models-lemma-normalization

Let $X$ be a model of a geometrically normal variety $V$ over $K$. Then the normalization $\nu : X^\nu \to X$ is finite and the base change of $X^\nu$ to the completion $R^\wedge$ is the normalization of the base change of $X$. Moreover, for each $x \in X^\nu$ the completion of $\mathcal{O}_{X^\nu, x}$ is normal.


models-lemma-regular

Let $X$ be a model of a smooth curve $C$ over $K$. Then there exists a resolution of singularities of $X$ and any resolution is a model of $C$.


models-lemma-pre-exists-minimal-model

\begin{slogan} A regular proper model of a curve is obtained by successive blowups from a minimal model \end{slogan} Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_C) = K$. If $X$ is a regular proper model for $C$, then there exists a sequence of morphisms $$ X = X_m \to X_{m - 1} \to \ldots \to X_1 \to X_0 $$ of proper regular models of $C$, such that each morphism is a contraction of an exceptional curve of the first kind, and such that $X_0$ is a minimal model.


models-lemma-divisor-special-fiber

Let $X$ be a regular model of a smooth curve $C$ over $K$. \begin{enumerate} \item the special fibre $X_k$ is an effective Cartier divisor on $X$, \item each irreducible component $C_i$ of $X_k$ is an effective Cartier divisor on $X$, \item $X_k = \sum m_i C_i$ (sum of effective Cartier divisors) where $m_i$ is the multiplicity of $C_i$ in $X_k$, \item $\mathcal{O}_X(X_k) \cong \mathcal{O}_X$. \end{enumerate}


models-lemma-gorenstein

Let $X$ be a regular model of a smooth curve $C$ over $K$. Then \begin{enumerate} \item $X \to \Spec(R)$ is a Gorenstein morphism of relative dimension $1$, \item each of the irreducible components $C_i$ of $X_k$ is Gorenstein. \end{enumerate}


models-lemma-regular-model-connected

In Situation \ref{situation-regular-model} the special fibre $X_k$ is connected.


models-lemma-regular-model-pic

In Situation \ref{situation-regular-model} there is an exact sequence $$ 0 \to \mathbf{Z} \to \mathbf{Z}^{\oplus n} \to \Pic(X) \to \Pic(C) \to 0 $$ where the first map sends $1$ to $(m_1, \ldots, m_n)$ and the second maps sends the $i$th basis vector to $\mathcal{O}_X(C_i)$.


models-lemma-intersection-pairing

In Situation \ref{situation-regular-model} given $\mathcal{L}$ an invertible $\mathcal{O}_X$-module and $a = (a_1, \ldots, a_n) \in \mathbf{Z}^{\oplus n}$ we define $$ \langle a, \mathcal{L} \rangle = \sum a_i\deg(\mathcal{L}|_{C_i}) $$ Then $\langle , \rangle$ is bilinear and for $b = (b_1, \ldots, b_n) \in \mathbf{Z}^{\oplus n}$ we have $$ \left\langle a, \mathcal{O}_X(\sum b_i C_i) \right\rangle = \left\langle b, \mathcal{O}_X(\sum a_i C_i) \right\rangle $$


models-lemma-properties-form

In Situation \ref{situation-regular-model} the symmetric bilinear form (\ref{equation-form}) has the following properties \begin{enumerate} \item $(C_i \cdot C_j) \geq 0$ if $i \not = j$ with equality if and only if $C_i \cap C_j = \emptyset$, \item $(\sum m_i C_i \cdot C_j) = 0$, \item there is no nonempty proper subset $I \subset \{1, \ldots, n\}$ such that $(C_i \cdot C_j) = 0$ for $i \in I$, $j \not \in I$. \item $(\sum a_i C_i \cdot \sum a_i C_i) \leq 0$ with equality if and only if there exists a $q \in \mathbf{Q}$ such that $a_i = qm_i$ for $i = 1, \ldots, n$, \end{enumerate}


models-lemma-multiple-fibre-normal-bundle

In Situation \ref{situation-regular-model} set $d = \gcd(m_1, \ldots, m_n)$ and let $D = \sum (m_i/d)C_i$ as an effective Cartier divisor. Then $\mathcal{O}_X(D)$ has order dividing $d$ in $\Pic(X)$ and $\mathcal{C}_{D/X}$ an invertible $\mathcal{O}_D$-module of order dividing $d$ in $\Pic(D)$.


models-lemma-regular-model-field

\begin{reference} \cite[Lemma 2.6]{Artin-Winters} \end{reference} In Situation \ref{situation-regular-model} let $d = \gcd(m_1, \ldots, m_n)$. Let $D = \sum (m_i/d) C_i$ as an effective Cartier divisor. Then there exists a sequence of effective Cartier divisors $$ (X_k)_{red} = Z_0 \subset Z_1 \subset \ldots \subset Z_m = D $$ such that $Z_j = Z_{j - 1} + C_{i_j}$ for some $i_j \in \{1, \ldots, n\}$ for $j = 1, \ldots, m$ and such that $H^0(Z_j, \mathcal{O}_{Z_j})$ is a field finite over $k$ for $j = 0, \ldots m$.


models-lemma-regular-model-genus

\begin{reference} \cite[Lemma 2.6]{Artin-Winters} \end{reference} In Situation \ref{situation-regular-model} let $d = \gcd(m_1, \ldots, m_n)$. Let $D = \sum (m_i/d) C_i$ as an effective Cartier divisor on $X$. Then $$ 1 - g_C = d [\kappa : k] (1 - g_D) $$ where $g_C$ is the genus of $C$, $g_D$ is the genus of $D$, and $\kappa = H^0(D, \mathcal{O}_D)$.


models-lemma-exceptional-curves-dont-meet

In Situation \ref{situation-regular-model} given a pair of indices $i, j$ such that $C_i$ and $C_j$ are exceptional curves of the first kind and $C_i \cap C_j \not = \emptyset$, then $n = 2$, $m_1 = m_2 = 1$, $C_1 \cong \mathbf{P}^1_k$, $C_2 \cong \mathbf{P}^1_k$, $C_1$ and $C_2$ meet in a $k$-rational point, and $C$ has genus $0$.


models-lemma-minimal-model-unique

Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_C) = K$ and genus $> 0$. There is a unique minimal model for $C$.


models-lemma-add-component

In Situation \ref{situation-regular-model} suppose we have an effective Cartier divisors $D, D' \subset X$ such that $D' = D + C_i$ for some $i \in \{1, \ldots, n\}$ and $D' \subset X_k$. Then $$ \chi(X_k, \mathcal{O}_{D'}) - \chi(X_k, \mathcal{O}_D) = \chi(X_k, \mathcal{O}_X(-D)|_{C_i}) = -(D \cdot C_i) + \chi(C_i, \mathcal{O}_{C_i}) $$


models-lemma-genus-formula

In Situation \ref{situation-regular-model} we have $$ g_C = 1 + \sum\nolimits_{i = 1, \ldots, n} m_i\left([\kappa_i : k] (g_i - 1) - \frac{1}{2}(C_i \cdot C_i)\right) $$ where $\kappa_i = H^0(C_i, \mathcal{O}_{C_i})$, $g_i$ is the genus of $C_i$, and $g_C$ is the genus of $C$.


