spaces-limits-lemma-closed-is-limit-closed-and-finite-presentation

\begin{slogan} Closed immersions of qcqs algebraic spaces can be approximated by finitely presented closed immersions. \end{slogan} Let $S$ be a scheme. Let $f : X \to Y$ be a closed immersion of algebraic spaces over $S$. Assume $Y$ quasi-compact and quasi-separated. Then $X$ can be written as a directed limit $X = \lim X_i$ where the transition maps are closed immersions and the morphisms $X_i \to Y$ are closed immersions of finite presentation.


spaces-limits-lemma-quasi-affine-closed-in-finite-presentation

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $f$ is locally of finite type and quasi-affine, and \item $Y$ is quasi-compact and quasi-separated. \end{enumerate} Then there exists a morphism of finite presentation $f' : X' \to Y$ and a closed immersion $X \to X'$ over $Y$.


spaces-limits-lemma-finite-type-closed-in-finite-presentation

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume: \begin{enumerate} \item $f$ is of locally of finite type. \item $X$ is quasi-compact and quasi-separated, and \item $Y$ is quasi-compact and quasi-separated. \end{enumerate} Then there exists a morphism of finite presentation $f' : X' \to Y$ and a closed immersion $X \to X'$ of algebraic spaces over $Y$.


spaces-limits-lemma-proper-limit-of-proper-finite-presentation

Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $S$ with $Y$ quasi-compact and quasi-separated. Then $X = \lim X_i$ is a directed limit of algebraic spaces $X_i$ proper and of finite presentation over $Y$ and with transition morphisms and morphisms $X \to X_i$ closed immersions.


spaces-limits-lemma-proper-limit-of-proper-finite-presentation-noetherian

Let $f : X \to Y$ be a proper morphism of algebraic spaces over $\mathbf{Z}$ with $Y$ quasi-compact and quasi-separated. Then there exists a directed set $I$, an inverse system $(f_i : X_i \to Y_i)$ of morphisms of algebraic spaces over $I$, such that the transition morphisms $X_i \to X_{i'}$ and $Y_i \to Y_{i'}$ are affine, such that $f_i$ is proper and of finite presentation, such that $Y_i$ is of finite presentation over $\mathbf{Z}$, and such that $(X \to Y) = \lim (X_i \to Y_i)$.


spaces-limits-lemma-eventually-proper-support

Assumptions and notation as in Situation \ref{situation-descent-property}. Let $\mathcal{F}_0$ be a quasi-coherent $\mathcal{O}_{X_0}$-module. Denote $\mathcal{F}$ and $\mathcal{F}_i$ the pullbacks of $\mathcal{F}_0$ to $X$ and $X_i$. Assume \begin{enumerate} \item $f_0$ is locally of finite type, \item $\mathcal{F}_0$ is of finite type, \item the scheme theoretic support of $\mathcal{F}$ is proper over $Y$. \end{enumerate} Then the scheme theoretic support of $\mathcal{F}_i$ is proper over $Y_i$ for some $i$.


spaces-limits-lemma-embedding-into-affine-over-ls-qs

Let $S$ be a scheme. Let $f : U \to X$ be a morphism of algebraic spaces over $S$. Assume $U$ is an affine scheme, $f$ is locally of finite type, and $X$ quasi-separated and locally separated. Then there exists an immersion $U \to \mathbf{A}^n_X$ over $X$.


spaces-limits-lemma-embedding-into-affine-over-qs

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Assume $X$ Noetherian and $f$ of finite presentation. Then there exists a dense open $V \subset Y$ and an immersion $V \to \mathbf{A}^n_X$.


spaces-limits-lemma-quasi-coherent-finite-type-ideals

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space. Let $U \subset X$ be an open subspace. The following are equivalent: \begin{enumerate} \item $U \to X$ is quasi-compact, \item $U$ is quasi-compact, and \item there exists a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ such that $|X| \setminus |U| = |V(\mathcal{I})|$. \end{enumerate}


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

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Consider the sheaf of $\mathcal{O}_X$-modules $\mathcal{F}'$ which associates to every object $U$ of $X_\etale$ the module $$ \mathcal{F}'(U) = \{s \in \mathcal{F}(U) \mid \mathcal{I}s = 0\} $$ Assume $\mathcal{I}$ is of finite type. Then \begin{enumerate} \item $\mathcal{F}'$ is a quasi-coherent sheaf of $\mathcal{O}_X$-modules, \item for affine $U$ in $X_\etale$ we have $\mathcal{F}'(U) = \{s \in \mathcal{F}(U) \mid \mathcal{I}(U)s = 0\}$, and \item $\mathcal{F}'_x = \{s \in \mathcal{F}_x \mid \mathcal{I}_x s = 0\}$. \end{enumerate}


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

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset and let $U \subset X$ be the open subspace such that $T \amalg |U| = |X|$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Consider the sheaf of $\mathcal{O}_X$-modules $\mathcal{F}'$ which associates to every object $\varphi : W \to X$ of $X_\etale$ the module $$ \mathcal{F}'(W) = \{s \in \mathcal{F}(W) \mid \text{the support of }s\text{ is contained in }|\varphi|^{-1}(T)\} $$ If $U \to X$ is quasi-compact, then \begin{enumerate} \item for $W$ affine there exist a finitely generated ideal $I \subset \mathcal{O}_X(W)$ such that $|\varphi|^{-1}(T) = V(I)$, \item for $W$ and $I$ as in (1) we have $\mathcal{F}'(W) = \{x \in \mathcal{F}(W) \mid I^nx = 0 \text{ for some } n\}$, \item $\mathcal{F}'$ is a quasi-coherent sheaf of $\mathcal{O}_X$-modules. \end{enumerate}


spaces-limits-lemma-affine

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is surjective and finite, and assume that $X$ is affine. Then $Y$ is affine.


spaces-limits-lemma-reduction-scheme

\begin{reference} \cite[3.1.12]{CLO} \end{reference} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X_{red}$ is a scheme, then $X$ is a scheme.


spaces-limits-lemma-integral-universally-bijective-scheme

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is integral and induces a bijection $|X| \to |Y|$. Then $X$ is a scheme if and only if $Y$ is a scheme.


spaces-limits-lemma-check-closed-infinitesimally

Let $S$ be a scheme. Let $f : X \to B$ and $B' \to B$ be morphisms of algebraic spaces over $S$. Assume \begin{enumerate} \item $B' \to B$ is a closed immersion, \item $|B'| \to |B|$ is bijective, \item $X \times_B B' \to B'$ is a closed immersion, and \item $X \to B$ is of finite type or $B' \to B$ is of finite presentation. \end{enumerate} Then $f : X \to B$ is a closed immersion.


spaces-limits-lemma-enough-local

Let $f: X \to S$ be a quasi-compact and quasi-separated morphism from an algebraic space to a scheme $S$. If for every $x \in |X|$ with image $s = f(x) \in S$ the algebraic space $X \times_S \Spec(\mathcal{O}_{S,s})$ is a scheme, then $X$ is a scheme.


spaces-limits-lemma-relative-glueing

Let $S = U \cup W$ be an open covering of a scheme. Then the functor $$ FP_S \longrightarrow FP_U \times_{FP_{U \cap W}} FP_W $$ given by base change is an equivalence where $FP_T$ is the category of algebraic spaces of finite presentation over the scheme $T$.


spaces-limits-lemma-glueing-near-closed-point

Let $S$ be a scheme. Let $s \in S$ be a closed point such that $U = S \setminus \{s\} \to S$ is quasi-compact. With $V = \Spec(\mathcal{O}_{S, s}) \setminus \{s\}$ there is an equivalence of categories $$ FP_S \longrightarrow FP_U \times_{FP_V} FP_{\Spec(\mathcal{O}_{S, s})} $$ where $FP_T$ is the category of algebraic spaces of finite presentation over $T$.


spaces-limits-lemma-glueing-near-multiple-closed-points

Let $S$ be a scheme. Let $s_1, \ldots, s_n \in S$ be pairwise distinct closed points such that $U = S \setminus \{s_1, \ldots, s_n\} \to S$ is quasi-compact. With $S_i = \Spec(\mathcal{O}_{S, s_i})$ and $U_i = S_i \setminus \{s_i\}$ there is an equivalence of categories $$ FP_S \longrightarrow FP_U \times_{(FP_{U_1} \times \ldots \times FP_{U_n})} (FP_{S_1} \times \ldots \times FP_{S_n}) $$ where $FP_T$ is the category of algebraic spaces of finite presentation over $T$.


spaces-limits-lemma-excision-modifications

Let $S$ be a scheme. Consider a separated \'etale morphism $f : V \to W$ of algebraic spaces over $S$. Assume there exists a closed subspace $T \subset W$ such that $f^{-1}T \to T$ is an isomorphism. Then, with $W^0 = W \setminus T$ and $V^0 = f^{-1}W^0$ the base change functor $$ \left\{ \begin{matrix} g : X \to W\text{ morphism of algebraic spaces} \\ g^{-1}(W^0) \to W^0\text{ is an isomorphism} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} h : Y \to V\text{ morphism of algebraic spaces} \\ h^{-1}(V^0) \to V^0\text{ is an isomorphism} \end{matrix} \right\} $$ is an equivalence of categories.


spaces-limits-lemma-excision-modifications-properties

Notation and assumptions as in Lemma \ref{lemma-excision-modifications}. Let $g : X \to W$ correspond to $h : Y \to V$ via the equivalence. Then $g$ is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, integral, finite, and add more here if and only if $h$ is so.


spaces-limits-lemma-separate

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Z \to Y$ be morphisms of algebraic spaces over $S$. Let $z \in |Z|$ and let $T \subset |X \times_Y Z|$ be a closed subset with $z \not \in \Im(T \to |Z|)$. If $f$ is quasi-compact, then there exists an \'etale neighbourhood $(V, v) \to (Z, z)$, a commutative diagram $$ \xymatrix{ V \ar[d] \ar[r]_a & Z' \ar[d]^b \\ Z \ar[r]^g & Y, } $$ and a closed subset $T' \subset |X \times_Y Z'|$ such that \begin{enumerate} \item the morphism $b : Z' \to Y$ is locally of finite presentation, \item with $z' = a(v)$ we have $z' \not \in \Im(T' \to |Z'|)$, and \item the inverse image of $T$ in $|X \times_Y V|$ maps into $T'$ via $|X \times_Y V| \to |X \times_Y Z'|$. \end{enumerate} Moreover, we may assume $V$ and $Z'$ are affine schemes and if $Z$ is a scheme we may assume $V$ is an affine open neighbourhood of $z$.


