sdga-lemma-characterize-graded-injectives-in-dg

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. There exists a set $T$ and for each $t \in T$ an injective map $\mathcal{M}_t \to \mathcal{M}'_t$ of acyclic differential graded $\mathcal{A}$-modules such that for an object $\mathcal{I}$ of $\text{Mod}_{(\mathcal{A}, \text{d})}$ the following are equivalent \begin{enumerate} \item $\mathcal{I}$ is graded injective, and \item for every solid diagram $$ \xymatrix{ \mathcal{M}_t \ar[r] \ar[d] & \mathcal{I} \\ \mathcal{M}'_t \ar@{..>}[ru] } $$ a dotted arrow exists in $\text{Mod}_{(\mathcal{A}, \text{d})}$ making the diagram commute. \end{enumerate}


sdga-lemma-small-acyclics

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. There exists a set $S$ and for each $s$ an acyclic differential graded $\mathcal{A}$-module $\mathcal{M}_s$ such that for every nonzero acyclic differential graded $\mathcal{A}$-module $\mathcal{M}$ there is an $s \in S$ and an injective map $\mathcal{M}_s \to \mathcal{M}$ in $\text{Mod}_{(\mathcal{A}, \text{d})}$.


sdga-lemma-product-K-injective

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $T$ be a set and for each $t \in T$ let $\mathcal{I}_t$ be a K-injective diffential graded $\mathcal{A}$-module. Then $\prod \mathcal{I}_t$ is a K-injective differential graded $\mathcal{A}$-module.


sdga-lemma-first-property-dg-injective

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{I}$ be a K-injective and graded injective object of $\text{Mod}_{(\mathcal{A}, \text{d})}$. For every solid diagram in $\text{Mod}_{(\mathcal{A}, \text{d})}$ $$ \xymatrix{ \mathcal{M} \ar[r]_a \ar[d]_b & \mathcal{I} \\ \mathcal{M}' \ar@{..>}[ru] } $$ where $b$ is injective and $\mathcal{M}$ is acyclic a dotted arrow exists making the diagram commute.


sdga-lemma-second-property-dg-injective

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{I}$ be a K-injective and graded injective object of $\text{Mod}_{(\mathcal{A}, \text{d})}$. For every solid diagram in $\text{Mod}_{(\mathcal{A}, \text{d})}$ $$ \xymatrix{ \mathcal{M} \ar[r]_a \ar[d]_b & \mathcal{I} \\ \mathcal{M}' \ar@{..>}[ru] } $$ where $b$ is a quasi-isomorphism a dotted arrow exists making the diagram commute up to homotopy.


sdga-lemma-better-set-of-monos

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. There exists a set $R$ and for each $r \in R$ an injective map $\mathcal{M}_r \to \mathcal{M}'_r$ of acyclic differential graded $\mathcal{A}$-modules such that for an object $\mathcal{I}$ of $\text{Mod}_{(\mathcal{A}, \text{d})}$ the following are equivalent \begin{enumerate} \item $\mathcal{I}$ is K-injective and graded injective, and \item for every solid diagram $$ \xymatrix{ \mathcal{M}_r \ar[r] \ar[d] & \mathcal{I} \\ \mathcal{M}'_r \ar@{..>}[ru] } $$ a dotted arrow exists in $\text{Mod}_{(\mathcal{A}, \text{d})}$ making the diagram commute. \end{enumerate}


sdga-lemma-functor-set-of-monos

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $R$ be a set and for each $r \in R$ let an injective map $\mathcal{M}_r \to \mathcal{M}'_r$ of acyclic differential graded $\mathcal{A}$-modules be given. There exists a functor $M : \text{Mod}_{(\mathcal{A}, \text{d})} \to \text{Mod}_{(\mathcal{A}, \text{d})}$ and a natural transformation $j : \text{id} \to M$ such that \begin{enumerate} \item $j_\mathcal{M} : \mathcal{M} \to M(\mathcal{M})$ is injective and a quasi-isomorphism, \item for every solid diagram $$ \xymatrix{ \mathcal{M}_r \ar[r] \ar[d] & \mathcal{M} \ar[d]^{j_\mathcal{M}} \\ \mathcal{M}'_r \ar@{..>}[r] & M(\mathcal{M}) } $$ a dotted arrow exists in $\text{Mod}_{(\mathcal{A}, \text{d})}$ making the diagram commute. \end{enumerate}


sdga-lemma-cohomology-homological

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The functor  $H^0 : \text{Mod}_{(\mathcal{A}, \text{d})} \to \textit{Mod}(\mathcal{O})$ of Section \ref{section-modules} factors through a functor $$ H^0 : K(\text{Mod}_{(\mathcal{A}, \text{d})}) \to \textit{Mod}(\mathcal{O}) $$ which is homological in the sense of Derived Categories, Definition \ref{derived-definition-homological}.


sdga-lemma-acyclics

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The full subcategory $\text{Ac}$ of the homotopy category $K(\text{Mod}_{(\mathcal{A}, \text{d})})$ consisting of acyclic modules is a strictly full saturated triangulated subcategory of $K(\text{Mod}_{(A, \text{d})})$.


sdga-lemma-qis

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Consider the subclass $\text{Qis} \subset \text{Arrows}(K(\text{Mod}_{(A, \text{d})}))$ consisting of quasi-isomorphisms. This is a saturated multiplicative system compatible with the triangulated structure on $K(\text{Mod}_{(A, \text{d})})$.


sdga-lemma-hom-derived

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ and $\mathcal{N}$ be differential graded $\mathcal{A}$-modules. Let $\mathcal{N} \to \mathcal{I}$ be a quasi-isomorphism with $\mathcal{I}$ a graded injective and K-injective differential graded $\mathcal{A}$-module. Then $$ \Hom_{D(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{N}) = \Hom_{K(\text{Mod}_{(\mathcal{A}, \text{d})})}(\mathcal{M}, \mathcal{I}) $$


sdga-lemma-derived-products

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Then \begin{enumerate} \item $D(\mathcal{A}, \text{d})$ has both direct sums and products, \item direct sums are obtained by taking direct sums of differential graded $\mathcal{A}$-modules, \item products are obtained by taking products of K-injective differential graded modules. \end{enumerate}


sdga-lemma-derived-canonical-delta-functor

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The localization functor $\text{Mod}_{(\mathcal{A}, \text{d})} \to D(\mathcal{A}, \text{d})$ has the natural structure of a $\delta$-functor, with $$ \delta_{\mathcal{K} \to \mathcal{L} \to \mathcal{M}} = - p \circ q^{-1} $$ with $p$ and $q$ as explained above.


sdga-lemma-derived-tensor-product

In the situation above, the functor (\ref{equation-pullback}) composed with the localization functor $K(\text{Mod}_{(\mathcal{A}', \text{d})}) \to D(\mathcal{A}', \text{d})$ has a left derived extension $D(\mathcal{B}, \text{d}) \to D(\mathcal{A}', \text{d})$ whose value on a good right differential graded $\mathcal{B}$-module $\mathcal{P}$ is $f^*\mathcal{P} \otimes_\mathcal{A} \mathcal{N}$.


sdga-lemma-compose-pullback-tensor

In Lemma \ref{lemma-derived-tensor-product} the functor $D(\mathcal{B}, \text{d}) \to D(\mathcal{A}', \text{d})$ is equal to $\mathcal{M} \mapsto Lf^*\mathcal{M} \otimes_\mathcal{A}^\mathbf{L} \mathcal{N}$.


sdga-lemma-compose-pullback

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$ and $(g, g^\sharp) : (\Sh(\mathcal{C}'), \mathcal{O}') \to (\Sh(\mathcal{C}''), \mathcal{O}'')$ be morphisms of ringed topoi. Let $\mathcal{A}$, $\mathcal{A}'$, and $\mathcal{A}''$ be a differential graded $\mathcal{O}$-algebra, $\mathcal{O}'$-algebra, and $\mathcal{O}''$-algebra. Let $\varphi : \mathcal{A}' \to f_*\mathcal{A}$ and $\varphi' : \mathcal{A}'' \to g_*\mathcal{A}'$ be a homomorphism of differential graded $\mathcal{O}'$-algebras and $\mathcal{O}''$-algebras. Then we have $L(g \circ f)^* = Lf^* \circ Lg^* : D(\mathcal{A}'', \text{d}) \to D(\mathcal{A}, \text{d})$.


sdga-lemma-tensor-symmetry

In the situation above, if $\mathcal{N} \to \mathcal{N}'$ is an isomorphism on cohomology sheaves, then $t$ is an isomorphism of functors $(- \otimes_\mathcal{A}^\mathbf{L} \mathcal{N}) \to (- \otimes_\mathcal{A}^\mathbf{L} \mathcal{N}')$.


sdga-lemma-good-on-other-side

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{B}$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N}$ be a  differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. If $\mathcal{N}$ is good as a left differential graded $\mathcal{A}$-module, then we have $\mathcal{M} \otimes_\mathcal{A}^\mathbf{L} \mathcal{N} = \mathcal{M} \otimes_\mathcal{A} \mathcal{N}$ for all differential graded $\mathcal{A}$-modules $\mathcal{M}$.


sdga-lemma-compose-tensor

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{A}'$, $\mathcal{A}''$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N}$ and $\mathcal{N}'$ be a differential graded $(\mathcal{A}, \mathcal{A}')$-bimodule and $(\mathcal{A}', \mathcal{A}'')$-bimodule. Assume that the canonical map $$ \mathcal{N} \otimes_{\mathcal{A}'}^\mathbf{L} \mathcal{N}' \longrightarrow \mathcal{N} \otimes_{\mathcal{A}'} \mathcal{N}' $$ in $D(\mathcal{A}'', \text{d})$ is a quasi-isomorphism. Then we have $$ (\mathcal{M} \otimes_\mathcal{A}^\mathbf{L} \mathcal{N}) \otimes_{\mathcal{A}'}^\mathbf{L} \mathcal{N}' = \mathcal{M} \otimes_\mathcal{A}^\mathbf{L} (\mathcal{N} \otimes_{\mathcal{A}'} \mathcal{N}') $$ as functors $D(\mathcal{A}, \text{d}) \to D(\mathcal{A}'', \text{d})$.


sdga-lemma-right-derived

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Then any exact functor $$ T : K(\text{Mod}_{(\mathcal{A}, \text{d})}) \longrightarrow \mathcal{D} $$ of triangulated categories has a right derived extension $RT : D(\mathcal{A}, \text{d}) \to \mathcal{D}$ whose value on a graded injective and K-injective differential graded $\mathcal{A}$-module $\mathcal{I}$ is $T(\mathcal{I})$.


sdga-lemma-derived-adjoint-tensor-hom

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{B}$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N}$ be a  differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. Then $$ R\SheafHom_\mathcal{B}(\mathcal{N}, -) : D(\mathcal{B}, \text{d}) \longrightarrow D(\mathcal{A}, \text{d}) $$ is right adjoint to $$ - \otimes_\mathcal{A}^\mathbf{L} \mathcal{N} : D(\mathcal{A}, \text{d}) \longrightarrow D(\mathcal{B}, \text{d}) $$


sdga-lemma-derived-adjoint-push-pull

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Let $\mathcal{A}$ be a differential graded $\mathcal{O}_\mathcal{C}$-algebra. Let $\mathcal{B}$ be a differential graded $\mathcal{O}_\mathcal{D}$-algebra. Let $\varphi : \mathcal{B} \to f_*\mathcal{A}$ be a homomorphism of differential graded $\mathcal{O}_\mathcal{D}$-algebras. Then $$ Rf_* :  D(\mathcal{A}, \text{d}) \longrightarrow D(\mathcal{B}, \text{d}) $$ is right adjoint to $$ Lf^* : D(\mathcal{B}, \text{d}) \longrightarrow D(\mathcal{A}, \text{d}) $$


sdga-lemma-compose-pushforward

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$ and $(g, g^\sharp) : (\Sh(\mathcal{C}'), \mathcal{O}') \to (\Sh(\mathcal{C}''), \mathcal{O}'')$ be morphisms of ringed topoi. Let $\mathcal{A}$, $\mathcal{A}'$, and $\mathcal{A}''$ be a differential graded $\mathcal{O}$-algebra, $\mathcal{O}'$-algebra, and $\mathcal{O}''$-algebra. Let $\varphi : \mathcal{A}' \to f_*\mathcal{A}$ and $\varphi' : \mathcal{A}'' \to g_*\mathcal{A}'$ be a homomorphism of differential graded $\mathcal{O}'$-algebras and $\mathcal{O}''$-algebras. Then we have $R(g \circ f)_* = Rg_* \circ Rf_* : D(\mathcal{A}, \text{d}) \to D(\mathcal{A}'', \text{d})$.


