stacks-perfect-lemma-shriek-derived

Let $\mathcal{X}$ be an algebraic stack. Notation as in Cohomology of Stacks, Lemmas \ref{stacks-cohomology-lemma-lisse-etale} and \ref{stacks-cohomology-lemma-lisse-etale-modules}. \begin{enumerate} \item The functor $g_! : \textit{Ab}(\mathcal{X}_{lisse,\etale}) \to \textit{Ab}(\mathcal{X}_\etale)$ has a left derived functor $$ Lg_! : D(\mathcal{X}_{lisse,\etale}) \longrightarrow D(\mathcal{X}_\etale) $$ which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \text{id}$. \item The functor $g_! :  \textit{Mod}(\mathcal{X}_{lisse,\etale}, \mathcal{O}_{\mathcal{X}_{lisse,\etale}}) \to \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_{\mathcal{X}})$ has a left derived functor $$ Lg_! : D(\mathcal{O}_{\mathcal{X}_{lisse,\etale}}) \longrightarrow D(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) $$ which is left adjoint to $g^*$ and such that $g^*Lg_! = \text{id}$. \item The functor $g_! : \textit{Ab}(\mathcal{X}_{flat,fppf}) \to \textit{Ab}(\mathcal{X}_{fppf})$ has a left derived functor $$ Lg_! : D(\mathcal{X}_{flat, fppf}) \longrightarrow D(\mathcal{X}_{fppf}) $$ which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \text{id}$. \item The functor $g_! : \textit{Mod}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \to \textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_{\mathcal{X}})$ has a left derived functor $$ Lg_! : D(\mathcal{O}_{\mathcal{X}_{flat, fppf}}) \longrightarrow D(\mathcal{O}_\mathcal{X}) $$ which is left adjoint to $g^*$ and such that $g^*Lg_! = \text{id}$. \end{enumerate} Warning: It is not clear (a priori) that $Lg_!$ on modules agrees with $Lg_!$ on abelian sheaves, see Cohomology on Sites, Remark \ref{sites-cohomology-remark-when-derived-shriek-equal}.


stacks-perfect-lemma-lisse-etale-functorial-derived

With assumptions and notation as in Cohomology of Stacks, Lemma \ref{stacks-cohomology-lemma-lisse-etale-functorial}. We have $$ g^{-1} \circ Rf_* = Rf'_* \circ (g')^{-1} \quad\text{and}\quad L(g')_! \circ (f')^{-1} = f^{-1} \circ Lg_! $$ on unbounded derived categories (both for the case of modules and for the case of abelian sheaves).


stacks-perfect-lemma-higher-shriek-quasi-coherent

Let $\mathcal{X}$ be an algebraic stack. Notation as in Cohomology of Stacks, Lemma \ref{stacks-cohomology-lemma-lisse-etale}. \begin{enumerate} \item Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_{\mathcal{X}_{lisse,\etale}}$-module  on the lisse-\'etale site of $\mathcal{X}$. For all $p \in \mathbf{Z}$ the sheaf $H^p(Lg_!\mathcal{H})$ is a locally quasi-coherent module with the flat base change property on $\mathcal{X}$. \item Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_{\mathcal{X}_{flat,fppf}}$-module  on the flat-fppf site of $\mathcal{X}$. For all $p \in \mathbf{Z}$ the sheaf $H^p(Lg_!\mathcal{H})$ is a locally quasi-coherent module with the flat base change property on $\mathcal{X}$. \end{enumerate}


stacks-perfect-lemma-compare-etale-fppf-QCoh

Let $\mathcal{X}$ be an algebraic stack. The comparison morphism $\epsilon : \mathcal{X}_{fppf} \to \mathcal{X}_\etale$ induces a commutative diagram $$ \xymatrix{ D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \ar[r] & D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \ar[r] & D(\mathcal{O}_\mathcal{X}) \\ D_{\mathcal{P}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) \ar[r] \ar[u]^{\epsilon^*} & D_{\mathcal{M}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) \ar[r] \ar[u]^{\epsilon^*} & D(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) \ar[u]^{\epsilon^*} } $$ Moreover, the left two vertical arrows are equivalences of triangulated categories, hence we also obtain an equivalence $$ D_{\mathcal{M}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) / D_{\mathcal{P}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) \longrightarrow D_\QCoh(\mathcal{O}_\mathcal{X}) $$


stacks-perfect-lemma-derived-quasi-coherent

Let $\mathcal{X}$ be an algebraic stack. \begin{enumerate} \item Let $\mathcal{F}^\bullet$ be an object of $D_{\mathcal{M}_\mathcal{X}}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$. With $g$ as in Cohomology of Stacks, Lemma \ref{stacks-cohomology-lemma-lisse-etale} for the lisse-\'etale site we have \begin{enumerate} \item $g^{-1}\mathcal{F}^\bullet$ is in $D_\QCoh(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$, \item $g^{-1}\mathcal{F}^\bullet = 0$ if and only if $\mathcal{F}^\bullet$ is in $D_{\mathcal{P}_\mathcal{X}}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$, \item $Lg_!\mathcal{H}^\bullet$ is in $D_{\mathcal{M}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ for $\mathcal{H}^\bullet$ in $D_\QCoh(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$, and \item the functors $g^{-1}$ and $Lg_!$ define mutually inverse functors $$ \xymatrix{ D_\QCoh(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} & D_\QCoh(\mathcal{O}_{\mathcal{X}_{lisse,\etale}}) \ar@<1ex>[l]^-{Lg_!} } $$ \end{enumerate} \item Let $\mathcal{F}^\bullet$ be an object of $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$. With $g$ as in Cohomology of Stacks, Lemma \ref{stacks-cohomology-lemma-lisse-etale} for the flat-fppf site we have \begin{enumerate} \item $g^{-1}\mathcal{F}^\bullet$ is in $D_\QCoh(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$, \item $g^{-1}\mathcal{F}^\bullet = 0$ if and only if $\mathcal{F}^\bullet$ is in $D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$, \item $Lg_!\mathcal{H}^\bullet$ is in $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ for $\mathcal{H}^\bullet$ in $D_\QCoh(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$, and \item the functors $g^{-1}$ and $Lg_!$ define mutually inverse functors $$ \xymatrix{ D_\QCoh(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} & D_\QCoh(\mathcal{O}_{\mathcal{X}_{flat,fppf}}) \ar@<1ex>[l]^-{Lg_!} } $$ \end{enumerate} \end{enumerate}


stacks-perfect-lemma-bousfield-colocalization

Let $\mathcal{X}$ be an algebraic stack. Let $E$ be an object of $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$. There exists a canonical distinguished triangle $$ E' \to E \to P \to E'[1] $$ in $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ such that $P$ is in $D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ and $$ \Hom_{D(\mathcal{O}_\mathcal{X})}(E', P') = 0 $$ for all $P'$ in $D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$.


stacks-perfect-proposition-derived-direct-image-quasi-coherent

Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor $Rf_*$ induces a commutative diagram $$ \xymatrix{ D^{+}_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \ar[r] \ar[d]^{Rf_*} & D^{+}_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \ar[r] \ar[d]^{Rf_*} & D(\mathcal{O}_\mathcal{X}) \ar[d]^{Rf_*} \\ D^{+}_{\mathcal{P}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y}) \ar[r] & D^{+}_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y}) \ar[r] & D(\mathcal{O}_\mathcal{Y}) } $$ and hence induces a functor $$ Rf_{\QCoh, *} : D^{+}_\QCoh(\mathcal{O}_\mathcal{X}) \longrightarrow D^{+}_\QCoh(\mathcal{O}_\mathcal{Y}) $$ on quotient categories. Moreover, the functor $R^if_\QCoh$ of Cohomology of Stacks, Proposition \ref{stacks-cohomology-proposition-direct-image-quasi-coherent} are equal to $H^i \circ Rf_{\QCoh, *}$ with $H^i$ as in (\ref{equation-Hi-quasi-coherent}).


spaces-more-morphisms-theorem-topological-invariance

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is integral, universally injective and surjective. The functor $$ V \longmapsto V_X = X \times_Y V $$ defines an equivalence of categories $Y_{spaces, \etale} \to X_{spaces, \etale}$.


spaces-more-morphisms-theorem-openness-flatness

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Assume $f$ is locally of finite presentation and that $\mathcal{F}$ is an $\mathcal{O}_X$-module which is locally of finite presentation. Then $$ \{x \in |X| : \mathcal{F}\text{ is flat over }Y\text{ at }x\} $$ is open in $|X|$.


spaces-more-morphisms-theorem-criterion-flatness-fibre

Let $S$ be a scheme. Let $f : X \to Y$ and $Y \to Z$ be morphisms of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume \begin{enumerate} \item $X$ is locally of finite presentation over $Z$, \item $\mathcal{F}$ an $\mathcal{O}_X$-module of finite presentation, and \item $Y$ is locally of finite type over $Z$. \end{enumerate} Let $x \in |X|$ and let $y \in |Y|$ and $z \in |Z|$ be the images of $x$. If $\mathcal{F}_{\overline{x}} \not = 0$, then the following are equivalent: \begin{enumerate} \item $\mathcal{F}$ is flat over $Z$ at $x$ and the restriction of $\mathcal{F}$ to its fibre over $z$ is flat at $x$ over the fibre of $Y$ over $z$, and \item $Y$ is flat over $Z$ at $y$ and $\mathcal{F}$ is flat over $Y$ at $x$. \end{enumerate} Moreover, the set of points $x$ where (1) and (2) hold is open in $\text{Supp}(\mathcal{F})$.


spaces-more-morphisms-theorem-criterion-flatness-fibre-Noetherian

Let $S$ be a scheme. Let $f : X \to Y$ and $Y \to Z$ be morphisms of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume \begin{enumerate} \item $X$, $Y$, $Z$ locally Noetherian, and \item $\mathcal{F}$ a coherent $\mathcal{O}_X$-module. \end{enumerate} Let $x \in |X|$ and let $y \in |Y|$ and $z \in |Z|$ be the images of $x$. If $\mathcal{F}_{\overline{x}} \not = 0$, then the following are equivalent: \begin{enumerate} \item $\mathcal{F}$ is flat over $Z$ at $x$ and the restriction of $\mathcal{F}$ to its fibre over $z$ is flat at $x$ over the fibre of $Y$ over $z$, and \item $Y$ is flat over $Z$ at $y$ and $\mathcal{F}$ is flat over $Y$ at $x$. \end{enumerate}


spaces-more-morphisms-theorem-stein-factorization-Noetherian

Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $S$ with $Y$ locally Noetherian. There exists a factorization $$ \xymatrix{ X \ar[rr]_{f'} \ar[rd]_f & & Y' \ar[dl]^\pi \\ & Y & } $$ with the following properties: \begin{enumerate} \item the morphism $f'$ is proper with connected geometric fibres, \item the morphism $\pi : Y' \to Y$ is finite, \item we have $f'_*\mathcal{O}_X = \mathcal{O}_{Y'}$, \item we have $Y' = \underline{\Spec}_Y(f_*\mathcal{O}_X)$, and \item $Y'$ is the normalization of $Y$ in $X$, see Morphisms, Definition \ref{morphisms-definition-normalization-X-in-Y}. \end{enumerate}


spaces-more-morphisms-theorem-stein-factorization-general

Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $S$. There exists a factorization $$ \xymatrix{ X \ar[rr]_{f'} \ar[rd]_f & & Y' \ar[dl]^\pi \\ & Y & } $$ with the following properties: \begin{enumerate} \item the morphism $f'$ is proper with connected geometric fibres, \item the morphism $\pi : Y' \to Y$ is integral, \item we have $f'_*\mathcal{O}_X = \mathcal{O}_{Y'}$, \item we have $Y' = \underline{\Spec}_Y(f_*\mathcal{O}_X)$, and \item $Y'$ is the normalization of $Y$ in $X$ (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-normalization-X-in-Y}). \end{enumerate}


spaces-more-morphisms-theorem-flatten-module

Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $X$ be an algebraic space over $B$. Let $\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \subset B$ be a quasi-compact open subspace. Assume \begin{enumerate} \item $X$ is quasi-compact, \item $X$ is locally of finite presentation over $B$, \item $\mathcal{F}$ is a module of finite type, \item $\mathcal{F}_U$ is of finite presentation, and \item $\mathcal{F}_U$ is flat over $U$. \end{enumerate} Then there exists a $U$-admissible blowup $B' \to B$ such that the strict transform $\mathcal{F}'$ of $\mathcal{F}$ is an $\mathcal{O}_{X \times_B B'}$-module of finite presentation and flat over $B'$.


spaces-more-morphisms-theorem-grothendieck-existence

In Situation \ref{situation-existence} the functor (\ref{equation-completion-functor-proper-over-A}) is an equivalence.


spaces-more-morphisms-lemma-radicial-implies-universally-injective

A radicial morphism of algebraic spaces is universally injective.