spaces-limits-lemma-test-universally-closed

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact morphism of algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is universally closed, \item for every morphism $Z \to Y$ which is locally of finite presentation the map $|X \times_Y Z| \to |Z|$ is closed, and \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that $|\mathbf{A}^n \times (X \times_Y V)| \to |\mathbf{A}^n \times V|$ is closed for all $n \geq 0$. \end{enumerate}


spaces-limits-lemma-refined-valuative-criterion-proper

Let $S$ be a scheme. Let $f : X \to Y$ and $h : U \to X$ be morphisms of algebraic spaces over $S$. Assume that $Y$ is locally Noetherian, that $f$ and $h$ are of finite type, that $f$ is separated, and that the image of $|h| : |U| \to |X|$ is dense in $|X|$. If given any commutative solid diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & U \ar[r]^h & X \ar[d]^f \\ \Spec(A) \ar[rr] \ar@{-->}[rru] & & Y } $$ where $A$ is a discrete valuation ring with field of fractions $K$, there exists a dotted arrow making the diagram commute, then $f$ is proper.


spaces-limits-lemma-refined-valuative-criterion-separated

Let $S$ be a scheme. Let $f : X \to Y$ and $h : U \to X$ be morphisms of algebraic spaces over $S$. Assume that $Y$ is locally Noetherian, that $f$ is locally of finite type and quasi-separated, that $h$ is of finite type, and that the image of $|h| : |U| \to |X|$ is dense in $|X|$. If given any commutative solid diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & U \ar[r]^h & X \ar[d]^f \\ \Spec(A) \ar[rr] \ar@{-->}[rru] & & Y } $$ where $A$ is a discrete valuation ring with field of fractions $K$, there exists at most one dotted arrow making the diagram commute, then $f$ is separated.


spaces-limits-lemma-good-diagram

In Situation \ref{situation-limit-noetherian}. Let $X \to B$ be a quasi-separated and finite type morphism of algebraic spaces. Then there exists an $i \in I$ and a diagram \begin{equation} \label{equation-good-diagram} \vcenter{ \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ B \ar[r] & B_i } } \end{equation} such that $W \to B_i$ is of finite type and such that the induced morphism $X \to B \times_{B_i} W$ is a closed immersion.


spaces-limits-lemma-limit-from-good-diagram

In Situation \ref{situation-limit-noetherian}. Let $X \to B$ be a quasi-separated and finite type morphism of algebraic spaces. Given $i \in I$ and a diagram $$ \vcenter{ \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ B \ar[r] & B_i } } $$ as in (\ref{equation-good-diagram}) for $i' \geq i$ let $X_{i'}$ be the scheme theoretic image of $X \to B_{i'} \times_{B_i} W$. Then $X = \lim_{i' \geq i} X_{i'}$.


spaces-limits-lemma-morphism-good-diagram

In Situation \ref{situation-limit-noetherian}. Let $f : X \to Y$ be a morphism of algebraic spaces quasi-separated and of finite type over $B$. Let $$ \vcenter{ \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ B \ar[r] & B_{i_1} } } \quad\text{and}\quad \vcenter{ \xymatrix{ Y \ar[r] \ar[d] & V \ar[d] \\ B \ar[r] & B_{i_2} } } $$ be diagrams as in (\ref{equation-good-diagram}). Let $X = \lim_{i \geq i_1} X_i$ and $Y = \lim_{i \geq i_2} Y_i$ be the corresponding limit descriptions as in Lemma \ref{lemma-limit-from-good-diagram}. Then there exists an $i_0 \geq \max(i_1, i_2)$ and a morphism $$ (f_i)_{i \geq i_0} : (X_i)_{i \geq i_0} \to (Y_i)_{i \geq i_0} $$ of inverse systems over $(B_i)_{i \geq i_0}$ such that such that $f = \lim_{i \geq i_0} f_i$. If $(g_i)_{i \geq i_0} : (X_i)_{i \geq i_0} \to (Y_i)_{i \geq i_0}$ is a second morphism of inverse systems over $(B_i)_{i \geq i_0}$ such that such that $f = \lim_{i \geq i_0} g_i$ then $f_i = g_i$ for all $i \gg i_0$.


spaces-limits-lemma-morphism-good-diagram-flat

Notation and assumptions as in Lemma \ref{lemma-morphism-good-diagram}. If $f$ is flat and of finite presentation, then there exists an $i_3 > i_0$ such that for $i \geq i_3$ we have $f_i$ is flat, $X_i = Y_i \times_{Y_{i_3}} X_{i_3}$, and $X = Y \times_{Y_{i_3}} X_{i_3}$.


spaces-limits-lemma-morphism-good-diagram-smooth

Notation and assumptions as in Lemma \ref{lemma-morphism-good-diagram}. If $f$ is smooth, then there exists an $i_3 > i_0$ such that for $i \geq i_3$ we have $f_i$ is smooth.


spaces-limits-lemma-morphism-good-diagram-proper

Notation and assumptions as in Lemma \ref{lemma-morphism-good-diagram}. If $f$ is proper, then there exists an $i_3 \geq i_0$ such that for $i \geq i_3$ we have $f_i$ is proper.


spaces-limits-lemma-good-diagram-fibre-product

In Situation \ref{situation-limit-noetherian} suppose that we have a cartesian diagram $$ \xymatrix{ X^1 \ar[r]_p \ar[d]_q & X^3 \ar[d]^a \\ X^2 \ar[r]^b & X^4 } $$ of algebraic spaces quasi-separated and of finite type over $B$. For each $j = 1, 2, 3, 4$ choose $i_j \in I$ and a diagram $$ \xymatrix{ X^j \ar[r] \ar[d] & W^j \ar[d] \\ B \ar[r] & B_{i_j} } $$ as in (\ref{equation-good-diagram}). Let $X^j = \lim_{i \geq i_j} X^j_i$ be the corresponding limit descriptions as in Lemma \ref{lemma-morphism-good-diagram}. Let $(a_i)_{i \geq i_5}$, $(b_i)_{i \geq i_6}$, $(p_i)_{i \geq i_7}$, and $(q_i)_{i \geq i_8}$ be the corresponding morphisms of inverse systems contructed in Lemma \ref{lemma-morphism-good-diagram}. Then there exists an $i_9 \geq \max(i_5, i_6, i_7, i_8)$ such that for $i \geq i_9$ we have $a_i \circ p_i = b_i \circ q_i$ and such that $$ (q_i, p_i) : X^1_i \longrightarrow X^2_i \times_{b_i, X^4_i, a_i} X^3_i $$ is a closed immersion. If $a$ and $b$ are flat and of finite presentation, then there exists an $i_{10} \geq \max(i_5, i_6, i_7, i_8, i_9)$ such that for $i \geq i_{10}$ the last displayed morphism is an isomorphism.


spaces-limits-proposition-characterize-locally-finite-presentation

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 a morphism of algebraic spaces which is locally of finite presentation, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-locally-finite-presentation}. \item The morphism $f : X \to Y$ is limit preserving as a transformation of functors, see Definition \ref{definition-locally-finite-presentation}. \end{enumerate}


spaces-limits-proposition-approximate

\begin{reference} Our proof follows closely the proof given in \cite[Theorem 1.2.2]{CLO}. \end{reference} Let $X$ be a quasi-compact and quasi-separated algebraic space over $\Spec(\mathbf{Z})$. There exist a directed set $I$ and an inverse system of algebraic spaces $(X_i, f_{ii'})$ over $I$ such that \begin{enumerate} \item the transition morphisms $f_{ii'}$ are affine \item each $X_i$ is quasi-separated and of finite type over $\mathbf{Z}$, and \item $X = \lim X_i$. \end{enumerate}


spaces-limits-proposition-separated-closed-in-finite-presentation

Let $S$ be a scheme. $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $f$ is of finite type and separated, and \item $Y$ is quasi-compact and quasi-separated. \end{enumerate} Then there exists a separated morphism of finite presentation $f' : X' \to Y$ and a closed immersion $X \to X'$ over $Y$.


spaces-limits-proposition-affine

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is surjective and integral, and assume that $X$ is affine. Then $Y$ is affine.


spaces-limits-proposition-there-is-a-scheme-finite-over

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. \begin{enumerate} \item There exists a surjective finite morphism $Y \to X$ of finite presentation where $Y$ is a scheme, \item given a surjective \'etale morphism $U \to X$ we may choose $Y \to X$ such that for every $y \in Y$ there is an open neighbourhood $V \subset Y$ such that $V \to X$ factors through $U$. \end{enumerate}


stacks-geometry-lemma-deformation-category

In Situation \ref{situation-versal} let $x_0 : \Spec(k) \to \mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. Then $\mathcal{F}_{\mathcal{X}, k, x_0}$ is a deformation category and $T\mathcal{F}_{\mathcal{X}, k, x_0}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0})$ are finite dimensional $k$-vector spaces.


stacks-geometry-lemma-versal-ring

In Situation \ref{situation-versal} let $x_0 : \Spec(k) \to \mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. Then a versal ring to $\mathcal{X}$ at $x_0$ exists. Given a pair $A$, $A'$ of these, then $A \cong A'[[t_1, \ldots, t_r]]$ or $A' \cong A[[t_1, \ldots, t_r]]$ as $S$-algebras for some $r$.


stacks-geometry-lemma-versal-ring-field-extension

In Situation \ref{situation-versal} let $x_0 : \Spec(k) \to \mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. Let $l/k$ be a finite extension of fields and denote $x_{l, 0} : \Spec(l) \to \mathcal{X}$ the induced morphism. Given a versal ring $A$ to $\mathcal{X}$ at $x_0$ there exists a versal ring $A'$ to $\mathcal{X}$ at $x_{l, 0}$ such that there is a $S$-algebra map $A \to A'$ which induces the given field extension $l/k$ and is formally smooth in the $\mathfrak m_{A'}$-adic topology.


stacks-geometry-lemma-compare-versal-ring-completion

In Situation \ref{situation-versal} let $x : U \to \mathcal{X}$ be a morphism where $U$ is a scheme locally of finite type over $S$. Let $u_0 \in U$ be a finite type point. Set $k = \kappa(u_0)$ and denote $x_0 : \Spec(k) \to \mathcal{X}$ the induced map. The following are equivalent \begin{enumerate} \item $x$ is versal at $u_0$ (Artin's Axioms, Definition \ref{artin-definition-versal}), \item $\hat x : \mathcal{F}_{U, k, u_0} \to \mathcal{F}_{\mathcal{X}, k, x_0}$ is smooth, \item the formal object associated to $x|_{\Spec(\mathcal{O}_{U, u_0}^\wedge)}$ is versal, and \item there is an open neighbourhood $U' \subset U$ of $x$ such that $x|_{U'} : U' \to \mathcal{X}$ is smooth. \end{enumerate} Moreover, in this case the completion $\mathcal{O}_{U, u_0}^\wedge$ is a versal ring to $\mathcal{X}$ at $x_0$.