sdga-lemma-pushforward-agrees

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Let $\mathcal{A}$ be a differential graded $\mathcal{O}_\mathcal{C}$-algebra. Let $\mathcal{B}$ be a differential graded $\mathcal{O}_\mathcal{D}$-algebra. Let $\varphi : \mathcal{B} \to f_*\mathcal{A}$ be a homomorphism of differential graded $\mathcal{O}_\mathcal{D}$-algebras. The diagram $$ \xymatrix{ D(\mathcal{A}, \text{d}) \ar[d]_{Rf_*} \ar[rr]_{forget} & & D(\mathcal{O}_\mathcal{C}) \ar[d]^{Rf_*} \\ D(\mathcal{B}, \text{d}) \ar[rr]^{forget} & & D(\mathcal{O}_\mathcal{D}) } $$ commutes.


sdga-lemma-special-good

In the situation above the differential graded $\mathcal{O}$-algebra $$ \mathcal{A} = \colim \mathcal{A}_i $$ has the following property: for any morphism $(f, f^\sharp) : (\Sh(\mathcal{C}'), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})$ of ringed topoi, the pullback $f^*\mathcal{A}$ is flat as a graded $\mathcal{O}'$-module and is K-flat as a complex of $\mathcal{O}'$-modules.


stacks-morphisms-theorem-quasi-DM

Let $\mathcal{X}$ be an algebraic stack. The following are equivalent \begin{enumerate} \item $\mathcal{X}$ is quasi-DM, and \item there exists a scheme $W$ and a surjective, flat, locally finitely presented, locally quasi-finite morphism $W \to \mathcal{X}$. \end{enumerate}


stacks-morphisms-theorem-DM

Let $\mathcal{X}$ be an algebraic stack. The following are equivalent \begin{enumerate} \item $\mathcal{X}$ is DM, \item $\mathcal{X}$ is Deligne-Mumford, and \item there exists a scheme $W$ and a surjective \'etale morphism $W \to \mathcal{X}$. \end{enumerate}


stacks-morphisms-lemma-isom-locally-finite-type

Let $\mathcal{X}$ be an algebraic stack. Let $T$ be a scheme and let $x, y$ be objects of the fibre category of $\mathcal{X}$ over $T$. Then the morphism $\mathit{Isom}_\mathcal{X}(x, y) \to T$ is locally of finite type.


stacks-morphisms-lemma-isom-pseudo-torsor-aut

Let $\mathcal{X}$ be an algebraic stack. Let $T$ be a scheme and let $x, y$ be objects of the fibre category of $\mathcal{X}$ over $T$. Then \begin{enumerate} \item $\mathit{Isom}_\mathcal{X}(y, y)$ is a group algebraic space over $T$, and \item $\mathit{Isom}_\mathcal{X}(x, y)$ is a pseudo torsor for $\mathit{Isom}_\mathcal{X}(y, y)$ over $T$. \end{enumerate}


stacks-morphisms-lemma-properties-diagonal

\begin{slogan} Diagonals of morphisms of algebraic stacks are representable by algebraic spaces and locally of finite type. \end{slogan} Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Then \begin{enumerate} \item $\Delta_f$ is representable by algebraic spaces, and \item $\Delta_f$ is locally of finite type. \end{enumerate}


stacks-morphisms-lemma-properties-diagonal-representable

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. Then \begin{enumerate} \item $\Delta_f$ is representable (by schemes), \item $\Delta_f$ is locally of finite type, \item $\Delta_f$ is a monomorphism, \item $\Delta_f$ is separated, and \item $\Delta_f$ is locally quasi-finite. \end{enumerate}


stacks-morphisms-lemma-representable-separated-diagonal-closed

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Then the following are equivalent \begin{enumerate} \item $f$ is separated, \item $\Delta_f$ is a closed immersion, \item $\Delta_f$ is proper, or \item $\Delta_f$ is universally closed. \end{enumerate}


stacks-morphisms-lemma-representable-quasi-separated-diagonal-quasi-compact

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Then the following are equivalent \begin{enumerate} \item $f$ is quasi-separated, \item $\Delta_f$ is quasi-compact, or \item $\Delta_f$ is of finite type. \end{enumerate}


stacks-morphisms-lemma-representable-locally-separated-diagonal-immersion

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Then the following are equivalent \begin{enumerate} \item $f$ is locally separated, and \item $\Delta_f$ is an immersion. \end{enumerate}


stacks-morphisms-lemma-trivial-implications

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. \begin{enumerate} \item If $f$ is separated, then $f$ is quasi-separated. \item If $f$ is DM, then $f$ is quasi-DM. \item If $f$ is representable by algebraic spaces, then $f$ is DM. \end{enumerate}


stacks-morphisms-lemma-base-change-separated

All of the separation axioms listed in Definition \ref{definition-separated} are stable under base change.


stacks-morphisms-lemma-check-separated-covering

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $W \to \mathcal{Y}$ be a surjective, flat, and locally of finite presentation where $W$ is an algebraic space. If the base change $W \times_\mathcal{Y} \mathcal{X} \to W$ has one of the separation properties of Definition \ref{definition-separated} then so does $f$.


stacks-morphisms-lemma-fibre-product-after-map

Let $f : \mathcal{X} \to \mathcal{Z}$, $g : \mathcal{Y} \to \mathcal{Z}$ and $\mathcal{Z} \to \mathcal{T}$ be morphisms of algebraic stacks. Consider the induced morphism $i : \mathcal{X} \times_\mathcal{Z} \mathcal{Y} \to \mathcal{X} \times_\mathcal{T} \mathcal{Y}$. Then \begin{enumerate} \item $i$ is representable by algebraic spaces and locally of finite type, \item if $\Delta_{\mathcal{Z}/\mathcal{T}}$ is quasi-separated, then $i$ is quasi-separated, \item if $\Delta_{\mathcal{Z}/\mathcal{T}}$ is separated, then $i$ is separated, \item if $\mathcal{Z} \to \mathcal{T}$ is DM, then $i$ is unramified, \item if $\mathcal{Z} \to \mathcal{T}$ is quasi-DM, then $i$ is locally quasi-finite, \item if $\mathcal{Z} \to \mathcal{T}$ is separated, then $i$ is proper, and \item if $\mathcal{Z} \to \mathcal{T}$ is quasi-separated, then $i$ is quasi-compact and quasi-separated. \end{enumerate}


stacks-morphisms-lemma-semi-diagonal

Let $\mathcal{T}$ be an algebraic stack. Let $g : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks over $\mathcal{T}$. Consider the graph $i : \mathcal{X} \to \mathcal{X} \times_\mathcal{T} \mathcal{Y}$ of $g$. Then \begin{enumerate} \item $i$ is representable by algebraic spaces and locally of finite type, \item if $\mathcal{Y} \to \mathcal{T}$ is DM, then $i$ is unramified, \item if $\mathcal{Y} \to \mathcal{T}$ is quasi-DM, then $i$ is locally quasi-finite, \item if $\mathcal{Y} \to \mathcal{T}$ is separated, then $i$ is proper, and \item if $\mathcal{Y} \to \mathcal{T}$ is quasi-separated, then $i$ is quasi-compact and quasi-separated. \end{enumerate}


stacks-morphisms-lemma-composition-separated

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


stacks-morphisms-lemma-separated-over-separated

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks over the base scheme $S$. \begin{enumerate} \item If $\mathcal{Y}$ is DM over $S$ and $f$ is DM, then $\mathcal{X}$ is DM over $S$. \item If $\mathcal{Y}$ is quasi-DM over $S$ and $f$ is quasi-DM, then $\mathcal{X}$ is quasi-DM over $S$. \item If $\mathcal{Y}$ is separated over $S$ and $f$ is separated, then $\mathcal{X}$ is separated over $S$. \item If $\mathcal{Y}$ is quasi-separated over $S$ and $f$ is quasi-separated, then $\mathcal{X}$ is quasi-separated over $S$. \item If $\mathcal{Y}$ is DM and $f$ is DM, then $\mathcal{X}$ is DM. \item If $\mathcal{Y}$ is quasi-DM and $f$ is quasi-DM, then $\mathcal{X}$ is quasi-DM. \item If $\mathcal{Y}$ is separated and $f$ is separated, then $\mathcal{X}$ is separated. \item If $\mathcal{Y}$ is quasi-separated and $f$ is quasi-separated, then $\mathcal{X}$ is quasi-separated. \end{enumerate}


stacks-morphisms-lemma-compose-after-separated

Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. \begin{enumerate} \item If $g \circ f$ is DM then so is $f$. \item If $g \circ f$ is quasi-DM then so is $f$. \item If $g \circ f$ is separated and $\Delta_g$ is separated, then $f$ is separated. \item If $g \circ f$ is quasi-separated and $\Delta_g$ is quasi-separated, then $f$ is quasi-separated. \end{enumerate}


stacks-morphisms-lemma-separated-implies-morphism-separated

Let $\mathcal{X}$ be an algebraic stack over the base scheme $S$. \begin{enumerate} \item $\mathcal{X}$ is DM $\Leftrightarrow$ $\mathcal{X}$ is DM over $S$. \item $\mathcal{X}$ is quasi-DM $\Leftrightarrow$ $\mathcal{X}$ is quasi-DM over $S$. \item If $\mathcal{X}$ is separated, then $\mathcal{X}$ is separated over $S$. \item If $\mathcal{X}$ is quasi-separated, then $\mathcal{X}$ is quasi-separated over $S$. \end{enumerate} Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks over the base scheme $S$. \begin{enumerate} \item[(5)] If $\mathcal{X}$ is DM over $S$, then $f$ is DM. \item[(6)] If $\mathcal{X}$ is quasi-DM over $S$, then $f$ is quasi-DM. \item[(7)] If $\mathcal{X}$ is separated over $S$ and $\Delta_{\mathcal{Y}/S}$ is separated, then $f$ is separated. \item[(8)] If $\mathcal{X}$ is quasi-separated over $S$ and $\Delta_{\mathcal{Y}/S}$ is quasi-separated, then $f$ is quasi-separated. \end{enumerate}


stacks-morphisms-lemma-properties-covering-imply-diagonal

Let $\mathcal{X}$ be an algebraic stack. Let $W$ be an algebraic space, and let $f : W \to \mathcal{X}$ be a surjective, flat, locally finitely presented morphism. \begin{enumerate} \item If $f$ is unramified (i.e., \'etale, i.e., $\mathcal{X}$ is Deligne-Mumford), then $\mathcal{X}$ is DM. \item If $f$ is locally quasi-finite, then $\mathcal{X}$ is quasi-DM. \end{enumerate}


stacks-morphisms-lemma-monomorphism-separated

A monomorphism of algebraic stacks is separated and DM. The same is true for immersions of algebraic stacks.


stacks-morphisms-lemma-separation-properties-residual-gerbe

Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. Assume the residual gerbe $\mathcal{Z}_x$ of $\mathcal{X}$ at $x$ exists. If $\mathcal{X}$ is DM, resp.\ quasi-DM, resp.\ separated, resp.\ quasi-separated, then so is $\mathcal{Z}_x$.


stacks-morphisms-lemma-inertia

Let $\mathcal{X}$ be an algebraic stack. Then the inertia stack $\mathcal{I}_\mathcal{X}$ is an algebraic stack as well. The morphism $$ \mathcal{I}_\mathcal{X} \longrightarrow \mathcal{X} $$ is representable by algebraic spaces and locally of finite type. More generally, let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Then the relative inertia $\mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is an algebraic stack and the morphism $$ \mathcal{I}_{\mathcal{X}/\mathcal{Y}} \longrightarrow \mathcal{X} $$ is representable by algebraic spaces and locally of finite type.


