wedge = A_{\{f\}}/\mathfrak p A_{\{f\}} $$


restricted-lemma-rig-closed-point-after-localization

Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$ which is adic and topologically of finite type. Let $\mathfrak q \subset B$ be rig-closed. There exists an $f \in A$ which maps to a unit in $B/\mathfrak q$ such that we obtain a diagram $$ \vcenter{ \xymatrix{ B \ar[r] & B_{\{f\}} \\ A \ar[r] \ar[u]_\varphi & A_{\{f\}} \ar[u]_{\varphi_{\{f\}}} } } \quad\text{with primes}\quad \vcenter{ \xymatrix{ \mathfrak q \ar@{-}[r] \ar@{-}[d] & \mathfrak q' \ar@{-}[d] \ar@{=}[r] & \mathfrak q B_{\{f\}} \\ \mathfrak p \ar@{-}[r] & \mathfrak p' } } $$ such that $\mathfrak p'$ is rig-closed, i.e., the map $A_{\{f\}} \to B_{\{f\}}$ and the prime ideals $\mathfrak q'$ and $\mathfrak p'$ satisfy the equivalent conditions of Lemma \ref{lemma-rig-closed-point-relative}.


restricted-lemma-rig-closed-point-variables

Let $A$ be a Noetherian adic topological ring. Denote $A\{x_1, \ldots, x_n\}$ the restricted power series over $A$. Let $\mathfrak q \subset A\{x_1, \ldots, x_n\}$ be a prime ideal. Set $\mathfrak q' = A[x_1, \ldots, x_n] \cap \mathfrak q$ and $\mathfrak p = A \cap \mathfrak q$. If $\mathfrak q$ and $\mathfrak p$ are rig-closed, then the map $$ A[x_1, \ldots, x_n]_{\mathfrak q'} \to A\{x_1, \ldots, x_n\}_\mathfrak q $$ defines an isomorphism on completions with respect to their maximal ideals.


restricted-lemma-rig-closed-point-etale

Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$. Assume $\varphi$ is adic, topologically of finite type, flat, and $A/I \to B/IB$ is \'etale for some (resp.\ any) ideal of definition $I \subset A$. Let $\mathfrak q \subset B$ be rig-closed such that $\mathfrak p = A \cap \mathfrak q$ is rig-closed as well. Then $\mathfrak p B_\mathfrak q = \mathfrak q B_\mathfrak q$.


restricted-lemma-fibre-regular

Let $A$ be an adic Noetherian topological ring. Let $\mathfrak p \subset A$ be a rig-closed prime. For any $n \geq 1$ the ring map $$ A/\mathfrak p \longrightarrow A\{x_1, \ldots, x_n\} \otimes_A A/\mathfrak p = A/\mathfrak p\{x_1, \ldots, x_n\} $$ is regular. In particular, the algebra $A\{x_1, \ldots, x_n\} \otimes_A \kappa(\mathfrak p)$ is geometrically regular over $\kappa(\mathfrak p)$.


restricted-lemma-naively-rig-flat-continuous

Let $\varphi : A \to B$ be a morphism in $\textit{WAdm}^{adic*}$ (Formal Spaces, Section \ref{formal-spaces-section-morphisms-rings}). Assume $\varphi$ is adic. The following are equivalent: \begin{enumerate} \item $B_f$ is flat over $A$ for all topologically nilpotent $f \in A$, \item $B_g$ is flat over $A$ for all topologically nilpotent $g \in B$, \item $B_\mathfrak q$ is flat over $A$ for all primes $\mathfrak q \subset B$ which do not contain an ideal of definition, \item $B_\mathfrak q$ is flat over $A$ for every rig-closed prime $\mathfrak q \subset B$, and \item add more here. \end{enumerate}


restricted-lemma-rig-flat-base-change

Let $\varphi : A \to B$ and $A \to C$ be arrows of $\textit{WAdm}^{Noeth}$. Assume $\varphi$ is rig-flat and $A \to C$ adic and topologically of finite type. Then $C \to B \widehat{\otimes}_A C$ is rig-flat.