stacks-geometry-lemma-characterize-smoothness

In Situation \ref{situation-versal}. Let $x_0 : \Spec(k) \to \mathcal{X}$ be a morphism such that $\Spec(k) \to S$ is of finite type with image $s$. Let $A$ be a versal ring to $\mathcal{X}$ at $x_0$. The following are equivalent \begin{enumerate} \item $x_0$ is in the smooth locus of $\mathcal{X} \to S$ (Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-where-smooth}), \item $\mathcal{O}_{S, s} \to A$ is formally smooth in the $\mathfrak m_A$-adic topology, and \item $\mathcal{F}_{\mathcal{X}, k, x_0}$ is unobstructed. \end{enumerate}


stacks-geometry-lemma-Artin-approximation-by-smooth-morphism

In Situation \ref{situation-versal}. Let $x_0 : \Spec(k) \to \mathcal{X}$ be a morphism such that $\Spec(k) \to S$ is of finite type with image $s$. Let $A$ be a versal ring to $\mathcal{X}$ at $x_0$. If $\mathcal{O}_{S, s}$ is a G-ring, then we may find a smooth morphism $U \to \mathcal{X}$ whose source is a scheme and a point $u_0 \in U$ with residue field $k$, such that \begin{enumerate} \item $\Spec(k) \to U \to \mathcal{X}$ coincides with the given morphism $x_0$, \item there is an isomorphism $\mathcal{O}_{U, u_0}^\wedge \cong A$. \end{enumerate}


stacks-geometry-lemma-versal-ring-flat

In Situation \ref{situation-versal} let $x_0 : \Spec(k) \to \mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. Let $A$ be a versal ring to $\mathcal{X}$ at $x_0$. Then the morphism $\Spec(A) \to \mathcal{X}$ of Remark \ref{remark-upgrade} is flat.


stacks-geometry-lemma-map-of-components

Let $f : U \to \mathcal{X}$ be a smooth morphism from a scheme to a locally Noetherian algebraic stack. The closure of the image of any irreducible component of $|U|$ is an irreducible component of $|\mathcal{X}|$. If $U \to \mathcal{X}$ is surjective, then all irreducible components of $|\mathcal{X}|$ are obtained in this way.


stacks-geometry-lemma-multiplicities

Let $U \to X$ be a smooth morphism of locally Noetherian schemes. Let $T'$ is an irreducible component of $U$. Let $T$ be the irreducible component of $X$ obtained as the closure of the image of $T'$. Then $m_{T', U} = m_{T, X}$.


stacks-geometry-lemma-branches

In the situation of Definition \ref{definition-formal-branches} there is a canonical surjection from the set of formal branches of $\mathcal{X}$ through $x_0$ to the set of irreducible components of $|\mathcal{X}|$ containing $x_0$ in $|\mathcal{X}|$.


stacks-geometry-lemma-behaviour-of-dimensions-wrt-smooth-morphisms

If $f: U \to X$ is a smooth morphism of locally Noetherian algebraic spaces, and if $u \in |U|$ with image $x \in |X|$, then $$ \dim_u (U) = \dim_x(X) + \dim_{u} (U_x) $$ where $\dim_u (U_x)$ is defined via Definition \ref{definition-relative-dimension}.


stacks-geometry-lemma-base-change-invariance-of-relative-dimension

Suppose given a Cartesian square of morphisms of locally Noetherian stacks $$ \xymatrix{ \mathcal{T}' \ar[d]\ar[r] & \mathcal{T} \ar[d] \\ \mathcal{X}' \ar[r] & \mathcal{X} } $$ in which the vertical morphisms are locally of finite type. If $t' \in |\mathcal{T}'|$, with images $t$, $x'$, and $x$ in $|\mathcal{T}|$, $|\mathcal{X}'|$, and $|\mathcal{X}|$ respectively, then $\dim_{t'}(\mathcal{T}'_{x'}) = \dim_{t}(\mathcal{T}_x).$


stacks-geometry-lemma-relative-dimension-is-semi-continuous

Let $f: \mathcal{T} \to \mathcal{X}$ be a locally of finite type morphism of algebraic stacks. \begin{enumerate} \item The function $t \mapsto \dim_t(\mathcal{T}_{f(t)})$ is upper semi-continuous on $|\mathcal{T}|$. \item If $f$ is smooth, then the function $t \mapsto \dim_t(\mathcal{T}_{f(t)})$ is locally constant on $|\mathcal{T}|$. \end{enumerate}


stacks-geometry-lemma-dimension-achieved-by-finite-type-point

If $X$ is a finite dimensional scheme, then there exists a closed (and hence finite type) point $x \in X$ such that $\dim_x X = \dim X$.


stacks-geometry-lemma-constancy-of-dimension

If $X$ is an irreducible, Jacobson, catenary, and locally Noetherian scheme of finite dimension, then $\dim U = \dim X$ for every non-empty open subset $U$ of $X$. Equivalently, $\dim_x X$ is a constant function on $X$.


stacks-geometry-lemma-irreducible-implies-equidimensional

If $\mathcal{X}$ is a Jacobson, pseudo-catenary, and locally Noetherian  algebraic stack for which $|\mathcal{X}|$ is irreducible, then $\dim_x(\mathcal{X})$ is a constant function on $|\mathcal{X}|$.


stacks-geometry-lemma-closed-immersions

If $\mathcal{Z} \hookrightarrow \mathcal{X}$ is a closed immersion of locally Noetherian schemes, and if $z \in |\mathcal{Z}|$ has image $x \in |\mathcal{X}|$, then $\dim_z (\mathcal{Z}) \leq \dim_x(\mathcal{X})$.


stacks-geometry-lemma-dims-of-images-two

Let $f: \mathcal{T} \to \mathcal{X}$ be a locally of finite type morphism of Jacobson, pseudo-catenary, and locally Noetherian algebraic stacks which is quasi-DM, whose source is irreducible and whose target is quasi-separated, and let $\mathcal{Z} \hookrightarrow \mathcal{X}$ denote the scheme-theoretic image of $\mathcal{T}$. Then $\dim \mathcal{Z} \leq \dim \mathcal{T}$, and furthermore, exactly one of the following two conditions holds: \begin{enumerate} \item for every finite type point $t \in |T|,$ we have $\dim_t(\mathcal{T}_{f(t)}) > 0,$ in which case $\dim \mathcal{Z} < \dim \mathcal{T}$; or \item   $\mathcal{T}$ and $\mathcal{Z}$ are of the same dimension. \end{enumerate}


stacks-geometry-lemma-dimension-local-ring

Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Let $x \in |\mathcal{X}|$ be a finite type point  Morphisms of Stacks, Definition \ref{stacks-morphisms-definition-finite-type-point}). Let $d \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item there exists a scheme $U$, a smooth morphism $U \to \mathcal{X}$, and a finite type point $u \in U$ mapping to $x$ such that $2\dim(\mathcal{O}_{U, \overline{u}}) - \dim(\mathcal{O}_{R, e(\overline{u})}) = d$, and \item for any scheme $U$, a smooth morphism $U \to \mathcal{X}$, and finite type point $u \in U$ mapping to $x$ we have $2\dim(\mathcal{O}_{U, \overline{u}}) - \dim(\mathcal{O}_{R, e(\overline{u})}) = d$. \end{enumerate} Here $R = U \times_\mathcal{X} U$ with projections $s, t : R \to U$ and diagonal $e : U \to R$ and $R_u$ is the fibre of $s : R \to U$ over $u$.


spaces-morphisms-lemma-properties-diagonal

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\Delta_{X/Y} : X \to X \times_Y X$ be the diagonal morphism. Then \begin{enumerate} \item $\Delta_{X/Y}$ is representable, \item $\Delta_{X/Y}$ is locally of finite type, \item $\Delta_{X/Y}$ is a monomorphism, \item $\Delta_{X/Y}$ is separated, and \item $\Delta_{X/Y}$ is locally quasi-finite. \end{enumerate}


spaces-morphisms-lemma-base-change-separated

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


spaces-morphisms-lemma-fibre-product-after-map

\begin{slogan} The top arrow of a ``magic diagram'' of algebraic spaces has nice immersion-like properties, and under separatedness hypotheses these get stronger. \end{slogan} Let $S$ be a scheme. Let $f : X \to Z$, $g : Y \to Z$ and $Z \to T$ be morphisms of algebraic spaces over $S$. Consider the induced morphism $i : X \times_Z Y \to X \times_T Y$. Then \begin{enumerate} \item $i$ is representable, locally of finite type, locally quasi-finite, separated and a monomorphism, \item if $Z \to T$ is locally separated, then $i$ is an immersion, \item if $Z \to T$ is separated, then $i$ is a closed immersion, and \item if $Z \to T$ is quasi-separated, then $i$ is quasi-compact. \end{enumerate}


spaces-morphisms-lemma-semi-diagonal

\begin{slogan} Properties of the graph of a morphism of algebraic spaces as a consequence of separation properties of the target. \end{slogan} Let $S$ be a scheme. Let $T$ be an algebraic space over $S$. Let $g : X \to Y$ be a morphism of algebraic spaces over $T$. Consider the graph $i : X \to X \times_T Y$ of $g$. Then \begin{enumerate} \item $i$ is representable, locally of finite type, locally quasi-finite, separated and a monomorphism, \item if $Y \to T$ is locally separated, then $i$ is an immersion, \item if $Y \to T$ is separated, then $i$ is a closed immersion, and \item if $Y \to T$ is quasi-separated, then $i$ is quasi-compact. \end{enumerate}


spaces-morphisms-lemma-composition-separated

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


spaces-morphisms-lemma-separated-over-separated

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


spaces-morphisms-lemma-compose-after-separated

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of algebraic spaces over $S$. \begin{enumerate} \item If $g \circ f$ is separated then so is $f$. \item If $g \circ f$ is locally separated then so is $f$. \item If $g \circ f$ is quasi-separated then so is $f$. \end{enumerate}


spaces-morphisms-lemma-separated-implies-morphism-separated

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item If $X$ is separated then $X$ is separated over $S$. \item If $X$ is locally separated then $X$ is locally separated over $S$. \item If $X$ is quasi-separated then $X$ is quasi-separated over $S$. \end{enumerate} Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item[(4)] If $X$ is separated over $S$ then $f$ is separated. \item[(5)] If $X$ is locally separated over $S$ then $f$ is locally separated. \item[(6)] If $X$ is quasi-separated over $S$ then $f$ is quasi-separated. \end{enumerate}


spaces-morphisms-lemma-separated-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{P}$ be any of the separation axioms of Definition \ref{definition-separated}. The following are equivalent \begin{enumerate} \item $f$ is $\mathcal{P}$, \item for every scheme $Z$ and morphism $Z \to Y$ the base change $Z \times_Y X \to Z$ of $f$ is $\mathcal{P}$, \item for every affine scheme $Z$ and every morphism $Z \to Y$ the base change $Z \times_Y X \to Z$ of $f$ is $\mathcal{P}$, \item for every affine scheme $Z$ and every morphism $Z \to Y$ the algebraic space $Z \times_Y X$ is $\mathcal{P}$ (see Properties of Spaces, Definition \ref{spaces-properties-definition-separated}), \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that the base change $V \times_Y X \to V$ has $\mathcal{P}$, and \item there exists a Zariski covering $Y = \bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \to Y_i$ has $\mathcal{P}$. \end{enumerate}


spaces-morphisms-lemma-surjective-representable

Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is surjective (in the sense of Section \ref{section-representable}) if and only if $|f| : |X| \to |Y|$ is surjective.