spaces-more-morphisms-lemma-when-universally-injective-radicial

Let $S$ be a scheme. Let $f : X \to Y$ be a universally injective morphism of algebraic spaces over $S$. \begin{enumerate} \item If $f$ is decent then $f$ is radicial. \item If $f$ is quasi-separated then $f$ is radicial. \item If $f$ is locally separated then $f$ is radicial. \end{enumerate}


spaces-more-morphisms-lemma-ui-case

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact monomorphism of algebraic spaces such that for every $T \to Y$ the map $$ \mathcal{O}_T \to f_{T,*}\mathcal{O}_{X \times_Y T} $$ is injective. Then $f$ is an isomorphism (and hence representable by schemes).


spaces-more-morphisms-lemma-etale-conormal

Let $S$ be a scheme. Let $i : Z \to X$ be an immersion. Let $\varphi : U \to X$ be an \'etale morphism where $U$ is a scheme. Set $Z_U = U \times_X Z$ which is a locally closed subscheme of $U$. Then $$ \mathcal{C}_{Z/X}|_{Z_U} = \mathcal{C}_{Z_U/U} $$ canonically and functorially in $U$.


spaces-more-morphisms-lemma-conormal-functorial

Let $S$ be a scheme. Let $$ \xymatrix{ Z \ar[r]_i \ar[d]_f & X \ar[d]^g \\ Z' \ar[r]^{i'} & X' } $$ be a commutative diagram of algebraic spaces over $S$. Assume $i$, $i'$ immersions. There is a canonical map of $\mathcal{O}_Z$-modules $$ f^*\mathcal{C}_{Z'/X'} \longrightarrow \mathcal{C}_{Z/X} $$


spaces-more-morphisms-lemma-conormal-functorial-more

Let $S$ be a scheme. The conormal sheaf of Definition \ref{definition-conormal-sheaf}, and its functoriality of Lemma \ref{lemma-conormal-functorial} satisfy the following properties: \begin{enumerate} \item If $Z \to X$ is an immersion of schemes over $S$, then the conormal sheaf agrees with the one from Morphisms, Definition \ref{morphisms-definition-conormal-sheaf}. \item If in Lemma \ref{lemma-conormal-functorial} all the spaces are schemes, then the map $f^*\mathcal{C}_{Z'/X'} \to \mathcal{C}_{Z/X}$ is the same as the one constructed in Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial}. \item Given a commutative diagram $$ \xymatrix{ Z \ar[r]_i \ar[d]_f & X \ar[d]^g \\ Z' \ar[r]^{i'} \ar[d]_{f'} & X' \ar[d]^{g'} \\ Z'' \ar[r]^{i''} & X'' } $$ then the map $(f' \circ f)^*\mathcal{C}_{Z''/X''} \to \mathcal{C}_{Z/X}$ is the same as the composition of $f^*\mathcal{C}_{Z'/X'} \to \mathcal{C}_{Z/X}$ with the pullback by $f$ of $(f')^*\mathcal{C}_{Z''/X''} \to \mathcal{C}_{Z'/X'}$ \end{enumerate}


spaces-more-morphisms-lemma-conormal-functorial-flat

Let $S$ be a scheme. Let $$ \xymatrix{ Z \ar[r]_i \ar[d]_f & X \ar[d]^g \\ Z' \ar[r]^{i'} & X' } $$ be a fibre product diagram of algebraic spaces over $S$. Assume $i$, $i'$ immersions. Then the canonical map $f^*\mathcal{C}_{Z'/X'} \to \mathcal{C}_{Z/X}$ of Lemma \ref{lemma-conormal-functorial} is surjective. If $g$ is flat, then it is an isomorphism.


spaces-more-morphisms-lemma-transitivity-conormal

Let $S$ be a scheme. Let $Z \to Y \to X$ be immersions of algebraic spaces. Then there is a canonical exact sequence $$ i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0 $$ where the maps come from Lemma \ref{lemma-conormal-functorial} and $i : Z \to Y$ is the first morphism.


spaces-more-morphisms-lemma-etale-conormal-algebra

Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. Let $\varphi : U \to X$ be an \'etale morphism where $U$ is a scheme. Set $Z_U = U \times_X Z$ which is a locally closed subscheme of $U$. Then $$ \mathcal{C}_{Z/X, *}|_{Z_U} = \mathcal{C}_{Z_U/U, *} $$ canonically and functorially in $U$.


spaces-more-morphisms-lemma-conormal-algebra-functorial

Let $S$ be a scheme. Let $$ \xymatrix{ Z \ar[r]_i \ar[d]_f & X \ar[d]^g \\ Z' \ar[r]^{i'} & X' } $$ be a commutative diagram of algebraic spaces over $S$. Assume $i$, $i'$ immersions. There is a canonical map of graded $\mathcal{O}_Z$-algebras $$ f^*\mathcal{C}_{Z'/X', *} \longrightarrow \mathcal{C}_{Z/X, *} $$


spaces-more-morphisms-lemma-localize-differentials

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Consider any commutative diagram $$ \xymatrix{ U \ar[d]_a \ar[r]_\psi & V \ar[d]^b \\ X \ar[r]^f & Y } $$ where the vertical arrows are \'etale morphisms of algebraic spaces. Then $$ \Omega_{X/Y}|_{U_\etale} = \Omega_{U/V} $$ In particular, if $U$, $V$ are schemes, then this is equal to the usual sheaf of differentials of the morphism of schemes $U \to V$.


spaces-more-morphisms-lemma-module-differentials-quasi-coherent

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then $\Omega_{X/Y}$ is a quasi-coherent $\mathcal{O}_X$-module.


spaces-more-morphisms-lemma-functoriality-differentials

Let $S$ be a scheme. Let $$ \xymatrix{ X' \ar[d] \ar[r]_f & X \ar[d] \\ Y' \ar[r] & Y } $$ be a commutative diagram of algebraic spaces. The map $f^\sharp : \mathcal{O}_X \to f_*\mathcal{O}_{X'}$ composed with the map $f_*\text{d}_{X'/Y'} : f_*\mathcal{O}_{X'} \to f_*\Omega_{X'/Y'}$ is a $Y$-derivation. Hence we obtain a canonical map of $\mathcal{O}_X$-modules $\Omega_{X/Y} \to f_*\Omega_{X'/Y'}$, and by adjointness of $f_*$ and $f^*$ a canonical $\mathcal{O}_{X'}$-module homomorphism $$ c_f : f^*\Omega_{X/Y} \longrightarrow \Omega_{X'/Y'}. $$ It is uniquely characterized by the property that $f^*\text{d}_{X/Y}(t)$ mapsto $\text{d}_{X'/Y'}(f^* t)$ for any local section $t$ of $\mathcal{O}_X$.


spaces-more-morphisms-lemma-check-functoriality-differentials

Let $S$ be a scheme. Let $$ \xymatrix{ X'' \ar[d] \ar[r]_g & X' \ar[d] \ar[r]_f & X \ar[d] \\ Y'' \ar[r] & Y' \ar[r] & Y } $$ be a commutative diagram of algebraic spaces over $S$. Then we have $$ c_{f \circ g} = c_g \circ g^* c_f $$ as maps $(f \circ g)^*\Omega_{X/Y} \to \Omega_{X''/Y''}$.


spaces-more-morphisms-lemma-triangle-differentials

Let $S$ be a scheme. Let $f : X \to Y$, $g : Y \to B$ be morphisms of algebraic spaces over $S$. Then there is a canonical exact sequence $$ f^*\Omega_{Y/B} \to \Omega_{X/B} \to \Omega_{X/Y} \to 0 $$ where the maps come from applications of Lemma \ref{lemma-functoriality-differentials}.


spaces-more-morphisms-lemma-differentials-relative-immersion

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $i : Z \to X$ be an immersion of algebraic spaces over $B$. There is a canonical exact sequence $$ \mathcal{C}_{Z/X} \to i^*\Omega_{X/B} \to \Omega_{Z/B} \to 0 $$ where the first arrow is induced by $\text{d}_{X/B}$ and the second arrow comes from Lemma \ref{lemma-functoriality-differentials}.


spaces-more-morphisms-lemma-base-change-differentials

Let $S$ be a scheme. Let $X \to Y$ be a morphism of algebraic spaces over $S$. Let $g : Y' \to Y$ be a morphism of algebraic spaces over $S$. Let $X' = X_{Y'}$ be the base change of $X$. Denote $g' : X' \to X$ the projection. Then the map $$ (g')^*\Omega_{X/Y} \to \Omega_{X'/Y'} $$ of Lemma \ref{lemma-functoriality-differentials} is an isomorphism.


spaces-more-morphisms-lemma-differential-product

Let $S$ be a scheme. Let $f : X \to B$ and $g : Y \to B$ be morphisms of algebraic spaces over $S$ with the same target. Let $p : X \times_B Y \to X$ and $q : X \times_B Y \to Y$ be the projection morphisms. The maps from Lemma \ref{lemma-functoriality-differentials} $$ p^*\Omega_{X/B} \oplus q^*\Omega_{Y/B} \longrightarrow \Omega_{X \times_B Y/B} $$ give an isomorphism.


spaces-more-morphisms-lemma-finite-presentation-differentials

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite presentation, then $\Omega_{X/Y}$ is an $\mathcal{O}_X$-module of finite presentation.


spaces-more-morphisms-lemma-smooth-omega-finite-locally-free

Let $S$ be a scheme. Let $f : X \to Y$ be a smooth morphism of algebraic spaces over $S$. Then the module of differentials $\Omega_{X/Y}$ is finite locally free.


spaces-more-morphisms-lemma-topological-invariance

With assumption and notation as in Theorem \ref{theorem-topological-invariance} the equivalence of categories $Y_{spaces, \etale} \to X_{spaces, \etale}$ restricts to equivalences of categories $Y_\etale \to X_\etale$ and $Y_{affine, \etale} \to X_{affine, \etale}$.


spaces-more-morphisms-lemma-first-order-thickening-maps

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X \subset X'$ and $Y \subset Y'$ be thickenings of algebraic spaces over $B$. Let $f : X \to Y$ be a morphism of algebraic spaces over $B$. Given any map of $\mathcal{O}_B$-algebras $$ \alpha : f_{spaces, \etale}^{-1}\mathcal{O}_{Y'} \to \mathcal{O}_{X'} $$ such that $$ \xymatrix{ f_{spaces, \etale}^{-1}\mathcal{O}_Y \ar[r]_-{f^\sharp} \ar[r] & \mathcal{O}_X \\ f_{spaces, \etale}^{-1}\mathcal{O}_{Y'} \ar[r]^-\alpha \ar[u]^{i_Y^\sharp} & \mathcal{O}_{X'} \ar[u]_{i_X^\sharp} } $$ commutes, there exists a unique morphism of $(f, f')$ of thickenings over $B$ such that $\alpha = (f')^\sharp$.


spaces-more-morphisms-lemma-first-order-thickening-surjective

Let $S$ be a scheme. Let $X \subset X'$ be a thickening of algebraic spaces over $S$. Let $U$ be an affine object of $X_{spaces, \etale}$. Then $$ \Gamma(U, \mathcal{O}_{X'}) \to \Gamma(U, \mathcal{O}_X) $$ is surjective where we think of $\mathcal{O}_{X'}$ as a sheaf on $X_{spaces, \etale}$ via (\ref{equation-fundamental-equivalence}).


spaces-more-morphisms-lemma-thickening-scheme

Let $S$ be a scheme. Let $X \subset X'$ be a thickening of algebraic spaces over $S$. If $X$ is (representable by) a scheme, then so is $X'$.


spaces-more-morphisms-lemma-thickening-equivalence

Let $S$ be a scheme. Let $X \subset X'$ be a thickening of algebraic spaces over $S$. The functor $$ V' \longmapsto V = X \times_{X'} V' $$ defines an equivalence of categories $X'_\etale \to X_\etale$.


spaces-more-morphisms-lemma-first-order-thickening

Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Consider a short exact sequence $$ 0 \to \mathcal{I} \to \mathcal{A} \to \mathcal{O}_X \to 0 $$ of sheaves on $X_\etale$ where $\mathcal{A}$ is a sheaf of $f^{-1}\mathcal{O}_B$-algebras, $\mathcal{A} \to \mathcal{O}_X$ is a surjection of sheaves of $f^{-1}\mathcal{O}_B$-algebras, and $\mathcal{I}$ is its kernel. If \begin{enumerate} \item $\mathcal{I}$ is an ideal of square zero in $\mathcal{A}$, and \item $\mathcal{I}$ is quasi-coherent as an $\mathcal{O}_X$-module \end{enumerate} then there exists a first order thickening $X \subset X'$ over $B$ and an isomorphism $\mathcal{O}_{X'} \to \mathcal{A}$ of $f^{-1}\mathcal{O}_B$-algebras compatible with the surjections to $\mathcal{O}_X$.


spaces-more-morphisms-lemma-base-change-thickening

Let $S$ be a scheme. Let $Y \subset Y'$ be a thickening of algebraic spaces over $S$. Let $X' \to Y'$ be a morphism and set $X = Y \times_{Y'} X'$. Then $(X \subset X') \to (Y \subset Y')$ is a morphism of thickenings. If $Y \subset Y'$ is a first (resp.\ finite order) thickening, then $X \subset X'$ is a first (resp.\ finite order) thickening.