stacks-morphisms-lemma-isom-pseudo-torsor-aut-over-space

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $Z$ be an algebraic space and let $x_i : Z \to \mathcal{X}$, $i = 1, 2$ be morphisms. Then \begin{enumerate} \item $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x_2, x_2)$ is a group algebraic space over $Z$, \item there is an exact sequence of groups $$ 0 \to \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x_2, x_2) \to \mathit{Isom}_\mathcal{X}(x_2, x_2) \to \mathit{Isom}_\mathcal{Y}(f \circ x_2, f \circ x_2) $$ \item there is a map of algebraic spaces $ \mathit{Isom}_\mathcal{X}(x_1, x_2) \to \mathit{Isom}_\mathcal{Y}(f \circ x_1, f \circ x_2) $ such that for any $2$-morphism $\alpha : f \circ x_1 \to f \circ x_2$ we obtain a cartesian diagram $$ \xymatrix{ \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha(x_1, x_2) \ar[d] \ar[r] & Z \ar[d]^\alpha \\ \mathit{Isom}_\mathcal{X}(x_1, x_2) \ar[r] & \mathit{Isom}_\mathcal{Y}(f \circ x_1, f \circ x_2) } $$ \item for any $2$-morphism $\alpha : f \circ x_1 \to f \circ x_2$ the algebraic space $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha(x_1, x_2)$ is a pseudo torsor for $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x_2, x_2)$ over $Z$. \end{enumerate}


stacks-morphisms-lemma-cartesian-square-inertia

Let $\pi : \mathcal{X} \to \mathcal{Y}$ and $f : \mathcal{Y}' \to \mathcal{Y}$ be morphisms of algebraic stacks. Set $\mathcal{X}' = \mathcal{X} \times_\mathcal{Y} \mathcal{Y}'$. Then both squares in the diagram $$ \xymatrix{ \mathcal{I}_{\mathcal{X}'/\mathcal{Y}'} \ar[r] \ar[d]_{ \text{Categories, Equation}\ (\ref{categories-equation-functorial}) } & \mathcal{X}' \ar[r]_{\pi'} \ar[d] & \mathcal{Y}' \ar[d]^f \\ \mathcal{I}_{\mathcal{X}/\mathcal{Y}} \ar[r] & \mathcal{X} \ar[r]^\pi & \mathcal{Y} } $$ are fibre product squares.


stacks-morphisms-lemma-monomorphism-cartesian-square-inertia

Let $f : \mathcal{X} \to \mathcal{Y}$ be a monomorphism of algebraic stacks. Then the diagram $$ \xymatrix{ \mathcal{I}_\mathcal{X} \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ \mathcal{I}_\mathcal{Y} \ar[r] & \mathcal{Y} } $$ is a fibre product square.


stacks-morphisms-lemma-presentation-inertia

Let $\mathcal{X}$ be an algebraic stack. Let $[U/R] \to \mathcal{X}$ be a presentation. Let $G/U$ be the stabilizer group algebraic space associated to the groupoid $(U, R, s, t, c)$. Then $$ \xymatrix{ G \ar[d] \ar[r] & U \ar[d] \\ \mathcal{I}_\mathcal{X} \ar[r] & \mathcal{X} } $$ is a fibre product diagram.


stacks-morphisms-lemma-diagonal-diagonal

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. \begin{enumerate} \item The following are equivalent \begin{enumerate} \item $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is separated, \item $\Delta_{f, 1} = \Delta_f : \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$ is separated, and \item $\Delta_{f, 2} = e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is a closed immersion. \end{enumerate} \item The following are equivalent \begin{enumerate} \item $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is quasi-separated, \item $\Delta_{f, 1} = \Delta_f : \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$ is quasi-separated, and \item $\Delta_{f, 2} = e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is a quasi-compact. \end{enumerate} \item The following are equivalent \begin{enumerate} \item $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is locally separated, \item $\Delta_{f, 1} = \Delta_f : \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$ is locally separated, and \item $\Delta_{f, 2} = e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is an immersion. \end{enumerate} \item The following are equivalent \begin{enumerate} \item $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is unramified, \item $f$ is DM. \end{enumerate} \item The following are equivalent \begin{enumerate} \item $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is locally quasi-finite, \item $f$ is quasi-DM. \end{enumerate} \end{enumerate}


stacks-morphisms-lemma-second-diagonal

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent: \begin{enumerate} \item the morphism $f$ is representable by algebraic spaces, \item the second diagonal of $f$ is an isomorphism, \item the group stack $ \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is trivial over $\mathcal X$, and \item for a scheme $T$ and a morphism $x : T \to \mathcal{X}$ the kernel of $\mathit{Isom}_\mathcal{X}(x, x) \to \mathit{Isom}_\mathcal{Y}(f(x), f(x))$ is trivial. \end{enumerate}


stacks-morphisms-lemma-hierarchy

A morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks is \begin{enumerate} \item a monomorphism if and only if $\Delta_{f, 1}$ is an isomorphism, and \item representable by algebraic spaces if and only if $\Delta_{f, 1}$ is a monomorphism. \end{enumerate} Moreover, the second diagonal $\Delta_{f, 2}$ is always a monomorphism.


stacks-morphisms-lemma-first-diagonal-separated-second-diagonal-closed

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Then \begin{enumerate} \item $\Delta_{f, 1}$ separated $\Leftrightarrow$ $\Delta_{f, 2}$ closed immersion $\Leftrightarrow$ $\Delta_{f, 2}$ proper $\Leftrightarrow$ $\Delta_{f, 2}$ universally closed, \item $\Delta_{f, 1}$ quasi-separated $\Leftrightarrow$ $\Delta_{f, 2}$ finite type $\Leftrightarrow$ $\Delta_{f, 2}$ quasi-compact, and \item $\Delta_{f, 1}$ locally separated $\Leftrightarrow$ $\Delta_{f, 2}$ immersion. \end{enumerate}


stacks-morphisms-lemma-separated-implies-isom

Let $f : \mathcal{X} \to \mathcal{Y}$ be a separated (resp.\ quasi-separated, resp.\ quasi-DM, resp.\ DM) morphism of algebraic stacks. Then \begin{enumerate} \item given algebraic spaces $T_i$, $i = 1, 2$ and morphisms $x_i : T_i \to \mathcal{X}$, with $y_i = f \circ x_i$ the morphism $$ T_1 \times_{x_1, \mathcal{X}, x_2} T_2 \longrightarrow T_1 \times_{y_1, \mathcal{Y}, y_2} T_2 $$ is proper (resp.\ quasi-compact and quasi-separated, resp.\ locally quasi-finite, resp.\ unramified), \item given an algebraic space $T$ and morphisms $x_i : T \to \mathcal{X}$, $i = 1, 2$, with $y_i = f \circ x_i$ the morphism $$ \mathit{Isom}_\mathcal{X}(x_1, x_2) \longrightarrow \mathit{Isom}_\mathcal{Y}(y_1, y_2) $$ is proper (resp.\ quasi-compact and quasi-separated, resp.\ locally quasi-finite, resp.\ unramified). \end{enumerate}


stacks-morphisms-lemma-base-change-quasi-compact

The base change of a quasi-compact morphism of algebraic stacks by any morphism of algebraic stacks is quasi-compact.


stacks-morphisms-lemma-composition-quasi-compact

The composition of a pair of quasi-compact morphisms of algebraic stacks is quasi-compact.


stacks-morphisms-lemma-closed-immersion-quasi-compact

A closed immersion of algebraic stacks is quasi-compact.


stacks-morphisms-lemma-surjection-from-quasi-compact

Let $$ \xymatrix{ \mathcal{X} \ar[rr]_f \ar[rd]_p & & \mathcal{Y} \ar[dl]^q \\ & \mathcal{Z} } $$ be a $2$-commutative diagram of morphisms of algebraic stacks. If $f$ is surjective and $p$ is quasi-compact, then $q$ is quasi-compact.


stacks-morphisms-lemma-quasi-compact-permanence

Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. If $g \circ f$ is quasi-compact and $g$ is quasi-separated then $f$ is quasi-compact.


stacks-morphisms-lemma-quasi-compact-quasi-separated-permanence

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. \begin{enumerate} \item If $\mathcal{X}$ is quasi-compact and $\mathcal{Y}$ is quasi-separated, then $f$ is quasi-compact. \item If $\mathcal{X}$ is quasi-compact and quasi-separated and $\mathcal{Y}$ is quasi-separated, then $f$ is quasi-compact and quasi-separated. \item A fibre product of quasi-compact and quasi-separated algebraic stacks is quasi-compact and quasi-separated. \end{enumerate}


stacks-morphisms-lemma-reach-points-scheme-theoretic-image

Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact morphism of algebraic stacks. Let $y \in |\mathcal{Y}|$ be a point in the closure of the image of $|f|$. There exists a valuation ring $A$ with fraction field $K$ and a commutative diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ \Spec(A) \ar[r] & \mathcal{Y} } $$ such that the closed point of $\Spec(A)$ maps to $y$.


stacks-morphisms-lemma-check-quasi-compact-covering

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $W \to \mathcal{Y}$ be surjective, flat, and locally of finite presentation where $W$ is an algebraic space. If the base change $W \times_\mathcal{Y} \mathcal{X} \to W$ is quasi-compact, then $f$ is quasi-compact.


stacks-morphisms-lemma-locally-closed-in-noetherian

Let $j : \mathcal{X} \to \mathcal{Y}$ be an immersion of algebraic stacks. \begin{enumerate} \item If $\mathcal{Y}$ is locally Noetherian, then $\mathcal{X}$ is locally Noetherian and $j$ is quasi-compact. \item If $\mathcal{Y}$ is Noetherian, then $\mathcal{X}$ is Noetherian. \end{enumerate}


stacks-morphisms-lemma-Noetherian-topology

Let $\mathcal{X}$ be an algebraic stack. \begin{enumerate} \item If $\mathcal{X}$ is locally Noetherian then $|\mathcal{X}|$ is a locally Noetherian topological space. \item If $\mathcal{X}$ is quasi-compact and locally Noetherian, then $|\mathcal{X}|$ is a Noetherian topological space. \end{enumerate}


stacks-morphisms-lemma-base-change-affine

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


stacks-morphisms-lemma-base-change-integral

Let $\mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $\mathcal{Z} \to \mathcal{Y}$ be an integral (or finite) morphism of algebraic stacks. Then $\mathcal{Z} \times_\mathcal{Y} \mathcal{X} \to \mathcal{X}$ is an integral (or finite) morphism of algebraic stacks.


stacks-morphisms-lemma-characterize-representable-universally-open

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent \begin{enumerate} \item $f$ is universally open (as in Properties of Stacks, Section \ref{stacks-properties-section-properties-morphisms}), and \item for every morphism of algebraic stacks $\mathcal{Z} \to \mathcal{Y}$ the morphism of topological spaces $|\mathcal{Z} \times_\mathcal{Y} \mathcal{X}| \to |\mathcal{Z}|$ is open. \end{enumerate}


stacks-morphisms-lemma-base-change-universally-open

The base change of a universally open morphism of algebraic stacks by any morphism of algebraic stacks is universally open.


stacks-morphisms-lemma-composition-universally-open

The composition of a pair of (universally) open morphisms of algebraic stacks is (universally) open.


stacks-morphisms-lemma-characterize-representable-universally-submersive

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent \begin{enumerate} \item $f$ is universally submersive (as in Properties of Stacks, Section \ref{stacks-properties-section-properties-morphisms}), and \item for every morphism of algebraic stacks $\mathcal{Z} \to \mathcal{Y}$ the morphism of topological spaces $|\mathcal{Z} \times_\mathcal{Y} \mathcal{X}| \to |\mathcal{Z}|$ is submersive. \end{enumerate}


stacks-morphisms-lemma-base-change-universally-submersive

The base change of a universally submersive morphism of algebraic stacks by any morphism of algebraic stacks is universally submersive.


stacks-morphisms-lemma-composition-universally-submersive

The composition of a pair of (universally) submersive morphisms of algebraic stacks is (universally) submersive.


stacks-morphisms-lemma-characterize-representable-universally-closed

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent \begin{enumerate} \item $f$ is universally closed (as in Properties of Stacks, Section \ref{stacks-properties-section-properties-morphisms}), and \item for every morphism of algebraic stacks $\mathcal{Z} \to \mathcal{Y}$ the morphism of topological spaces $|\mathcal{Z} \times_\mathcal{Y} \mathcal{X}| \to |\mathcal{Z}|$ is closed. \end{enumerate}


stacks-morphisms-lemma-base-change-universally-closed

The base change of a universally closed morphism of algebraic stacks by any morphism of algebraic stacks is universally closed.


stacks-morphisms-lemma-composition-universally-closed

The composition of a pair of (universally) closed morphisms of algebraic stacks is (universally) closed.