restricted-lemma-rig-flat-local-etale

Consider a commutative diagram $$ \xymatrix{ B \ar[r] & B' \\ A \ar[r] \ar[u]^\varphi & A' \ar[u]_{\varphi'} } $$ in $\textit{WAdm}^{Noeth}$ with all arrows adic and topologically of finite type. Assume $A \to A'$ and $B \to B'$ are flat. Let $I \subset A$ be an ideal of definition. If $\varphi$ is rig-flat and $A/I \to A'/IA'$ is \'etale, then $\varphi'$ is rig-flat.


restricted-lemma-rig-flat-local-down

Consider a commutative diagram $$ \xymatrix{ B \ar[r] & B' \\ A \ar[r] \ar[u]^\varphi & A' \ar[u]_{\varphi'} } $$ in $\textit{WAdm}^{Noeth}$ with all arrows adic and topologically of finite type. Assume $A \to A'$ flat and $B \to B'$ faithfully flat. If $\varphi'$ is rig-flat, then $\varphi$ is rig-flat.


restricted-lemma-rig-flat-axioms

The property $P(\varphi)=$``$\varphi$ is rig-flat'' on arrows of $\textit{WAdm}^{adic*}$ is a local property as defined in Formal Spaces, Remark \ref{formal-spaces-remark-variant-adic-star}.


restricted-lemma-composition-rig-flat-continuous

The property $P(\varphi)=$``$\varphi$ is rig-flat'' on arrows of $\textit{WAdm}^{Noeth}$ is stable under composition as defined in Formal Spaces, Remark \ref{formal-spaces-remark-composition-variant-Noetherian}.


restricted-lemma-rig-smooth-continuous

Let $A \to B$ be a morphism in $\textit{WAdm}^{Noeth}$ (Formal Spaces, Section \ref{formal-spaces-section-morphisms-rings}). The following are equivalent: \begin{enumerate} \item[(a)] $A \to B$ satisfies the equivalent conditions of Lemma \ref{lemma-finite-type} and there exists an ideal of definition $I \subset B$ such that $B$ is rig-smooth over $(A, I)$, and \item[(b)] $A \to B$ satisfies the equivalent conditions of Lemma \ref{lemma-finite-type} and for all ideals of definition $I \subset A$ the algebra $B$ is rig-smooth over $(A, I)$. \end{enumerate}


restricted-lemma-rig-smooth-axioms

The property $P(\varphi)=$``$\varphi$ is rig-smooth'' on arrows of $\textit{WAdm}^{Noeth}$ is a local property as defined in Formal Spaces, Remark \ref{formal-spaces-remark-variant-Noetherian}.


restricted-lemma-base-change-rig-smooth-continuous

Consider the properties $P(\varphi)=$``$\varphi$ is rig-smooth'' and $Q(\varphi)$=``$\varphi$ is adic'' on arrows of $\textit{WAdm}^{Noeth}$. Then $P$ is stable under base change by $Q$ as defined in Formal Spaces, Remark \ref{formal-spaces-remark-base-change-variant-variant-Noetherian}.


restricted-lemma-composition-rig-smooth-continuous

The property $P(\varphi)=$``$\varphi$ is rig-smooth'' on arrows of $\textit{WAdm}^{Noeth}$ is stable under composition as defined in Formal Spaces, Remark \ref{formal-spaces-remark-composition-variant-Noetherian}.


restricted-lemma-rig-smooth-rig-flat

Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$. If $\varphi$ is rig-smooth, then $\varphi$ is rig-flat, and for any presentation $B = A\{x_1, \ldots, x_n\}/J$ and prime $J \subset \mathfrak q \subset A\{x_1, \ldots, x_n\}$ not containing an ideal of definition the ideal $J_\mathfrak q \subset A\{x_1, \ldots, x_n\}_\mathfrak q$ is generated by a regular sequence.