models-lemma-numerical-type-of-model

In Situation \ref{situation-regular-model} with $\kappa_i = H^0(C_i, \mathcal{O}_{C_i})$ and $g_i$ the genus of $C_i$ the data $$ n, m_i, (C_i \cdot C_j), [\kappa_i : k], g_i $$ is a numerical type of genus equal to the genus of $C$.


models-lemma-numerical-type-minimal-model

In Situation \ref{situation-regular-model}. The following are equivalent \begin{enumerate} \item $X$ is a minimal model, and \item the numerical type associated to $X$ is minimal. \end{enumerate}


models-lemma-genus-reduction-bigger-than

In Situation \ref{situation-regular-model} assume $X_k$ has a $k$-rational point $x$ which is a smooth point of $X_k \to \Spec(k)$. Then $$ \dim_k H^1((X_k)_{red}, \mathcal{O}_{(X_k)_{red}}) \geq g_{top} + g_{geom}(X_k/k) $$ where $g_{geom}$ is as in Algebraic Curves, Section \ref{curves-section-genus-geometric-genus} and $g_{top}$ is the topological genus (Definition \ref{definition-top-genus}) of the numerical type associated to $X_k$ (Definition \ref{definition-numerical-type-model}).


models-lemma-blowdown-regular-model

In Situation \ref{situation-regular-model} assume that $C_n$ is an exceptional curve of the first kind. Let $f : X \to X'$ be the contraction of $C_n$. Let $C'_i = f(C_i)$. Write $X'_k = \sum m'_i C'_i$. Then $X'$, $C'_i$, $i = 1, \ldots, n' = n - 1$, and $m'_i = m_i$ is as in Situation \ref{situation-regular-model} and we have \begin{enumerate} \item for $i, j < n$ we have $(C'_i \cdot C'_j) = (C_i \cdot C_j) - (C_i \cdot C_n) (C_j \cdot C_n) /(C_n \cdot C_n)$, \item for $i < n$ if $C_i \cap C_n \not = \emptyset$, then there are maps $\kappa_i \leftarrow \kappa'_i \rightarrow \kappa_n$. \end{enumerate} Here $\kappa_i = H^0(C_i, \mathcal{O}_{C_i})$ and $\kappa'_i = H^0(C'_i, \mathcal{O}_{C'_i})$.


models-lemma-nonuniqueness

Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_C) = K$ and genus $0$. If there is more than one minimal model for $C$, then the special fibre of every minimal model is isomorphic to $\mathbf{P}^1_k$.


models-lemma-characterize-trivial

In Situation \ref{situation-regular-model} let $d = \gcd(m_1, \ldots, m_n)$. If $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module which \begin{enumerate} \item restricts to the trivial invertible module on $C$, and \item has degree $0$ on each $C_i$, \end{enumerate} then $\mathcal{L}^{\otimes d} \cong \mathcal{O}_X$.


models-lemma-etale-local-at-worst-nodal

Let $R$ be a discrete valuation ring. Let $X$ be a scheme which is at-worst-nodal of relative dimension $1$ over $R$. Let $x \in X$ be a point of the special fibre of $X$ over $R$. Then there exists a commutative diagram $$ \xymatrix{ X \ar[d] & U \ar[r] \ar[d] \ar[l] & \Spec(A) \ar[dl] \\ \Spec(R) & \Spec(R') \ar[l] } $$ where $R \subset R'$ is an \'etale extension of discrete valuation rings, the morphism $U \to X$ is \'etale, the morphism $U \to \Spec(A)$ is \'etale, there is a point $x' \in U$ mapping to $x$, and $$ A = R'[u, v]/(uv) \quad\text{or}\quad A = R'[u, v]/(uv - \pi^n) $$ where $n \geq 0$ and $\pi \in R'$ is a uniformizer.


models-lemma-blowup-at-worst-nodal

Let $R$ be a discrete valuation ring. Let $X$ be a quasi-compact scheme which is at-worst-nodal of relative dimension $1$ with smooth generic fibre over $R$. Then there exists $m \geq 0$ and a sequence $$ X_m \to \ldots \to X_1 \to X_0 = X $$ such that \begin{enumerate} \item $X_{i + 1} \to X_i$ is the blowing up of a closed point $x_i$ where $X_i$ is singular, \item $X_i \to \Spec(R)$ is at-worst-nodal of relative dimension $1$, \item $X_m$ is regular. \end{enumerate}


models-lemma-blowdown-at-worst-nodal

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Assume $X \to \Spec(R)$ is at-worst-nodal of relative dimension $1$ over $R$. Let $X \to X'$ be the contraction of an exceptional curve $E \subset X$ of the first kind. Then $X'$ is at-worst-nodal of relative dimension $1$ over $R$.


models-lemma-semistable

Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_C) = K$. The following are equivalent \begin{enumerate} \item there exists a proper model of $C$ which is at-worst-nodal of relative dimension $1$ over $R$, \item there exists a minimal model of $C$ which is at-worst-nodal of relative dimension $1$ over $R$, and \item any minimal model of $C$ is at-worst-nodal of relative dimension $1$ over $R$. \end{enumerate}


models-lemma-good

Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_C) = K$. The following are equivalent \begin{enumerate} \item there exists a proper smooth model for $C$, \item there exists a minimal model for $C$ which is smooth over $R$, \item any minimal model is smooth over $R$. \end{enumerate}


models-proposition-classify-subgraphs

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$. Let $I \subset \{1, \ldots, n\}$ be a proper subset of cardinality $\geq 2$ consisting of $(-2)$-indices such that there does not exist a nonempty proper subset $I' \subset I$ with $a_{i'i} = 0$ for $i' \in I$, $i \in I \setminus I'$. Then up to reordering the $m_i$'s, $a_{ij}$'s, $w_i$'s for $i, j \in I$ are as listed in Lemmas \ref{lemma-two-by-two}, \ref{lemma-three-by-three}, \ref{lemma-four-by-four}, \ref{lemma-D4}, \ref{lemma-five-by-five}, \ref{lemma-D5}, \ref{lemma-long}, \ref{lemma-Dn}, \ref{lemma-E6}, \ref{lemma-E7}, or \ref{lemma-E8}.


models-proposition-exists-minimal-model

Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_C) = K$. A minimal model exists.


spaces-groupoids-lemma-restrict-relation

Let $B \to S$ as in Section \ref{section-notation}. Let $U$ be an algebraic space over $B$. Let $j : R \to U \times_B U$ be a pre-relation. Let $g : U' \to U$ be a morphism of algebraic spaces over $B$. Finally, set $$ R' = (U' \times_B U')\times_{U \times_B U} R \xrightarrow{j'} U' \times_B U' $$ Then $j'$ is a pre-relation on $U'$ over $B$. If $j$ is a relation, then $j'$ is a relation. If $j$ is a pre-equivalence relation, then $j'$ is a pre-equivalence relation. If $j$ is an equivalence relation, then $j'$ is an equivalence relation.