spaces-more-morphisms-lemma-composition-thickening

Let $S$ be a scheme. If $X \subset X'$ and $X' \subset X''$ are thickenings of algebraic spaces over $S$, then so is $X \subset X''$.


spaces-more-morphisms-lemma-descending-property-thickening

The property of being a thickening is fpqc local. Similarly for first order thickenings.


spaces-more-morphisms-lemma-thicken-property-morphisms

Let $S$ be a scheme. Let $(f, f') : (X \subset X') \to (Y \subset Y')$ be a morphism of thickenings of algebraic spaces over $S$. Then \begin{enumerate} \item $f$ is an affine morphism if and only if $f'$ is an affine morphism, \item $f$ is a surjective morphism if and only if $f'$ is a surjective morphism, \item $f$ is quasi-compact if and only if $f'$ quasi-compact, \item $f$ is universally closed if and only if $f'$ is universally closed, \item $f$ is integral if and only if $f'$ is integral, \item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated, \item $f$ is universally injective if and only if $f'$ is universally injective, \item $f$ is universally open if and only if $f'$ is universally open, \item $f$ is representable if and only if $f'$ is representable, and \item add more here. \end{enumerate}


spaces-more-morphisms-lemma-thicken-property-morphisms-cartesian

Let $S$ be a scheme. Let $(f, f') : (X \subset X') \to (Y \subset Y')$ be a morphism of thickenings of algebraic spaces over $S$ such that $X = Y \times_{Y'} X'$. If $X \subset X'$ is a finite order thickening, then \begin{enumerate} \item $f$ is a closed immersion if and only if $f'$ is a closed immersion, \item $f$ is locally of finite type if and only if $f'$ is locally of finite type, \item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \item $f$ is locally of finite type of relative dimension $d$ if and only if $f'$ is locally of finite type of relative dimension $d$, \item $\Omega_{X/Y} = 0$ if and only if $\Omega_{X'/Y'} = 0$, \item $f$ is unramified if and only if $f'$ is unramified, \item $f$ is proper if and only if $f'$ is proper, \item $f$ is a finite morphism if and only if $f'$ is an finite morphism, \item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \item $f$ is an immersion if and only if $f'$ is an immersion, and \item add more here. \end{enumerate}


spaces-more-morphisms-lemma-properties-that-extend-over-thickenings

Let $S$ be a scheme. Let $(f, f') : (X \subset X') \to (Y \to Y')$ be a morphism of thickenings of algebraic spaces over $S$. Assume $f$ and $f'$ are locally of finite type and $X = Y \times_{Y'} X'$. Then \begin{enumerate} \item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \item $f$ is finite if and only if $f'$ is finite, \item $f$ is a closed immersion if and only if $f'$ is a closed immersion, \item $\Omega_{X/Y} = 0$ if and only if $\Omega_{X'/Y'} = 0$, \item $f$ is unramified if and only if $f'$ is unramified, \item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \item $f$ is an immersion if and only if $f'$ is an immersion, \item $f$ is proper if and only if $f'$ is proper, and \item add more here. \end{enumerate}


spaces-more-morphisms-lemma-picard-group-first-order-thickening

Let $S$ be a scheme. Let $X \subset X'$ be a first order thickening of algebraic spaces over $S$ with ideal sheaf $\mathcal{I}$. Then there is a canonical exact sequence $$ \xymatrix{ 0 \ar[r] & H^0(X, \mathcal{I}) \ar[r] & H^0(X', \mathcal{O}_{X'}^*) \ar[r] & H^0(X, \mathcal{O}^*_X) \ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\ & H^1(X, \mathcal{I}) \ar[r] & \Pic(X') \ar[r] & \Pic(X) \ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\ & H^2(X, \mathcal{I}) \ar[r] & \ldots \ar[r] & \ldots } $$ of abelian groups.


spaces-more-morphisms-lemma-first-order-infinitesimal-neighbourhood

Let $i : Z \to X$ be an immersion of algebraic spaces. The first order infinitesimal neighbourhood $Z'$ of $Z$ in $X$ has the following universal property: Given any commutative diagram $$ \xymatrix{ Z \ar[d]_i & T \ar[l]^a \ar[d] \\ X & T' \ar[l]_b } $$ where $T \subset T'$ is a first order thickening over $X$, there exists a unique morphism $(a', a) : (T \subset T') \to (Z \subset Z')$ of thickenings over $X$.


spaces-more-morphisms-lemma-infinitesimal-neighbourhood-conormal

Let $i : Z \to X$ be an immersion of algebraic spaces. Let $Z \subset Z'$ be the first order infinitesimal neighbourhood of $Z$ in $X$. Then the diagram $$ \xymatrix{ Z \ar[r] \ar[d] & Z' \ar[d] \\ Z \ar[r] & X } $$ induces a map of conormal sheaves $\mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Z'}$ by Lemma \ref{lemma-conormal-functorial}. This map is an isomorphism.


spaces-more-morphisms-lemma-formally-etale-is-combination

Let $S$ be a scheme. Let $a : F \to G$ be a transformation of functors $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Then $a$ is formally \'etale if and only if $a$ is both formally smooth and formally unramified.


spaces-more-morphisms-lemma-composition-formally-smooth-etale-unramified

Composition. \begin{enumerate} \item A composition of formally smooth transformations of functors is formally smooth. \item A composition of formally \'etale transformations of functors is formally \'etale. \item A composition of formally unramified transformations of functors is formally unramified. \end{enumerate}


spaces-more-morphisms-lemma-base-change-formally-smooth-etale-unramified

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : H \to G$ be transformations of functors. Consider the fibre product diagram $$ \xymatrix{ H \times_{b, G, a} F \ar[r]_-{b'} \ar[d]_{a'} & F \ar[d]^a \\ H \ar[r]^b & G } $$ \begin{enumerate} \item If $a$ is formally smooth, then the base change $a'$ is formally smooth. \item If $a$ is formally \'etale, then the base change $a'$ is formally \'etale. \item If $a$ is formally unramified, then the base change $a'$ is formally unramified. \end{enumerate}


spaces-more-morphisms-lemma-representable-property-formally-property

Let $S$ be a scheme. Let $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$ be a representable transformation of functors. \begin{enumerate} \item If $a$ is smooth then $a$ is formally smooth. \item If $a$ is \'etale, then $a$ is formally \'etale. \item If $a$ is unramified, then $a$ is formally unramified. \end{enumerate}


spaces-more-morphisms-lemma-etale-on-top

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : G \to H$ be transformations of functors. Assume that $a$ is representable, surjective, and \'etale. \begin{enumerate} \item If $b$ is formally smooth, then $b \circ a$ is formally smooth. \item If $b$ is formally \'etale, then $b \circ a$ is formally \'etale. \item If $b$ is formally unramified, then $b \circ a$ is formally unramified. \end{enumerate} Conversely, consider a solid commutative diagram $$ \xymatrix{ G \ar[d]_b & T \ar[d]^i \ar[l] \\ H & T' \ar[l] \ar@{-->}[lu] } $$ with $T'$ an affine scheme over $S$ and $i : T \to T'$ a closed immersion defined by an ideal of square zero. \begin{enumerate} \item[(4)] If $b \circ a$ is formally smooth, then for every $t \in T$ there exists an \'etale morphism of affines $U' \to T'$ and a morphism $U' \to G$ such that $$ \xymatrix{ G \ar[d]_b & T \ar[l] & T \times_{T'} U' \ar[d] \ar[l]\\ H & T' \ar[l] & U' \ar[llu] \ar[l] } $$ commutes and $t$ is in the image of $U' \to T'$. \item[(5)] If $b \circ a$ is formally unramified, then there exists at most one dotted arrow in the diagram above, i.e., $b$ is formally unramified. \item[(6)] If $b \circ a$ is formally \'etale, then there exists exactly one dotted arrow in the diagram above, i.e., $b$ is formally \'etale. \end{enumerate}


spaces-more-morphisms-lemma-formally-permanence

Let $S$ be a scheme. Let $F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : G \to H$ be transformations of functors. Assume $b$ is formally unramified. \begin{enumerate} \item If $b \circ a$ is formally unramified then $a$ is formally unramified. \item If $b \circ a$ is formally \'etale then $a$ is formally \'etale. \item If $b \circ a$ is formally smooth then $a$ is formally smooth. \end{enumerate}


spaces-more-morphisms-lemma-formally-unramified

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is formally unramified, \item for every diagram $$ \xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^f & Y } $$ where $U$ and $V$ are schemes and the vertical arrows are \'etale the morphism of schemes $\psi$ is formally unramified (as in More on Morphisms, Definition \ref{more-morphisms-definition-formally-unramified}), and \item for one such diagram with surjective vertical arrows the morphism $\psi$ is formally unramified. \end{enumerate}


spaces-more-morphisms-lemma-formally-unramified-not-affine

Let $S$ be a scheme. If $f : X \to Y$ is a formally unramified morphism of algebraic spaces over $S$, then given any solid commutative diagram $$ \xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l] \\ S & T' \ar[l] \ar@{-->}[lu] } $$ where $T \subset T'$ is a first order thickening of algebraic spaces over $S$ there exists at most one dotted arrow making the diagram commute. In other words, in Definition \ref{definition-formally-unramified} the condition that $T$ be an affine scheme may be dropped.


spaces-more-morphisms-lemma-composition-formally-unramified

A composition of formally unramified morphisms is formally unramified.


spaces-more-morphisms-lemma-base-change-formally-unramified

A base change of a formally unramified morphism is formally unramified.


spaces-more-morphisms-lemma-characterize-formally-unramified

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is formally unramified, and \item $\Omega_{X/Y} = 0$. \end{enumerate}


spaces-more-morphisms-lemma-unramified-formally-unramified

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item The morphism $f$ is unramified, \item the morphism $f$ is locally of finite type and $\Omega_{X/Y} = 0$, and \item the morphism $f$ is locally of finite type and formally unramified. \end{enumerate}


spaces-more-morphisms-lemma-universally-injective-unramified

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is unramified and a monomorphism, \item $f$ is unramified and universally injective, \item $f$ is locally of finite type and a monomorphism, \item $f$ is universally injective, locally of finite type, and formally unramified. \end{enumerate} Moreover, in this case $f$ is also representable, separated, and locally quasi-finite.


spaces-more-morphisms-lemma-characterize-closed-immersion

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is a closed immersion, \item $f$ is universally closed, unramified, and a monomorphism, \item $f$ is universally closed, unramified, and universally injective, \item $f$ is universally closed, locally of finite type, and a monomorphism, \item $f$ is universally closed, universally injective, locally of finite type, and formally unramified. \end{enumerate}


spaces-more-morphisms-lemma-check-universal-first-order-thickening

Let $S$ be a scheme. Let $h : Z \to X$ be a morphism of algebraic spaces over $S$. Let $Z \subset Z'$ be a first order thickening over $X$. The following are equivalent \begin{enumerate} \item $Z \subset Z'$ is a universal first order thickening, \item for any diagram (\ref{equation-universal-first-order-thickening}) with $T'$ a scheme a unique dotted arrow exists making the diagram commute, and \item for any diagram (\ref{equation-universal-first-order-thickening}) with $T'$ an affine scheme a unique dotted arrow exists making the diagram commute. \end{enumerate}


spaces-more-morphisms-lemma-universal-thickening-over-formally-etale

Let $S$ be a scheme. Let $Z \to Y \to X$ be morphisms of algebraic spaces over $S$. If $Z \subset Z'$ is a universal first order thickening of $Z$ over $Y$ and $Y \to X$ is formally \'etale, then $Z \subset Z'$ is a universal first order thickening of $Z$ over $X$.


spaces-more-morphisms-lemma-etale-morphism-of-universal-thickenings

Let $S$ be a scheme. Let $Z \to Y \to X$ be morphisms of algebraic spaces over $S$. Assume $Z \to Y$ is \'etale. \begin{enumerate} \item If $Y \subset Y'$ is a universal first order thickening of $Y$ over $X$, then the unique \'etale morphism $Z' \to Y'$ such that $Z = Y \times_{Y'} Z'$ (see Theorem \ref{theorem-topological-invariance}) is a universal first order thickening of $Z$ over $X$. \item If $Z \to Y$ is surjective and $(Z \subset Z') \to (Y \subset Y')$ is an \'etale morphism of first order thickenings over $X$ and $Z'$ is a universal first order thickening of $Z$ over $X$, then $Y'$ is a universal first order thickening of $Y$ over $X$. \end{enumerate}


spaces-more-morphisms-lemma-universal-thickening

Let $S$ be a scheme. Let $h : Z \to X$ be a formally unramified morphism of algebraic spaces over $S$. There exists a universal first order thickening $Z \subset Z'$ of $Z$ over $X$.


spaces-more-morphisms-lemma-universal-thickening-unramified

Let $S$ be a scheme. Let $Z \to X$ be a formally unramified morphism of algebraic spaces over $S$. Then the universal first order thickening $Z'$ is formally unramified over $X$.