spaces-morphisms-lemma-surjective-local

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


spaces-morphisms-lemma-composition-surjective

The composition of surjective morphisms is surjective.


spaces-morphisms-lemma-base-change-surjective

The base change of a surjective morphism is surjective.


spaces-morphisms-lemma-characterize-representable-universally-open

Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is universally open (in the sense of Section \ref{section-representable}), and \item for every morphism of algebraic spaces $Z \to Y$ the morphism of topological spaces $|Z \times_Y X| \to |Z|$ is open. \end{enumerate}


spaces-morphisms-lemma-base-change-universally-open

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


spaces-morphisms-lemma-composition-open

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


spaces-morphisms-lemma-universally-open-local

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


spaces-morphisms-lemma-space-over-field-universally-open

Let $S$ be a scheme. Let $p : X \to \Spec(k)$ be a morphism of algebraic spaces over $S$ where $k$ is a field. Then $p : X \to \Spec(k)$ is universally open.


spaces-morphisms-lemma-characterize-representable-universally-submersive

Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is universally submersive (in the sense of Section \ref{section-representable}), and \item for every morphism of algebraic spaces $Z \to Y$ the morphism of topological spaces $|Z \times_Y X| \to |Z|$ is submersive. \end{enumerate}


spaces-morphisms-lemma-base-change-universally-submersive

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


spaces-morphisms-lemma-composition-universally-submersive

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


spaces-morphisms-lemma-quasi-compact-is-quasi-compact

Let $S$ be a scheme. If $f : X \to Y$ is a quasi-compact morphism of algebraic spaces over $S$, then the underlying map $|f| : |X| \to |Y|$ of topological space is quasi-compact.


spaces-morphisms-lemma-base-change-quasi-compact

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


spaces-morphisms-lemma-composition-quasi-compact

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


spaces-morphisms-lemma-surjection-from-quasi-compact

\begin{slogan} The image of a quasi-compact algebraic space under a surjective morphism is quasi-compact. \end{slogan} Let $S$ be a scheme. \begin{enumerate} \item If $X \to Y$ is a surjective morphism of algebraic spaces over $S$, and $X$ is quasi-compact then $Y$ is quasi-compact. \item If $$ \xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & Z } $$ is a commutative diagram of morphisms of algebraic spaces over $S$ and $f$ is surjective and $p$ is quasi-compact, then $q$ is quasi-compact. \end{enumerate}


spaces-morphisms-lemma-descent-quasi-compact

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


spaces-morphisms-lemma-quasi-compact-local

\begin{slogan} Quasi-compact morphisms of algebraic spaces are preserved under pullback and local on the target. \end{slogan} 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 quasi-compact, \item for every scheme $Z$ and any morphism $Z \to Y$ the morphism of algebraic spaces $Z \times_Y X \to Z$ is quasi-compact, \item for every affine scheme $Z$ and any morphism $Z \to Y$ the algebraic space $Z \times_Y X$ is quasi-compact, \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that $V \times_Y X \to V$ is a quasi-compact morphism of algebraic spaces, and \item there exists a surjective \'etale morphism $Y' \to Y$ of algebraic spaces such that $Y' \times_Y X \to Y'$ is a quasi-compact morphism of algebraic spaces, and \item there exists a Zariski covering $Y = \bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \to Y_i$ is quasi-compact. \end{enumerate}


spaces-morphisms-lemma-quasi-compact-permanence

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of algebraic spaces over $S$. If $g \circ f$ is quasi-compact and $g$ is quasi-separated then $f$ is quasi-compact.


spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence

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


spaces-morphisms-lemma-characterize-representable-universally-closed

Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is universally closed (in the sense of Section \ref{section-representable}), and \item for every morphism of algebraic spaces $Z \to Y$ the morphism of topological spaces $|Z \times_Y X| \to |Z|$ is closed. \end{enumerate}


spaces-morphisms-lemma-base-change-universally-closed

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


spaces-morphisms-lemma-composition-universally-closed

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


spaces-morphisms-lemma-universally-closed-local

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


spaces-morphisms-lemma-universally-closed-quasi-compact

Let $S$ be a scheme. A universally closed morphism of algebraic spaces over $S$ is quasi-compact.


spaces-morphisms-lemma-image-universally-closed-separated

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $f : X \to Y$ be a surjective universally closed morphism of algebraic spaces over $B$. \begin{enumerate} \item If $X$ is quasi-separated, then $Y$ is quasi-separated. \item If $X$ is separated, then $Y$ is separated. \item If $X$ is quasi-separated over $B$, then $Y$ is quasi-separated over $B$. \item If $X$ is separated over $B$, then $Y$ is separated over $B$. \end{enumerate}


spaces-morphisms-lemma-monomorphism

Let $S$ be a scheme. Let $j : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $j$ is a monomorphism (as in Definition \ref{definition-monomorphism}), \item $j$ is a monomorphism in the category of algebraic spaces over $S$, and \item the diagonal morphism $\Delta_{X/Y} : X \to X \times_Y X$ is an isomorphism. \end{enumerate}


spaces-morphisms-lemma-monomorphism-separated

A monomorphism of algebraic spaces is separated.


spaces-morphisms-lemma-composition-monomorphism

A composition of monomorphisms is a monomorphism.


spaces-morphisms-lemma-base-change-monomorphism

The base change of a monomorphism is a monomorphism.


spaces-morphisms-lemma-immersions-monomorphisms

An immersion of algebraic spaces is a monomorphism. In particular, any immersion is separated.


spaces-morphisms-lemma-monomorphism-toward-field

Let $S$ be a scheme. Let $k$ be a field and let $Z \to \Spec(k)$ be a monomorphism of algebraic spaces over $S$. Then either $Z = \emptyset$ or $Z = \Spec(k)$.


spaces-morphisms-lemma-monomorphism-injective-points

Let $S$ be a scheme. If $X \to Y$ is a monomorphism of algebraic spaces over $S$, then $|X| \to |Y|$ is injective.


spaces-morphisms-lemma-compute-pushforward

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $U \to X$ be a surjective \'etale morphism from a scheme to $X$. Set $R = U \times_X U$ and denote $t, s : R \to U$ the projection morphisms as usual. Denote $a : U \to Y$ and $b : R \to Y$ the induced morphisms. For any object $\mathcal{F}$ of $\textit{Mod}(\mathcal{O}_X)$ there exists an exact sequence $$ 0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_U) \to b_*(\mathcal{F}|_R) $$ where the second arrow is the difference $t^* - s^*$.


spaces-morphisms-lemma-pushforward

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is quasi-compact and quasi-separated, then $f_*$ transforms quasi-coherent $\mathcal{O}_X$-modules into quasi-coherent $\mathcal{O}_Y$-modules.


spaces-morphisms-lemma-closed-immersion-local

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


spaces-morphisms-lemma-immersion-when-closed

Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. Then $|i| : |Z| \to |X|$ is a homeomorphism onto a locally closed subset, and $i$ is a closed immersion if and only if the image $|i|(|Z|) \subset |X|$ is a closed subset.


spaces-morphisms-lemma-factor-the-other-way

Let $S$ be a scheme. Let $Z \to X$ be an immersion of algebraic spaces over $S$. Assume $Z \to X$ is quasi-compact. There exists a factorization $Z \to \overline{Z} \to X$ where $Z \to \overline{Z}$ is an open immersion and $\overline{Z} \to X$ is a closed immersion.


spaces-morphisms-lemma-closed-immersion-ideals

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For every closed immersion $i : Z \to X$ the sheaf $i_*\mathcal{O}_Z$ is a quasi-coherent $\mathcal{O}_X$-module, the map $i^\sharp : \mathcal{O}_X \to i_*\mathcal{O}_Z$ is surjective and its kernel is a quasi-coherent sheaf of ideals. The rule $Z \mapsto \Ker(\mathcal{O}_X \to i_*\mathcal{O}_Z)$ defines an inclusion reversing bijection $$ \begin{matrix} \text{closed subspaces}\\ Z \subset X \end{matrix} \longrightarrow \begin{matrix} \text{quasi-coherent sheaves}\\ \text{of ideals }\mathcal{I} \subset \mathcal{O}_X \end{matrix} $$ Moreover, given a closed subscheme $Z$ corresponding to the quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ a morphism of algebraic spaces $h : Y \to X$ factors through $Z$ if and only if the map $h^*\mathcal{I} \to h^*\mathcal{O}_X = \mathcal{O}_Y$ is zero.


spaces-morphisms-lemma-closed-immersion-quasi-compact

A closed immersion of algebraic spaces is quasi-compact.


spaces-morphisms-lemma-closed-immersion-separated

A closed immersion of algebraic spaces is separated.


spaces-morphisms-lemma-closed-immersion-push-pull

Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. \begin{enumerate} \item The functor $$ i_{small, *} : \Sh(Z_\etale) \longrightarrow \Sh(X_\etale) $$ is fully faithful and its essential image is those sheaves of sets $\mathcal{F}$ on $X_\etale$ whose restriction to $X \setminus Z$ is isomorphic to $*$, and \item the functor $$ i_{small, *} : \textit{Ab}(Z_\etale) \longrightarrow \textit{Ab}(X_\etale) $$ is fully faithful and its essential image is those abelian sheaves on $X_\etale$ whose support is contained in $|Z|$. \end{enumerate} In both cases $i_{small}^{-1}$ is a left inverse to the functor $i_{small, *}$.