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

Let $B \to S$ as in Section \ref{section-notation}. Let $j : R \to U \times_B U$ be a pre-relation of algebraic spaces over $B$. Consider the relation on $|U|$ defined by the rule $$ x \sim y \Leftrightarrow \exists\ r \in |R| : t(r) = x, s(r) = y. $$ If $j$ is a pre-equivalence relation then this is an equivalence relation.


spaces-groupoids-lemma-base-change-group-space

Let $B \to S$ as in Section \ref{section-notation}. Let $(G, m)$ be a group algebraic space over $B$. Let $B' \to B$ be a morphism of algebraic spaces. The pullback $(G_{B'}, m_{B'})$ is a group algebraic space over $B'$.


spaces-groupoids-lemma-group-scheme-separated

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


spaces-groupoids-lemma-group-scheme-unramified-or-lqf

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Assume $G \to B$ is locally of finite type. Then $G \to B$ is unramified (resp.\ locally quasi-finite) if and only if $G \to B$ is unramified (resp.\ quasi-finite) at $e(b)$ for all $b \in |B|$.


spaces-groupoids-lemma-open-over-which-unramified-or-lqf

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Assume $G \to B$ is locally of finite type. \begin{enumerate} \item There exists a maximal open subspace $U \subset B$ such that $G_U \to U$ is unramified and formation of $U$ commutes with base change. \item There exists a maximal open subspace $U \subset B$ such that $G_U \to U$ is locally quasi-finite and formation of $U$ commutes with base change. \end{enumerate}


spaces-groupoids-lemma-free-action

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


spaces-groupoids-lemma-characterize-trivial-pseudo-torsors

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


spaces-groupoids-lemma-torsor

Let $S$ be a scheme. Let $(G, m)$ be a group algebraic space over $S$. Let $X$ be an algebraic space over $S$, and let $a : G \times_S X \to X$ be an action of $G$ on $X$. Then $X$ is a $G$-torsor in the $fppf$-topology in the sense of Definition \ref{definition-principal-homogeneous-space} if and only if $X$ is a $G$-torsor on $(\Sch/S)_{fppf}$ in the sense of Cohomology on Sites, Definition \ref{sites-cohomology-definition-torsor}.


spaces-groupoids-lemma-pseudo-torsor-implications

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Let $X$ be a pseudo $G$-torsor over $B$. Assume $G$ and $X$ locally of finite type over $B$. \begin{enumerate} \item If $G \to B$ is unramified, then $X \to B$ is unramified. \item If $G \to B$ is locally quasi-finite, then $X \to B$ is locally quasi-finite. \end{enumerate}


spaces-groupoids-lemma-pullback-equivariant

Let $B \to S$ as in Section \ref{section-notation}. Let $G$ be a group algebraic space over $B$. Let $f : X \to Y$ be a $G$-equivariant morphism between algebraic spaces over $B$ endowed with $G$-actions. Then pullback $f^*$ given by $(\mathcal{F}, \alpha) \mapsto (f^*\mathcal{F}, (1_G \times f)^*\alpha)$ defines a functor from the category of $G$-equivariant sheaves on $X$ to the category of quasi-coherent $G$-equivariant sheaves on $Y$.


spaces-groupoids-lemma-groupoid-pre-equivalence

Let $B \to S$ as in Section \ref{section-notation}. Given a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$ the morphism $j : R \to U \times_B U$ is a pre-equivalence relation.


spaces-groupoids-lemma-equivalence-groupoid

Let $B \to S$ as in Section \ref{section-notation}. Given an equivalence relation $j : R \to U \times_B U$ over $B$ there is a unique way to extend it to a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$.


spaces-groupoids-lemma-diagram

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


spaces-groupoids-lemma-base-change-groupoid

Let $B \to S$ be as in Section \ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $B' \to B$ be a morphism of algebraic spaces. Then the base changes $U' = B' \times_B U$, $R' = B' \times_B R$ endowed with the base changes $s'$, $t'$, $c'$ of the morphisms $s, t, c$ form a groupoid in algebraic spaces $(U', R', s', t', c')$ over $B'$ and the projections determine a morphism $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoids in algebraic spaces over $B$.


spaces-groupoids-lemma-pullback

Let $B \to S$ as in Section \ref{section-notation}. Consider a morphism $f : (U, R, s, t, c) \to (U', R', s', t', c')$ of groupoid in algebraic spaces over $B$. Then pullback $f^*$ given by $$ (\mathcal{F}, \alpha) \mapsto (f^*\mathcal{F}, f^*\alpha) $$ defines a functor from the category of quasi-coherent sheaves on $(U', R', s', t', c')$ to the category of quasi-coherent sheaves on $(U, R, s, t, c)$.


spaces-groupoids-lemma-abelian

Let $B \to S$ as in Section \ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. If $s$, $t$ are flat, then the category of quasi-coherent modules on $(U, R, s, t, c)$ is abelian.


spaces-groupoids-lemma-crystals-in-quasi-coherent-modules

In the situation above, if all the morphisms $f_\phi$ are flat, then there exists a cardinal $\kappa$ such that every object $(\{\mathcal{F}_i\}_{i \in I}, \{\alpha_\phi\}_{\phi \in \Phi})$ of $\textit{CQC}(X)$ is the directed colimit of its $\kappa$-generated submodules.


spaces-groupoids-lemma-set-generators

Let $B \to S$ as in Section \ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. If $s$, $t$ are flat, then there exists a set $T$ and a family of objects $(\mathcal{F}_t, \alpha_t)_{t \in T}$ of $\QCoh(U, R, s, t, c)$ such that every object $(\mathcal{F}, \alpha)$ is the directed colimit of its submodules isomorphic to one of the objects $(\mathcal{F}_t, \alpha_t)$.


spaces-groupoids-lemma-groupoid-from-action

Let $B \to S$ as in Section \ref{section-notation}. Let $(G, m)$ be a group algebraic space over $B$ with identity $e_G$ and inverse $i_G$. Let $X$ be an algebraic space over $B$ and let $a : G \times_B X \to X$ be an action of $G$ on $X$ over $B$. Then we get a groupoid in algebraic spaces $(U, R, s, t, c, e, i)$ over $B$ in the following manner: \begin{enumerate} \item We set $U = X$, and $R = G \times_B X$. \item We set $s : R \to U$ equal to $(g, x) \mapsto x$. \item We set $t : R \to U$ equal to $(g, x) \mapsto a(g, x)$. \item We set $c : R \times_{s, U, t} R \to R$ equal to $((g, x), (g', x')) \mapsto (m(g, g'), x')$. \item We set $e : U \to R$ equal to $x \mapsto (e_G(x), x)$. \item We set $i : R \to R$ equal to $(g, x) \mapsto (i_G(g), a(g, x))$. \end{enumerate}


spaces-groupoids-lemma-groupoid-stabilizer

Let $B \to S$ as in Section \ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The algebraic space $G$ defined by the cartesian square $$ \xymatrix{ G \ar[r] \ar[d] & R \ar[d]^{j = (t, s)} \\ U \ar[r]^-{\Delta} & U \times_B U } $$ is a group algebraic space over $U$ with composition law $m$ induced by the composition law $c$.