restricted-lemma-exact-sequence-NL-rig-smooth

Let $A \to B \to C$ be arrows in $\textit{WAdm}^{Noeth}$ which are adic and topologically of finite type. If $B \to C$ is rig-smooth, then the kernel of the map $$ H^{-1}(\NL_{B/A}^\wedge \otimes_B C) \to H^{-1}(\NL_{C/A}^\wedge) $$ (see Lemma \ref{lemma-exact-sequence-NL}) is annihilated by an ideal of definition.


restricted-lemma-rig-smooth-morphisms

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


restricted-lemma-base-change-rig-smooth

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


restricted-lemma-rig-etale-continuous

Let $A \to B$ be a morphism in $\textit{WAdm}^{Noeth}$ (Formal Spaces, Section \ref{formal-spaces-section-morphisms-rings}). The following are equivalent: \begin{enumerate} \item[(a)] $A \to B$ satisfies the equivalent conditions of Lemma \ref{lemma-finite-type} and there exists an ideal of definition $I \subset B$ such that $B$ is rig-\'etale over $(A, I)$, and \item[(b)] $A \to B$ satisfies the equivalent conditions of Lemma \ref{lemma-finite-type} and for all ideals of definition $I \subset A$ the algebra $B$ is rig-\'etale over $(A, I)$. \end{enumerate}


restricted-lemma-rig-etale-axioms

The property $P(\varphi)=$``$\varphi$ is rig-\'etale'' on arrows of $\textit{WAdm}^{Noeth}$ is a local property as defined in Formal Spaces, Remark \ref{formal-spaces-remark-variant-Noetherian}.


restricted-lemma-base-change-rig-etale-continuous

Consider the properties $P(\varphi)=$``$\varphi$ is rig-\'etale'' and $Q(\varphi)$=``$\varphi$ is adic'' on arrows of $\textit{WAdm}^{Noeth}$. Then $P$ is stable under base change by $Q$ as defined in Formal Spaces, Remark \ref{formal-spaces-remark-base-change-variant-variant-Noetherian}.


restricted-lemma-composition-rig-etale-continuous

The property $P(\varphi)=$``$\varphi$ is rig-\'etale'' on arrows of $\textit{WAdm}^{Noeth}$ is stable under composition as defined in Formal Spaces, Remark \ref{formal-spaces-remark-composition-variant-Noetherian}.


restricted-lemma-permanence-rig-etale-continuous

The property $P(\varphi)=$``$\varphi$ is rig-\'etale'' on arrows of $\textit{WAdm}^{Noeth}$ has the cancellation property as defined in Formal Spaces, Remark \ref{formal-spaces-remark-permanence-variant-Noetherian}.


restricted-lemma-rig-etale-morphisms

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


restricted-lemma-rig-etale-rig-smooth-morphism

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


restricted-lemma-base-change-rig-etale

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


restricted-lemma-rig-etale-permanence

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


restricted-lemma-rig-etale-alternative-permanence

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of locally Noetherian formal algebraic spaces over $S$. If $g \circ f$ is rig-\'etale and $g$ is an adic monomorphism, then $f$ is rig-\'etale.


restricted-lemma-closed-immersion-rig-smooth

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces. Assume that $X$ and $Y$ are locally Noetherian and $f$ is a closed immersion. The following are equivalent \begin{enumerate} \item $f$ is rig-smooth, \item $f$ is rig-\'etale, \item for every affine formal algebraic space $V$ and every morphism $V \to Y$ which is representable by algebraic spaces and \'etale the morphism $X \times_Y V \to V$ corresponds to a surjective morphism $B \to A$ in $\textit{WAdm}^{Noeth}$ whose kernel $J$ has the following property: $I(J/J^2) = 0$ for some ideal of definition $I$ of $B$. \end{enumerate}


restricted-lemma-composition-rig-surjective

\begin{slogan} Rig-surjectivity of locally finite type morphisms is preserved under composition \end{slogan} Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of formal algebraic spaces over $S$. Assume $X$, $Y$, $Z$ are locally Noetherian and $f$ and $g$ locally of finite type. Then if $f$ and $g$ are rig-surjective, so is $g \circ f$.