spaces-more-morphisms-lemma-universal-thickening-functorial

Let $S$ be a scheme Consider a commutative diagram of algebraic spaces over $S$ $$ \xymatrix{ Z \ar[r]_h \ar[d]_f & X \ar[d]^g \\ W \ar[r]^{h'} & Y } $$ with $h$ and $h'$ formally unramified. Let $Z \subset Z'$ be the universal first order thickening of $Z$ over $X$. Let $W \subset W'$ be the universal first order thickening of $W$ over $Y$. There exists a canonical morphism $(f, f') : (Z, Z') \to (W, W')$ of thickenings over $Y$ which fits into the following commutative diagram $$ \xymatrix{ & & & Z' \ar[ld] \ar[d]^{f'} \\ Z \ar[rr] \ar[d]_f \ar[rrru] & & X \ar[d] & W' \ar[ld] \\ W \ar[rrru]|!{[rr];[rruu]}\hole \ar[rr] & & Y } $$ In particular the morphism $(f, f')$ of thickenings induces a morphism of conormal sheaves $f^*\mathcal{C}_{W/Y} \to \mathcal{C}_{Z/X}$.


spaces-more-morphisms-lemma-universal-thickening-fibre-product

Let $S$ be a scheme. Let $$ \xymatrix{ Z \ar[r]_h \ar[d]_f & X \ar[d]^g \\ W \ar[r]^{h'} & Y } $$ be a fibre product diagram of algebraic spaces over $S$ with $h'$ formally unramified. Then $h$ is formally unramified and if $W \subset W'$ is the universal first order thickening of $W$ over $Y$, then $Z = X \times_Y W \subset X \times_Y W'$ is the universal first order thickening of $Z$ over $X$. In particular the canonical map $f^*\mathcal{C}_{W/Y} \to \mathcal{C}_{Z/X}$ of Lemma \ref{lemma-universal-thickening-functorial} is surjective.


spaces-more-morphisms-lemma-universal-thickening-localize

Taking the universal first order thickenings commutes with \'etale localization. More precisely, let $h : Z \to X$ be a formally unramified morphism of algebraic spaces over a base scheme $S$. Let $$ \xymatrix{ V \ar[d] \ar[r] & U \ar[d] \\ Z \ar[r] & X } $$ be a commutative diagram with \'etale vertical arrows. Let $Z'$ be the universal first order thickening of $Z$ over $X$. Then $V \to U$ is formally unramified and the universal first order thickening $V'$ of $V$ over $U$ is \'etale over $Z'$. In particular, $\mathcal{C}_{Z/X}|_V = \mathcal{C}_{V/U}$.


spaces-more-morphisms-lemma-differentials-universally-unramified

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $h : Z \to X$ be a formally unramified morphism of algebraic spaces over $B$. Let $Z \subset Z'$ be the universal first order thickening of $Z$ over $X$ with structure morphism $h' : Z' \to X$. The canonical map $$ \text{d}h' : (h')^*\Omega_{X/B} \to \Omega_{Z'/B} $$ induces an isomorphism $h^*\Omega_{X/B} \to \Omega_{Z'/B} \otimes \mathcal{O}_Z$.


spaces-more-morphisms-lemma-universally-unramified-differentials-sequence

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $h : Z \to X$ be a formally unramified morphism of algebraic spaces over $B$. There is a canonical exact sequence $$ \mathcal{C}_{Z/X} \to h^*\Omega_{X/B} \to \Omega_{Z/B} \to 0. $$ The first arrow is induced by $\text{d}_{Z'/B}$ where $Z'$ is the universal first order neighbourhood of $Z$ over $X$.


spaces-more-morphisms-lemma-two-unramified-morphisms

Let $S$ be a scheme. Let $$ \xymatrix{ Z \ar[r]_i \ar[rd]_j & X \ar[d] \\ & Y } $$ be a commutative diagram of algebraic spaces over $S$ where $i$ and $j$ are formally unramified. Then there is a canonical exact sequence $$ \mathcal{C}_{Z/Y} \to \mathcal{C}_{Z/X} \to i^*\Omega_{X/Y} \to 0 $$ where the first arrow comes from Lemma \ref{lemma-universal-thickening-functorial} and the second from Lemma \ref{lemma-universally-unramified-differentials-sequence}.


spaces-more-morphisms-lemma-formally-etale

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is formally \'etale, \item for every diagram $$ \xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^f & Y } $$ where $U$ and $V$ are schemes and the vertical arrows are \'etale the morphism of schemes $\psi$ is formally \'etale (as in More on Morphisms, Definition \ref{more-morphisms-definition-formally-etale}), and \item for one such diagram with surjective vertical arrows the morphism $\psi$ is formally \'etale. \end{enumerate}


spaces-more-morphisms-lemma-composition-formally-etale

A composition of formally \'etale morphisms is formally \'etale.


spaces-more-morphisms-lemma-base-change-formally-etale

A base change of a formally \'etale morphism is formally \'etale.


spaces-more-morphisms-lemma-etale-formally-etale

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item The morphism $f$ is \'etale, and \item the morphism $f$ is locally of finite presentation and formally \'etale. \end{enumerate}


spaces-more-morphisms-lemma-difference-derivation

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X \subset X'$ and $Y \subset Y'$ be two first order thickenings of algebraic spaces over $B$. Let $(a, a'), (b, b') : (X \subset X') \to (Y \subset Y')$ be two morphisms of thickenings over $B$. Assume that \begin{enumerate} \item $a = b$, and \item the two maps $a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ (Lemma \ref{lemma-conormal-functorial}) are equal. \end{enumerate} Then the map $(a')^\sharp - (b')^\sharp$ factors as $$ \mathcal{O}_{Y'} \to \mathcal{O}_Y \xrightarrow{D} a_*\mathcal{C}_{X/X'} \to a_*\mathcal{O}_{X'} $$ where $D$ is an $\mathcal{O}_B$-derivation.


spaces-more-morphisms-lemma-action-by-derivations

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(a, a') : (X \subset X') \to (Y \subset Y')$ be a morphism of first order thickenings over $B$. Let $$ \theta : a^*\Omega_{Y/B} \to \mathcal{C}_{X/X'} $$ be an $\mathcal{O}_X$-linear map. Then there exists a unique morphism of pairs $(b, b') : (X \subset X') \to (Y \subset Y')$ such that (1) and (2) of Lemma \ref{lemma-difference-derivation} hold and the derivation $D$ and $\theta$ are related by Equation (\ref{equation-D}).


spaces-more-morphisms-lemma-sheaf

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X \subset X'$ and $Y \subset Y'$ be first order thickenings over $B$. Assume given a morphism $a : X \to Y$ and a map $A : a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ of $\mathcal{O}_X$-modules. For an object $U'$ of $(X')_{spaces, \etale}$ with $U = X \times_{X'} U'$ consider morphisms $a' : U' \to Y'$ such that \begin{enumerate} \item $a'$ is a morphism over $B$, \item $a'|_U = a|_U$, and \item the induced map $a^*\mathcal{C}_{Y/Y'}|_U \to \mathcal{C}_{X/X'}|_U$ is the restriction of $A$ to $U$. \end{enumerate} Then the rule \begin{equation} \label{equation-sheaf} U' \mapsto \{a' : U' \to Y'\text{ such that (1), (2), (3) hold.}\} \end{equation} defines a sheaf of sets on $(X')_{spaces, \etale}$.


spaces-more-morphisms-lemma-action-sheaf

Same notation and assumptions as in Lemma \ref{lemma-sheaf}. We identify sheaves on $X$ and $X'$ via (\ref{equation-equivalence-etale-spaces}). There is an action of the sheaf $$ \SheafHom_{\mathcal{O}_X}(a^*\Omega_{Y/B}, \mathcal{C}_{X/X'}) $$ on the sheaf (\ref{equation-sheaf}). Moreover, the action is simply transitive for any object $U'$ of $(X')_{spaces, \etale}$ over which the sheaf (\ref{equation-sheaf}) has a section.


spaces-more-morphisms-lemma-action-by-derivations-etale-localization

Let $S$ be a scheme. Consider a commutative diagram of first order thickenings $$ \vcenter{ \xymatrix{ (T_2 \subset T_2') \ar[d]_{(h, h')} \ar[rr]_{(a_2, a_2')} & & (X_2 \subset X_2') \ar[d]^{(f, f')} \\ (T_1 \subset T_1') \ar[rr]^{(a_1, a_1')} & & (X_1 \subset X_1') } } \quad \begin{matrix} \text{and a commutative} \\ \text{diagram} \end{matrix} \quad \vcenter{ \xymatrix{ X_2' \ar[r] \ar[d] & B_2 \ar[d] \\ X_1' \ar[r] & B_1 } } $$ of algebraic spaces over $S$ with $X_2 \to X_1$ and $B_2 \to B_1$ \'etale. For any $\mathcal{O}_{T_1}$-linear map $\theta_1 : a_1^*\Omega_{X_1/B_1} \to \mathcal{C}_{T_1/T'_1}$ let $\theta_2$ be the composition $$ \xymatrix{ a_2^*\Omega_{X_2/B_2} \ar@{=}[r] & h^*a_1^*\Omega_{X_1/B_1} \ar[r]^-{h^*\theta_1} & h^*\mathcal{C}_{T_1/T'_1} \ar[r] & \mathcal{C}_{T_2/T'_2} } $$ (equality sign is explained in the proof). Then the diagram $$ \xymatrix{ T_2' \ar[rr]_{\theta_2 \cdot a_2'} \ar[d] & & X'_2 \ar[d] \\ T_1' \ar[rr]^{\theta_1 \cdot a_1'} & & X'_1 } $$ commutes where the actions $\theta_2 \cdot a_2'$ and $\theta_1 \cdot a_1'$ are as in Remark \ref{remark-action-by-derivations}.


spaces-more-morphisms-lemma-deform

Let $S$ be a scheme. Let $(f, f') : (X \subset X') \to (Y \subset Y')$ be a morphism of first order thickenings of algebraic spaces over $S$. Assume that $f$ is flat. Then the following are equivalent \begin{enumerate} \item $f'$ is flat and $X = Y \times_{Y'} X'$, and \item the canonical map $f^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ is an isomorphism. \end{enumerate}


spaces-more-morphisms-lemma-deform-property

Let $S$ be a scheme. Consider a commutative diagram $$ \xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (B \subset B') } $$ of thickenings of algebraic spaces over $S$. Assume $B \subset B'$ is a finite order thickening, $X'$ flat over $B'$, $X = B \times_{B'} X'$, and $Y = B \times_{B'} Y'$. Then \begin{enumerate} \item $f$ is representable if and only if $f'$ is representable, \label{item-representable} \item $f$ is flat if and only if $f'$ is flat, \label{item-flat} \item $f$ is an isomorphism if and only if $f'$ is an isomorphism, \label{item-isomorphism} \item $f$ is an open immersion if and only if $f'$ is an open immersion, \label{item-open-immersion} \item $f$ is quasi-compact if and only if $f'$ is quasi-compact, \label{item-quasi-compact} \item $f$ is universally closed if and only if $f'$ is universally closed, \label{item-universally-closed} \item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated, \label{item-separated} \item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \label{item-monomorphism} \item $f$ is surjective if and only if $f'$ is surjective, \label{item-surjective} \item $f$ is universally injective if and only if $f'$ is universally injective, \label{item-universally-injective} \item $f$ is affine if and only if $f'$ is affine, \label{item-affine} \item \label{item-finite-type} $f$ is locally of finite type if and only if $f'$ is locally of finite type, \item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \label{item-quasi-finite} \item \label{item-finite-presentation} $f$ is locally of finite presentation if and only if $f'$ is locally of finite presentation, \item \label{item-relative-dimension-d} $f$ is locally of finite type of relative dimension $d$ if and only if $f'$ is locally of finite type of relative dimension $d$, \item $f$ is universally open if and only if $f'$ is universally open, \label{item-universally-open} \item $f$ is syntomic if and only if $f'$ is syntomic, \label{item-syntomic} \item $f$ is smooth if and only if $f'$ is smooth, \label{item-smooth} \item $f$ is unramified if and only if $f'$ is unramified, \label{item-unramified} \item $f$ is \'etale if and only if $f'$ is \'etale, \label{item-etale} \item $f$ is proper if and only if $f'$ is proper, \label{item-proper} \item $f$ is integral if and only if $f'$ is integral, \label{item-integral} \item $f$ is finite if and only if $f'$ is finite, \label{item-finite} \item \label{item-finite-locally-free} $f$ is finite locally free (of rank $d$) if and only if $f'$ is finite locally free (of rank $d$), and \item add more here. \end{enumerate}


spaces-more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft

Let $S$ be a scheme. Consider a commutative diagram $$ \xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (B \subset B') } $$ of thickenings of algebraic spaces over $S$. Assume \begin{enumerate} \item $Y' \to B'$ is locally of finite type, \item $X' \to B'$ is flat and locally of finite presentation, \item $f$ is flat, and \item $X = B \times_{B'} X'$ and $Y = B \times_{B'} Y'$. \end{enumerate} Then $f'$ is flat and for all $y' \in |Y'|$ in the image of $|f'|$ the morphism $Y' \to B'$ is flat at $y'$.