spaces-groupoids-lemma-groupoid-action-stabilizer

Let $B \to S$ as in Section \ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$, and let $G/U$ be its stabilizer. Denote $R_t/U$ the algebraic space $R$ seen as an algebraic space over $U$ via the morphism $t : R \to U$. There is a canonical left action $$ a : G \times_U R_t \longrightarrow R_t $$ induced by the composition law $c$.


spaces-groupoids-lemma-restrict-groupoid

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


spaces-groupoids-lemma-restrict-groupoid-relation

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


spaces-groupoids-lemma-criterion-quotient-representable

In the situation of Definition \ref{definition-quotient-sheaf}. Assume there is an algebraic space $M$ over $S$, and a morphism $U \to M$ such that \begin{enumerate} \item the morphism $U \to M$ equalizes $s, t$, \item the map $U \to M$ is a surjection of sheaves, and \item the induced map $(t, s) : R \to U \times_M U$ is a surjection of sheaves. \end{enumerate} In this case $M$ represents the quotient sheaf $U/R$.


spaces-groupoids-lemma-quotient-pre-equivalence

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $j : R \to U \times_B U$ be a pre-equivalence relation over $B$. For a scheme $S'$ over $S$ and $a, b \in U(S')$ the following are equivalent: \begin{enumerate} \item $a$ and $b$ map to the same element of $(U/R)(S')$, and \item there exists an fppf covering $\{f_i : S_i \to S'\}$ of $S'$ and morphisms $r_i : S_i \to R$ such that $a \circ f_i = s \circ r_i$ and $b \circ f_i = t \circ r_i$. \end{enumerate} In other words, in this case the map of sheaves $$ R \longrightarrow U \times_{U/R} U $$ is surjective.


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

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $j : R \to U \times_B U$ be a pre-relation over $B$ and $g : U' \to U$ a morphism of algebraic spaces over $B$. Let $j' : R' \to U' \times_B U'$ be the restriction of $j$ to $U'$. The map of quotient sheaves $$ U'/R' \longrightarrow U/R $$ is injective. If $U' \to U$ is surjective as a map of sheaves, for example if $\{g : U' \to U\}$ is an fppf covering (see Topologies on Spaces, Definition \ref{spaces-topologies-definition-fppf-covering}), then $U'/R' \to U/R$ is an isomorphism of sheaves.


spaces-groupoids-lemma-quotient-stack-arrows

Assume $B \to S$ and $(U, R, s, t, c)$ as in Definition \ref{definition-quotient-stack} (1). There are canonical $1$-morphisms $\pi : \mathcal{S}_U \to [U/R]$, and $[U/R] \to \mathcal{S}_B$ of stacks in groupoids over $(\Sch/S)_{fppf}$. The composition $\mathcal{S}_U \to \mathcal{S}_B$ is the $1$-morphism associated to the structure morphism $U \to B$.


spaces-groupoids-lemma-quotient-stack-2-arrow

Assumptions and notation as in Lemma \ref{lemma-quotient-stack-arrows}. There exists a canonical $2$-morphism $\alpha : \pi \circ s \to \pi \circ t$ making the diagram $$ \xymatrix{ \mathcal{S}_R \ar[r]_s \ar[d]_t & \mathcal{S}_U \ar[d]^\pi \\ \mathcal{S}_U \ar[r]^-\pi & [U/R] } $$ $2$-commutative.


spaces-groupoids-lemma-quotient-stack-functorial

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $f : (U, R, s, t, c) \to (U', R', s', t', c')$ be a morphism of groupoids in algebraic spaces over $B$. Then $f$ induces a canonical $1$-morphism of quotient stacks $$ [f] : [U/R] \longrightarrow [U'/R']. $$


spaces-groupoids-lemma-cartesian-square-of-morphism

Notation and assumption as in Lemma \ref{lemma-quotient-stack-functorial}. Let $(U'', R'', s'', t'', c'')$ be the groupoid in algebraic spaces over $B$ constructed above. There is a $2$-commutative square $$ \xymatrix{ [U''/R''] \ar[d] \ar[r]_{[g]} & [U/R] \ar[d]^{[f]} \\ \mathcal{S}_{U'} \ar[r] & [U'/R'] } $$ which identifies $[U''/R'']$ with the $2$-fibre product.


spaces-groupoids-lemma-quotient-stack-morphisms

Assume $B \to S$, $(U, R, s, t, c)$ and $\pi : \mathcal{S}_U \to [U/R]$ are as in Lemma \ref{lemma-quotient-stack-arrows}. Let $S'$ be a scheme over $S$. Let $x, y \in \Ob([U/R]_{S'})$ be objects of the quotient stack over $S'$. If $x = \pi(x')$ and $y = \pi(y')$ for some morphisms $x', y' : S' \to U$, then $$ \mathit{Isom}(x, y) = S' \times_{(y', x'), U \times_S U} R $$ as sheaves over $S'$.


spaces-groupoids-lemma-quotient-stack-2-cartesian

Assume $B \to S$, $(U, R, s, t, c)$, and $\pi : \mathcal{S}_U \to [U/R]$ are as in Lemma \ref{lemma-quotient-stack-arrows}. The $2$-commutative square $$ \xymatrix{ \mathcal{S}_R \ar[r]_s \ar[d]_t & \mathcal{S}_U \ar[d]^\pi \\ \mathcal{S}_U \ar[r]^-\pi & [U/R] } $$ of Lemma \ref{lemma-quotient-stack-2-arrow} is a $2$-fibre product of stacks in groupoids of $(\Sch/S)_{fppf}$.


spaces-groupoids-lemma-quotient-stack-isom

Assume $B \to S$ and $(U, R, s, t, c)$ are as in Definition \ref{definition-quotient-stack} (1). For any scheme $T$ over $S$ and objects $x, y$ of $[U/R]$ over $T$ the sheaf $\mathit{Isom}(x, y)$ on $(\Sch/T)_{fppf}$ has the following property: There exists a fppf covering $\{T_i \to T\}_{i \in I}$ such that $\mathit{Isom}(x, y)|_{(\Sch/T_i)_{fppf}}$ is representable by an algebraic space.


spaces-groupoids-lemma-quotient-stack-cocycle

Assumptions and notation as in Lemmas \ref{lemma-quotient-stack-arrows} and \ref{lemma-quotient-stack-2-arrow}. The vertical composition of $$ \xymatrix@C=15pc{ \mathcal{S}_{R \times_{s, U, t} R} \ruppertwocell^{\pi \circ s \circ \text{pr}_1 = \pi \circ s \circ c}{\ \ \ \ \ \ \alpha \star \text{id}_{\text{pr}_1}} \ar[r]_(.3){\pi \circ t \circ \text{pr}_1 = \pi \circ s \circ \text{pr}_0} \rlowertwocell_{\pi \circ t \circ \text{pr}_0 = \pi \circ t \circ c}{\ \ \ \ \ \ \alpha \star \text{id}_{\text{pr}_0}} & [U/R] } $$ is the $2$-morphism $\alpha \star \text{id}_c$. In a formula $\alpha \star \text{id}_c = (\alpha \star \text{id}_{\text{pr}_0}) \circ (\alpha \star \text{id}_{\text{pr}_1}) $.