restricted-lemma-base-change-rig-surjective

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


restricted-lemma-rig-surjective-alternative-permanence

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


restricted-lemma-permanence-rig-surjective

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


restricted-lemma-etale-covering-rig-surjective

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces which is representable by algebraic spaces, \'etale, and surjective. Then $f$ is rig-surjective.


restricted-lemma-upshot

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces which is locally of finite type. Let $\{g_i : Y_i \to Y\}$ be a family of morphisms of formal algebraic spaces which are representable by algebraic spaces and \'etale such that $\coprod g_i$ is surjective. Then $f$ is rig-surjective if and only if each $f_i : X \times_Y Y_i \to Y_i$ is rig-surjective.


restricted-lemma-faithfully-flat-rig-surjective

Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Let $B$ be an $I$-adically complete $A$-algebra. If $A/I^n \to B/I^nB$ is of finite type and flat for all $n$ and faithfully flat for $n = 1$, then $\text{Spf}(B) \to \text{Spf}(A)$ is rig-surjective.


restricted-lemma-monomorphism-rig-surjective

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces. Assume $X$ and $Y$ are locally Noetherian, $f$ locally of finite type, and $f$ a monomorphism. Then $f$ is rig surjective if and only if every adic morphism $\text{Spf}(R) \to Y$ where $R$ is a complete discrete valuation ring factors through $X$.


restricted-lemma-closed-immersion-rig-surjective

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces. Assume that $X$ and $Y$ are locally Noetherian and $f$ is a closed immersion. The following are equivalent \begin{enumerate} \item $f$ is rig-surjective, and \item for every affine formal algebraic space $V$ and every morphism $V \to Y$ which is representable by algebraic spaces and \'etale the morphism $X \times_Y V \to V$ corresponds to a surjective morphism $B \to A$ in $\textit{WAdm}^{Noeth}$ whose kernel $J$ has the following property: $IJ^n = 0$ for some ideal of definition $I$ of $B$ and some $n \geq 1$. \end{enumerate}


restricted-lemma-closed-immersion-rig-smooth-rig-surjective

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces. Assume that $X$ and $Y$ are locally Noetherian and $f$ is a closed immersion. The following are equivalent \begin{enumerate} \item $f$ is rig-smooth and rig-surjective, \item $f$ is rig-\'etale and rig-surjective, and \item for every affine formal algebraic space $V$ and every morphism $V \to Y$ which is representable by algebraic spaces and \'etale the morphism $X \times_Y V \to V$ corresponds to a surjective morphism $B \to A$ in $\textit{WAdm}^{Noeth}$ whose kernel $J$ has the following property: $IJ = 0$ for some ideal of definition $I$ of $B$. \end{enumerate}


restricted-lemma-rig-etale-descent

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of locally Noetherian formal algebraic spaces over $S$. Assume \begin{enumerate} \item $g$ is locally of finite type, \item $f$ is rig-smooth (resp.\ rig-\'etale) and rig-surjective, \item $g \circ f$ is rig-smooth (resp.\ rig-\'etale) \end{enumerate} then $g$ is rig-smooth (resp.\ rig-\'etale).


restricted-lemma-flat-locus

Let $X$ be a locally Noetherian formal algebraic space over a complete discrete valuation ring $A$. Then there exists a closed immersion $X' \to X$ of formal algebraic spaces such that $X'$ is flat over $A$ and such that any morphism $Y \to X$ of locally Noetherian formal algebraic spaces with $Y$ flat over $A$ factors through $X'$.