spaces-more-morphisms-lemma-deform-property-fp-over-ft

Let $S$ be a scheme. Consider a commutative diagram $$ \xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (B \subset B') } $$ of thickenings of algebraic spaces over $S$. Assume $Y' \to B'$ locally of finite type, $X' \to B'$ flat and locally of finite presentation, $X = B \times_{B'} X'$, and $Y = B \times_{B'} Y'$. Then \begin{enumerate} \item $f$ is representable if and only if $f'$ is representable, \label{item-representable-fp-over-ft} \item $f$ is flat if and only if $f'$ is flat, \label{item-flat-fp-over-ft} \item $f$ is an isomorphism if and only if $f'$ is an isomorphism, \label{item-isomorphism-fp-over-ft} \item $f$ is an open immersion if and only if $f'$ is an open immersion, \label{item-open-immersion-fp-over-ft} \item $f$ is quasi-compact if and only if $f'$ is quasi-compact, \label{item-quasi-compact-fp-over-ft} \item $f$ is universally closed if and only if $f'$ is universally closed, \label{item-universally-closed-fp-over-ft} \item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated, \label{item-separated-fp-over-ft} \item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \label{item-monomorphism-fp-over-ft} \item $f$ is surjective if and only if $f'$ is surjective, \label{item-surjective-fp-over-ft} \item $f$ is universally injective if and only if $f'$ is universally injective, \label{item-universally-injective-fp-over-ft} \item $f$ is affine if and only if $f'$ is affine, \label{item-affine-fp-over-ft} \item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \label{item-quasi-finite-fp-over-ft} \item \label{item-relative-dimension-d-fp-over-ft} $f$ is locally of finite type of relative dimension $d$ if and only if $f'$ is locally of finite type of relative dimension $d$, \item $f$ is universally open if and only if $f'$ is universally open, \label{item-universally-open-fp-over-ft} \item $f$ is syntomic if and only if $f'$ is syntomic, \label{item-syntomic-fp-over-ft} \item $f$ is smooth if and only if $f'$ is smooth, \label{item-smooth-fp-over-ft} \item $f$ is unramified if and only if $f'$ is unramified, \label{item-unramified-fp-over-ft} \item $f$ is \'etale if and only if $f'$ is \'etale, \label{item-etale-fp-over-ft} \item $f$ is proper if and only if $f'$ is proper, \label{item-proper-fp-over-ft} \item $f$ is finite if and only if $f'$ is finite, \label{item-finite-fp-over-ft} \item \label{item-finite-locally-free-fp-over-ft} $f$ is finite locally free (of rank $d$) if and only if $f'$ is finite locally free (of rank $d$), and \item add more here. \end{enumerate}


spaces-more-morphisms-lemma-composition-formally-smooth

A composition of formally smooth morphisms is formally smooth.


spaces-more-morphisms-lemma-formally-etale-unramified-smooth

Let $f : X \to S$ be a morphism of schemes. Then $f$ is formally \'etale if and only if $f$ is formally smooth and formally unramified.


spaces-more-morphisms-lemma-helper-formally-smooth

Let $S$ be a scheme. Let $$ \xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^f & Y } $$ be a commutative diagram of morphisms of algebraic spaces over $S$. If the vertical arrows are \'etale and $f$ is formally smooth, then $\psi$ is formally smooth.


spaces-more-morphisms-lemma-smooth-formally-smooth

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item The morphism $f$ is smooth. \item The morphism $f$ is locally of finite presentation, and formally smooth. \end{enumerate}


spaces-more-morphisms-lemma-smooth-strong-lift

Let $S$ be a scheme. Consider a commutative diagram $$ \xymatrix{ X \ar[d] & T \ar[l] \ar[d] \\ Y & T' \ar[l] } $$ of algebraic spaces over $S$ where $X \to Y$ is smooth and $T \to T'$ is a thickening. Then there exists an \'etale covering $\{T'_i \to T'\}$ such that we can find the dotted arrow in $$ \xymatrix{ X \ar[d] & T \ar[l] \ar[d] & T \times_{T'} T'_i \ar[l] \ar[d] \\ Y & T' \ar[l] & T'_i \ar[l] \ar@{..>}[llu] } $$ making the diagram commute (for all $i$).


spaces-more-morphisms-lemma-formally-smooth-sheaf-differentials

Let $S$ be a scheme. Let $f : X \to Y$ be a formally smooth morphism of algebraic spaces over $S$. Then $\Omega_{X/Y}$ is locally projective on $X$.


spaces-more-morphisms-lemma-h1-is-zero

Let $T$ be an affine scheme. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent $\mathcal{O}_T$-modules on $T_\etale$. Consider the internal hom sheaf $\mathcal{H} = \SheafHom_{\mathcal{O}_T}(\mathcal{F}, \mathcal{G})$ on $T_\etale$. If $\mathcal{F}$ is locally projective, then $H^1(T_\etale, \mathcal{H}) = 0$.


spaces-more-morphisms-lemma-formally-smooth

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is formally smooth, \item for every diagram $$ \xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^f & Y } $$ where $U$ and $V$ are schemes and the vertical arrows are \'etale the morphism of schemes $\psi$ is formally smooth (as in More on Morphisms, Definition \ref{more-morphisms-definition-formally-unramified}), and \item for one such diagram with surjective vertical arrows the morphism $\psi$ is formally smooth. \end{enumerate}


spaces-more-morphisms-lemma-lifting-along-artinian-at-point

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$. Assume that $Y$ is locally Noetherian and $f$ locally of finite type. The following are equivalent: \begin{enumerate} \item $f$ is smooth at $x$, \item for every solid commutative diagram $$ \xymatrix{ X \ar[d]_f & \Spec(B) \ar[d]^i \ar[l]^-\alpha \\ Y & \Spec(B') \ar[l]_-{\beta} \ar@{-->}[lu] } $$ where $B' \to B$ is a surjection of local rings with $\Ker(B' \to B)$ of square zero, and $\alpha$ mapping the closed point of $\Spec(B)$ to $x$ there exists a dotted arrow making the diagram commute, and \item same as in (2) but with $B' \to B$ ranging over small extensions (see Algebra, Definition \ref{algebra-definition-small-extension}). \end{enumerate}


spaces-more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ locally Noetherian. Let $Z \subset Y$ be a closed subspace with $n$th infinitesimal neighbourhood $Z_n \subset Y$. Set $X_n = Z_n \times_Y X$. \begin{enumerate} \item If $X_n \to Z_n$ is smooth for all $n$, then $f$ is smooth at every point of $f^{-1}(Z)$. \item If $X_n \to Z_n$ is \'etale for all $n$, then $f$ is \'etale at every point of $f^{-1}(Z)$. \end{enumerate}


spaces-more-morphisms-lemma-NL-etale-localization

Let $S$ be a scheme. Consider a commutative diagram $$ \xymatrix{ U \ar[d]_p \ar[r]_g & V \ar[d]^q \\ X \ar[r]^f & Y } $$ of algebraic spaces over $S$ with $p$ and $q$ \'etale. Then there is a canonical identification $\NL_{X/Y}|_{U_\etale} = \NL_{U/V}$ in $D(\mathcal{O}_U)$.


spaces-more-morphisms-lemma-NL-compare-spaces-schemes

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $X$ and $Y$ representable by schemes $X_0$ and $Y_0$. Then there is a canonical identification $\NL_{X/Y} = \epsilon^*\NL_{X_0/Y_0}$ in $D(\mathcal{O}_X)$ where $\epsilon$ is as in Derived Categories of Spaces, Section \ref{spaces-perfect-section-derived-quasi-coherent-etale} and $\NL_{X_0/Y_0}$ is as in More on Morphisms, Definition \ref{more-morphisms-definition-netherlander}.


spaces-more-morphisms-lemma-netherlander-quasi-coherent

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The cohomology sheaves of the complex $\NL_{X/Y}$ are quasi-coherent, zero outside degrees $-1$, $0$ and equal to $\Omega_{X/Y}$ in degree $0$.


spaces-more-morphisms-lemma-netherlander-fp

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite presentation, then $\NL_{X/Y}$ is \'etale locally on $X$ quasi-isomorphic to a complex $$ \ldots \to 0 \to \mathcal{F}^{-1} \to \mathcal{F}^0 \to 0 \to \ldots $$ of quasi-coherent $\mathcal{O}_X$-modules with $\mathcal{F}^0$ of finite presentation and $\mathcal{F}^{-1}$ of finite type.


spaces-more-morphisms-lemma-NL-formally-smooth

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is formally smooth, \item $H^{-1}(\NL_{X/Y}) = 0$ and $H^0(\NL_{X/Y}) = \Omega_{X/Y}$ is locally projective. \end{enumerate}


spaces-more-morphisms-lemma-flat-locus-base-change

Let $S$ be a scheme. Let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ be a cartesian diagram of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume $g$ is flat, $f$ is locally of finite presentation, and $\mathcal{F}$ is locally of finite presentation. Then $$ \{x' \in |X'| : (g')^*\mathcal{F}\text{ is flat over }Y'\text{ at }x'\} $$ is the inverse image of the open subset of Theorem \ref{theorem-openness-flatness} under the continuous map $|g'| : |X'| \to |X|$.


spaces-more-morphisms-lemma-flat-on-fibres-at-point

In the situation above the following are equivalent \begin{enumerate} \item Pick a geometric point $\overline{x}$ of $X$ lying over $x$. Set $\overline{y} = f \circ \overline{x}$ and $\overline{z} = g \circ \overline{x}$. Then the module $\mathcal{F}_{\overline{x}}/ \mathfrak m_{\overline{z}}\mathcal{F}_{\overline{x}}$ is flat over $\mathcal{O}_{Y, \overline{y}}/ \mathfrak m_{\overline{z}}\mathcal{O}_{Y, \overline{y}}$. \item Pick a morphism $x : \Spec(K) \to X$ in the equivalence class of $x$. Set $z = g \circ x$, $X_z = \Spec(K) \times_{z, Z} X$, $Y_z = \Spec(K) \times_{z, Z} Y$, and $\mathcal{F}_z$ the pullback of $\mathcal{F}$ to $X_z$. Then $\mathcal{F}_z$ is flat at $x$ over $Y_z$ (as defined in Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-flat-module}). \item Pick a commutative diagram $$ \xymatrix{ & & & U \ar[llld]_a \ar[rr] \ar[dr] & & V \ar[llld]_>>>>>>>b \ar[dl] \\ X \ar[rr]_f \ar[dr]_g & & Y \ar[dl]^h &  & W \ar[llld]_c \\ & Z } $$ where $U, V, W$ are schemes, and $a, b, c$ are \'etale, and a point $u \in U$ mapping to $x$. Let $w \in W$ be the image of $u$. Let $\mathcal{F}_w$ be the pullback of $\mathcal{F}$ to the fibre $U_w$ of $U \to W$ at $w$. Then $\mathcal{F}_w$ is flat over $V_w$ at $u$. \end{enumerate}


spaces-more-morphisms-lemma-base-change-criterion-flatness-fibre

Let $S$ be a scheme. Let $f : X \to Y$ and $Y \to Z$ be morphisms of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume \begin{enumerate} \item $X$ is locally of finite presentation over $Z$, \item $\mathcal{F}$ an $\mathcal{O}_X$-module of finite presentation, \item $\mathcal{F}$ is flat over $Z$, and \item $Y$ is locally of finite type over $Z$. \end{enumerate} Then the set $$ A = \{x \in |X| : \mathcal{F} \text{ flat at }x \text{ over }Y\}. $$ is open in $|X|$ and its formation commutes with arbitrary base change: If $Z' \to Z$ is a morphism of algebraic spaces, and $A'$ is the set of points of $X' = X \times_Z Z'$ where $\mathcal{F}' = \mathcal{F} \times_Z Z'$ is flat over $Y' = Y \times_Z Z'$, then $A'$ is the inverse image of $A$ under the continuous map $|X'| \to |X|$.


spaces-more-morphisms-lemma-base-change-flatness-fibres

Let $S$ be a scheme. Let $f : X \to Y$ and $Y \to Z$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $X$ is locally of finite presentation over $Z$, \item $X$ is flat over $Z$, and \item $Y$ is locally of finite type over $Z$. \end{enumerate} Then the set $$ \{x \in |X| : X\text{ flat at }x \text{ over }Y\}. $$ is open in $|X|$ and its formation commutes with arbitrary base change $Z' \to Z$.