spaces-groupoids-lemma-quotient-stack-2-coequalizer

Assumptions and notation as in Lemmas \ref{lemma-quotient-stack-arrows} and \ref{lemma-quotient-stack-2-arrow}. The $2$-commutative diagram of Lemma \ref{lemma-quotient-stack-2-arrow} is a $2$-coequalizer in the following sense: Given \begin{enumerate} \item a stack in groupoids $\mathcal{X}$ over $(\Sch/S)_{fppf}$, \item a $1$-morphism $f : \mathcal{S}_U \to \mathcal{X}$, and \item a $2$-arrow $\beta : f \circ s \to f \circ t$ \end{enumerate} such that $$ \beta \star \text{id}_c = (\beta \star \text{id}_{\text{pr}_0}) \circ (\beta \star \text{id}_{\text{pr}_1}) $$ then there exists a $1$-morphism $[U/R] \to \mathcal{X}$ which makes the diagram $$ \xymatrix{ \mathcal{S}_R \ar[r]_s \ar[d]^t & \mathcal{S}_U \ar[d] \ar[ddr]^f \\ \mathcal{S}_U \ar[r] \ar[rrd]_f & [U/R] \ar[rd] \\ & & \mathcal{X} } $$ $2$-commute.


spaces-groupoids-lemma-quotient-stack-objects

Assume $B \to S$ and $(U, R, s, t, c)$ are as in Definition \ref{definition-quotient-stack} (1). Let $\pi : \mathcal{S}_U \to [U/R]$ be as in Lemma \ref{lemma-quotient-stack-arrows}. Let $T$ be a scheme over $S$. \begin{enumerate} \item for every object $x$ of the fibre category $[U/R]_T$ there exists an fppf covering $\{f_i : T_i \to T\}_{i \in I}$ such that $f_i^*x \cong \pi(u_i)$ for some $u_i \in U(T_i)$, \item the composition of the isomorphisms $$ \pi(u_i \circ \text{pr}_0) = \text{pr}_0^*\pi(u_i) \cong \text{pr}_0^*f_i^*x \cong \text{pr}_1^*f_j^*x \cong \text{pr}_1^*\pi(u_j) = \pi(u_j \circ \text{pr}_1) $$ are of the form $\pi(r_{ij})$ for certain morphisms $r_{ij} : T_i \times_T T_j \to R$, \item the system $(u_i, r_{ij})$ forms a $[U/R]$-descent datum as defined above, \item any $[U/R]$-descent datum $(u_i, r_{ij})$ arises in this manner, \item if $x$ corresponds to $(u_i, r_{ij})$ as above, and $y \in \Ob([U/R]_T)$ corresponds to $(u'_i, r'_{ij})$ then there is a canonical bijection $$ \Mor_{[U/R]_T}(x, y) \longleftrightarrow \left\{ \begin{matrix} \text{morphisms }(u_i, r_{ij}) \to (u'_i, r'_{ij})\\ \text{of }[U/R]\text{-descent data} \end{matrix} \right\} $$ \item this correspondence is compatible with refinements of fppf coverings. \end{enumerate}


spaces-groupoids-lemma-quotient-stack-restrict

Notation and assumption as in Lemma \ref{lemma-quotient-stack-functorial}. The morphism of quotient stacks $$ [f] : [U/R] \longrightarrow [U'/R'] $$ is fully faithful if and only if $R$ is the restriction of $R'$ via the morphism $f : U \to U'$.


spaces-groupoids-lemma-quotient-stack-restrict-equivalence

Notation and assumption as in Lemma \ref{lemma-quotient-stack-functorial}. The morphism of quotient stacks $$ [f] : [U/R] \longrightarrow [U'/R'] $$ is an equivalence if and only if \begin{enumerate} \item $(U, R, s, t, c)$ is the restriction of $(U', R', s', t', c')$ via $f : U \to U'$, and \item the map $$ \xymatrix{ U \times_{f, U', t'} R' \ar[r]_-{\text{pr}_1} \ar@/^3ex/[rr]^h & R' \ar[r]_{s'} & U' } $$ is a surjection of sheaves. \end{enumerate} Part (2) holds for example if $\{h : U \times_{f, U', t'} R' \to U'\}$ is an fppf covering, or if $f : U \to U'$ is a surjection of sheaves, or if $\{f : U \to U'\}$ is an fppf covering.


spaces-groupoids-lemma-criterion-fibre-product

Notation and assumption as in Lemma \ref{lemma-quotient-stack-functorial}. Assume that $$ \xymatrix{ R \ar[d]_s \ar[r]_f & R' \ar[d]^{s'} \\ U \ar[r]^f & U' } $$ is cartesian. Then $$ \xymatrix{ \mathcal{S}_U \ar[d] \ar[r] & [U/R] \ar[d]^{[f]} \\ \mathcal{S}_{U'} \ar[r] & [U'/R'] } $$ is a $2$-fibre product square.


spaces-groupoids-lemma-presentation-inertia

Assume $B \to S$ and $(U, R, s, t, c)$ as in Definition \ref{definition-quotient-stack} (1). Let $G/U$ be the stabilizer group algebraic space of the groupoid $(U, R, s, t, c, e, i)$, see Definition \ref{definition-stabilizer-groupoid}. Set $R' = R \times_{s, U} G$ and set \begin{enumerate} \item $s' : R' \to G$, $(r, g) \mapsto g$, \item $t' : R' \to G$, $(r, g) \mapsto c(r, c(g, i(r)))$, \item $c' : R' \times_{s', G, t'} R' \to R'$, $((r_1, g_1), (r_2, g_2) \mapsto (c(r_1, r_2), g_1)$. \end{enumerate} Then $(G, R', s', t', c')$ is a groupoid in algebraic spaces over $B$ and $$ \mathcal{I}_{[U/R]} = [G/ R']. $$ i.e., the associated quotient stack is the inertia stack of $[U/R]$.


spaces-groupoids-lemma-2-cartesian-inertia

Assume $B \to S$ and $(U, R, s, t, c)$ as in Definition \ref{definition-quotient-stack} (1). Let $G/U$ be the stabilizer group algebraic space of the groupoid $(U, R, s, t, c, e, i)$, see Definition \ref{definition-stabilizer-groupoid}. There is a canonical $2$-cartesian diagram $$ \xymatrix{ \mathcal{S}_G \ar[r] \ar[d] & \mathcal{S}_U \ar[d] \\ \mathcal{I}_{[U/R]} \ar[r] & [U/R] } $$ of stacks in groupoids of $(\Sch/S)_{fppf}$.


spaces-groupoids-lemma-when-gerbe

Notation and assumption as in Lemma \ref{lemma-quotient-stack-functorial}. The morphism of quotient stacks $$ [f] : [U/R] \longrightarrow [U'/R'] $$ turns $[U/R]$ into a gerbe over $[U'/R']$ if $f : U \to U'$ and $R \to R'|_U$ are surjective maps of fppf sheaves. Here $R'|_U$ is the restriction of $R'$ to $U$ via $f : U \to U'$.