stacks-morphisms-lemma-universally-closed-local

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \begin{enumerate} \item $f$ is universally closed, \item for every scheme $Z$ and every morphism $Z \to \mathcal{Y}$ the projection $|Z \times_\mathcal{Y} \mathcal{X}| \to |Z|$ is closed, \item for every affine scheme $Z$ and every morphism $Z \to \mathcal{Y}$ the projection $|Z \times_\mathcal{Y} \mathcal{X}| \to |Z|$ is closed, and \item there exists an algebraic space $V$ and a surjective smooth morphism $V \to \mathcal{Y}$ such that $V \times_\mathcal{Y} \mathcal{X} \to V$ is a universally closed morphism of algebraic stacks. \end{enumerate}


stacks-morphisms-lemma-characterize-representable-universally-injective

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent \begin{enumerate} \item $f$ is universally injective (as in Properties of Stacks, Section \ref{stacks-properties-section-properties-morphisms}), and \item for every morphism of algebraic stacks $\mathcal{Z} \to \mathcal{Y}$ the map $|\mathcal{Z} \times_\mathcal{Y} \mathcal{X}| \to |\mathcal{Z}|$ is injective. \end{enumerate}


stacks-morphisms-lemma-base-change-universally-injective

The base change of a universally injective morphism of algebraic stacks by any morphism of algebraic stacks is universally injective.


stacks-morphisms-lemma-composition-universally-injective

The composition of a pair of universally injective morphisms of algebraic stacks is universally injective.


stacks-morphisms-lemma-universally-injective

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \begin{enumerate} \item $f$ is universally injective, \item $\Delta : \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$ is surjective, and \item for an algebraically closed field, for $x_1, x_2 : \Spec(k) \to \mathcal{X}$, and for a $2$-arrow $\beta : f \circ x_1 \to f \circ x_2$ there is a $2$-arrow $\alpha : x_1 \to x_2$ with $\beta = \text{id}_f \star \alpha$. \end{enumerate}


stacks-morphisms-lemma-universally-injective-point

Let $f : \mathcal{X} \to \mathcal{Y}$ be a universally injective morphism of algebraic stacks. Let $y : \Spec(k) \to \mathcal{Y}$ be a morphism where $k$ is an algebraically closed field. If $y$ is in the image of $|\mathcal{X}| \to |\mathcal{Y}|$, then there is a morphism $x : \Spec(k) \to \mathcal{X}$ with $y = f \circ x$.


stacks-morphisms-lemma-universally-injective-local

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent: \begin{enumerate} \item $f$ is universally injective, \item for every affine scheme $Z$ and any morphism $Z \to \mathcal{Y}$ the morphism $Z \times_\mathcal{Y} \mathcal{X} \to Z$ is universally injective, and \item add more here. \end{enumerate}


stacks-morphisms-lemma-check-universally-injective-covering

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $W \to \mathcal{Y}$ be surjective, flat, and locally of finite presentation where $W$ is an algebraic space. If the base change $W \times_\mathcal{Y} \mathcal{X} \to W$ is universally injective, then $f$ is universally injective.


stacks-morphisms-lemma-characterize-representable-universal-homeomorphism

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent \begin{enumerate} \item $f$ is a universal homeomorphism (Properties of Stacks, Section \ref{stacks-properties-section-properties-morphisms}), and \item for every morphism of algebraic stacks $\mathcal{Z} \to \mathcal{Y}$ the map of topological spaces $|\mathcal{Z} \times_\mathcal{Y} \mathcal{X}| \to |\mathcal{Z}|$ is a homeomorphism. \end{enumerate}


stacks-morphisms-lemma-base-change-universal-homeomorphism

The base change of a universal homeomorphism of algebraic stacks by any morphism of algebraic stacks is a universal homeomorphism.


stacks-morphisms-lemma-composition-universal-homeomorphism

The composition of a pair of universal homeomorphisms of algebraic stacks is a universal homeomorphism.


stacks-morphisms-lemma-check-universal-homeomorphism-covering

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $W \to \mathcal{Y}$ be surjective, flat, and locally of finite presentation where $W$ is an algebraic space. If the base change $W \times_\mathcal{Y} \mathcal{X} \to W$ is a universal homeomorphism, then $f$ is a universal homeomorphism.


stacks-morphisms-lemma-local-source-target

Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is smooth local on the source-and-target. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Consider commutative diagrams $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ \mathcal{X} \ar[r]^f & \mathcal{Y} } $$ where $U$ and $V$ are algebraic spaces and the vertical arrows are smooth. The following are equivalent \begin{enumerate} \item for any diagram as above such that in addition $U \to \mathcal{X} \times_\mathcal{Y} V$ is smooth the morphism $h$ has property $\mathcal{P}$, and \item for some diagram as above with $a : U \to \mathcal{X}$ surjective the morphism $h$ has property $\mathcal{P}$. \end{enumerate} If $\mathcal{X}$ and $\mathcal{Y}$ are representable by algebraic spaces, then this is also equivalent to $f$ (as a morphism of algebraic spaces) having property $\mathcal{P}$. If $\mathcal{P}$ is also preserved under any base change, and fppf local on the base, then for morphisms $f$ which are representable by algebraic spaces this is also equivalent to $f$ having property $\mathcal{P}$ in the sense of Properties of Stacks, Section \ref{stacks-properties-section-properties-morphisms}.


stacks-morphisms-lemma-composition-finite-type

The composition of finite type morphisms is of finite type. The same holds for locally of finite type.


stacks-morphisms-lemma-base-change-finite-type

A base change of a finite type morphism is finite type. The same holds for locally of finite type.


stacks-morphisms-lemma-immersion-locally-finite-type

An immersion is locally of finite type.


stacks-morphisms-lemma-locally-finite-type-locally-noetherian

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. If $f$ is locally of finite type and $\mathcal{Y}$ is locally Noetherian, then $\mathcal{X}$ is locally Noetherian.


stacks-morphisms-lemma-check-finite-type-covering

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $W \to \mathcal{Y}$ be a surjective, flat, and locally of finite presentation where $W$ is an algebraic space. If the base change $W \times_\mathcal{Y} \mathcal{X} \to W$ is locally of finite type, then $f$ is locally of finite type.


stacks-morphisms-lemma-check-finite-type-precompose

Let $\mathcal{X} \to \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. Assume $\mathcal{X} \to \mathcal{Z}$ is locally of finite type and that $\mathcal{X} \to \mathcal{Y}$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then $\mathcal{Y} \to \mathcal{Z}$ is locally of finite type.


stacks-morphisms-lemma-finite-type-permanence

Let $f : \mathcal{X} \to \mathcal{Y}$, $g : \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. If $g \circ f : \mathcal{X} \to \mathcal{Z}$ is locally of finite type, then $f : \mathcal{X} \to \mathcal{Y}$ is locally of finite type.


stacks-morphisms-lemma-point-finite-type

Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. The following are equivalent: \begin{enumerate} \item There exists a morphism $\Spec(k) \to \mathcal{X}$ which is locally of finite type and represents $x$. \item There exists a scheme $U$, a closed point $u \in U$, and a smooth morphism $\varphi : U \to \mathcal{X}$ such that $\varphi(u) = x$. \end{enumerate}


stacks-morphisms-lemma-identify-finite-type-points

Let $\mathcal{X}$ be an algebraic stack. We have $$ \mathcal{X}_{\text{ft-pts}} = \bigcup\nolimits_{\varphi : U \to \mathcal{X}\text{ smooth}} |\varphi|(U_0) $$ where $U_0$ is the set of closed points of $U$. Here we may let $U$ range over all schemes smooth over $\mathcal{X}$ or over all affine schemes smooth over $\mathcal{X}$.


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

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


stacks-morphisms-lemma-enough-finite-type-points

Let $\mathcal{X}$ be an algebraic stack. For any locally closed subset $T \subset |\mathcal{X}|$ we have $$ T \not = \emptyset \Rightarrow T \cap \mathcal{X}_{\text{ft-pts}} \not = \emptyset. $$ In particular, for any closed subset $T \subset |\mathcal{X}|$ we see that $T \cap \mathcal{X}_{\text{ft-pts}}$ is dense in $T$.


stacks-morphisms-lemma-point-finite-type-monomorphism

Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. The following are equivalent: \begin{enumerate} \item $x$ is a finite type point, \item there exists an algebraic stack $\mathcal{Z}$ whose underlying topological space $|\mathcal{Z}|$ is a singleton, and a morphism $f : \mathcal{Z} \to \mathcal{X}$ which is locally of finite type such that $\{x\} = |f|(|\mathcal{Z}|)$, and \item the residual gerbe $\mathcal{Z}_x$ of $\mathcal{X}$ at $x$ exists and the inclusion morphism $\mathcal{Z}_x \to \mathcal{X}$ is locally of finite type. \end{enumerate}


stacks-morphisms-lemma-automorphism-group-scheme

In the situation above $G_x$ is a scheme if one of the following holds \begin{enumerate} \item $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is quasi-separated \item $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is locally separated, \item $\mathcal{X}$ is quasi-DM, \item $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is quasi-separated, \item $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is locally separated, or \item $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is locally quasi-finite. \end{enumerate}


stacks-morphisms-lemma-property-automorphism-groups

Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$ be a point. Let $P$ be a property of algebraic spaces over fields which is invariant under ground field extensions; for example $P(X/k) = X \to \Spec(k)\text{ is finite}$. The following are equivalent \begin{enumerate} \item for some morphism $x : \Spec(k) \to \mathcal{X}$ in the class of $x$ the automorphism group algebraic space $G_x/k$ has $P$, and \item for any morphism $x : \Spec(k) \to \mathcal{X}$ in the class of $x$ the automorphism group algebraic space $G_x/k$ has $P$. \end{enumerate}


stacks-morphisms-lemma-iso-automorphism-groups

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $x \in |\mathcal{X}|$ be a point. The following are equivalent \begin{enumerate} \item for some morphism $x : \Spec(k) \to \mathcal{X}$ in the class of $x$ setting $y = f \circ x$ the map $G_x \to G_y$ of automorphism group algebraic spaces is an isomorphism, and \item for any morphism $x : \Spec(k) \to \mathcal{X}$ in the class of $x$ setting $y = f \circ x$ the map $G_x \to G_y$ of automorphism group algebraic spaces is an isomorphism. \end{enumerate}


stacks-morphisms-lemma-properties-diagonal-from-presentation

Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces such that $s, t : R \to U$ are flat and locally of finite presentation. Consider the algebraic stack $\mathcal{X} = [U/R]$ (see above). \begin{enumerate} \item If $R \to U \times U$ is separated, then $\Delta_\mathcal{X}$ is separated. \item If $U$, $R$ are separated, then $\Delta_\mathcal{X}$ is separated. \item If $R \to U \times U$ is locally quasi-finite, then $\mathcal{X}$ is quasi-DM. \item If $s, t : R \to U$ are locally quasi-finite, then $\mathcal{X}$ is quasi-DM. \item If $R \to U \times U$ is proper, then $\mathcal{X}$ is separated. \item If $s, t : R \to U$ are proper and $U$ is separated, then $\mathcal{X}$ is separated. \item Add more here. \end{enumerate}


stacks-morphisms-lemma-points-presentation

Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces such that $s, t : R \to U$ are flat and locally of finite presentation. Consider the algebraic stack $\mathcal{X} = [U/R]$ (see above). Then the image of $|R| \to |U| \times |U|$ is an equivalence relation and $|\mathcal{X}|$ is the quotient of $|U|$ by this equivalence relation.


stacks-morphisms-lemma-slice

Let $\mathcal{X}$ be an algebraic stack. Consider a cartesian diagram $$ \xymatrix{ U \ar[d] & F \ar[l]^p \ar[d] \\ \mathcal{X} & \Spec(k) \ar[l] } $$ where $U$ is an algebraic space, $k$ is a field, and $U \to \mathcal{X}$ is flat and locally of finite presentation. Let $f_1, \ldots, f_r \in \Gamma(U, \mathcal{O}_U)$ and $z \in |F|$ such that $f_1, \ldots, f_r$ map to a regular sequence in the local ring $\mathcal{O}_{F, \overline{z}}$. Then, after replacing $U$ by an open subspace containing $p(z)$, the morphism $$ V(f_1, \ldots, f_r) \longrightarrow \mathcal{X} $$ is flat and locally of finite presentation.


stacks-morphisms-lemma-quasi-finite-at-point

Let $\mathcal{X}$ be an algebraic stack. Consider a cartesian diagram $$ \xymatrix{ U \ar[d] & F \ar[l]^p \ar[d] \\ \mathcal{X} & \Spec(k) \ar[l] } $$ where $U$ is an algebraic space, $k$ is a field, and $U \to \mathcal{X}$ is locally of finite type. Let $z \in |F|$ be such that $\dim_z(F) = 0$. Then, after replacing $U$ by an open subspace containing $p(z)$, the morphism $$ U \longrightarrow \mathcal{X} $$ is locally quasi-finite.


stacks-morphisms-lemma-DM-residual-gerbe

Let $\mathcal{Z}$ be a DM, locally Noetherian, reduced algebraic stack with $|\mathcal{Z}|$ a singleton. Then there exists a field $k$ and a surjective \'etale morphism $\Spec(k) \to \mathcal{Z}$.