spaces-more-morphisms-lemma-flat-and-free-at-point-fibre

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite presentation. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_X$-module. Let $x \in |X|$ with image $y \in |Y|$. If $\mathcal{F}$ is flat at $x$ over $Y$, then the following are equivalent \begin{enumerate} \item $(\mathcal{F}_{\overline{y}})_{\overline{x}}$ is a flat $\mathcal{O}_{X_{\overline{y}}, \overline{x}}$-module, \item $(\mathcal{F}_{\overline{y}})_{\overline{x}}$ is a free $\mathcal{O}_{X_{\overline{y}}, \overline{x}}$-module, \item $\mathcal{F}_{\overline{y}}$ is finite free in an \'etale neighbourhood of $\overline{x}$ in $X_{\overline{y}}$, and \item $\mathcal{F}$ is finite free in an \'etale neighbourhood of $x$ in $X$. \end{enumerate} Here $\overline{x}$ is a geometric point of $X$ lying over $x$ and $\overline{y} = f \circ \overline{x}$.


spaces-more-morphisms-lemma-finite-free-open

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite presentation. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_X$-module flat over $Y$. Then the set $$ \{x \in |X| : \mathcal{F}\text{ free in an \'etale neighbourhood of }x\} $$ is open in $|X|$ and its formation commutes with arbitrary base change $Y' \to Y$.


spaces-more-morphisms-lemma-flatness-over-Noetherian-ring

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $X$ be an algebraic space locally of finite presentation over $S = \Spec(A)$. For $n \geq 1$ set $S_n = \Spec(A/I^n)$ and $X_n = S_n \times_S X$. Let $\mathcal{F}$ be coherent $\mathcal{O}_X$-module. If for every $n \geq 1$ the pullback $\mathcal{F}_n$ of $\mathcal{F}$ to $X$ is flat over $S_n$, then the (open) locus where $\mathcal{F}$ is flat over $X$ contains the inverse image of $V(I)$ under $X \to S$.


spaces-more-morphisms-lemma-integral-closure-smooth-pullback

Let $S$ be a scheme. Let $f : Y \to X$ be a smooth morphism of algebraic spaces over $S$. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-algebras. The integral closure of $\mathcal{O}_Y$ in $f^*\mathcal{A}$ is equal to $f^*\mathcal{A}'$ where $\mathcal{A}' \subset \mathcal{A}$ is the integral closure of $\mathcal{O}_X$ in $\mathcal{A}$.


spaces-more-morphisms-lemma-normalization-smooth-localization

Let $S$ be a scheme. Let $$ \xymatrix{ Y_2 \ar[r] \ar[d] & Y_1 \ar[d]^f \\ X_2 \ar[r]^\varphi & X_1 } $$ be a fibre square of algebraic spaces over $S$. Assume $f$ is quasi-compact and quasi-separated and $\varphi$ is smooth. Let $Y_i \to X_i' \to X_i$ be the normalization of $X_i$ in $Y_i$. Then $X_2' \cong X_2 \times_{X_1} X_1'$.


spaces-more-morphisms-lemma-CM-local-ring-fibre

The property of morphisms of germs of schemes \begin{align*} & \mathcal{P}((X, x) \to (S, s)) = \\ & \text{the local ring } \mathcal{O}_{X_s, x} \text{ of the fibre is Noetherian and Cohen-Macaulay} \end{align*} is \'etale local on the source-and-target (Descent, Definition \ref{descent-definition-local-source-target-at-point}).


spaces-more-morphisms-lemma-CM

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume the fibres of $f$ are locally Noetherian. The following are equivalent \begin{enumerate} \item $f$ is Cohen-Macaulay, \item $f$ is flat and for some surjective \'etale morphism $V \to Y$ where $V$ is a scheme, the fibres of $X_V \to V$ are Cohen-Macaulay algebraic spaces, and \item $f$ is flat and for any \'etale morphism $V \to Y$ where $V$ is a scheme, the fibres of $X_V \to V$ are Cohen-Macaulay algebraic spaces. \end{enumerate} Given $x \in |X|$ with image $y \in |Y|$ the following are equivalent \begin{enumerate} \item[(a)] $f$ is Cohen-Macaulay at $x$, and \item[(b)] $\mathcal{O}_{Y, \overline{y}} \to \mathcal{O}_{X, \overline{x}}$ is flat and $\mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{y}}\mathcal{O}_{X, \overline{x}}$ is Cohen-Macaulay. \end{enumerate}


spaces-more-morphisms-lemma-composition-CM

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of algebraic spaces over $S$. Assume that the fibres of $f$, $g$, and $g \circ f$ are locally Noetherian. Let $x \in |X|$ with images $y \in |Y|$ and $z \in |Z|$. \begin{enumerate} \item If $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$, then $g \circ f$ is Cohen-Macaulay at $x$. \item If $f$ and $g$ are Cohen-Macaulay, then $g \circ f$ is Cohen-Macaulay. \item If $g \circ f$ is Cohen-Macaulay at $x$ and $f$ is flat at $x$, then $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$. \item If $f \circ g$ is Cohen-Macaulay and $f$ is flat, then $f$ is Cohen-Macaulay and $g$ is Cohen-Macaulay at every point in the image of $f$. \end{enumerate}


spaces-more-morphisms-lemma-flat-finite-presentation-CM-open

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. Let $$ W = \{x \in |X| : f\text{ is Cohen-Macaulay at }x\} $$ Then $W$ is open in $|X|$ and the formation of $W$ commutes with arbitrary base change of $f$: For any morphism $g : Y' \to Y$, consider the base change $f' : X' \to Y'$ of $f$ and the projection $g' : X' \to X$. Then the corresponding set $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$.


spaces-more-morphisms-lemma-lfp-CM-relative-dimension

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is locally of finite presentation and Cohen-Macaulay. Then there exist open and closed subschemes $X_d \subset X$ such that $X = \coprod_{d \geq 0} X_d$ and $f|_{X_d} : X_d \to Y$ has relative dimension $d$.


spaces-more-morphisms-lemma-gorenstein-local-ring-fibre

The property of morphisms of germs of schemes \begin{align*} & \mathcal{P}((X, x) \to (S, s)) = \\ & \text{the local ring } \mathcal{O}_{X_s, x} \text{ of the fibre is Noetherian and Gorenstein} \end{align*} is \'etale local on the source-and-target (Descent, Definition \ref{descent-definition-local-source-target-at-point}).


spaces-more-morphisms-lemma-gorenstein

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume the fibres of $f$ are locally Noetherian. The following are equivalent \begin{enumerate} \item $f$ is Gorenstein, \item $f$ is flat and for some surjective \'etale morphism $V \to Y$ where $V$ is a scheme, the fibres of $X_V \to V$ are Gorenstein algebraic spaces, and \item $f$ is flat and for any \'etale morphism $V \to Y$ where $V$ is a scheme, the fibres of $X_V \to V$ are Gorenstein algebraic spaces. \end{enumerate} Given $x \in |X|$ with image $y \in |Y|$ the following are equivalent \begin{enumerate} \item[(a)] $f$ is Gorenstein at $x$, and \item[(b)] $\mathcal{O}_{Y, \overline{y}} \to \mathcal{O}_{X, \overline{x}}$ is flat and $\mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{y}}\mathcal{O}_{X, \overline{x}}$ is Gorenstein. \end{enumerate}


spaces-more-morphisms-lemma-composition-gorenstein

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of algebraic spaces over $S$. Assume that the fibres of $f$, $g$, and $g \circ f$ are locally Noetherian. Let $x \in |X|$ with images $y \in |Y|$ and $z \in |Z|$. \begin{enumerate} \item If $f$ is Gorenstein at $x$ and $g$ is Gorenstein at $f(x)$, then $g \circ f$ is Gorenstein at $x$. \item If $f$ and $g$ are Gorenstein, then $g \circ f$ is Gorenstein. \item If $g \circ f$ is Gorenstein at $x$ and $f$ is flat at $x$, then $f$ is Gorenstein at $x$ and $g$ is Gorenstein at $f(x)$. \item If $f \circ g$ is Gorenstein and $f$ is flat, then $f$ is Gorenstein and $g$ is Gorenstein at every point in the image of $f$. \end{enumerate}


spaces-more-morphisms-lemma-flat-finite-presentation-gorenstein-open

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. Let $$ W = \{x \in |X| : f\text{ is Gorenstein at }x\} $$ Then $W$ is open in $|X|$ and the formation of $W$ commutes with arbitrary base change of $f$: For any morphism $g : Y' \to Y$, consider the base change $f' : X' \to Y'$ of $f$ and the projection $g' : X' \to X$. Then the corresponding set $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$.


spaces-more-morphisms-lemma-slice

Let $S$ be a scheme. Consider a cartesian diagram $$ \xymatrix{ X \ar[d] & F \ar[l]^p \ar[d] \\ Y & \Spec(k) \ar[l] } $$ where $X \to Y$ is a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation, and where $k$ is a field over $S$. Let $f_1, \ldots, f_r \in \Gamma(X, \mathcal{O}_X)$ 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 $X$ by an open subspace containing $p(z)$, the morphism $$ V(f_1, \ldots, f_r) \longrightarrow Y $$ is flat and locally of finite presentation.


spaces-more-morphisms-lemma-geometrically-reduced-fibre

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $y \in |Y|$. The following are equivalent \begin{enumerate} \item for some morphism $\Spec(k) \to Y$ in the equivalence class of $y$ the algebraic space $X_k$ is geometrically reduced over $k$, \item for every morphism $\Spec(k) \to Y$ in the equivalence class of $y$ the algebraic space $X_k$ is geometrically reduced over $k$, \item for every morphism $\Spec(k) \to Y$ in the equivalence class of $y$ the algebraic space $X_k$ is reduced. \end{enumerate}


spaces-more-morphisms-lemma-base-change-fibres-geometrically-reduced

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y' \to Y$ be morphisms of algebraic spaces over $S$. Denote $f' : X' \to Y'$ the base change of $f$ by $g$. Then \begin{align*} \{y' \in |Y'| : \text{the fibre of }f' : X' \to Y'\text{ at }y' \text{ is geometrically reduced}\} \\ = g^{-1}(\{y \in |Y| : \text{the fibre of }f : X \to Y\text{ at }y \text{ is geometrically reduced}\}). \end{align*}


spaces-more-morphisms-lemma-geometrically-reduced-constructible

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is quasi-compact and locally of finite presentation. Then the set $$ E = \{y \in |Y| : \text{the fibre of }f : X \to Y\text{ at }y \text{ is geometrically reduced}\} $$ is \'etale locally constructible.


spaces-more-morphisms-lemma-proper-flat-over-dvr-reduced-fibre

Let $X$ be an algebraic space over a discrete valuation ring $R$ whose structure morphism $X \to \Spec(R)$ is proper and flat. If the special fibre is reduced, then both $X$ and the generic fibre $X_\eta$ are reduced.


spaces-more-morphisms-lemma-geometrically-reduced-open

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is flat, proper, and of finite presentation, then the set $$ E = \{y \in |Y| : \text{the fibre of }f : X \to Y\text{ at }y \text{ is geometrically reduced}\} $$ is open in $|Y|$.


spaces-more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $$ n_{X/Y} : |Y| \to \{0, 1, 2, 3, \ldots, \infty\} $$ be the function which associates to $y \in Y$ the number of connected components of $X_k$ where $\Spec(k) \to Y$ is in the equivalence class of $y$ with $k$ algebraically closed. This is well defined and if $g : Y' \to Y$ is a morphism then $$ n_{X'/Y'} = n_{X/Y} \circ g $$ where $X' \to Y'$ is the base change of $f$.


spaces-more-morphisms-lemma-dimension-fibre

Let $S$ be a scheme. Let $f : X \to Y$ be a finite type morphism of algebraic spaces over $S$. Let $y \in |Y|$. The following quantities are the same \begin{enumerate} \item the minimal integer $d$ such that $f$ has relative dimension $\leq d$ at every $x \in |X|$ mapping to $y$, \item the dimension of the algebraic space $X_k = \Spec(k) \times_Y X$ for any morphism $\Spec(k) \to Y$ in the equivalence class defining $y$. \end{enumerate}


spaces-more-morphisms-lemma-base-change-dimension-fibres

Let $S$ be a scheme. Let $f : X \to Y$ be a finite type morphism of algebraic spaces over $S$. Let $$ n_{X/Y} : |Y| \to \{0, 1, 2, 3, \ldots, \infty\} $$ be the function which associates to $y \in |Y|$ the integer discussed in Lemma \ref{lemma-dimension-fibre}. If $g : Y' \to Y$ is a morphism then $$ n_{X'/Y'} = n_{X/Y} \circ |g| $$ where $X' \to Y'$ is the base change of $f$.


spaces-more-morphisms-lemma-dimension-fibres-flat

Let $S$ be a scheme. Let $f : X \to Y$ be a flat morphism of finite presentation of algebraic spaces over $S$. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma \ref{lemma-base-change-dimension-fibres}. Then $n_{X/Y}$ is lower semi-continuous.


spaces-more-morphisms-lemma-dimension-fibres-proper

Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $S$. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma \ref{lemma-base-change-dimension-fibres}. Then $n_{X/Y}$ is upper semi-continuous.