restricted-lemma-flat-and-diagonal-rig-surjective

Let $X$ be a locally Noetherian formal algebraic space which is locally of finite type over a complete discrete valuation ring $A$. Let $X' \subset X$ be as in Lemma \ref{lemma-flat-locus}. If $X \to X \times_{\text{Spf}(A)} X$ is rig-\'etale and rig-surjective, then $X' = \text{Spf}(A)$ or $X' = \emptyset$.


restricted-lemma-rig-monomorphism-rig-surjective

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces. Assume \begin{enumerate} \item $X$ and $Y$ are locally Noetherian, \item $f$ locally of finite type, \item $\Delta_f : X \to X \times_Y X$ is rig-\'etale and rig-surjective. \end{enumerate} Then $f$ is rig surjective if and only if every adic morphism $\text{Spf}(R) \to Y$ where $R$ is a complete discrete valuation ring lifts to a morphism $\text{Spf}(R) \to X$.


restricted-lemma-map-completions-finite-type

In the situation above. If $f$ is locally of finite type, then $f_{/T}$ is locally of finite type.


restricted-lemma-map-completions-etale

In the situation above. If $f$ is \'etale, then $f_{/T}$ is \'etale.


restricted-lemma-closed-immersion-gives-closed-immersion

In the situation above. If $f$ is a closed immersion, then $f_{/T}$ is a closed immersion.


restricted-lemma-proper-gives-proper

In the situation above. If $f$ is proper, then $f_{/T}$ is proper.


restricted-lemma-quasi-compact-gives-quasi-compact

In the situation above. If $f$ is quasi-compact, then $f_{/T}$ is quasi-compact.


restricted-lemma-quasi-separated-gives-quasi-separated

In the situation above. If $f$ is (quasi-)separated, then $f_{/T}$ is too.


restricted-lemma-smooth-gives-rig-smooth

In the situation above. If $X$ is locally Noetherian, $f$ is locally of finite type, and $U' \to U$ is smooth, then $f_{/T}$ is rig-smooth.


restricted-lemma-etale-gives-rig-etale

In the situation above. If $X$ is locally Noetherian, $f$ is locally of finite type, and $U' \to U$ is \'etale, then $f_{/T}$ is rig-\'etale.


restricted-lemma-completion-proper-surjective-rig-surjective

In the situation above. If $X$ is locally Noetherian, $f$ is proper, and $U' \to U$ is surjective, then $f_{/T}$ is rig-surjective.


restricted-lemma-separated-mono-open-diagonal-rig-surjective

In the situation above. If $X$ is locally Noetherian, $f$ is separated and locally of finite type, and $U' \to U$ is a monomorphism, then $\Delta_{f_{/T}}$ is rig-surjective.


restricted-lemma-modification-gives-formal-modification

Let $S$, $f : X' \to X$, $T \subset |X|$, $U \subset X$, $T' \subset |X'|$, and $U' \subset X'$ be as in Section \ref{section-completion-functor}. If $X$ is locally Noetherian, $f$ is proper, and $U' \to U$ is an isomorphism, then $f_{/T} : X'_{/T'} \to X_{/T}$ is a formal modification.


restricted-lemma-base-change-formal-modification

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$ which is a formal modification. Then for any adic morphism $Y' \to Y$ of locally Noetherian formal algebraic spaces, the base change $f' : X \times_Y Y' \to Y'$ is a formal modification.


restricted-lemma-algebraize-rig-etale-affine

Let $T \subset X$ be a closed subset of a Noetherian affine scheme $X$. Let $W$ be a Noetherian affine formal algebraic space. Let $g : W \to X_{/T}$ be a rig-\'etale morphism. Then there exists an affine scheme $X'$ and a finite type morphism $f : X' \to X$ \'etale over $X \setminus T$ such that there is an isomorphism $X'_{/f^{-1}T} \cong W$ compatible with $f_{/T}$ and $g$. Moreover, if $W \to X_{/T}$ is \'etale, then $X' \to X$ is \'etale.