stacks-morphisms-lemma-etale-at-point

Let $\mathcal{X}$ be an algebraic stack. Consider a cartesian diagram $$ \xymatrix{ U \ar[d] & F \ar[l]^p \ar[d] \\ \mathcal{X} & \Spec(k) \ar[l] } $$ where $U$ is an algebraic space, $k$ is a field, and $U \to \mathcal{X}$ is flat and locally of finite presentation. Let $z \in |F|$ be such that $F \to \Spec(k)$ is unramified at $z$. Then, after replacing $U$ by an open subspace containing $p(z)$, the morphism $$ U \longrightarrow \mathcal{X} $$ is \'etale.


stacks-morphisms-lemma-DM

Let $f : \mathcal{X} \to \mathcal{Y}$ be a DM morphism of algebraic stacks. Then \begin{enumerate} \item For every DM algebraic stack $\mathcal{Z}$ and morphism $\mathcal{Z} \to \mathcal{Y}$ there exists a scheme and a surjective \'etale morphism $U \to \mathcal{X} \times_\mathcal{Y} \mathcal{Z}$. \item For every algebraic space $Z$ and morphism $Z \to \mathcal{Y}$ there exists a scheme and a surjective \'etale morphism $U \to \mathcal{X} \times_\mathcal{Y} Z$. \end{enumerate}


stacks-morphisms-lemma-open-DM-locus

Let $\mathcal{X}$ be an algebraic stack. There exist open substacks $$ \mathcal{X}'' \subset \mathcal{X}' \subset \mathcal{X} $$ such that $\mathcal{X}''$ is DM, $\mathcal{X}'$ is quasi-DM, and such that these are the largest open substacks with these properties.


stacks-morphisms-lemma-points-DM-locus

Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$ correspond to $x : \Spec(k) \to \mathcal{X}$. Let $G_x/k$ be the automorphism group algebraic space of $x$. Then \begin{enumerate} \item  $x$ is in the DM locus of $\mathcal{X}$ if and only if $G_x \to \Spec(k)$ is unramified, and \item $x$ is in the quasi-DM locus of $\mathcal{X}$ if and only if $G_x \to \Spec(k)$ is locally quasi-finite. \end{enumerate}


stacks-morphisms-lemma-base-change-locally-quasi-finite

A base change of a locally quasi-finite morphism is locally quasi-finite.


stacks-morphisms-lemma-locally-quasi-finite-over-field

Let $\mathcal{X} \to \Spec(k)$ be a locally quasi-finite morphism where $\mathcal{X}$ is an algebraic stack and $k$ is a field. Let $f : V \to \mathcal{X}$ be a locally quasi-finite morphism where $V$ is a scheme. Then $V \to \Spec(k)$ is locally quasi-finite.


stacks-morphisms-lemma-composition-locally-quasi-finite

A composition of a locally quasi-finite morphisms is locally quasi-finite.


stacks-morphisms-lemma-characterize-quasi-DM

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


stacks-morphisms-lemma-locally-quasi-finite-permanence

Let $\mathcal{X} \to \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. Assume that $\mathcal{X} \to \mathcal{Z}$ is locally quasi-finite and $\mathcal{Y} \to \mathcal{Z}$ is quasi-DM. Then $\mathcal{X} \to \mathcal{Y}$ is locally quasi-finite.


stacks-morphisms-lemma-composition-flat

The composition of flat morphisms is flat.


stacks-morphisms-lemma-base-change-flat

A base change of a flat morphism is flat.


stacks-morphisms-lemma-descent-flat

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


stacks-morphisms-lemma-flat-permanence

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


stacks-morphisms-lemma-lift-valuation-ring-through-flat-morphism

Let $f : \mathcal{X} \to \mathcal{Y}$ be a flat morphism of algebraic stacks. Let $\Spec(A) \to \mathcal{Y}$ be a morphism where $A$ is a valuation ring. If the closed point of $\Spec(A)$ maps to a point of $|\mathcal{Y}|$ in the image of $|\mathcal{X|} \to |\mathcal{Y}|$, then there exists a commutative diagram $$ \xymatrix{ \Spec(A') \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ \Spec(A) \ar[r] & \mathcal{Y} } $$ where $A \to A'$ is an extension of valuation rings (More on Algebra, Definition \ref{more-algebra-definition-extension-valuation-rings}).


stacks-morphisms-lemma-flat-at-point

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $x \in |\mathcal{X}|$. Consider commutative diagrams $$ \vcenter{ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ \mathcal{X} \ar[r]^f & \mathcal{Y} } } \quad\text{with points} \vcenter{ \xymatrix{ u \in |U| \ar[d] \\ x \in |\mathcal{X}| } } $$ where $U$ and $V$ are algebraic spaces, $b$ is flat, and $(a, h) : U \to \mathcal{X} \times_\mathcal{Y} V$ is flat. The following are equivalent \begin{enumerate} \item $h$ is flat at $u$ for one diagram as above, \item $h$ is flat at $u$ for every diagram as above. \end{enumerate}


stacks-morphisms-lemma-composition-finite-presentation

The composition of finitely presented morphisms is of finite presentation. The same holds for morphisms which are locally of finite presentation.


stacks-morphisms-lemma-base-change-finite-presentation

A base change of a finitely presented morphism is of finite presentation. The same holds for morphisms which are locally of finite presentation.


stacks-morphisms-lemma-noetherian-finite-type-finite-presentation

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. \begin{enumerate} \item If $\mathcal{Y}$ is locally Noetherian and $f$ locally of finite type then $f$ is locally of finite presentation. \item If $\mathcal{Y}$ is locally Noetherian and $f$ of finite type and quasi-separated then $f$ is of finite presentation. \end{enumerate}


stacks-morphisms-lemma-finite-presentation-permanence

Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks If $g \circ f$ is locally of finite presentation and $g$ is locally of finite type, then $f$ is locally of finite presentation.


stacks-morphisms-lemma-diagonal-morphism-finite-type

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks with diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$. If $f$ is locally of finite type then $\Delta$ is locally of finite presentation. If $f$ is quasi-separated and locally of finite type, then $\Delta$ is of finite presentation.


stacks-morphisms-lemma-check-property-after-fppf-base-change

Let $P$ be a property of morphisms of algebraic spaces which is fppf local on the target and preserved by arbitrary base change. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Let $\mathcal{Z} \to \mathcal{Y}$ be a morphism of algebraic stacks which is surjective, flat, and locally of finite presentation. Set $\mathcal{W} = \mathcal{Z} \times_\mathcal{Y} \mathcal{X}$. Then $$ (f\text{ has }P) \Leftrightarrow (\text{the projection }\mathcal{W} \to \mathcal{Z}\text{ has }P). $$ For the meaning of this statement see Properties of Stacks, Section \ref{stacks-properties-section-properties-morphisms}.


stacks-morphisms-lemma-descent-property

Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is smooth local on the source-and-target and fppf local on the target. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $\mathcal{Z} \to \mathcal{Y}$ be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change $\mathcal{Z} \times_\mathcal{Y} \mathcal{X} \to \mathcal{Z}$ has $\mathcal{P}$, then $f$ has $\mathcal{P}$.


stacks-morphisms-lemma-descent-finite-presentation

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $\mathcal{Z} \to \mathcal{Y}$ be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change $\mathcal{Z} \times_\mathcal{Y} \mathcal{X} \to \mathcal{Z}$ is locally of finite presentation, then $f$ is locally of finite presentation.


stacks-morphisms-lemma-flat-finite-presentation-permanence

Let $\mathcal{X} \to \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. If $\mathcal{X} \to \mathcal{Z}$ is locally of finite presentation and $\mathcal{X} \to \mathcal{Y}$ is surjective, flat, and locally of finite presentation, then $\mathcal{Y} \to \mathcal{Z}$ is locally of finite presentation.


stacks-morphisms-lemma-surjective-flat-locally-finite-presentation

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is surjective, flat, and locally of finite presentation. Then for every scheme $U$ and object $y$ of $\mathcal{Y}$ over $U$ there exists an fppf covering $\{U_i \to U\}$ and objects $x_i$ of $\mathcal{X}$ over $U_i$ such that $f(x_i) \cong y|_{U_i}$ in $\mathcal{Y}_{U_i}$.


stacks-morphisms-lemma-surjective-family-flat-locally-finite-presentation

Let $f_j : \mathcal{X}_j \to \mathcal{X}$, $j \in J$ be a family of morphisms of algebraic stacks which are each flat and locally of finite presentation and which are jointly surjective, i.e., $|\mathcal{X}| = \bigcup |f_j|(|\mathcal{X}_j|)$. Then for every scheme $U$ and object $x$ of $\mathcal{X}$ over $U$ there exists an fppf covering $\{U_i \to U\}_{i \in I}$, a map $a : I \to J$, and objects $x_i$ of $\mathcal{X}_{a(i)}$ over $U_i$ such that $f_{a(i)}(x_i) \cong y|_{U_i}$ in $\mathcal{X}_{U_i}$.


stacks-morphisms-lemma-fppf-open

Let $f : \mathcal{X} \to \mathcal{Y}$ be flat and locally of finite presentation. Then $|f| : |\mathcal{X}| \to |\mathcal{Y}|$ is open.


stacks-morphisms-lemma-descent-quasi-compact

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $\mathcal{Z} \to \mathcal{Y}$ be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change $\mathcal{Z} \times_\mathcal{Y} \mathcal{X} \to \mathcal{Z}$ is quasi-compact, then $f$ is quasi-compact.


stacks-morphisms-lemma-check-separated-on-ui-cover

Let $f : \mathcal{X} \to \mathcal{Y}$, $g : \mathcal{Y} \to \mathcal{Z}$ be composable morphisms of algebraic stacks with composition $h = g \circ f : \mathcal{X} \to \mathcal{Z}$. If $f$ is surjective, flat, locally of finite presentation, and universally injective and if $h$ is separated, then $g$ is separated.


stacks-morphisms-lemma-gerbe-over-iso-classes

Let $\mathcal{X}$ be an algebraic stack. If $\mathcal{X}$ is a gerbe, then the sheafification of the presheaf $$ (\Sch/S)_{fppf}^{opp} \to \textit{Sets}, \quad U \mapsto \Ob(\mathcal{X}_U)/\!\!\cong $$ is an algebraic space and $\mathcal{X}$ is a gerbe over it.


stacks-morphisms-lemma-base-change-gerbe

Let $$ \xymatrix{ \mathcal{X}' \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ \mathcal{Y}' \ar[r] & \mathcal{Y} } $$ be a fibre product of algebraic stacks. If $\mathcal{X}$ is a gerbe over $\mathcal{Y}$, then $\mathcal{X}'$ is a gerbe over $\mathcal{Y}'$.


stacks-morphisms-lemma-gerbe-descent

Let $$ \xymatrix{ \mathcal{X}' \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ \mathcal{Y}' \ar[r] & \mathcal{Y} } $$ be a fibre product of algebraic stacks. If $\mathcal{Y}' \to \mathcal{Y}$ is surjective, flat, and locally of finite presentation and $\mathcal{X}'$ is a gerbe over $\mathcal{Y}'$, then $\mathcal{X}$ is a gerbe over $\mathcal{Y}$.


stacks-morphisms-lemma-local-structure-gerbe

Let $\pi : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \begin{enumerate} \item $\mathcal{X}$ is a gerbe over $\mathcal{Y}$, and \item there exists an algebraic space $U$, a group algebraic space $G$ flat and locally of finite presentation over $U$, and a surjective, flat, and locally finitely presented morphism $U \to \mathcal{Y}$ such that $\mathcal{X} \times_\mathcal{Y} U \cong [U/G]$ over $U$. \end{enumerate}


stacks-morphisms-lemma-gerbe-fppf

Let $\pi : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. If $\mathcal{X}$ is a gerbe over $\mathcal{Y}$, then $\pi$ is surjective, flat, and locally of finite presentation.


stacks-morphisms-lemma-gerbe-isom-fppf

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which makes $\mathcal{X}$ a gerbe over $\mathcal{Y}$. Then \begin{enumerate} \item $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is flat and locally of finite presentation, \item $\mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$ is surjective, flat, and locally of finite presentation, \item given algebraic spaces $T_i$, $i = 1, 2$ and morphisms $x_i : T_i \to \mathcal{X}$, with $y_i = f \circ x_i$ the morphism $$ T_1 \times_{x_1, \mathcal{X}, x_2} T_2 \longrightarrow T_1 \times_{y_1, \mathcal{Y}, y_2} T_2 $$ is surjective, flat, and locally of finite presentation, \item given an algebraic space $T$ and morphisms $x_i : T \to \mathcal{X}$, $i = 1, 2$, with $y_i = f \circ x_i$ the morphism $$ \mathit{Isom}_\mathcal{X}(x_1, x_2) \longrightarrow \mathit{Isom}_\mathcal{Y}(y_1, y_2) $$ is surjective, flat, and locally of finite presentation. \end{enumerate}


stacks-morphisms-lemma-gerbe-bijection-points

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. If $\mathcal{X}$ is a gerbe over $\mathcal{Y}$ then $f$ is a universal homeomorphism.