spaces-more-morphisms-lemma-dimension-fibres-proper-flat

Let $S$ be a scheme. Let $f : X \to Y$ be a proper, flat, finitely presented morphism of algebraic spaces over $S$. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma \ref{lemma-base-change-dimension-fibres}. Then $n_{X/Y}$ is locally constant.


spaces-more-morphisms-lemma-construct-glueing

Let $S$ be a scheme. Let $f : X \to Y$ be an integral morphism of algebraic spaces over $S$. Let $y \in |Y|$ be a point which can be represented by a closed immersion $y : \Spec(k) \to Y$. Then there exists a factorization $X \to X' \to Y$ of $f$ such that \begin{enumerate} \item $X' \to Y$ is integral, \item $X \to X'$ is an isomorphism over $X' \setminus X'_y$, \item $X'_y$ has a unique point $x'$ with $\kappa(x') = k$. \end{enumerate} Moreover, if $f$ is finite and $Y$ is locally Noetherian, then $X' \to Y$ is finite.


spaces-more-morphisms-lemma-etale-splits-off-quasi-finite-part

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $y \in |Y|$. Let $x_1, \ldots, x_n \in |X|$ mapping to $y$. Assume that \begin{enumerate} \item $f$ is locally of finite type, \item $f$ is separated, \item $f$ is quasi-finite at $x_1, \ldots, x_n$, and \item $f$ is quasi-compact or $Y$ is decent. \end{enumerate} Then there exists an \'etale morphism $(U, u) \to (Y, y)$ of pointed algebraic spaces and a decomposition $$ U \times_Y X = W \amalg V $$ into open and closed subspaces such that the morphism $V \to U$ is finite, every point of the fibre of $|V| \to |U|$ over $u$ maps to an $x_i$, and the fibre of $|W| \to |U|$ over $u$ contains no point mapping to an $x_i$.


spaces-more-morphisms-lemma-finite-type-separated

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is of finite type and separated. Let $Y'$ be the normalization of $Y$ in $X$. Picture: $$ \xymatrix{ X \ar[rd]_f \ar[rr]_{f'} & & Y' \ar[ld]^\nu \\ & Y & } $$ Then there exists an open subspace $U' \subset Y'$ such that \begin{enumerate} \item $(f')^{-1}(U') \to U'$ is an isomorphism, and \item $(f')^{-1}(U') \subset X$ is the set of points at which $f$ is quasi-finite. \end{enumerate}


spaces-more-morphisms-lemma-quasi-finite-separated-quasi-affine

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-finite and separated. Let $Y'$ be the normalization of $Y$ in $X$. Picture: $$ \xymatrix{ X \ar[rd]_f \ar[rr]_{f'} & & Y' \ar[ld]^\nu \\ & Y & } $$ Then $f'$ is a quasi-compact open immersion and $\nu$ is integral. In particular $f$ is quasi-affine.


spaces-more-morphisms-lemma-quasi-finite-separated-pass-through-finite

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-finite and separated and assume that $Y$ is quasi-compact and quasi-separated. Then there exists a factorization $$ \xymatrix{ X \ar[rd]_f \ar[rr]_j & & T \ar[ld]^\pi \\ & Y & } $$ where $j$ is a quasi-compact open immersion and $\pi$ is finite.


spaces-more-morphisms-lemma-quasi-finite-separated-pass-through-finite-addendum

With notation and hypotheses as in Lemma \ref{lemma-quasi-finite-separated-pass-through-finite}. Assume moreover that $f$ is locally of finite presentation. Then we can choose the factorization such that $T$ is finite and of finite presentation over $Y$.


spaces-more-morphisms-lemma-characterize-finite

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is finite, \item $f$ is proper and locally quasi-finite, \item $f$ is proper and $|X_k|$ is a discrete space for every morphism $\Spec(k) \to Y$ where $k$ is a field, \item $f$ is universally closed, separated, locally of finite type and $|X_k|$ is a discrete space for every morphism $\Spec(k) \to Y$ where $k$ is a field. \end{enumerate}


spaces-more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $y \in |Y|$. Assume \begin{enumerate} \item $f$ is proper, and \item $f$ is quasi-finite at all $x \in |X|$ lying over $y$ (Decent Spaces, Lemma \ref{decent-spaces-lemma-conditions-on-fibre-and-qf}). \end{enumerate} Then there exists an open neighbourhood $V \subset Y$ of $y$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is finite.


spaces-more-morphisms-lemma-flat-proper-family-cannot-collapse-fibre

\begin{slogan} Collapsing a fibre of a proper family forces nearby ones to collapse too. \end{slogan} Let $S$ be a scheme. Let $$ \xymatrix{ X \ar[rr]_h \ar[rd]_f & & Y \ar[ld]^g \\ & B } $$ be a commutative diagram of morphism of algebraic spaces over $S$. Let $b \in B$ and let $\Spec(k) \to B$ be a morphism in the equivalence class of $b$. Assume \begin{enumerate} \item $X \to B$ is a proper morphism, \item $Y \to B$ is separated and locally of finite type, \item one of the following is true \begin{enumerate} \item the image of $|X_k| \to |Y_k|$ is finite, \item the image of $|f|^{-1}(\{b\})$ in $|Y|$ is finite and $B$ is decent. \end{enumerate} \end{enumerate} Then there is an open subspace $B' \subset B$ containing $b$ such that $X_{B'} \to Y_{B'}$ factors through a closed subspace $Z \subset Y_{B'}$ finite over $B'$.


spaces-more-morphisms-lemma-stein-universally-closed

Let $S$ be a scheme. Let $f : X \to Y$ be a universally closed and quasi-separated morphism of algebraic spaces over $S$. There exists a factorization $$ \xymatrix{ X \ar[rr]_{f'} \ar[rd]_f & & Y' \ar[dl]^\pi \\ & Y & } $$ with the following properties: \begin{enumerate} \item the morphism $f'$ is universally closed, quasi-compact, quasi-separated, and surjective, \item the morphism $\pi : Y' \to Y$ is integral, \item we have $f'_*\mathcal{O}_X = \mathcal{O}_{Y'}$, \item we have $Y' = \underline{\Spec}_Y(f_*\mathcal{O}_X)$, and \item $Y'$ is the normalization of $Y$ in $X$ as defined in Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-normalization-X-in-Y}. \end{enumerate} Formation of the factorization $f = \pi \circ f'$ commutes with flat base change.


spaces-more-morphisms-lemma-stein-universally-closed-residue-fields

In Lemma \ref{lemma-stein-universally-closed} assume in addition that $f$ is locally of finite type and $Y$ affine. Then for $y \in Y$ the fibre $\pi^{-1}(\{y\}) = \{y_1, \ldots, y_n\}$ is finite and the field extensions $\kappa(y_i)/\kappa(y)$ are finite.


spaces-more-morphisms-lemma-characterize-geometrically-connected-fibres

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\overline{y}$ be a geometric point of $Y$. Then $X_{\overline{y}}$ is connected, if and only if for every \'etale neighbourhood $(V, \overline{v}) \to (Y, \overline{y})$ where $V$ is a scheme the base change $X_V \to V$ has connected fibre $X_v$.


spaces-more-morphisms-lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous

Let $S$ be a scheme. Let $X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is proper, flat, and of finite presentation, then the function $n_{X/Y} : |Y| \to \mathbf{Z}$ counting the number of geometric connected components of fibres of $f$ (Lemma \ref{lemma-base-change-fibres-nr-geometrically-connected-components}) is lower semi-continuous.


spaces-more-morphisms-lemma-proper-flat-geom-red

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $f$ is proper, flat, and of finite presentation, and \item the geometric fibres of $f$ are reduced. \end{enumerate} Then the function $n_{X/S} : |Y| \to \mathbf{Z}$ counting the numbers of geometric connected components of fibres of $f$ (Lemma \ref{lemma-base-change-fibres-nr-geometrically-connected-components}) is locally constant.


spaces-more-morphisms-lemma-stein-factorization-etale

Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $S$. Let $X \to Y' \to Y$ be the Stein factorization of $f$ (Theorem \ref{theorem-stein-factorization-general}). If $f$ is of finite presentation, flat, with geometrically reduced fibres (Definition \ref{definition-geometrically-reduced-fibre}), then $Y' \to Y$ is finite \'etale.


spaces-more-morphisms-lemma-split-off-proper-part-henselian

Let $(A, I)$ be a henselian pair. Let $X$ be an algebraic space separated and of finite type over $A$. Set $X_0 = X \times_{\Spec(A)} \Spec(A/I)$. Let $Y \subset X_0$ be an open and closed subspace such that $Y \to \Spec(A/I)$ is proper. Then there exists an open and closed subspace $W \subset X$ which is proper over $A$ with $W \times_{\Spec(A)} \Spec(A/I) = Y$.


spaces-more-morphisms-lemma-flat-finite-type-finitely-presented-over-dense-open

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $V \subset Y$ be an open subspace. Assume \begin{enumerate} \item $f$ is locally of finite presentation, \item $\mathcal{F}$ is of finite type and flat over $Y$, \item $V \to Y$ is quasi-compact and scheme theoretically dense, \item $\mathcal{F}|_{f^{-1}V}$ is of finite presentation. \end{enumerate} Then $\mathcal{F}$ is of finite presentation.


spaces-more-morphisms-lemma-flat-finite-type-finitely-presented-over-dense-open-X

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $V \subset Y$ be an open subspace. Assume \begin{enumerate} \item $f$ is locally of finite type and flat, \item $V \to Y$ is quasi-compact and scheme theoretically dense, \item $f|_{f^{-1}V} : f^{-1}V \to V$ is locally of finite presentation. \end{enumerate} Then $f$ is of locally of finite presentation.


spaces-more-morphisms-lemma-flat-fp-dimension-over-dense-open

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite type. Let $V \subset Y$ be an open subspace such that $|V| \subset |Y|$ is dense and such that $X_V \to V$ has relative dimension $\leq d$. If also either \begin{enumerate} \item $f$ is locally of finite presentation, or \item $V \to Y$ is quasi-compact, \end{enumerate} then $f : X \to Y$ has relative dimension $\leq d$.


spaces-more-morphisms-lemma-proper-flat-finite-over-dense-open

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and proper. Let $V \to Y$ be an open subspace with $|V| \subset |Y|$ dense such that $X_V \to V$ is finite. If also either $f$ is locally of finite presentation or $V \to Y$ is quasi-compact, then $f$ is finite.


spaces-more-morphisms-lemma-zariski

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $V \subset Y$ be an open subspace. If \begin{enumerate} \item $f$ is separated, locally of finite type, and flat, \item $f^{-1}(V) \to V$ is an isomorphism, and \item $V \to Y$ is quasi-compact and scheme theoretically dense, \end{enumerate} then $f$ is an open immersion.


spaces-more-morphisms-lemma-push-ideal

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\varphi : W \to X$ be a quasi-compact separated \'etale morphism. Let $U \subset X$ be a quasi-compact open subspace. Let $\mathcal{I} \subset \mathcal{O}_U$ be a finite type quasi-coherent sheaf of ideals such that $V(\mathcal{I}) \cap \varphi^{-1}(U) = \emptyset$. Then there exists a finite type quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_X$ such that \begin{enumerate} \item $V(\mathcal{J}) \cap U = \emptyset$, and \item $\varphi^{-1}(\mathcal{J})\mathcal{O}_W = \mathcal{I} \mathcal{I}'$ for some finite type quasi-coherent ideal $\mathcal{I}' \subset \mathcal{O}_W$. \end{enumerate}


spaces-more-morphisms-lemma-flat-after-blowing-up

Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $X$ be an algebraic space over $B$. Let $U \subset B$ be a quasi-compact open subspace. Assume \begin{enumerate} \item $X \to B$ is of finite type and quasi-separated, and \item $X_U \to U$ is flat and locally of finite presentation. \end{enumerate} Then there exists a $U$-admissible blowup $B' \to B$ such that the strict transform of $X$ is flat and of finite presentation over $B'$.


spaces-more-morphisms-lemma-zariski-after-blowup

Let $S$ be a scheme. Let $\varphi : X \to B$ be a morphism of algebraic spaces over $S$. Assume $\varphi$ is of finite type with $B$ quasi-compact and quasi-separated. Let $U \subset B$ be a quasi-compact open subspace such that $\varphi^{-1}U \to U$ is an isomorphism. Then there exists a $U$-admissible blowup $B' \to B$ such that $U$ is scheme theoretically dense in $B'$ and such that the strict transform $X'$ of $X$ is isomorphic to an open subspace of $B'$.


spaces-more-morphisms-lemma-dominate-modification-by-blowup

Let $S$ be a scheme. Let $\varphi : X \to B$ be a proper morphism of algebraic spaces over $S$. Assume $B$ quasi-compact and quasi-separated. Let $U \subset B$ be a quasi-compact open subspace such that $\varphi^{-1}U \to U$ is an isomorphism. Then there exists a $U$-admissible blowup $B' \to B$ which dominates $X$, i.e., such that there exists a factorization $B' \to X \to B$ of the blowup morphism.