restricted-lemma-algebraize-morphism-rig-etale

Assume we have \begin{enumerate} \item Noetherian affine schemes $X$, $X'$, and $Y$, \item a closed subset $T \subset |X|$, \item a morphism $f : X' \to X$ locally of finite type and \'etale over $X \setminus T$, \item a morphism $h : Y \to X$, \item a morphism $\alpha : Y_{/T} \to X'_{/T}$ over $X_{/T}$ (see proof for notation). \end{enumerate} Then there exists an \'etale morphism $b : Y' \to Y$ of affine schemes which induces an isomorphism $b_{/T} : Y'_{/T} \to Y_{/T}$ and a morphism $a : Y' \to X'$ over $X$ such that $\alpha = a_{/T} \circ b_{/T}^{-1}$.


restricted-lemma-factor

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Z \to Y$ be morphisms of algebraic spaces. Let $T \subset |X|$ be closed. Assume that \begin{enumerate} \item $X$ is locally Noetherian, \item $g$ is a monomorphism and locally of finite type, \item $f|_{X \setminus T} : X \setminus T \to Y$ factors through $g$, and \item $f_{/T} : X_{/T} \to Y$ factors through $g$, \end{enumerate} then $f$ factors through $g$.


restricted-lemma-faithful-general

Let $S$ be a scheme. Let $X$, $W$ be algebraic spaces over $S$ with $X$ locally Noetherian. Let $T \subset |X|$ be a closed subset. Let $a, b : X \to W$ be morphisms of algebraic spaces over $S$ such that $a|_{X \setminus T} = b|_{X \setminus T}$ and such that $a_{/T} = b_{/T}$ as morphisms $X_{/T} \to W$. Then $a = b$.


restricted-lemma-faithful

Let $S$ be a scheme. Let $X$, $Y$ be locally Noetherian algebraic spaces over $S$. Let $T \subset |X|$ and $T' \subset |Y|$ be closed subsets. Let $a, b : X \to Y$ be morphisms of algebraic spaces over $S$ such that $a|_{X \setminus T} = b|_{X \setminus T}$, such that $|a|(T) \subset T'$ and $|b|(T) \subset T'$, and such that $a_{/T} = b_{/T}$ as morphisms $X_{/T} \to Y_{/T'}$. Then $a = b$.


restricted-lemma-equivalence-relation

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $T \subset |X|$ be a closed subset. Let $s, t : R \to U$ be two morphisms of algebraic spaces over $X$. Assume \begin{enumerate} \item $R$, $U$ are locally of finite type over $X$, \item the base change of $s$ and $t$ to $X \setminus T$ is an \'etale equivalence relation, and \item the formal completion $(t_{/T}, s_{/T}) : R_{/T} \to U_{/T} \times_{X_{/T}} U_{/T}$ is an equivalence relation too (see proof for notation). \end{enumerate} Then $(t, s) : R \to U \times_X U$ is an \'etale equivalence relation.


restricted-lemma-smash-away-from-T

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$ and let $T \subset |X|$ be a closed subset. Let $f : X' \to X$ be a morphism of algebraic spaces which is locally of finite type and \'etale outside of $T$. There exists a factorization $$ X' \longrightarrow X'' \longrightarrow X $$ of $f$ with the following properties: $X'' \to X$ is locally of finite type, $X'' \to X$ is an isomorphism over $X \setminus T$, and $X'_{/T} \to X''_{/T}$ is an isomorphism (see proof for notation).