stacks-morphisms-lemma-gerbe-diagonal-quasi-compact

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks such that $\mathcal{X}$ is a gerbe over $\mathcal{Y}$. If $\Delta_\mathcal{X}$ is quasi-compact, so is $\Delta_\mathcal{Y}$.


stacks-morphisms-lemma-gerbe-residual-gerbe-exists

Let $\mathcal{X}$ be an algebraic stack. If $\mathcal{X}$ is a gerbe then for every $x \in |\mathcal{X}|$ the residual gerbe of $\mathcal{X}$ at $x$ exists.


stacks-morphisms-lemma-every-point-in-a-stratum

Let $\mathcal{X}$ be an algebraic stack such that $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is quasi-compact. Then there exists a well-ordered index set $I$ and for every $i \in I$ a reduced locally closed substack $\mathcal{U}_i \subset \mathcal{X}$ such that \begin{enumerate} \item each $\mathcal{U}_i$ is a gerbe, \item we have $|\mathcal{X}| = \bigcup_{i \in I} |\mathcal{U}_i|$, \item $T_i = |\mathcal{X}| \setminus \bigcup_{i' < i} |\mathcal{U}_{i'}|$ is closed in $|\mathcal{X}|$ for all $i \in I$, and \item $|\mathcal{U}_i|$ is open in $T_i$. \end{enumerate} We can moreover arrange it so that either (a) $|\mathcal{U}_i| \subset T_i$ is dense, or (b) $\mathcal{U}_i$ is quasi-compact. In case (a), if we choose $\mathcal{U}_i$ as large as possible (see proof for details), then the stratification is canonical.


stacks-morphisms-lemma-spectral-qc-diagonal-qc

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


stacks-morphisms-lemma-every-point-residual-gerbe

Let $\mathcal{X}$ be an algebraic stack such that $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is quasi-compact. Then the residual gerbe of $\mathcal{X}$ at $x$ exists for every $x \in |\mathcal{X}|$.


stacks-morphisms-lemma-every-point-residual-gerbe-quasi-DM

Let $\mathcal{X}$ be a quasi-DM algebraic stack. Then the residual gerbe of $\mathcal{X}$ at $x$ exists for every $x \in |\mathcal{X}|$.


stacks-morphisms-lemma-quotient-etale

Let $Y$ be an algebraic space. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $Y$. Assume $U \to Y$ is flat and locally of finite presentation and $R \to U \times_Y U$ an open immersion. Then $X = [U/R] = U/R$ is an algebraic space and $X \to Y$ is \'etale.


stacks-morphisms-lemma-quasi-splitting-etale

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Let $P \subset R$ be an open subspace such that $(U, P, s|_P, t|_P, c|_{P \times_{s, U, t} P})$ is a groupoid in algebraic spaces over $S$. Then $$ [U/P] \longrightarrow [U/R] $$ is a morphism of algebraic stacks which is representable by algebraic spaces, surjective, and \'etale.


stacks-morphisms-lemma-etale-local-quasi-DM

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


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

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


stacks-morphisms-lemma-composition-smooth

The composition of smooth morphisms is smooth.


stacks-morphisms-lemma-base-change-smooth

A base change of a smooth morphism is smooth.


stacks-morphisms-lemma-descent-smooth

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $\mathcal{Z} \to \mathcal{Y}$ be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change $\mathcal{Z} \times_\mathcal{Y} \mathcal{X} \to \mathcal{Z}$ is smooth, then $f$ is smooth.


stacks-morphisms-lemma-smooth-locally-finite-presentation

A smooth morphism of algebraic stacks is locally of finite presentation.


stacks-morphisms-lemma-where-smooth

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. There is a maximal open substack $\mathcal{U} \subset \mathcal{X}$ such that $f|_\mathcal{U} : \mathcal{U} \to \mathcal{Y}$ is smooth. Moreover, formation of this open commutes with \begin{enumerate} \item precomposing by smooth morphisms, \item base change by morphisms which are flat and locally of finite presentation, \item base change by flat morphisms provided $f$ is locally of finite presentation. \end{enumerate}


stacks-morphisms-lemma-smooth-quotient-stack

Let $X \to Y$ be a smooth morphism of algebraic spaces. Let $G$ be a group algebraic space over $Y$ which is flat and locally of finite presentation over $Y$. Let $G$ act on $X$ over $Y$. Then the quotient stack $[X/G]$ is smooth over $Y$.


stacks-morphisms-lemma-gerbe-smooth

Let $\pi : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. If $\mathcal{X}$ is a gerbe over $\mathcal{Y}$, then $\pi$ is surjective and smooth.


stacks-morphisms-lemma-etale-smooth-local-source-target

Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is \'etale-smooth local on the source-and-target. Let $f : \mathcal{X} \to \mathcal{Y}$ be a DM morphism of algebraic stacks. Consider commutative diagrams $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ \mathcal{X} \ar[r]^f & \mathcal{Y} } $$ where $U$ and $V$ are algebraic spaces, $V \to \mathcal{Y}$ is smooth, and $U \to \mathcal{X} \times_\mathcal{Y} V$ is \'etale. The following are equivalent \begin{enumerate} \item for any diagram as above the morphism $h$ has property $\mathcal{P}$, and \item for some diagram as above with $a : U \to \mathcal{X}$ surjective the morphism $h$ has property $\mathcal{P}$. \end{enumerate} If $\mathcal{X}$ and $\mathcal{Y}$ are representable by algebraic spaces, then this is also equivalent to $f$ (as a morphism of algebraic spaces) having property $\mathcal{P}$. If $\mathcal{P}$ is also preserved under any base change, and fppf local on the base, then for morphisms $f$ which are representable by algebraic spaces this is also equivalent to $f$ having property $\mathcal{P}$ in the sense of Properties of Stacks, Section \ref{stacks-properties-section-properties-morphisms}.


stacks-morphisms-lemma-base-change-etale

A base change of an \'etale morphism is \'etale.


stacks-morphisms-lemma-etale-permanence

Let $\mathcal{X}, \mathcal{Y}$ be algebraic stacks \'etale over an algebraic stack $\mathcal{Z}$. Any morphism $\mathcal{X} \to \mathcal{Y}$ over $\mathcal{Z}$ is \'etale.


stacks-morphisms-lemma-characterize-unramified

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \begin{enumerate} \item $f$ is unramified, and \item $f$ is locally of finite type and its diagonal is \'etale. \end{enumerate}


stacks-morphisms-lemma-universally-closed-permanence

Consider a commutative diagram $$ \xymatrix{ \mathcal{X} \ar[rr] \ar[rd] & & \mathcal{Y} \ar[ld] \\ & \mathcal{Z} & } $$ of algebraic stacks. \begin{enumerate} \item If $\mathcal{X} \to \mathcal{Z}$ is universally closed and $\mathcal{Y} \to \mathcal{Z}$ is separated, then the morphism $\mathcal{X} \to \mathcal{Y}$ is universally closed. In particular, the image of $|\mathcal{X}|$ in $|\mathcal{Y}|$ is closed. \item If $\mathcal{X} \to \mathcal{Z}$ is proper and $\mathcal{Y} \to \mathcal{Z}$ is separated, then the morphism $\mathcal{X} \to \mathcal{Y}$ is proper. \end{enumerate}


stacks-morphisms-lemma-image-proper-is-proper

Let $\mathcal{Z}$ be an algebraic stack. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks over $\mathcal{Z}$. If $\mathcal{X}$ is universally closed over $\mathcal{Z}$ and $f$ is surjective then $\mathcal{Y}$ is universally closed over $\mathcal{Z}$. In particular, if also $\mathcal{Y}$ is separated and of finite type over $\mathcal{Z}$, then $\mathcal{Y}$ is proper over $\mathcal{Z}$.


stacks-morphisms-lemma-cover-upstairs

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $g : \mathcal{W} \to \mathcal{X}$ be a morphism of algebraic stacks which is surjective, flat, and locally of finite presentation. Then the scheme theoretic image of $f$ exists if and only if the scheme theoretic image of $f \circ g$ exists and if so then these scheme theoretic images are the same.


stacks-morphisms-lemma-scheme-theoretic-image-existence

Let $f : \mathcal{Y} \to \mathcal{X}$ be a morphism of algebraic stacks. Then the scheme theoretic image of $f$ exists.


stacks-morphisms-lemma-factor-factor

Let $$ \xymatrix{ \mathcal{X}_1 \ar[d] \ar[r]_{f_1} & \mathcal{Y}_1 \ar[d] \\ \mathcal{X}_2 \ar[r]^{f_2} & \mathcal{Y}_2 } $$ be a commutative diagram of algebraic stacks. Let $\mathcal{Z}_i \subset \mathcal{Y}_i$, $i = 1, 2$ be the scheme theoretic image of $f_i$. Then the morphism $\mathcal{Y}_1 \to \mathcal{Y}_2$ induces a morphism $\mathcal{Z}_1 \to \mathcal{Z}_2$ and a commutative diagram $$ \xymatrix{ \mathcal{X}_1 \ar[r] \ar[d] & \mathcal{Z}_1 \ar[d] \ar[r] & \mathcal{Y}_1 \ar[d] \\ \mathcal{X}_2 \ar[r] & \mathcal{Z}_2 \ar[r] & \mathcal{Y}_2 } $$


stacks-morphisms-lemma-existence-plus-flat-base-change

Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact morphism of algebraic stacks. Then formation of the scheme theoretic image commutes with flat base change.


stacks-morphisms-lemma-topology-scheme-theoretic-image

Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact morphism of algebraic stacks. Let $\mathcal{Z} \subset \mathcal{Y}$ be the scheme theoretic image of $f$. Then $|\mathcal{Z}|$ is the closure of the image of $|f|$.


stacks-morphisms-lemma-cat-dotted-arrows

In the situation of Definition \ref{definition-fill-in-diagram} the category of dotted arrows is a groupoid. If $\Delta_f$ is separated, then it is a setoid.


stacks-morphisms-lemma-cat-dotted-arrows-independent

In Definition \ref{definition-fill-in-diagram} assume $\mathcal{I}_\mathcal{Y} \to \mathcal{Y}$ is proper (for example if $\mathcal{Y}$ is separated or if $\mathcal{Y}$ is separated over an algebraic space). Then the category of dotted arrows is independent (up to noncanonical equivalence) of the choice of $\gamma$ and the existence of a dotted arrow (for some and hence equivalently all $\gamma$) is equivalent to the existence of a diagram $$ \xymatrix{ \Spec(K) \ar[r]_-x \ar[d]_j & \mathcal{X} \ar[d]^f \\ \Spec(A) \ar[r]^-y \ar[ru]_a & \mathcal{Y} } $$ with $2$-commutative triangles (without checking the $2$-morphisms compose correctly).


stacks-morphisms-lemma-cat-dotted-arrows-base-change

Assume given a $2$-commutative diagram $$ \xymatrix{ \Spec(K) \ar[r]_-{x'} \ar[d]_j & \mathcal{X}' \ar[d]^p \ar[r]_q & \mathcal{X} \ar[d]^f \\ \Spec(A) \ar[r]^-{y'} & \mathcal{Y}' \ar[r]^g & \mathcal{Y} } $$ with the right square $2$-cartesian. Choose a $2$-arrow $\gamma' : y' \circ j \to p \circ x'$. Set $x = q \circ x'$, $y = g \circ y'$ and let $\gamma : y \circ j \to f \circ x$ be the composition of $\gamma'$ with the $2$-arrow implicit in the $2$-commutativity of the right square. Then the category of dotted arrows for the left square and $\gamma'$ is equivalent to the category of dotted arrows for the outer rectangle and $\gamma$.