spaces-groupoids-lemma-group-quotient-gerbe

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Endow $B$ with the trivial action of $G$. The morphism $$ [B/G] \longrightarrow \mathcal{S}_B $$ (Lemma \ref{lemma-quotient-stack-arrows}) turns $[B/G]$ into a gerbe over $B$.


spaces-groupoids-lemma-quotient-stack-change-big-site

Suppose given big sites $\Sch_{fppf}$ and $\Sch'_{fppf}$. Assume that $\Sch_{fppf}$ is contained in $\Sch'_{fppf}$, see Topologies, Section \ref{topologies-section-change-alpha}. Let $S \in \Ob(\Sch_{fppf})$. Let $B, U, R \in \Sh((\Sch/S)_{fppf})$ be algebraic spaces, and let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $f : (\Sch'/S)_{fppf} \to (\Sch/S)_{fppf}$ the morphism of sites corresponding to the inclusion functor $u : \Sch_{fppf} \to \Sch'_{fppf}$. Then we have a canonical equivalence $$ [f^{-1}U/f^{-1}R] \longrightarrow f^{-1}[U/R] $$ of stacks in groupoids over $(\Sch'/S)_{fppf}$.


spaces-groupoids-lemma-diagram-diagonal

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


spaces-groupoids-lemma-diagonal

Let $B \to S$ be as in Section \ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $G \to U$ be the stabilizer group algebraic space. \begin{enumerate} \item The following are equivalent \begin{enumerate} \item $j : R \to U \times_B U$ is separated, \item $G \to U$ is separated, and \item $e : U \to G$ is a closed immersion. \end{enumerate} \item The following are equivalent \begin{enumerate} \item $j : R \to U \times_B U$ is locally separated, \item $G \to U$ is locally separated, and \item $e : U \to G$ is an immersion. \end{enumerate} \item The following are equivalent \begin{enumerate} \item $j : R \to U \times_B U$ is quasi-separated, \item $G \to U$ is quasi-separated, and \item $e : U \to G$ is quasi-compact. \end{enumerate} \end{enumerate}


spaces-descent-lemma-map-families

Let $S$ be a scheme. Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ and $\mathcal{V} = \{V_j \to V\}_{j \in J}$ be families of morphisms of algebraic spaces over $S$ with fixed targets. Let $(g, \alpha : I \to J, (g_i)) : \mathcal{U} \to \mathcal{V}$ be a morphism of families of maps with fixed target, see Sites, Definition \ref{sites-definition-morphism-coverings}. Let $(\mathcal{F}_j, \varphi_{jj'})$ be a descent datum for quasi-coherent sheaves with respect to the family $\{V_j \to V\}_{j \in J}$. Then \begin{enumerate} \item The system $$ \left(g_i^*\mathcal{F}_{\alpha(i)}, (g_i \times g_{i'})^*\varphi_{\alpha(i)\alpha(i')}\right) $$ is a descent datum with respect to the family $\{U_i \to U\}_{i \in I}$. \item This construction is functorial in the descent datum $(\mathcal{F}_j, \varphi_{jj'})$. \item Given a second morphism $(g', \alpha' : I \to J, (g'_i))$ of families of maps with fixed target with $g = g'$ there exists a functorial isomorphism of descent data $$ (g_i^*\mathcal{F}_{\alpha(i)}, (g_i \times g_{i'})^*\varphi_{\alpha(i)\alpha(i')}) \cong ((g'_i)^*\mathcal{F}_{\alpha'(i)}, (g'_i \times g'_{i'})^*\varphi_{\alpha'(i)\alpha'(i')}). $$ \end{enumerate}


spaces-descent-lemma-zariski-descent-effective

Let $S$ be a scheme. Let $U$ be an algebraic space over $S$. Let $\{U_i \to U\}$ be a Zariski covering of $U$, see Topologies on Spaces, Definition \ref{spaces-topologies-definition-zariski-covering}. Any descent datum on quasi-coherent sheaves for the family $\mathcal{U} = \{U_i \to U\}$ is effective. Moreover, the functor from the category of quasi-coherent $\mathcal{O}_U$-modules to the category of descent data with respect to $\{U_i \to U\}$ is fully faithful.


spaces-descent-lemma-finite-presentation-descends

Let $X$ be an algebraic space over a scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that each $f_i^*\mathcal{F}$ is an $\mathcal{O}_{X_i}$-module of finite presentation. Then $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation.


spaces-descent-lemma-locally-projective-descends

Let $X$ be an algebraic space over a scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that each $f_i^*\mathcal{F}$ is a locally projective $\mathcal{O}_{X_i}$-module. Then $\mathcal{F}$ is a locally projective $\mathcal{O}_X$-module.


spaces-descent-lemma-finite-finitely-presented-module

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume $f$ is finite and of finite presentation. Then $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation if and only if $f_*\mathcal{F}$ is an $\mathcal{O}_Y$-module of finite presentation.


spaces-descent-lemma-open-fpqc-covering

Let $S$ be a scheme. Let $\{f_i : T_i \to T\}_{i \in I}$ be an fpqc covering of algebraic spaces over $S$. Suppose that for each $i$ we have an open subspace $W_i \subset T_i$ such that for all $i, j \in I$ we have $\text{pr}_0^{-1}(W_i) = \text{pr}_1^{-1}(W_j)$ as open subspaces of $T_i \times_T T_j$. Then there exists a unique open subspace $W \subset T$ such that $W_i = f_i^{-1}(W)$ for each $i$.


spaces-descent-lemma-fpqc-universal-effective-epimorphisms

Let $S$ be a scheme. Let $\{T_i \to T\}$ be an fpqc covering of algebraic spaces over $S$, see Topologies on Spaces, Definition \ref{spaces-topologies-definition-fpqc-covering}. Then given an algebraic space $B$ over $S$ the sequence $$ \xymatrix{ \Mor_S(T, B) \ar[r] & \prod\nolimits_i \Mor_S(T_i, B) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod\nolimits_{i, j} \Mor_S(T_i \times_T T_j, B) } $$ is an equalizer diagram. In other words, every representable functor on the category of algebraic spaces over $S$ satisfies the sheaf condition for fpqc coverings.


spaces-descent-lemma-curiosity

Let $S$ be a scheme. Let $X \to Y \to Z$ be morphism of algebraic spaces. Let $P$ be one of the following properties of morphisms of algebraic spaces over $S$: flat, locally finite type, locally finite presentation. Assume that $X \to Z$ has $P$ and that $X \to Y$ is a surjection of sheaves on $(\Sch/S)_{fppf}$. Then $Y \to Z$ is $P$.


spaces-descent-lemma-flat-finitely-presented-permanence

Let $S$ be a scheme. Let $$ \xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & B } $$ be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that $f$ is surjective, flat, and locally of finite presentation and assume that $p$ is locally of finite presentation (resp.\ locally of finite type). Then $q$ is locally of finite presentation (resp.\ locally of finite type).