spaces-morphisms-lemma-stalk-push-closed

Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. Let $\overline{z}$ be a geometric point of $Z$ with image $\overline{x}$ in $X$. Then $(i_{small, *}\mathcal{F})_{\overline{z}} = \mathcal{F}_{\overline{x}}$ for any sheaf $\mathcal{F}$ on $Z_\etale$.


spaces-morphisms-lemma-closed-immersion-rings

Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. Let $\mathcal{A}$ be a sheaf of rings on $X_\etale$. Let $\mathcal{B}$ be a sheaf of rings on $Z_\etale$. Let $\varphi : \mathcal{A} \to i_{small, *}\mathcal{B}$ be a homomorphism of sheaves of rings so that we obtain a morphism of ringed topoi $$ f : (\Sh(Z_\etale), \mathcal{B}) \longrightarrow (\Sh(X_\etale), \mathcal{A}). $$ For a sheaf of $\mathcal{A}$-modules $\mathcal{F}$ and a sheaf of $\mathcal{B}$-modules $\mathcal{G}$ the canonical map $$ \mathcal{F} \otimes_\mathcal{A} f_*\mathcal{G} \longrightarrow f_*(f^*\mathcal{F} \otimes_\mathcal{B} \mathcal{G}). $$ is an isomorphism.


spaces-morphisms-lemma-i-star-equivalence

Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ be the quasi-coherent sheaf of ideals cutting out $Z$. \begin{enumerate} \item For any $\mathcal{O}_X$-module $\mathcal{F}$ the adjunction map $\mathcal{F} \to i_*i^*\mathcal{F}$ induces an isomorphism $\mathcal{F}/\mathcal{I}\mathcal{F} \cong i_*i^*\mathcal{F}$. \item The functor $i^*$ is a left inverse to $i_*$, i.e., for any $\mathcal{O}_Z$-module $\mathcal{G}$ the adjunction map $i^*i_*\mathcal{G} \to \mathcal{G}$ is an isomorphism. \item The functor $$ i_* : \QCoh(\mathcal{O}_Z) \longrightarrow \QCoh(\mathcal{O}_X) $$ is exact, fully faithful, with essential image those quasi-coherent $\mathcal{O}_X$-modules $\mathcal{F}$ such that $\mathcal{I}\mathcal{F} = 0$. \end{enumerate}


spaces-morphisms-lemma-largest-quasi-coherent-subsheaf

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{G} \subset \mathcal{F}$ be a $\mathcal{O}_X$-submodule. There exists a unique quasi-coherent $\mathcal{O}_X$-submodule $\mathcal{G}' \subset \mathcal{G}$ with the following property: For every quasi-coherent $\mathcal{O}_X$-module $\mathcal{H}$ the map $$ \Hom_{\mathcal{O}_X}(\mathcal{H}, \mathcal{G}') \longrightarrow \Hom_{\mathcal{O}_X}(\mathcal{H}, \mathcal{G}) $$ is bijective. In particular $\mathcal{G}'$ is the largest quasi-coherent $\mathcal{O}_X$-submodule of $\mathcal{F}$ contained in $\mathcal{G}$.


spaces-morphisms-lemma-scheme-theoretic-union

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Y, Z \subset X$ be closed subspaces. Let $Y \cup Z$ be the scheme theoretic union of $Y$ and $Z$. Let $Y \cap Z$ be the scheme theoretic intersection of $Y$ and $Z$. Then $Y \to Y \cup Z$ and $Z \to Y \cup Z$ are closed immersions, there is a short exact sequence $$ 0 \to \mathcal{O}_{Y \cup Z} \to \mathcal{O}_Y \times \mathcal{O}_Z \to \mathcal{O}_{Y \cap Z} \to 0 $$ of $\mathcal{O}_Z$-modules, and the diagram $$ \xymatrix{ Y \cap Z \ar[r] \ar[d] & Y \ar[d] \\ Z \ar[r] & Y \cup Z } $$ is cocartesian in the category of algebraic spaces over $S$, i.e., $Y \cup Z = Y \amalg_{Y \cap Z} Z$.


spaces-morphisms-lemma-support-covering

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $U$ be a scheme and let $\varphi : U \to X$ be an \'etale morphism. Then $$ \text{Supp}(\varphi^*\mathcal{F}) = |\varphi|^{-1}(\text{Supp}(\mathcal{F})) $$ where the left hand side is the support of $\varphi^*\mathcal{F}$ as a quasi-coherent module on the scheme $U$.


spaces-morphisms-lemma-support-finite-type

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Then \begin{enumerate} \item The support of $\mathcal{F}$ is closed. \item For a geometric point $\overline{x}$ lying over $x \in |X|$ we have $$ x \in \text{Supp}(\mathcal{F}) \Leftrightarrow \mathcal{F}_{\overline{x}} \not = 0 \Leftrightarrow \mathcal{F}_{\overline{x}} \otimes_{\mathcal{O}_{X, \overline{x}}} \kappa(\overline{x}) \not = 0. $$ \item For any morphism of algebraic spaces $f : Y \to X$ the pullback $f^*\mathcal{F}$ is of finite type as well and we have $\text{Supp}(f^*\mathcal{F}) = f^{-1}(\text{Supp}(\mathcal{F}))$. \end{enumerate}


spaces-morphisms-lemma-scheme-theoretic-support

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. There exists a smallest closed subspace $i : Z \to X$ such that there exists a quasi-coherent $\mathcal{O}_Z$-module $\mathcal{G}$ with $i_*\mathcal{G} \cong \mathcal{F}$. Moreover: \begin{enumerate} \item If $U$ is a scheme and $\varphi : U \to X$ is an \'etale morphism then $Z \times_X U$ is the scheme theoretic support of $\varphi^*\mathcal{F}$. \item The quasi-coherent sheaf $\mathcal{G}$ is unique up to unique isomorphism. \item The quasi-coherent sheaf $\mathcal{G}$ is of finite type. \item The support of $\mathcal{G}$ and of $\mathcal{F}$ is $|Z|$. \end{enumerate}


spaces-morphisms-lemma-scheme-theoretic-image

\begin{slogan} The scheme-theoretic image of a morphism of algebraic spaces exists. \end{slogan} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. There exists a closed subspace $Z \subset Y$ such that $f$ factors through $Z$ and such that for any other closed subspace $Z' \subset Y$ such that $f$ factors through $Z'$ we have $Z \subset Z'$.


spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $Z \subset Y$ be the scheme theoretic image of $f$. If $f$ is quasi-compact then \begin{enumerate} \item the sheaf of ideals $\mathcal{I} = \Ker(\mathcal{O}_Y \to f_*\mathcal{O}_X)$ is quasi-coherent, \item the scheme theoretic image $Z$ is the closed subspace corresponding to $\mathcal{I}$, \item for any \'etale morphism $V \to Y$ the scheme theoretic image of $X \times_Y V \to V$ is equal to $Z \times_Y V$, and \item the image $|f|(|X|) \subset |Z|$ is a dense subset of $|Z|$. \end{enumerate}


spaces-morphisms-lemma-scheme-theoretic-image-reduced

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $X$ is reduced. Then \begin{enumerate} \item the scheme theoretic image $Z$ of $f$ is the reduced induced algebraic space structure on $\overline{|f|(|X|)}$, and \item for any \'etale morphism $V \to Y$ the scheme theoretic image of $X \times_Y V \to V$ is equal to $Z \times_Y V$. \end{enumerate}


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

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact morphism of algebraic spaces over $S$. Let $Z$ be the scheme theoretic image of $f$. Let $z \in |Z|$. There exists a valuation ring $A$ with fraction field $K$ and a commutative diagram $$ \xymatrix{ \Spec(K) \ar[rr] \ar[d] & & X \ar[d] \ar[ld] \\ \Spec(A) \ar[r] & Z \ar[r] & Y } $$ such that the closed point of $\Spec(A)$ maps to $z$.


spaces-morphisms-lemma-factor-factor

Let $S$ be a scheme. Let $$ \xymatrix{ X_1 \ar[d] \ar[r]_{f_1} & Y_1 \ar[d] \\ X_2 \ar[r]^{f_2} & Y_2 } $$ be a commutative diagram of algebraic spaces over $S$. Let $Z_i \subset Y_i$, $i = 1, 2$ be the scheme theoretic image of $f_i$. Then the morphism $Y_1 \to Y_2$ induces a morphism $Z_1 \to Z_2$ and a commutative diagram $$ \xymatrix{ X_1 \ar[r] \ar[d] & Z_1 \ar[d] \ar[r] & Y_1 \ar[d] \\ X_2 \ar[r] & Z_2 \ar[r] & Y_2 } $$


spaces-morphisms-lemma-scheme-theoretically-dense-representable

Let $S$ be a scheme. Let $W \subset S$ be a scheme theoretically dense open subscheme (Morphisms, Definition \ref{morphisms-definition-scheme-theoretically-dense}). Let $f : X \to S$ be a morphism of schemes which is flat, locally of finite presentation, and locally quasi-finite. Then $f^{-1}(W)$ is scheme theoretically dense in $X$.


spaces-morphisms-lemma-scheme-theoretically-dense

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U \subset X$ be an open subspace. The following are equivalent \begin{enumerate} \item for every \'etale morphism $\varphi : V \to X$ (of algebraic spaces) the scheme theoretic closure of $\varphi^{-1}(U)$ in $V$ is equal to $V$, \item there exists a scheme $V$ and a surjective \'etale morphism $\varphi : V \to X$ such that the scheme theoretic closure of $\varphi^{-1}(U)$ in $V$ is equal to $V$, \end{enumerate}


spaces-morphisms-lemma-characterize-scheme-theoretically-dense

Let $S$ be a scheme. Let $j : U \to X$ be an open immersion of algebraic spaces over $S$. Then $U$ is scheme theoretically dense in $X$ if and only if $\mathcal{O}_X \to j_*\mathcal{O}_U$ is injective.


spaces-morphisms-lemma-quasi-compact-immersion

Let $S$ be a scheme. Let $h : Z \to X$ be an immersion of algebraic spaces over $S$. Assume either $Z \to X$ is quasi-compact or $Z$ is reduced. Let $\overline{Z} \subset X$ be the scheme theoretic image of $h$. Then the morphism $Z \to \overline{Z}$ is an open immersion which identifies $Z$ with a scheme theoretically dense open subspace of $\overline{Z}$. Moreover, $Z$ is topologically dense in $\overline{Z}$.