spaces-more-morphisms-lemma-find-common-blowups

Let $S$ be a scheme. Let $Y$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U \to X_1$ and $U \to X_2$ be open immersions of algebraic spaces over $Y$ and assume $U$, $X_1$, $X_2$ of finite type and separated over $Y$. Then there exists a commutative diagram $$ \xymatrix{ X_1' \ar[d] \ar[r] & X & X_2' \ar[l] \ar[d] \\ X_1 & U \ar[l] \ar[lu] \ar[u] \ar[ru] \ar[r] & X_2 } $$ of algebraic spaces over $Y$ where $X_i' \to X_i$ is a $U$-admissible blowup, $X_i' \to X$ is an open immersion, and $X$ is separated and finite type over $Y$.


spaces-more-morphisms-lemma-blowup-to-find-embedding

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $U \subset X$ be an open subspace. Assume \begin{enumerate} \item $U$ is quasi-compact, \item $Y$ is quasi-compact and quasi-separated, \item there exists an immersion $U \to \mathbf{P}^n_Y$ over $Y$, \item $f$ is of finite type and separated. \end{enumerate} Then there exists a commutative diagram $$ \xymatrix{ & U \ar[ld] \ar[d] \ar[rd] \ar[rrd] \\ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & Z' \ar[ld] \ar[r] & Z \ar[ld] \\ & Y & \mathbf{P}^n_Y \ar[l] } $$ where the arrows with source $U$ are open immersions, $X' \to X$ is a $U$-admissible blowup, $X' \to Z'$ is an open immersion, $Z' \to Y$ is a proper and representable morphism of algebraic spaces. More precisely, $Z' \to Z$ is a $U$-admissible blowup and $Z \to \mathbf{P}^n_Y$ is a closed immersion.


spaces-more-morphisms-lemma-chow-noetherian

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ separated, of finite type, and $Y$ Noetherian. Then there exists a dense open subspace $U \subset X$ and a commutative diagram $$ \xymatrix{ & U \ar[ld] \ar[d] \ar[rd] \ar[rrd] \\ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & Z' \ar[ld] \ar[r] & Z \ar[ld] \\ & Y & \mathbf{P}^n_Y \ar[l] } $$ where the arrows with source $U$ are open immersions, $X' \to X$ is a $U$-admissible blowup, $X' \to Z'$ is an open immersion, $Z' \to Y$ is a proper and representable morphism of algebraic spaces. More precisely, $Z' \to Z$ is a $U$-admissible blowup and $Z \to \mathbf{P}^n_Y$ is a closed immersion.


spaces-more-morphisms-lemma-chow-noetherian-separated

\begin{reference} \cite[IV Theorem 3.1]{Kn} \end{reference} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ separated of finite type, and $Y$ separated and Noetherian. Then there exists a commutative diagram $$ \xymatrix{ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & \mathbf{P}^n_Y \ar[ld] \\ & Y } $$ where $X' \to X$ is a $U$-admissible blowup for some dense open $U \subset X$ and the morphism $X' \to \mathbf{P}^n_Y$ is an immersion.


spaces-more-morphisms-lemma-inverse-systems-abelian

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. \begin{enumerate} \item The category $\textit{Coh}(X, \mathcal{I})$ is abelian. \item Exactness in $\textit{Coh}(X, \mathcal{I})$ can be checked \'etale locally. \item For any flat morphism $f : X' \to X$ of Noetherian algebraic spaces the functor $f^* : \textit{Coh}(X, \mathcal{I}) \to \textit{Coh}(X', f^{-1}\mathcal{I}\mathcal{O}_{X'})$ is exact. \end{enumerate}


spaces-more-morphisms-lemma-exact

The functor (\ref{equation-completion-functor}) is exact.


spaces-more-morphisms-lemma-completion-internal-hom

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. Set $\mathcal{H} = \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$. Then $$ \lim H^0(X, \mathcal{H}/\mathcal{I}^n\mathcal{H}) = \Mor_{\textit{Coh}(X, \mathcal{I})} (\mathcal{F}^\wedge, \mathcal{G}^\wedge). $$


spaces-more-morphisms-lemma-fully-faithful

In Situation \ref{situation-existence}. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. Assume that the intersection of the supports of $\mathcal{F}$ and $\mathcal{G}$ is proper over $\Spec(A)$. Then the map $$ \Mor_{\textit{Coh}(\mathcal{O}_X)}(\mathcal{F}, \mathcal{G}) \longrightarrow \Mor_{\textit{Coh}(X, \mathcal{I})} (\mathcal{F}^\wedge, \mathcal{G}^\wedge) $$ coming from (\ref{equation-completion-functor}) is a bijection. In particular, (\ref{equation-completion-functor-proper-over-A}) is fully faithful.


spaces-more-morphisms-lemma-existence-easy

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{G}$ be a coherent $\mathcal{O}_X$-module, $(\mathcal{F}_n)$ an object of $\textit{Coh}(X, \mathcal{I})$, and $\alpha : (\mathcal{F}_n) \to \mathcal{G}^\wedge$ a map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$. Then there exists a unique (up to unique isomorphism) triple $(\mathcal{F}, a, \beta)$ where \begin{enumerate} \item $\mathcal{F}$ is a coherent $\mathcal{O}_X$-module, \item $a : \mathcal{F} \to \mathcal{G}$ is an $\mathcal{O}_X$-module map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$, \item $\beta : (\mathcal{F}_n) \to \mathcal{F}^\wedge$ is an isomorphism, and \item $\alpha = a^\wedge \circ \beta$. \end{enumerate}


spaces-more-morphisms-lemma-existence-tricky

In Situation \ref{situation-existence}. Let $\mathcal{K} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $X_e \subset X$ be the closed subspace cut out by $\mathcal{K}^e$. Let $\mathcal{I}_e = \mathcal{I}\mathcal{O}_{X_e}$. Let $(\mathcal{F}_n)$ be an object of $\textit{Coh}_{\text{support proper over } A}(X, \mathcal{I})$. Assume \begin{enumerate} \item the functor $\textit{Coh}_{\text{support proper over } A}(\mathcal{O}_{X_e}) \to \textit{Coh}_{\text{support proper over } A}(X_e, \mathcal{I}_e)$ is an equivalence for all $e \geq 1$, and \item there exists an object $\mathcal{H}$ of $\textit{Coh}_{\text{support proper over } A}(\mathcal{O}_X)$ and a map $\alpha : (\mathcal{F}_n) \to \mathcal{H}^\wedge$ whose kernel and cokernel are annihilated by a power of $\mathcal{K}$. \end{enumerate} Then $(\mathcal{F}_n)$ is in the essential image of (\ref{equation-completion-functor-proper-over-A}).


spaces-more-morphisms-lemma-inverse-systems-push-pull

Let $S$ be a scheme. Let $f : X \to Y$ be a representable proper morphism of Noetherian algebraic spaces over $S$. Let $\mathcal{J}, \mathcal{K} \subset \mathcal{O}_Y$ be quasi-coherent sheaves of ideals. Assume $f$ is an isomorphism over $V = Y \setminus V(\mathcal{K})$. Set $\mathcal{I} = f^{-1}\mathcal{J} \mathcal{O}_X$. Let $(\mathcal{G}_n)$ be an object of $\textit{Coh}(Y, \mathcal{J})$, let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module, and let $\beta : (f^*\mathcal{G}_n)  \to \mathcal{F}^\wedge$ be an isomorphism in $\textit{Coh}(X, \mathcal{I})$. Then there exists a map $$ \alpha : (\mathcal{G}_n) \longrightarrow (f_*\mathcal{F})^\wedge $$ in $\textit{Coh}(Y, \mathcal{J})$ whose kernel and cokernel are annihilated by a power of $\mathcal{K}$.


spaces-more-morphisms-lemma-algebraize-formal-closed-subscheme

Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \Spec(A)$ and $S_n = \Spec(A/I^n)$. Let $X \to S$ be a morphism of algebraic spaces that is separated and of finite type. For $n \geq 1$ we set $X_n = X \times_S S_n$. Suppose given a commutative diagram $$ \xymatrix{ Z_1 \ar[r] \ar[d] & Z_2 \ar[r] \ar[d] & Z_3 \ar[r] \ar[d] & \ldots \\ X_1 \ar[r]^{i_1} & X_2 \ar[r]^{i_2} & X_3 \ar[r] & \ldots } $$ of algebraic spaces with cartesian squares. Assume that \begin{enumerate} \item $Z_1 \to X_1$ is a closed immersion, and \item $Z_1 \to S_1$ is proper. \end{enumerate} Then there exists a closed immersion of algebraic spaces $Z \to X$ such that $Z_n = Z \times_S S_n$ for all $n \geq 1$. Moreover, $Z$ is proper over $S$.


spaces-more-morphisms-lemma-algebraize-morphism

Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \Spec(A)$ and $S_n = \Spec(A/I^n)$. Let $X$, $Y$ be algebraic spaces over $S$. For $n \geq 1$ we set $X_n = X \times_S S_n$ and $Y_n = Y \times_S S_n$. Suppose given a compatible system of commutative diagrams $$ \xymatrix{ & & X_{n + 1} \ar[rd] \ar[rr]_{g_{n + 1}} & & Y_{n + 1} \ar[ld] \\ X_n \ar[rru] \ar[rd] \ar[rr]_{g_n} & & Y_n \ar[rru] \ar[ld] & S_{n + 1} \\ & S_n \ar[rru] } $$ Assume that \begin{enumerate} \item $X \to S$ is proper, and \item $Y \to S$ is separated of finite type. \end{enumerate} Then there exists a unique morphism of algebraic spaces $g : X \to Y$ over $S$ such that $g_n$ is the base change of $g$ to $S_n$.


spaces-more-morphisms-lemma-formal-algebraic-space-proper-reldim-1

Let $(A, \mathfrak m, \kappa)$ be a complete local Noetherian ring. Set $S = \Spec(A)$ and $S_n = \Spec(A/\mathfrak m^n)$. Consider a commutative diagram $$ \xymatrix{ X_1 \ar[r]_{i_1} \ar[d] & X_2 \ar[r]_{i_2} \ar[d] & X_3 \ar[r] \ar[d] & \ldots \\ S_1 \ar[r] & S_2 \ar[r] & S_3 \ar[r] & \ldots } $$ of algebraic spaces with cartesian squares. If $\dim(X_1) \leq 1$, then there exists a projective morphism of schemes $X \to S$ and isomorphisms $X_n \cong X \times_S S_n$ compatible with $i_n$.


spaces-more-morphisms-lemma-projective-over-complete

Let $(A, \mathfrak m, \kappa)$ be a complete Noetherian local ring. Let $X$ be an algebraic space over $\Spec(A)$. If $X \to \Spec(A)$ is proper and $\dim(X_\kappa) \leq 1$, then $X$ is a scheme projective over $A$.


spaces-more-morphisms-lemma-representable-etale-local-target

Let $\mathcal{P}$ be a property of morphisms of schemes which is \'etale local on the target. Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. Consider commutative diagrams $$ \xymatrix{ X \times_Y V \ar[d] \ar[r] & V \ar[d] \\ X \ar[r]^f & Y } $$ where $V$ is a scheme and $V \to Y$ is \'etale. The following are equivalent \begin{enumerate} \item for any diagram as above the projection $X \times_Y V \to V$ has property $\mathcal{P}$, and \item for some diagram as above with $V \to Y$ surjective the projection $X \times_Y V \to V$ has property $\mathcal{P}$. \end{enumerate} If $X$ and $Y$ are representable, then this is also equivalent to $f$ (as a morphism of schemes) having property $\mathcal{P}$.


spaces-more-morphisms-lemma-regular-quasi-regular-immersion

Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. We have the following implications: $i$ is Koszul-regular $\Rightarrow$ $i$ is $H_1$-regular $\Rightarrow$ $i$ is quasi-regular.


spaces-more-morphisms-lemma-quasi-regular-immersion

Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. Then $i$ is a quasi-regular immersion if and only if the following conditions are satisfied \begin{enumerate} \item $i$ is locally of finite presentation, \item the conormal sheaf $\mathcal{C}_{Z/X}$ is finite locally free, and \item the map (\ref{equation-conormal-algebra-quotient}) is an isomorphism. \end{enumerate}


spaces-more-morphisms-lemma-qcoh-relative-pseudo-coherence-characterize

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $m \in \mathbf{Z}$. Let $E \in D_\QCoh(\mathcal{O}_X)$. With notation as explained in Remark \ref{remark-match-relative-pseudo-coherence} the following are equivalent: \begin{enumerate} \item for every commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are schemes and the vertical arrows are \'etale, the complex $E|_U$ is $m$-pseudo-coherent relative to $V$, \item for some commutative diagram as in (1) with $U \to X$ surjective, the complex $E|_U$ is $m$-pseudo-coherent relative to $V$, \item for every commutative diagram as in (1) with $U$ and $V$ affine the complex $R\Gamma(U, E)$ of $\mathcal{O}_X(U)$-modules is $m$-pseudo-coherent relative to $\mathcal{O}_Y(V)$. \end{enumerate}