spaces-descent-lemma-syntomic-smooth-etale-permanence

Let $S$ be a scheme. Let $$ \xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & B } $$ be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that \begin{enumerate} \item $f$ is surjective, and syntomic (resp.\ smooth, resp.\ \'etale), \item $p$ is syntomic (resp.\ smooth, resp.\ \'etale). \end{enumerate} Then $q$ is syntomic (resp.\ smooth, resp.\ \'etale).


spaces-descent-lemma-smooth-permanence

Let $S$ be a scheme. Let $$ \xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & B } $$ be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that \begin{enumerate} \item $f$ is surjective, flat, and locally of finite presentation, \item $p$ is smooth (resp.\ \'etale). \end{enumerate} Then $q$ is smooth (resp.\ \'etale).


spaces-descent-lemma-descend-unibranch

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$. If $f$ is flat at $x$ and $X$ is geometrically unibranch at $x$, then $Y$ is geometrically unibranch at $f(x)$.


spaces-descent-lemma-descend-reduced

\begin{slogan} A flat and surjective morphism of algebraic spaces with a reduced source has a reduced target. \end{slogan} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is flat and surjective and $X$ is reduced, then $Y$ is reduced.


spaces-descent-lemma-descend-locally-Noetherian

Let $f : X \to Y$ be a morphism of algebraic spaces. If $f$ is locally of finite presentation, flat, and surjective and $X$ is locally Noetherian, then $Y$ is locally Noetherian.


spaces-descent-lemma-descend-regular

Let $f : X \to Y$ be a morphism of algebraic spaces. If $f$ is locally of finite presentation, flat, and surjective and $X$ is regular, then $Y$ is regular.


spaces-descent-lemma-reduced-local-smooth

Let $f : X \to Y$ be a smooth morphism of algebraic spaces. If $Y$ is reduced, then $X$ is reduced. If $f$ is surjective and $X$ is reduced, then $Y$ is reduced.


spaces-descent-lemma-pullback-property-local-target

Let $S$ be a scheme. Let $\tau \in \{fpqc, fppf, syntomic, smooth, \etale\}$. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is $\tau$ local on the target. Let $f : X \to Y$ have property $\mathcal{P}$. For any morphism $Y' \to Y$ which is flat, resp.\ flat and locally of finite presentation, resp.\ syntomic, resp.\ \'etale, the base change $f' : Y' \times_Y X \to Y'$ of $f$ has property $\mathcal{P}$.


spaces-descent-lemma-largest-open-of-the-base

Let $S$ be a scheme. Let $\tau \in \{fppf, syntomic, smooth, \etale\}$. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is $\tau$ local on the target. For any morphism of algebraic spaces $f : X \to Y$ over $S$ there exists a largest open subspace $W(f) \subset Y$ such that the restriction $X_{W(f)} \to W(f)$ has $\mathcal{P}$. Moreover, \begin{enumerate} \item if $g : Y' \to Y$ is a morphism of algebraic spaces which is flat and locally of finite presentation, syntomic, smooth, or \'etale and the base change $f' : X_{Y'} \to Y'$ has $\mathcal{P}$, then $g$ factors through $W(f)$, \item if $g : Y' \to Y$ is flat and locally of finite presentation, syntomic, smooth, or \'etale, then $W(f') = g^{-1}(W(f))$, and \item if $\{g_i : Y_i \to Y\}$ is a $\tau$-covering, then $g_i^{-1}(W(f)) = W(f_i)$, where $f_i$ is the base change of $f$ by $Y_i \to Y$. \end{enumerate}


spaces-descent-lemma-descending-properties-morphisms

Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Assume \begin{enumerate} \item if $X_i \to Y_i$, $i = 1, 2$ have property $\mathcal{P}$ so does $X_1 \amalg X_2 \to Y_1 \amalg Y_2$, \item a morphism of algebraic spaces $f : X \to Y$ has property $\mathcal{P}$ if and only if for every affine scheme $Z$ and morphism $Z \to Y$ the base change $Z \times_Y X \to Z$ of $f$ has property $\mathcal{P}$, and \item for any surjective flat morphism of affine schemes $Z' \to Z$ over $S$ and a morphism $f : X \to Z$ from an algebraic space to $Z$ we have $$ f' : Z' \times_Z X \to Z'\text{ has }\mathcal{P} \Rightarrow f\text{ has }\mathcal{P}. $$ \end{enumerate} Then $\mathcal{P}$ is fpqc local on the base.


spaces-descent-lemma-descending-property-quasi-compact

Let $S$ be a scheme. The property $\mathcal{P}(f) =$``$f$ is quasi-compact'' is fpqc local on the base on algebraic spaces over $S$.


spaces-descent-lemma-descending-property-quasi-separated

Let $S$ be a scheme. The property $\mathcal{P}(f) =$``$f$ is quasi-separated'' is fpqc local on the base on algebraic spaces over $S$.


spaces-descent-lemma-descending-property-universally-closed

Let $S$ be a scheme. The property $\mathcal{P}(f) =$``$f$ is universally closed'' is fpqc local on the base on algebraic spaces over $S$.


spaces-descent-lemma-descending-property-universally-open

Let $S$ be a scheme. The property $\mathcal{P}(f) =$``$f$ is universally open'' is fpqc local on the base on algebraic spaces over $S$.


spaces-descent-lemma-descending-property-surjective

The property $\mathcal{P}(f) =$``$f$ is surjective'' is fpqc local on the base.


spaces-descent-lemma-descending-property-universally-injective

The property $\mathcal{P}(f) =$``$f$ is universally injective'' is fpqc local on the base.


spaces-descent-lemma-descending-property-locally-finite-type

The property $\mathcal{P}(f) =$``$f$ is locally of finite type'' is fpqc local on the base.


spaces-descent-lemma-descending-property-locally-finite-presentation

The property $\mathcal{P}(f) =$``$f$ is locally of finite presentation'' is fpqc local on the base.


spaces-descent-lemma-descending-property-finite-type

The property $\mathcal{P}(f) =$``$f$ is of finite type'' is fpqc local on the base.


spaces-descent-lemma-descending-property-finite-presentation

The property $\mathcal{P}(f) =$``$f$ is of finite presentation'' is fpqc local on the base.


spaces-descent-lemma-descending-property-flat

The property $\mathcal{P}(f) =$``$f$ is flat'' is fpqc local on the base.


spaces-descent-lemma-descending-property-open-immersion

The property $\mathcal{P}(f) =$``$f$ is an open immersion'' is fpqc local on the base.


spaces-descent-lemma-descending-property-isomorphism

The property $\mathcal{P}(f) =$``$f$ is an isomorphism'' is fpqc local on the base.


spaces-descent-lemma-descending-property-affine

The property $\mathcal{P}(f) =$``$f$ is affine'' is fpqc local on the base.


spaces-descent-lemma-descending-property-closed-immersion

The property $\mathcal{P}(f) =$``$f$ is a closed immersion'' is fpqc local on the base.


spaces-descent-lemma-descending-property-separated

The property $\mathcal{P}(f) =$``$f$ is separated'' is fpqc local on the base.