restricted-lemma-functoriality-completion-functor

In the situation above, let $X_1 \to X$ be a morphism of algebraic spaces with $X_1$ locally Noetherian. Denote $T_1 \subset |X_1|$ the inverse image of $T$ and $U_1 \subset X_1$ the inverse image of $U$. We denote \begin{enumerate} \item $\mathcal{C}_{X, T}$ the category whose objects are morphisms of algebraic spaces $f : X' \to X$ which are locally of finite type and such that $U' = f^{-1}U \to U$ is an isomorphism, \item $\mathcal{C}_{X_1, T_1}$ the category whose objects are morphisms of algebraic spaces $f_1 : X_1' \to X_1$ which are locally of finite type and such that $f_1^{-1}U_1 \to U_1$ is an isomorphism, \item $\mathcal{C}_{X_{/T}}$ the category whose objects are morphisms $g : W \to X_{/T}$ of formal algebraic spaces with $W$ locally Noetherian and $g$ rig-\'etale, \item $\mathcal{C}_{X_{1, /T_1}}$ the category whose objects are morphisms $g_1 : W_1 \to X_{1, /T_1}$ of formal algebraic spaces with $W_1$ locally Noetherian and $g_1$ rig-\'etale. \end{enumerate} Then the diagram $$ \xymatrix{ \mathcal{C}_{X, T} \ar[d] \ar[r] & \mathcal{C}_{X_{/T}} \ar[d] \\ \mathcal{C}_{X_1, T_1} \ar[r] & \mathcal{C}_{X_{1, /T_1}} } $$ is commutative where the horizonal arrows are given by (\ref{equation-completion-functor}) and the vertical arrows by base change along $X_1 \to X$ and along $X_{1, /T_1} \to X_{/T}$.


restricted-lemma-completion-functor-fully-faithful

In the situation above. Let $f : X' \to X$ be a morphism of algebraic spaces which is locally of finite type and an isomorphism over $U$. Let $g : Y \to X$ be a morphism with $Y$ locally Noetherian. Then completion defines a bijection $$ \Mor_X(Y, X') \longrightarrow \Mor_{X_{/T}}(Y_{/T}, X'_{/T}) $$ In particular, the functor (\ref{equation-completion-functor}) is fully faithful.


restricted-lemma-dilatations-affine

In the situation above. Assume $X$ is affine. Then the functor (\ref{equation-completion-functor}) is an equivalence.


restricted-lemma-output-quasi-compact

With assumptions and notation as in Theorem \ref{theorem-dilatations-general} let $f : X' \to X$ correspond to $g : W \to X_{/T}$. Then $f$ is quasi-compact if and only if $g$ is quasi-compact.


restricted-lemma-output-quasi-separated

With assumptions and notation as in Theorem \ref{theorem-dilatations-general} let $f : X' \to X$ correspond to $g : W \to X_{/T}$. Then $f$ is quasi-separated if and only if $g$ is so.


restricted-lemma-output-separated

With assumptions and notation as in Theorem \ref{theorem-dilatations-general} let $f : X' \to X$ correspond to $g : W \to X_{/T}$. Then $f$ is separated $\Leftrightarrow$ $g$ is separated and $\Delta_g : W \to W \times_{X_{/T}} W$ is rig-surjective.


restricted-lemma-output-proper

With assumptions and notation as in Theorem \ref{theorem-dilatations-general} let $f : X' \to X$ correspond to $g : W \to X_{/T}$. Then $f$ is proper if and only if $g$ is a formal modification (Definition \ref{definition-formal-modification}).


restricted-lemma-output-etale

With assumptions and notation as in Theorem \ref{theorem-dilatations-general} let $f : X' \to X$ correspond to $g : W \to X_{/T}$. Then $f$ is \'etale if and only if $g$ is \'etale.


restricted-lemma-Noetherian-local-ring

Let $A \to B$ be a ring homomorphism of Noetherian rings inducing an isomorphism on $I$-adic completions for some ideal $I \subset A$ (for example if $B$ is the $I$-adic completion of $A$). Then base change defines an equivalence of categories between the category (\ref{equation-modification}) for $(A, I)$ with the category (\ref{equation-modification}) for $(B, IB)$.


restricted-lemma-equivalence-to-completion

Let $A \to B$ be a local map of local Noetherian rings such that \begin{enumerate} \item $A \to B$ is flat, \item $\mathfrak m_B = \mathfrak m_A B$, and \item $\kappa(\mathfrak m_A) = \kappa(\mathfrak m_B)$ \end{enumerate} Then the base change functor from the category (\ref{equation-modification}) for $(A, \mathfrak m_A)$ to the category (\ref{equation-modification}) for $(B, \mathfrak m_B)$ is an equivalence.