spaces-morphisms-lemma-equality-of-morphisms

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $f, g : X \to Y$ be morphisms of algebraic spaces over $B$. Let $U \subset X$ be an open subspace such that $f|_U = g|_U$. If the scheme theoretic closure of $U$ in $X$ is $X$ and $Y \to B$ is separated, then $f = g$.


spaces-morphisms-lemma-universally-injective

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 map $X(K) \to Y(K)$ is injective for every field $K$ over $S$ \item for every morphism $Y' \to Y$ of algebraic spaces over $S$ the induced map $|Y' \times_Y X| \to |Y'|$ is injective, and \item the diagonal morphism $X \to X \times_Y X$ is surjective. \end{enumerate}


spaces-morphisms-lemma-base-change-universally-injective

The base change of a universally injective morphism is universally injective.


spaces-morphisms-lemma-universally-injective-local

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


spaces-morphisms-lemma-composition-universally-injective

A composition of universally injective morphisms is universally injective.


spaces-morphisms-lemma-affine-local

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


spaces-morphisms-lemma-composition-affine

The composition of affine morphisms is affine.


spaces-morphisms-lemma-base-change-affine

The base change of an affine morphism is affine.


spaces-morphisms-lemma-closed-immersion-affine

A closed immersion is affine.


spaces-morphisms-lemma-affine-equivalence-algebras

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There is an anti-equivalence of categories $$ \begin{matrix} \text{algebraic spaces} \\ \text{affine over }X \end{matrix} \longleftrightarrow \begin{matrix} \text{quasi-coherent sheaves} \\ \text{of }\mathcal{O}_X\text{-algebras} \end{matrix} $$ which associates to $f : Y \to X$ the sheaf $f_*\mathcal{O}_Y$. Moreover, this equivalence is compatible with arbitrary base change.


spaces-morphisms-lemma-affine-equivalence-modules

Let $S$ be a scheme. Let $f : Y \to X$ be an affine morphism of algebraic spaces over $S$. Let $\mathcal{A} = f_*\mathcal{O}_Y$. The functor $\mathcal{F} \mapsto f_*\mathcal{F}$ induces an equivalence of categories $$ \left\{ \begin{matrix} \text{category of quasi-coherent}\\ \mathcal{O}_Y\text{-modules} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{category of quasi-coherent}\\ \mathcal{A}\text{-modules} \end{matrix} \right\} $$ Moreover, an $\mathcal{A}$-module is quasi-coherent as an $\mathcal{O}_X$-module if and only if it is quasi-coherent as an $\mathcal{A}$-module.


spaces-morphisms-lemma-affine-permanence

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Suppose $g : X \to Y$ is a morphism of algebraic spaces over $B$. \begin{enumerate} \item If $X$ is affine over $B$ and $\Delta : Y \to Y \times_B Y$ is affine, then $g$ is affine. \item If $X$ is affine over $B$ and $Y$ is separated over $B$, then $g$ is affine. \item A morphism from an affine scheme to an algebraic space with affine diagonal is affine. \item A morphism from an affine scheme to a separated algebraic space is affine. \end{enumerate}


spaces-morphisms-lemma-Artinian-affine

Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$. Let $A$ be an Artinian ring. Any morphism $\Spec(A) \to X$ is affine.


spaces-morphisms-lemma-quasi-affine-local

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


spaces-morphisms-lemma-composition-quasi-affine

The composition of quasi-affine morphisms is quasi-affine.


spaces-morphisms-lemma-base-change-quasi-affine

The base change of a quasi-affine morphism is quasi-affine.


spaces-morphisms-lemma-characterize-quasi-affine

Let $S$ be a scheme. A quasi-compact and quasi-separated morphism of algebraic spaces $f : Y \to X$ is quasi-affine if and only if the canonical factorization $Y \to \underline{\Spec}_X(f_*\mathcal{O}_Y)$ (Remark \ref{remark-factorization-quasi-compact-quasi-separated}) is an open immersion.


spaces-morphisms-lemma-local-source-target

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


spaces-morphisms-lemma-local-source-target-at-point

Let $\mathcal{Q}$ be a property of morphisms of germs which is \'etale local on the source-and-target. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$ be a point of $X$. Consider the diagrams $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y } \quad\quad \xymatrix{ u \ar[d] \ar[r] & v \ar[d] \\ x \ar[r] & y } $$ where $U$ and $V$ are schemes, $a, b$ are \'etale, and $u, v, x, y$ are points of the corresponding spaces. The following are equivalent \begin{enumerate} \item for any diagram as above we have $\mathcal{Q}((U, u) \to (V, v))$, and \item for some diagram as above we have $\mathcal{Q}((U, u) \to (V, v))$. \end{enumerate} If $X$ and $Y$ are representable, then this is also equivalent to $\mathcal{Q}((X, x) \to (Y, y))$.


spaces-morphisms-lemma-local-source-target-global-implies-local

Let $\mathcal{P}$ be a property of morphisms of schemes which is \'etale local on the source-and-target. Consider the property $\mathcal{Q}$ of morphisms of germs associated to $\mathcal{P}$ in Descent, Lemma \ref{descent-lemma-local-source-target-global-implies-local}. Then \begin{enumerate} \item $\mathcal{Q}$ is \'etale local on the source-and-target. \item given a morphism of algebraic spaces $f : X \to Y$ and $x \in |X|$ the following are equivalent \begin{enumerate} \item $f$ has $\mathcal{Q}$ at $x$, and \item there is an open neighbourhood $X' \subset X$ of $x$ such that $X' \to Y$ has $\mathcal{P}$. \end{enumerate} \item given a morphism of algebraic spaces $f : X \to Y$ the following are equivalent: \begin{enumerate} \item $f$ has $\mathcal{P}$, \item for every $x \in |X|$ the morphism $f$ has $\mathcal{Q}$ at $x$. \end{enumerate} \end{enumerate}


spaces-morphisms-lemma-composition-finite-type

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


spaces-morphisms-lemma-base-change-finite-type

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


spaces-morphisms-lemma-finite-type-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is locally of finite type, \item for every $x \in |X|$ the morphism $f$ is of finite type at $x$, \item for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is locally of finite type, \item for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is locally of finite type, \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that $V \times_Y X \to V$ is locally of finite type, \item there exists a scheme $U$ and a surjective \'etale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi$ is locally of finite type, \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 top horizontal arrow is locally of finite type, \item there exists a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \'etale, $U \to X$ is surjective, and the top horizontal arrow is locally of finite type, and \item there exist Zariski coverings $Y = \bigcup_{i \in I} Y_i$, and $f^{-1}(Y_i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_i$ is locally of finite type. \end{enumerate}


spaces-morphisms-lemma-locally-finite-type-locally-noetherian

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type and $Y$ is locally Noetherian, then $X$ is locally Noetherian.


spaces-morphisms-lemma-permanence-finite-type

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


spaces-morphisms-lemma-immersion-locally-finite-type

An immersion is locally of finite type.


spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points

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. The following are equivalent: \begin{enumerate} \item $f$ is surjective, and \item for every algebraically closed field $k$ over $S$ the induced map $X(k) \to Y(k)$ is surjective. \end{enumerate}


spaces-morphisms-lemma-large-enough

Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. \begin{enumerate} \item As $k$ ranges over all algebraically closed fields over $S$ the collection of geometric points $\overline{y} \in Y(k)$ cover all of $|Y|$. \item As $k$ ranges over all algebraically closed fields over $S$ with $|k| \geq \lambda(Y)$ and $|k| > \lambda(X)$ the geometric points $\overline{y} \in Y(k)$ cover all of $|Y|$. \item For any geometric point $\overline{s} : \Spec(k) \to S$ where $k$ has cardinality $> \lambda(X)$ the map $$ X(k) \longrightarrow |X_s| $$ is surjective. \item Let $X \to Y$ be a morphism of algebraic spaces over $S$. For any geometric point $\overline{s} : \Spec(k) \to S$ where $k$ has cardinality $> \lambda(X)$ the map $$ X(k) \longrightarrow |X| \times_{|Y|} Y(k) $$ is surjective. \item Let $X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item the map $X \to Y$ is surjective, \item for all algebraically closed fields $k$ over $S$ with $|k| > \lambda(X)$, and $|k| \geq \lambda(Y)$ the map $X(k) \to Y(k)$ is surjective. \end{enumerate} \end{enumerate}


spaces-morphisms-lemma-point-finite-type

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


spaces-morphisms-lemma-identify-finite-type-points

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


spaces-morphisms-lemma-finite-type-points-morphism

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type, then $f(X_{\text{ft-pts}}) \subset Y_{\text{ft-pts}}$.


spaces-morphisms-lemma-finite-type-points-surjective-morphism

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type and surjective, then $f(X_{\text{ft-pts}}) = Y_{\text{ft-pts}}$.


spaces-morphisms-lemma-enough-finite-type-points

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


spaces-morphisms-lemma-finite-type-nagata

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $Y$ is Nagata and $f$ locally of finite type then $X$ is Nagata.


spaces-morphisms-lemma-base-change-quasi-finite-locus

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 $g$. Denote $g' : X' \to X$ the projection. Assume $f$ is locally of finite type. Let $W \subset |X|$, resp.\ $W' \subset |X'|$ be the set of points where $f$, resp.\ $f'$ is quasi-finite. \begin{enumerate} \item $W \subset |X|$ and $W' \subset |X'|$ are open, \item $W' = (g')^{-1}(W)$, i.e., formation of the locus where $f$ is quasi-finite commutes with base change, \item the base change of a locally quasi-finite morphism is locally quasi-finite, and \item the base change of a quasi-finite morphism is quasi-finite. \end{enumerate}


spaces-morphisms-lemma-composition-quasi-finite

The composition of quasi-finite morphisms is quasi-finite. The same holds for locally quasi-finite.


spaces-morphisms-lemma-base-change-quasi-finite

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


spaces-morphisms-lemma-locally-quasi-finite

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


spaces-morphisms-lemma-quasi-finite-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is locally quasi-finite, \item for every $x \in |X|$ the morphism $f$ is quasi-finite at $x$, \item for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is locally quasi-finite, \item for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is locally quasi-finite, \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that $V \times_Y X \to V$ is locally quasi-finite, \item there exists a scheme $U$ and a surjective \'etale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi$ is locally quasi-finite, \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 top horizontal arrow is locally quasi-finite, \item there exists a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \'etale, and $U \to X$ is surjective such that the top horizontal arrow is locally quasi-finite, and \item there exist Zariski coverings $Y = \bigcup_{i \in I} Y_i$, and $f^{-1}(Y_i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_i$ is locally quasi-finite. \end{enumerate}


spaces-morphisms-lemma-immersion-quasi-finite

An immersion is locally quasi-finite.