stacks-morphisms-lemma-cat-dotted-arrows-composition

Assume given a $2$-commutative diagram $$ \xymatrix{ \Spec(K) \ar[r]_-x \ar[dd]_j & \mathcal{X} \ar[d]^f \\ & \mathcal{Y} \ar[d]^g \\ \Spec(A) \ar[r]^-z & \mathcal{Z} } $$ Choose a $2$-arrow $\gamma : z \circ j \to g \circ f \circ x$. Let $\mathcal{C}$ be the category of dotted arrows for the outer rectangle and $\gamma$. Let $\mathcal{C}'$ be the category of dotted arrows for the square $$ \xymatrix{ \Spec(K) \ar[r]_-{f \circ x} \ar[d]_j & \mathcal{Y} \ar[d]^g \\ \Spec(A) \ar[r]^-z & \mathcal{Z} } $$ and $\gamma$. There is a canonical functor $\mathcal{C} \to \mathcal{C}'$ which turns $\mathcal{C}$ into a category fibred in groupoids over $\mathcal{C}'$ and whose fibre categories are categories of dotted arrows for certain squares of the form $$ \xymatrix{ \Spec(K) \ar[r]_-x \ar[d]_j & \mathcal{X} \ar[d]^f \\ \Spec(A) \ar[r]^-y & \mathcal{Y} } $$ and some choice of $y \circ j \to f \circ x$.


stacks-morphisms-lemma-uniqueness-representable

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. Then the following are equivalent \begin{enumerate} \item $f$ satisfies the uniqueness part of the valuative criterion, \item for every scheme $T$ and morphism $T \to \mathcal{Y}$ the morphism $\mathcal{X} \times_\mathcal{Y} T \to T$ satisfies the uniqueness part of the valuative criterion as a morphism of algebraic spaces. \end{enumerate}


stacks-morphisms-lemma-base-change-existence

The base change of a morphism of algebraic stacks which satisfies the existence part of the valuative criterion by any morphism of algebraic stacks is a morphism of algebraic stacks which satisfies the existence part of the valuative criterion.


stacks-morphisms-lemma-composition-existence

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


stacks-morphisms-lemma-existence-representable

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. Then the following are equivalent \begin{enumerate} \item $f$ satisfies the existence part of the valuative criterion, \item for every scheme $T$ and morphism $T \to \mathcal{Y}$ the morphism $\mathcal{X} \times_\mathcal{Y} T \to T$ satisfies the existence part of the valuative criterion as a morphism of algebraic spaces. \end{enumerate}


stacks-morphisms-lemma-closed-immersion-valuative-criteria

A closed immersion of algebraic stacks satisfies both the existence and uniqueness part of the valuative criterion.


stacks-morphisms-lemma-setoids-and-diagonal

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. If $\Delta_f$ is quasi-separated and if for every diagram (\ref{equation-diagram}) and choice of $\gamma$ as in Definition \ref{definition-fill-in-diagram} the category of dotted arrows is a setoid, then $\Delta_f$ is separated.


stacks-morphisms-lemma-helper-diagonal

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Consider a $2$-commutative solid diagram $$ \xymatrix{ \Spec(K) \ar[rr]_-x \ar[d]_j & & \mathcal{X} \ar[d]^{\Delta_f} \\ \Spec(A) \ar[rr]^{(a_1, a_2, \varphi)} \ar@{..>}[rru] & & \mathcal{X} \times_\mathcal{Y} \mathcal{X} } $$ where $A$ is a valuation ring with field of fractions $K$. Let $\gamma : (a_1, a_2, \varphi) \circ j \longrightarrow \Delta_f \circ x$ be a $2$-morphism witnessing the $2$-commutativity of the diagram. Then \begin{enumerate} \item Writing $\gamma = (\alpha_1, \alpha_2)$ with $\alpha_i : a_i \circ j \to x$ we obtain two dotted arrows $(a_1, \alpha_1, \text{id})$ and $(a_2, \alpha_2, \varphi)$ in the diagram $$ \xymatrix{ \Spec(K) \ar[r]_-x \ar[d]_j & \mathcal{X} \ar[d]^f \\ \Spec(A) \ar[r]^-{f \circ a_1} \ar@{..>}[ru] & \mathcal{Y} } $$ \item The category of dotted arrows for the original diagram and $\gamma$ is a setoid whose set of isomorphism classes of objects equal to the set of morphisms $(a_1, \alpha_1, \text{id}) \to (a_2, \alpha_2, \varphi)$ in the category of dotted arrows. \end{enumerate}


stacks-morphisms-lemma-uniqueness-and-diagonal

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ is quasi-separated. If $f$ satisfies the uniqueness part of the valuative criterion, then $f$ is separated.


stacks-morphisms-lemma-converse-uniqueness-and-diagonal

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. If $f$ is separated, then $f$ satisfies the uniqueness part of the valuative criterion.


stacks-morphisms-lemma-quasi-compact-existence-universally-closed

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume \begin{enumerate} \item $f$ is quasi-compact, and \item $f$ satisfies the existence part of the valuative criterion. \end{enumerate} Then $f$ is universally closed.


stacks-morphisms-lemma-converse-existence-universally-closed

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume \begin{enumerate} \item $f$ is quasi-separated, and \item $f$ is universally closed. \end{enumerate} Then $f$ satisfies the existence part of the valuative criterion.


stacks-morphisms-lemma-criterion-proper

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ is of finite type and quasi-separated. Then the following are equivalent \begin{enumerate} \item $f$ is proper, and \item $f$ satisfies both the uniqueness and existence parts of the valuative criterion. \end{enumerate}


stacks-morphisms-lemma-lci-permanence

Let $$ \xymatrix{ \mathcal{X} \ar[rr]_f \ar[rd] & & \mathcal{Y} \ar[ld] \\ & \mathcal{Z} } $$ be a commutative diagram of morphisms of algebraic stacks. Assume $\mathcal{Y} \to \mathcal{Z}$ is smooth and $\mathcal{X} \to \mathcal{Z}$ is a local complete intersection morphism. Then $f : \mathcal{X} \to \mathcal{Y}$ is a local complete intersection morphism.


stacks-morphisms-lemma-stabilizer-preserving

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. If $\mathcal{I}_\mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{I}_\mathcal{Y}$ is an isomorphism, then $f$ is representable by algebraic spaces.


stacks-morphisms-lemma-aut-iso-unramified

Let $f : \mathcal{X} \to \mathcal{Y}$ be an unramified morphism of algebraic stacks. The following are equivalent \begin{enumerate} \item $\mathcal{I}_\mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{I}_\mathcal{Y}$ is an isomorphism, and \item $f$ induces an isomorphism between automorphism groups at $x$ and $f(x)$ (Remark \ref{remark-identify-automorphism-groups}) for all $x \in |\mathcal{X}|$. \end{enumerate}


stacks-morphisms-lemma-stabilizer-preserving-unramified

\begin{reference} \cite[Proposition 3.5]{rydh_quotients} and \cite[Proposition 2.5]{alper_quotient} \end{reference} Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume \begin{enumerate} \item $f$ is representable by algebraic spaces and unramified, and \item $\mathcal{I}_\mathcal{Y} \to \mathcal{Y}$ is proper. \end{enumerate} Then the set of $x \in |\mathcal{X}|$ such that $f$ induces an isomorphism between automorphism groups at $x$ and $f(x)$ (Remark \ref{remark-identify-automorphism-groups}) is open. Letting $\mathcal{U} \subset \mathcal{X}$ be the corresponding open substack, the morphism $\mathcal{I}_\mathcal{U} \to \mathcal{U} \times_\mathcal{Y} \mathcal{I}_\mathcal{Y}$ is an isomorphism.


stacks-morphisms-lemma-base-change-stabilizer-preserving

Let $$ \xymatrix{ \mathcal{X}' \ar[r] \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{Y}' \ar[r] & \mathcal{Y} } $$ be a cartesian diagram of algebraic stacks. \begin{enumerate} \item Let $x' \in |\mathcal{X}'|$ with image $x \in |\mathcal{X}|$. If $f$ induces an isomorphism between automorphism groups at $x$ and $f(x)$ (Remark \ref{remark-identify-automorphism-groups}), then $f'$ induces an isomorphism between automorphism groups at $x'$ and $f(x')$. \item If $\mathcal{I}_\mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{I}_\mathcal{Y}$ is an isomorphism, then $\mathcal{I}_{\mathcal{X}'} \to \mathcal{X}' \times_{\mathcal{Y}'} \mathcal{I}_{\mathcal{Y}'}$ is an isomorphism. \end{enumerate}


stacks-morphisms-lemma-stabilizer-preserving-points-cartesian

Let $$ \xymatrix{ \mathcal{X}' \ar[r] \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{Y}' \ar[r]^g & \mathcal{Y} } $$ be a cartesian diagram of algebraic stacks. If $f$ induces an isomorphism between automorphism groups at points (Remark \ref{remark-identify-automorphism-groups}), then $$ \Mor(\Spec(k), \mathcal{X}') \longrightarrow \Mor(\Spec(k), \mathcal{Y}') \times \Mor(\Spec(k), \mathcal{X}) $$ is injective on isomorphism classes for any field $k$.


stacks-morphisms-lemma-etale-iso

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ is separated, \'etale, $f$ induces an isomorphism between automorphism groups at points (Remark \ref{remark-identify-automorphism-groups}) and for every algebraically closed field $k$ the functor $$ f : \Mor(\Spec(k), \mathcal{X}) \longrightarrow \Mor(\Spec(k), \mathcal{Y}) $$ is an equivalence. Then $f$ is an isomorphism.


stacks-morphisms-proposition-when-gerbe

Let $\mathcal{X}$ be an algebraic stack. The following are equivalent \begin{enumerate} \item $\mathcal{X}$ is a gerbe, and \item $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is flat and locally of finite presentation. \end{enumerate}


stacks-morphisms-proposition-when-gerbe-over

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \begin{enumerate} \item $\mathcal{X}$ is a gerbe over $\mathcal{Y}$, and \item $f : \mathcal{X} \to \mathcal{Y}$ and $\Delta : \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$ are surjective, flat, and locally of finite presentation. \end{enumerate}


stacks-morphisms-proposition-open-stratum

Let $\mathcal{X}$ be a reduced algebraic stack such that $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is quasi-compact. Then there exists a dense open substack $\mathcal{U} \subset \mathcal{X}$ which is a gerbe.


schemes-lemma-isomorphism-locally-ringed

\begin{slogan} An isomorphism of ringed spaces between locally ringed spaces is an isomorphism of locally ringed spaces. \end{slogan} Let $X$, $Y$ be locally ringed spaces. If $f : X \to Y$ is an isomorphism of ringed spaces, then $f$ is an isomorphism of locally ringed spaces.


schemes-lemma-open-immersion

Let $f : X \to Y$ be an open immersion of locally ringed spaces. Let $j : V = f(X) \to Y$ be the open subspace of $Y$ associated to the image of $f$. There is a unique isomorphism $f' : X \cong V$ of locally ringed spaces such that $f = j \circ f'$.


schemes-lemma-restrict-map-to-opens

Let $f : X \to Y$ be a morphism of locally ringed spaces. Let $U \subset X$, and $V \subset Y$ be open subsets. Suppose that $f(U) \subset V$. There exists a unique morphism of locally ringed spaces $f|_U : U \to V$ such that the following diagram is a commutative square of locally ringed spaces $$ \xymatrix{ U \ar[d]_{f|_U} \ar[r] & X \ar[d]^f \\ V \ar[r] & Y } $$


schemes-lemma-closed-local-target

Let $f : Z \to X$ be a morphism of locally ringed spaces. In order for $f$ to be a closed immersion it suffices that there exists an open covering $X = \bigcup U_i$ such that each $f : f^{-1}U_i \to U_i$ is a closed immersion.


schemes-lemma-closed-immersion

Let $f : X \to Y$ be a closed immersion of locally ringed spaces. Let $\mathcal{I}$ be the kernel of the map $\mathcal{O}_Y \to f_*\mathcal{O}_X$. Let $i : Z \to Y$ be the closed subspace of $Y$ associated to $\mathcal{I}$. There is a unique isomorphism $f' : X \cong Z$ of locally ringed spaces such that $f = i \circ f'$.


schemes-lemma-characterize-closed-subspace

Let $X$, $Y$ be locally ringed spaces. Let $\mathcal{I} \subset \mathcal{O}_X$ be a sheaf of ideals locally generated by sections. Let $i : Z \to X$ be the associated closed subspace. A morphism $f : Y \to X$ factors through $Z$ if and only if the map $f^*\mathcal{I} \to f^*\mathcal{O}_X = \mathcal{O}_Y$ is zero. If this is the case the morphism $g : Y \to Z$ such that $f = i \circ g$ is unique.