restricted-proposition-approximate

Let $I$ be an ideal of a Noetherian G-ring $A$. Let $B$ be an object of (\ref{equation-C-prime}). If $B$ is rig-smooth over $(A, I)$, then there exists a finite type $A$-algebra $C$ and an isomorphism $B \cong C^\wedge$ of $A$-algebras.


restricted-proposition-glue-modification

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $T \subset |X|$ be a closed subset with complementary open subspace $U \subset X$. Let $f : X' \to X$ be a proper morphism of algebraic spaces such that $f^{-1}(U) \to U$ is an isomorphism. For any algebraic space $W$ over $S$ the map $$ \Mor_S(X, W) \longrightarrow \Mor_S(X', W) \times_{\Mor_S(X'_{/T}, W)} \Mor_S(X_{/T}, W) $$ is bijective.


more-groupoids-lemma-sheaf-differentials

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. The sheaf of differentials of $R$ seen as a scheme over $U$ via $t$ is a quotient of the pullback via $t$ of the conormal sheaf of the immersion $e : U \to R$. In a formula: there is a canonical surjection $t^*\mathcal{C}_{U/R} \to \Omega_{R/U}$. If $s$ is flat, then this map is an isomorphism.


more-groupoids-lemma-first-order-structure-c

The map $I/I^2 \to J/J^2$ induced by $c$ is the composition $$ I/I^2 \xrightarrow{(1, 1)} I/I^2 \oplus I/I^2 \to J/J^2 $$ where the second arrow comes from the equality $J = (I \otimes B + B \otimes I)C$. The map $i : B \to B$ induces the map $-1 : I/I^2 \to I/I^2$.


more-groupoids-lemma-local-source

Let $S$ be a scheme. Let $(U, R, s, t, c, e, i)$ be a groupoid over $S$. Let $g : U' \to U$ be a morphism of schemes. Denote $h$ the composition $$ \xymatrix{ h : U' \times_{g, U, t} R \ar[r]_-{\text{pr}_1} & R \ar[r]_s & U. } $$ Let $\mathcal{P}, \mathcal{Q}, \mathcal{R}$ be properties of morphisms of schemes. Assume \begin{enumerate} \item $\mathcal{R} \Rightarrow \mathcal{Q}$, \item $\mathcal{Q}$ is preserved under base change and composition, \item for any morphism $f : X \to Y$ which has $\mathcal{Q}$ there exists a largest open $W(\mathcal{P}, f) \subset X$ such that $f|_{W(\mathcal{P}, f)}$ has $\mathcal{P}$, and \item for any morphism $f : X \to Y$ which has $\mathcal{Q}$, and any morphism $Y' \to Y$ which has $\mathcal{R}$ we have $Y' \times_Y W(\mathcal{P}, f) = W(\mathcal{P}, f')$, where $f' : X_{Y'} \to Y'$ is the base change of $f$. \end{enumerate} If $s, t$ have $\mathcal{R}$ and $g$ has $\mathcal{Q}$, then there exists an open subscheme $W \subset U'$ such that $W \times_{g, U, t} R = W(\mathcal{P}, h)$.


more-groupoids-lemma-property-invariant

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$\footnote{The fact that $fpqc$ is missing is not a typo.}. Let $\mathcal{P}$ be a property of morphisms of schemes which is $\tau$-local on the target (Descent, Definition \ref{descent-definition-property-morphisms-local}). Assume $\{s : R \to U\}$ and $\{t : R \to U\}$ are coverings for the $\tau$-topology. Let $W \subset U$ be the maximal open subscheme such that $s|_{s^{-1}(W)} : s^{-1}(W) \to W$ has property $\mathcal{P}$. Then $W$ is $R$-invariant, see Groupoids, Definition \ref{groupoids-definition-invariant-open}.