spaces-morphisms-lemma-permanence-quasi-finite

Let $S$ be a scheme. Let $X \to Y \to Z$ be morphisms of algebraic spaces over $S$. If $X \to Z$ is locally quasi-finite, then $X \to Y$ is locally quasi-finite.


spaces-morphisms-lemma-quasi-finite-at-a-finite-number-of-points

Let $S$ be a scheme. Let $f : X \to Y$ be a finite type morphism of algebraic spaces over $S$. Let $y \in |Y|$. There are at most finitely many points of $|X|$ lying over $y$ at which $f$ is quasi-finite.


spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type and a monomorphism, then $f$ is separated and locally quasi-finite.


spaces-morphisms-lemma-composition-finite-presentation

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


spaces-morphisms-lemma-base-change-finite-presentation

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


spaces-morphisms-lemma-finite-presentation-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is locally of finite presentation, \item for every $x \in |X|$ the morphism $f$ is of finite presentation at $x$, \item for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is locally of finite presentation, \item for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is locally of finite presentation, \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that $V \times_Y X \to V$ is locally of finite presentation, \item there exists a scheme $U$ and a surjective \'etale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi$ is locally of finite presentation, \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 top horizontal arrow is locally of finite presentation, \item there exists a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \'etale, and $U \to X$ is surjective such that the top horizontal arrow is locally of finite presentation, and \item there exist Zariski coverings $Y = \bigcup_{i \in I} Y_i$, and $f^{-1}(Y_i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_i$ is locally of finite presentation. \end{enumerate}


spaces-morphisms-lemma-finite-presentation-finite-type

A morphism which is locally of finite presentation is locally of finite type. A morphism of finite presentation is of finite type.


spaces-morphisms-lemma-finite-presentation-noetherian

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is of finite presentation and $Y$ is Noetherian, then $X$ is Noetherian.


spaces-morphisms-lemma-noetherian-finite-type-finite-presentation

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


spaces-morphisms-lemma-finite-presentation-quasi-compact-quasi-separated

Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$ which is quasi-compact and quasi-separated. If $X$ is of finite presentation over $Y$, then $X$ is quasi-compact and quasi-separated.


spaces-morphisms-lemma-finite-presentation-permanence

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


spaces-morphisms-lemma-diagonal-morphism-finite-type

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ with diagonal $\Delta : X \to X \times_Y X$. If $f$ is locally of finite type then $\Delta$ is locally of finite presentation. If $f$ is quasi-separated and locally of finite type, then $\Delta$ is of finite presentation.


spaces-morphisms-lemma-open-immersion-locally-finite-presentation

An open immersion of algebraic spaces is locally of finite presentation.


spaces-morphisms-lemma-closed-immersion-finite-presentation

A closed immersion $i : Z \to X$ is of finite presentation if and only if the associated quasi-coherent sheaf of ideals $\mathcal{I} = \Ker(\mathcal{O}_X \to i_*\mathcal{O}_Z)$ is of finite type (as an $\mathcal{O}_X$-module).


spaces-morphisms-lemma-composition-flat

The composition of flat morphisms is flat.


spaces-morphisms-lemma-base-change-flat

The base change of a flat morphism is flat.


spaces-morphisms-lemma-flat-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is flat, \item for every $x \in |X|$ the morphism $f$ is flat at $x$, \item for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is flat, \item for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is flat, \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that $V \times_Y X \to V$ is flat, \item there exists a scheme $U$ and a surjective \'etale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi$ is flat, \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 top horizontal arrow is flat, \item there exists a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \'etale, and $U \to X$ is surjective such that the top horizontal arrow is flat, and \item there exists a Zariski coverings $Y = \bigcup Y_i$ and $f^{-1}(Y_i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_i$ is flat. \end{enumerate}


spaces-morphisms-lemma-fppf-open

A flat morphism locally of finite presentation is universally open.


spaces-morphisms-lemma-flat-at-point-etale-local-rings

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\overline{x}$ be a geometric point of $X$ lying over the point $x \in |X|$. Let $\overline{y} = f \circ \overline{x}$. The following are equivalent \begin{enumerate} \item $f$ is flat at $x$, and \item the map on \'etale local rings $\mathcal{O}_{Y, \overline{y}} \to \mathcal{O}_{X, \overline{x}}$ is flat. \end{enumerate}


spaces-morphisms-lemma-flat-morphism-sites

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then $f$ is flat if and only if the morphism of sites $ (f_{small}, f^\sharp) : (X_\etale, \mathcal{O}_X) \to (Y_\etale, \mathcal{O}_Y) $ associated to $f$ is flat.


spaces-morphisms-lemma-flat-pullback-support

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module with scheme theoretic support $Z \subset X$. If $f$ is flat, then $f^{-1}(Z)$ is the scheme theoretic support of $f^*\mathcal{F}$.


spaces-morphisms-lemma-flat-morphism-scheme-theoretically-dense-open

Let $S$ be a scheme. Let $f : X \to Y$ be a flat morphism of algebraic spaces over $S$. Let $V \to Y$ be a quasi-compact open immersion. If $V$ is scheme theoretically dense in $Y$, then $f^{-1}V$ is scheme theoretically dense in $X$.


spaces-morphisms-lemma-flat-base-change-scheme-theoretic-image

Let $S$ be a scheme. Let $f : X \to Y$ be a flat morphism of algebraic spaces over $S$. Let $g : V \to Y$ be a quasi-compact morphism of algebraic spaces. Let $Z \subset Y$ be the scheme theoretic image of $g$ and let $Z' \subset X$ be the scheme theoretic image of the base change $V \times_Y X \to X$. Then $Z' = f^{-1}Z$.


spaces-morphisms-lemma-flat-at-point

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 $X$. Let $x \in |X|$. The following are equivalent \begin{enumerate} \item for some commutative diagram $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y } $$ where $U$ and $V$ are schemes, $a, b$ are \'etale, and $u \in U$ mapping to $x$ the module $a^*\mathcal{F}$ is flat at $u$ over $V$, \item the stalk $\mathcal{F}_{\overline{x}}$ is flat over the \'etale local ring $\mathcal{O}_{Y, \overline{y}}$ where $\overline{x}$ is any geometric point lying over $x$ and $\overline{y} = f \circ \overline{x}$. \end{enumerate}


spaces-morphisms-lemma-base-change-module-flat

Let $S$ be a scheme. Let $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ be a cartesian diagram of algebraic spaces over $S$. Let $x' \in |X'|$ with image $x \in |X|$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$ and denote $\mathcal{F}' = (g')^*\mathcal{F}$. \begin{enumerate} \item If $\mathcal{F}$ is flat at $x$ over $Y$ then $\mathcal{F}'$ is flat at $x'$ over $Y'$. \item If $g$ is flat at $f'(x')$ and $\mathcal{F}'$ is flat at $x'$ over $Y'$, then $\mathcal{F}$ is flat at $x$ over $Y$. \end{enumerate} In particular, if $\mathcal{F}$ is flat over $Y$, then $\mathcal{F}'$ is flat over $Y'$.


spaces-morphisms-lemma-composition-module-flat

Let $S$ be a scheme. Let $X \to Y \to Z$ be morphisms of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in |X|$ with image $y \in |Y|$. \begin{enumerate} \item If $\mathcal{F}$ is flat at $x$ over $Y$ and $Y$ is flat at $y$ over $Z$, then $\mathcal{F}$ is flat at $x$ over $Z$. \item Let $x : \Spec(K) \to X$ be a representative of $x$. If \begin{enumerate} \item $\mathcal{F}$ is flat at $x$ over $Y$, \item $x^*\mathcal{F} \not = 0$, and \item $\mathcal{F}$ is flat at $x$ over $Z$, \end{enumerate} then $Y$ is flat at $y$ over $Z$. \item Let $\overline{x}$ be a geometric point of $X$ lying over $x$ with image $\overline{y}$ in $Y$. If $\mathcal{F}_{\overline{x}}$ is a faithfully flat $\mathcal{O}_{Y, \overline{y}}$-module and $\mathcal{F}$ is flat at $x$ over $Z$, then $Y$ is flat at $y$ over $Z$. \end{enumerate}


spaces-morphisms-lemma-flat-permanence

Let $S$ be a scheme. Let $f : X \to Y$, $g : Y \to Z$ be morphisms of algebraic spaces over $S$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $Y$. Let $x \in |X|$ with image $y \in |Y|$. If $f$ is flat at $x$, then $$ \mathcal{G}\text{ flat over }Z\text{ at }y \Leftrightarrow f^*\mathcal{G}\text{ flat over }Z\text{ at }x. $$ In particular: If $f$ is surjective and flat, then $\mathcal{G}$ is flat over $Z$, if and only if $f^*\mathcal{G}$ is flat over $Z$.


spaces-morphisms-lemma-compare-tr-deg

Let $S$ be a scheme. Let $X \to Y \to Z$ be morphisms of algebraic spaces over $S$. Let $x \in |X|$ and let $y \in |Y|$, $z \in |Z|$ be the images. Assume $X \to Y$ is locally quasi-finite and $Y \to Z$ locally of finite type. Then the transcendence degree of $x/z$ is equal to the transcendence degree of $y/z$.


spaces-morphisms-lemma-jacobson-finite-type-points

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type, $Y$ is Jacobson (Properties of Spaces, Remark \ref{spaces-properties-remark-list-properties-local-etale-topology}), and $x \in |X|$ is a finite type point of $X$, then the transcendence degree of $x/f(x)$ is $0$.


spaces-morphisms-lemma-dimension-fibre-at-a-point

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$. Assume $f$ is locally of finite type. Then we have $$ \begin{matrix} \text{relative dimension of }f\text{ at }x \\ = \\ \text{dimension of local ring of the fibre of }f\text{ at }x \\ + \\ \text{transcendence degree of }x/f(x) \end{matrix} $$ where the notation is as in Definition \ref{definition-dimension-fibre}.