schemes-lemma-restrict-map-to-closed

Let $f : X \to Y$ be a morphism of locally ringed spaces. Let $\mathcal{I} \subset \mathcal{O}_Y$ be a sheaf of ideals which is locally generated by sections. Let $i : Z \to Y$ be the closed subspace associated to the sheaf of ideals $\mathcal{I}$. Let $\mathcal{J}$ be the image of the map $f^*\mathcal{I} \to f^*\mathcal{O}_Y = \mathcal{O}_X$. Then this ideal is locally generated by sections. Moreover, let $i' : Z' \to X$ be the associated closed subspace of $X$. There exists a unique morphism of locally ringed spaces $f' : Z' \to Z$ such that the following diagram is a commutative square of locally ringed spaces $$ \xymatrix{ Z' \ar[d]_{f'} \ar[r]_{i'} & X \ar[d]^f \\ Z \ar[r]^{i} & Y } $$ Moreover, this diagram is a fibre square in the category of locally ringed spaces.


schemes-lemma-standard-open

Let $R$ be a ring. Let $f \in R$. \begin{enumerate} \item If $g\in R$ and $D(g) \subset D(f)$, then \begin{enumerate} \item $f$ is invertible in $R_g$, \item $g^e = af$ for some $e \geq 1$ and $a \in R$, \item there is a canonical ring map $R_f \to R_g$, and \item there is a canonical $R_f$-module map $M_f \to M_g$ for any $R$-module $M$. \end{enumerate} \item Any open covering of $D(f)$ can be refined to a finite open covering of the form $D(f) = \bigcup_{i = 1}^n D(g_i)$. \item If $g_1, \ldots, g_n \in R$, then $D(f) \subset \bigcup D(g_i)$ if and only if $g_1, \ldots, g_n$ generate the unit ideal in $R_f$. \end{enumerate}


schemes-lemma-spec-sheaves

Let $R$ be a ring. Let $M$ be an $R$-module. Let $\widetilde M$ be the sheaf of $\mathcal{O}_{\Spec(R)}$-modules associated to $M$. \begin{enumerate} \item We have $\Gamma(\Spec(R), \mathcal{O}_{\Spec(R)}) = R$. \item We have $\Gamma(\Spec(R), \widetilde M) = M$ as an $R$-module. \item For every $f \in R$ we have $\Gamma(D(f), \mathcal{O}_{\Spec(R)}) = R_f$. \item For every $f\in R$ we have $\Gamma(D(f), \widetilde M) = M_f$ as an $R_f$-module. \item Whenever $D(g) \subset D(f)$ the restriction mappings on $\mathcal{O}_{\Spec(R)}$ and $\widetilde M$ are the maps $R_f \to R_g$ and $M_f \to M_g$ from Lemma \ref{lemma-standard-open}. \item Let $\mathfrak p$ be a prime of $R$, and let $x \in \Spec(R)$ be the corresponding point. We have $\mathcal{O}_{\Spec(R), x} = R_{\mathfrak p}$. \item Let $\mathfrak p$ be a prime of $R$, and let $x \in \Spec(R)$ be the corresponding point. We have $\widetilde M_x = M_{\mathfrak p}$ as an $R_{\mathfrak p}$-module. \end{enumerate} Moreover, all these identifications are functorial in the $R$ module $M$. In particular, the functor $M \mapsto \widetilde M$ is an exact functor from the category of $R$-modules to the category of $\mathcal{O}_{\Spec(R)}$-modules.


schemes-lemma-morphism-into-affine-where-point-goes

Let $X$ be a locally ringed space. Let $Y$ be an affine scheme. Let $f \in \Mor(X, Y)$ be a morphism of locally ringed spaces. Given a point $x \in X$ consider the ring maps $$ \Gamma(Y, \mathcal{O}_Y) \xrightarrow{f^\sharp} \Gamma(X, \mathcal{O}_X) \to \mathcal{O}_{X, x} $$ Let $\mathfrak p \subset \Gamma(Y, \mathcal{O}_Y)$ denote the inverse image of $\mathfrak m_x$. Let $y \in Y$ be the corresponding point. Then $f(x) = y$.


schemes-lemma-f-open

Let $X$ be a locally ringed space. Let $f \in \Gamma(X, \mathcal{O}_X)$. The set $$ D(f) = \{x \in X \mid \text{image }f \not\in \mathfrak m_x\} $$ is open. Moreover $f|_{D(f)}$ has an inverse.


schemes-lemma-f-open-affine

In Lemma \ref{lemma-f-open} above, if $X$ is an affine scheme, then the open $D(f)$ agrees with the standard open $D(f)$ defined previously (in Algebra, Definition \ref{algebra-definition-spectrum-ring}).


schemes-lemma-morphism-into-affine

\begin{reference} A reference for this fact is \cite[II, Err 1, Prop. 1.8.1]{EGA} where it is attributed to J. Tate. \end{reference} Let $X$ be a locally ringed space. Let $Y$ be an affine scheme. The map $$ \Mor(X, Y) \longrightarrow \Hom(\Gamma(Y, \mathcal{O}_Y), \Gamma(X, \mathcal{O}_X)) $$ which maps $f$ to $f^\sharp$ (on global sections) is bijective.


schemes-lemma-category-affine-schemes

The category of affine schemes is equivalent to the opposite of the category of rings. The equivalence is given by the functor that associates to an affine scheme the global sections of its structure sheaf.


schemes-lemma-standard-open-affine

Let $Y$ be an affine scheme. Let $f \in \Gamma(Y, \mathcal{O}_Y)$. The open subspace $D(f)$ is an affine scheme.


schemes-lemma-fibre-product-affine-schemes

The category of affine schemes has finite products, and fibre products. In other words, it has finite limits. Moreover, the products and fibre products in the category of affine schemes are the same as in the category of locally ringed spaces. In a formula, we have (in the category of locally ringed spaces) $$ \Spec(R) \times \Spec(S) = \Spec(R \otimes_{\mathbf{Z}} S) $$ and given ring maps $R \to A$, $R \to B$ we have $$ \Spec(A) \times_{\Spec(R)} \Spec(B) = \Spec(A \otimes_R B). $$


schemes-lemma-disjoint-union-affines

Let $X$ be a locally ringed space. Assume $X = U \amalg V$ with $U$ and $V$ open and such that $U$, $V$ are affine schemes. Then $X$ is an affine scheme.


schemes-lemma-compare-constructions

Let $(X, \mathcal{O}_X) = (\Spec(R), \mathcal{O}_{\Spec(R)})$ be an affine scheme. Let $M$ be an $R$-module. There exists a canonical isomorphism between the sheaf $\widetilde M$ associated to the $R$-module $M$ (Definition \ref{definition-structure-sheaf}) and the sheaf $\mathcal{F}_M$ associated to the $R$-module $M$ (Modules, Definition \ref{modules-definition-sheaf-associated}). This isomorphism is functorial in $M$. In particular, the sheaves $\widetilde M$ are quasi-coherent. Moreover, they are characterized by the following mapping property $$ \Hom_{\mathcal{O}_X}(\widetilde M, \mathcal{F}) = \Hom_R(M, \Gamma(X, \mathcal{F})) $$ for any sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$. Here a map $\alpha : \widetilde M \to \mathcal{F}$ corresponds to its effect on global sections.


schemes-lemma-widetilde-pullback

Let $(X, \mathcal{O}_X) = (\Spec(S), \mathcal{O}_{\Spec(S)})$, $(Y, \mathcal{O}_Y) = (\Spec(R), \mathcal{O}_{\Spec(R)})$ be affine schemes. Let $\psi : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of affine schemes, corresponding to the ring map $\psi^\sharp : R \to S$ (see Lemma \ref{lemma-category-affine-schemes}). \begin{enumerate} \item We have $\psi^* \widetilde M = \widetilde{S \otimes_R M}$ functorially in the $R$-module $M$. \item We have $\psi_* \widetilde N = \widetilde{N_R}$ functorially in the $S$-module $N$. \end{enumerate}


schemes-lemma-quasi-coherent-affine

Let $(X, \mathcal{O}_X) = (\Spec(R), \mathcal{O}_{\Spec(R)})$ be an affine scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\mathcal{F}$ is isomorphic to the sheaf associated to the $R$-module $\Gamma(X, \mathcal{F})$.


schemes-lemma-equivalence-quasi-coherent

Let $(X, \mathcal{O}_X) = (\Spec(R), \mathcal{O}_{\Spec(R)})$ be an affine scheme. The functors $M \mapsto \widetilde M$ and $\mathcal{F} \mapsto \Gamma(X, \mathcal{F})$ define quasi-inverse equivalences of categories $$ \xymatrix{ \QCoh(\mathcal{O}_X) \ar@<1ex>[r] & \text{Mod-}R \ar@<1ex>[l] } $$ between the category of quasi-coherent $\mathcal{O}_X$-modules and the category of $R$-modules.


schemes-lemma-extension-quasi-coherent

Let $(X, \mathcal{O}_X) = (\Spec(R), \mathcal{O}_{\Spec(R)})$ be an affine scheme. Suppose that $$ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 $$ is a short exact sequence of sheaves of $\mathcal{O}_X$-modules. If two out of three are quasi-coherent then so is the third.


schemes-lemma-closed-immersion-affine-case

\begin{slogan} For affine schemes, closed immersions correspond to ideals. \end{slogan} Let $(X, \mathcal{O}_X) = (\Spec(R), \mathcal{O}_{\Spec(R)})$ be an affine scheme. Let $i : Z \to X$ be any closed immersion of locally ringed spaces. Then there exists a unique ideal $I \subset R$ such that the morphism $i : Z \to X$ can be identified with the closed immersion $\Spec(R/I) \to \Spec(R)$ constructed in Example \ref{example-closed-immersion-affines} above.


schemes-lemma-open-subspace-scheme

Let $X$ be a scheme. Let $j : U \to X$ be an open immersion of locally ringed spaces. Then $U$ is a scheme. In particular, any open subspace of $X$ is a scheme.


schemes-lemma-closed-subspace-scheme

Let $X$ be a scheme. Let $i : Z \to X$ be a closed immersion of locally ringed spaces. \begin{enumerate} \item The locally ringed space $Z$ is a scheme, \item the kernel $\mathcal{I}$ of the map $\mathcal{O}_X \to i_*\mathcal{O}_Z$ is a quasi-coherent sheaf of ideals, \item for any affine open $U = \Spec(R)$ of $X$ the morphism $i^{-1}(U) \to U$ can be identified with $\Spec(R/I) \to \Spec(R)$ for some ideal $I \subset R$, and \item we have $\mathcal{I}|_U = \widetilde I$. \end{enumerate} In particular, any sheaf of ideals locally generated by sections is a quasi-coherent sheaf of ideals (and vice versa), and any closed subspace of $X$ is a scheme.


schemes-lemma-immersion-when-closed

Let $f : Y \to X$ be an immersion of schemes. Then $f$ is a closed immersion if and only if $f(Y) \subset X$ is a closed subset.


schemes-lemma-scheme-sober

Let $X$ be a scheme. Any irreducible closed subset of $X$ has a unique generic point. In other words, $X$ is a sober topological space, see Topology, Definition \ref{topology-definition-generic-point}.


schemes-lemma-basis-affine-opens

Let $X$ be a scheme. The collection of affine opens of $X$ forms a basis for the topology on $X$.


schemes-lemma-standard-open-two-affines

Let $X$ be a scheme. Let $U, V$ be affine opens of $X$, and let $x \in U \cap V$. There exists an affine open neighbourhood $W$ of $x$ such that $W$ is a standard open of both $U$ and $V$.


schemes-lemma-good-subcover

Let $X$ be a scheme. Let $X = \bigcup_i U_i$ be an affine open covering. Let $V \subset X$ be an affine open. There exists a standard open covering $V = \bigcup_{j = 1, \ldots, m} V_j$ (see Definition \ref{definition-standard-covering}) such that each $V_j$ is a standard open in one of the $U_i$.


schemes-lemma-sheaf-on-affines

Let $X$ be a scheme. Let $\mathcal{B}$ be the set of affine opens of $X$. Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{B}$, see Sheaves, Definition \ref{sheaves-definition-presheaf-basis}. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is the restriction of a sheaf on $X$ to $\mathcal{B}$, \item $\mathcal{F}$ is a sheaf on $\mathcal{B}$, and \item $\mathcal{F}(\emptyset)$ is a singleton and whenever $U = V \cup W$ with $U, V, W \in \mathcal{B}$ and $V, W \subset U$ standard open (Algebra, Definition \ref{algebra-definition-Zariski-topology}) the map $$ \mathcal{F}(U) \longrightarrow \mathcal{F}(V) \times \mathcal{F}(W) $$ is injective with image the set of pairs $(s, t)$ such that $s|_{V \cap W} = t|_{V \cap W}$. \end{enumerate}