spaces-morphisms-lemma-dimension-fibre-at-a-point-additive

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of algebraic spaces over $S$. Let $x \in |X|$ and set $y = f(x)$. Assume $f$ and $g$ locally of finite type. Then \begin{enumerate} \item $$ \begin{matrix} \text{relative dimension of }g \circ f\text{ at }x \\ \leq \\ \text{relative dimension of }f\text{ at }x \\ + \\ \text{relative dimension of }g\text{ at }y \end{matrix} $$ \item equality holds in (1) if for some morphism $\Spec(k) \to Z$ from the spectrum of a field in the class of $g(f(x)) = g(y)$ the morphism $X_k \to Y_k$ is flat at $x$, for example if $f$ is flat at $x$, \item $$ \begin{matrix} \text{transcendence degree of }x/g(f(x)) \\ = \\ \text{transcendence degree of }x/f(x) \\ + \\ \text{transcendence degree of }f(x)/g(f(x)) \end{matrix} $$ \end{enumerate}


spaces-morphisms-lemma-dimension-fibre-after-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 fibre product diagram of algebraic spaces over $S$. Let $x' \in |X'|$. Set $x = g'(x')$. Assume $f$ locally of finite type. Then \begin{enumerate} \item $$ \begin{matrix} \text{relative dimension of }f\text{ at }x \\ = \\ \text{relative dimension of }f'\text{ at }x' \end{matrix} $$ \item we have $$ \begin{matrix} \text{dimension of local ring of the fibre of }f'\text{ at }x' \\ - \\ \text{dimension of local ring of the fibre of }f\text{ at }x \\ = \\ \text{transcendence degree of }x/f(x) \\ - \\ \text{transcendence degree of }x'/f'(x') \end{matrix} $$ and the common value is $\geq 0$, \item given $x$ and $y' \in |Y'|$ mapping to the same $y \in |Y|$ there exists a choice of $x'$ such that the integer in (2) is $0$. \end{enumerate}


spaces-morphisms-lemma-openness-bounded-dimension-fibres

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $n \geq 0$. Assume $f$ is locally of finite type. The set $$ W_n = \{x \in |X| \text{ such that the relative dimension of }f\text{ at } x \leq n\} $$ is open in $|X|$.


spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0

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. Then $f$ is locally quasi-finite if and only if $f$ has relative dimension $0$ at each $x \in |X|$.


spaces-morphisms-lemma-locally-finite-type-quasi-finite-part

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. Then there exists a canonical open subspace $X' \subset X$ such that $f|_{X'} : X' \to Y$ is locally quasi-finite, and such that the relative dimension of $f$ at any $x \in |X|$, $x \not \in |X'|$ is $\geq 1$. Formation of $X'$ commutes with arbitrary base change.


spaces-morphisms-lemma-quasi-finite-at-point

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 locally of finite type and where $k$ is a field over $S$. Let $z \in |F|$ be such that $\dim_z(F) = 0$. Then, after replacing $X$ by an open subspace containing $p(z)$, the morphism $$ X \longrightarrow Y $$ is locally quasi-finite.


spaces-morphisms-lemma-dimension-formula-general

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $Y$ is locally Noetherian and $f$ locally of finite type. Let $x \in |X|$ with image $y \in |Y|$. Then we have \begin{align*} & \text{the dimension of the local ring of }X\text{ at }x \leq \\ & \text{the dimension of the local ring of }Y\text{ at }y + E - \\ & \text{ the transcendence degree of }x/y \end{align*} Here $E$ is the maximum of the transcendence degrees of $\xi/f(\xi)$ where $\xi \in |X|$ runs over the points specializing to $x$ at which the local ring of $X$ has dimension $0$.


spaces-morphisms-lemma-alteration-dimension-general

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $Y$ is locally Noetherian and $f$ is locally of finite type. Then $$ \dim(X) \leq \dim(Y) + E $$ where $E$ is the supremum of the transcendence degrees of $\xi/f(\xi)$ where $\xi$ runs through the points at which the local ring of $X$ has dimension $0$.


spaces-morphisms-lemma-syntomic-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is syntomic, \item for every $x \in |X|$ the morphism $f$ is syntomic at $x$, \item for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is syntomic, \item for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is syntomic, \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that $V \times_Y X \to V$ is a syntomic morphism, \item there exists a scheme $U$ and a surjective \'etale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi$ is syntomic, \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 top horizontal arrow is syntomic, \item there exists a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \'etale, and $U \to X$ is surjective such that the top horizontal arrow is syntomic, and \item there exist Zariski coverings $Y = \bigcup_{i \in I} Y_i$, and $f^{-1}(Y_i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_i$ is syntomic. \end{enumerate}


spaces-morphisms-lemma-syntomic-locally-finite-presentation

A syntomic morphism is locally of finite presentation.


spaces-morphisms-lemma-syntomic-flat

A syntomic morphism is flat.


spaces-morphisms-lemma-composition-smooth

The composition of smooth morphisms is smooth.


spaces-morphisms-lemma-base-change-smooth

The base change of a smooth morphism is smooth.


spaces-morphisms-lemma-smooth-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is smooth, \item for every $x \in |X|$ the morphism $f$ is smooth at $x$, \item for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is smooth, \item for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is smooth, \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that $V \times_Y X \to V$ is a smooth morphism, \item there exists a scheme $U$ and a surjective \'etale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi$ is smooth, \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 top horizontal arrow is smooth, \item there exists a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \'etale, and $U \to X$ is surjective such that the top horizontal arrow is smooth, and \item there exist Zariski coverings $Y = \bigcup_{i \in I} Y_i$, and $f^{-1}(Y_i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_i$ is smooth. \end{enumerate}


spaces-morphisms-lemma-smooth-locally-finite-presentation

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


spaces-morphisms-lemma-smooth-locally-finite-type

A smooth morphism of algebraic spaces is locally of finite type.


spaces-morphisms-lemma-smooth-flat

A smooth morphism of algebraic spaces is flat.


spaces-morphisms-lemma-smooth-syntomic

A smooth morphism of algebraic spaces is syntomic.


spaces-morphisms-lemma-where-smooth

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


spaces-morphisms-lemma-smoothness-dimension-spaces

Let $X$ and $Y$ be locally Noetherian algebraic spaces over a scheme $S$, and let $f : X \to Y$ be a smooth morphism. For every point $x \in |X|$ with image $y \in |Y|$, $$ \dim_x(X) = \dim_y(Y) + \dim_x(X_y) $$ where $\dim_x(X_y)$ is the relative dimension of $f$ at $x$ as in Definition \ref{definition-dimension-fibre}.


spaces-morphisms-lemma-composition-unramified

The composition of unramified morphisms is unramified.


spaces-morphisms-lemma-base-change-unramified

The base change of an unramified morphism is unramified.


spaces-morphisms-lemma-unramified-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is unramified, \item for every $x \in |X|$ the morphism $f$ is unramified at $x$, \item for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is unramified, \item for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is unramified, \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that $V \times_Y X \to V$ is an unramified morphism, \item there exists a scheme $U$ and a surjective \'etale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi$ is unramified, \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 top horizontal arrow is unramified, \item there exists a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \'etale, and $U \to X$ is surjective such that the top horizontal arrow is unramified, and \item there exist Zariski coverings $Y = \bigcup_{i \in I} Y_i$, and $f^{-1}(Y_i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_i$ is unramified. \end{enumerate}


spaces-morphisms-lemma-unramified-locally-finite-type

An unramified morphism of algebraic spaces is locally of finite type.


spaces-morphisms-lemma-unramified-quasi-finite

If $f$ is unramified at $x$ then $f$ is quasi-finite at $x$. In particular, an unramified morphism is locally quasi-finite.


spaces-morphisms-lemma-immersion-unramified

An immersion of algebraic spaces is unramified.


spaces-morphisms-lemma-diagonal-unramified-morphism

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item If $f$ is unramified, then the diagonal morphism $\Delta_{X/Y} : X \to X \times_Y X$ is an open immersion. \item If $f$ is locally of finite type and $\Delta_{X/Y}$ is an open immersion, then $f$ is unramified. \end{enumerate}


spaces-morphisms-lemma-where-unramified

Let $S$ be a scheme. Consider a commutative diagram $$ \xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[ld]^q \\ & Z } $$ of algebraic spaces over $S$. Assume that $X \to Z$ is locally of finite type. Then there exists an open subspace $U(f) \subset X$ such that $|U(f)| \subset |X|$ is the set of points where $f$ is unramified. Moreover, for any morphism of algebraic spaces $Z' \to Z$, if $f' : X' \to Y'$ is the base change of $f$ by $Z' \to Z$, then $U(f')$ is the inverse image of $U(f)$ under the projection $X' \to X$.


spaces-morphisms-lemma-permanence-unramified

Let $S$ be a scheme. Let $X \to Y \to Z$ be morphisms of algebraic spaces over $S$. If $X \to Z$ is unramified, then $X \to Y$ is unramified.


spaces-morphisms-lemma-etale-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is \'etale, \item for every $x \in |X|$ the morphism $f$ is \'etale at $x$, \item for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is \'etale, \item for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is \'etale, \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that $V \times_Y X \to V$ is an \'etale morphism, \item there exists a scheme $U$ and a surjective \'etale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi$ is \'etale, \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 top horizontal arrow is \'etale, \item there exists a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \'etale, and $U \to X$ surjective such that the top horizontal arrow is \'etale, and \item there exist Zariski coverings $Y = \bigcup Y_i$ and $f^{-1}(Y_i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_i$ is \'etale. \end{enumerate}


spaces-morphisms-lemma-composition-etale

The composition of two \'etale morphisms of algebraic spaces is \'etale.


spaces-morphisms-lemma-base-change-etale

The base change of an \'etale morphism of algebraic spaces by any morphism of algebraic spaces is \'etale.


spaces-morphisms-lemma-etale-locally-quasi-finite

An \'etale morphism of algebraic spaces is locally quasi-finite.


spaces-morphisms-lemma-etale-smooth

An \'etale morphism of algebraic spaces is smooth.


spaces-morphisms-lemma-etale-flat

An \'etale morphism of algebraic spaces is flat.


spaces-morphisms-lemma-etale-locally-finite-presentation

\begin{slogan} \'Etale implies locally of finite presentation. \end{slogan} An \'etale morphism of algebraic spaces is locally of finite presentation.


spaces-morphisms-lemma-etale-locally-finite-type

An \'etale morphism of algebraic spaces is locally of finite type.


spaces-morphisms-lemma-etale-unramified

An \'etale morphism of algebraic spaces is unramified.


spaces-morphisms-lemma-etale-permanence

Let $S$ be a scheme. Let $X, Y$ be algebraic spaces \'etale over an algebraic space $Z$. Any morphism $X \to Y$ over $Z$ is \'etale.


spaces-morphisms-lemma-unramified-flat-lfp-etale

A locally finitely presented, flat, unramified morphism of algebraic spaces is \'etale.