spaces-descent-lemma-descending-property-proper

The property $\mathcal{P}(f) =$``$f$ is proper'' is fpqc local on the base.


spaces-descent-lemma-descending-property-quasi-affine

The property $\mathcal{P}(f) =$``$f$ is quasi-affine'' is fpqc local on the base.


spaces-descent-lemma-descending-property-integral

The property $\mathcal{P}(f) =$``$f$ is integral'' is fpqc local on the base.


spaces-descent-lemma-descending-property-finite

The property $\mathcal{P}(f) =$``$f$ is finite'' is fpqc local on the base.


spaces-descent-lemma-descending-property-quasi-finite

The properties $\mathcal{P}(f) =$``$f$ is locally quasi-finite'' and $\mathcal{P}(f) =$``$f$ is quasi-finite'' are fpqc local on the base.


spaces-descent-lemma-descending-property-syntomic

The property $\mathcal{P}(f) =$``$f$ is syntomic'' is fpqc local on the base.


spaces-descent-lemma-descending-property-smooth

The property $\mathcal{P}(f) =$``$f$ is smooth'' is fpqc local on the base.


spaces-descent-lemma-descending-property-unramified

The property $\mathcal{P}(f) =$``$f$ is unramified'' is fpqc local on the base.


spaces-descent-lemma-descending-property-etale

The property $\mathcal{P}(f) =$``$f$ is \'etale'' is fpqc local on the base.


spaces-descent-lemma-descending-property-monomorphism

The property $\mathcal{P}(f) =$``$f$ is a monomorphism'' is fpqc local on the base.


spaces-descent-lemma-descending-fppf-property-immersion

The property $\mathcal{P}(f) =$``$f$ is an immersion'' is fppf local on the base.


spaces-descent-lemma-descending-property-ample

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $\{g_i : Y_i \to Y\}_{i \in I}$ be an fpqc covering. Let $f_i : X_i \to Y_i$ be the base change of $f$ and let $\mathcal{L}_i$ be the pullback of $\mathcal{L}$ to $X_i$. The following are equivalent \begin{enumerate} \item $\mathcal{L}$ is ample on $X/Y$, and \item $\mathcal{L}_i$ is ample on $X_i/Y_i$ for every $i \in I$. \end{enumerate}


spaces-descent-lemma-ample-in-neighbourhood

Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. There exists an open subspace $V \subset Y$ characterized by the following property: A morphism $Y' \to Y$ of algebraic spaces factors through $V$ if and only if the pullback $\mathcal{L}'$ of $\mathcal{L}$ to $X' = Y' \times_Y X$ is ample on $X'/Y'$ (as in Divisors on Spaces, Definition \ref{spaces-divisors-definition-relatively-ample}).


spaces-descent-lemma-precompose-property-local-source

Let $S$ be a scheme. Let $\tau \in \{fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] \etale\}$. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is $\tau$ local on the source. Let $f : X \to Y$ have property $\mathcal{P}$. For any morphism $a : X' \to X$ which is flat, resp.\ flat and locally of finite presentation, resp.\ syntomic, resp.\ smooth, resp.\ \'etale, the composition $f \circ a : X' \to Y$ has property $\mathcal{P}$.


spaces-descent-lemma-transfer-from-schemes

Let $S$ be a scheme. Let $\tau \in \{fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] \etale\}$. Suppose that $\mathcal{P}$ is a property of morphisms of schemes over $S$ which is \'etale local on the source-and-target. Denote $\mathcal{P}_{spaces}$ the corresponding property of morphisms of algebraic spaces over $S$, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-P}. If $\mathcal{P}$ is local on the source for the $\tau$-topology, then $\mathcal{P}_{spaces}$ is local on the source for the $\tau$-topology.


spaces-descent-lemma-flat-fpqc-local-source

The property $\mathcal{P}(f)=$``$f$ is flat'' is fpqc local on the source.


spaces-descent-lemma-locally-finite-presentation-fppf-local-source

The property $\mathcal{P}(f)=$``$f$ is locally of finite presentation'' is fppf local on the source.


spaces-descent-lemma-locally-finite-type-fppf-local-source

The property $\mathcal{P}(f)=$``$f$ is locally of finite type'' is fppf local on the source.


spaces-descent-lemma-universally-open-fppf-local-source

The property $\mathcal{P}(f)=$``$f$ is universally open'' is fppf local on the source.


spaces-descent-lemma-syntomic-syntomic-local-source

The property $\mathcal{P}(f)=$``$f$ is syntomic'' is syntomic local on the source.


spaces-descent-lemma-smooth-smooth-local-source

The property $\mathcal{P}(f)=$``$f$ is smooth'' is smooth local on the source.


spaces-descent-lemma-etale-etale-local-source

The property $\mathcal{P}(f)=$``$f$ is \'etale'' is \'etale local on the source.


spaces-descent-lemma-locally-quasi-finite-etale-local-source

The property $\mathcal{P}(f)=$``$f$ is locally quasi-finite'' is \'etale local on the source.


spaces-descent-lemma-unramified-etale-local-source

The property $\mathcal{P}(f)=$``$f$ is unramified'' is \'etale local on the source.


spaces-descent-lemma-local-source-target-implies

Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is smooth local on source-and-target. Then \begin{enumerate} \item $\mathcal{P}$ is smooth local on the source, \item $\mathcal{P}$ is smooth local on the target, \item $\mathcal{P}$ is stable under postcomposing with smooth morphisms: if $f : X \to Y$ has $\mathcal{P}$ and $g : Y \to Z$ is smooth, then $g \circ f$ has $\mathcal{P}$. \end{enumerate}


spaces-descent-lemma-local-source-target-characterize

Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is smooth local on source-and-target. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item[(a)] $f$ has property $\mathcal{P}$, \item[(b)] for every $x \in |X|$ there exists a smooth morphism of pairs $a : (U, u) \to (X, x)$, a smooth morphism $b : V \to Y$, and a morphism $h : U \to V$ such that $f \circ a = b \circ h$ and $h$ has $\mathcal{P}$, \item[(c)] for some commutative diagram $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y } $$ with $a$, $b$ smooth and $a$ surjective the morphism $h$ has $\mathcal{P}$, \item[(d)] for any commutative diagram $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y } $$ with $b$ smooth and $U \to X \times_Y V$ smooth the morphism $h$ has $\mathcal{P}$, \item[(e)] there exists a smooth covering $\{Y_i \to Y\}_{i \in I}$ such that each base change $Y_i \times_Y X \to Y_i$ has $\mathcal{P}$, \item[(f)] there exists a smooth covering $\{X_i \to X\}_{i \in I}$ such that each composition $X_i \to Y$ has $\mathcal{P}$, \item[(g)] there exists a smooth covering $\{Y_i \to Y\}_{i \in I}$ and for each $i \in I$ a smooth covering $\{X_{ij} \to Y_i \times_Y X\}_{j \in J_i}$ such that each morphism $X_{ij} \to Y_i$ has $\mathcal{P}$. \end{enumerate}


spaces-descent-lemma-smooth-local-source-target

Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Assume \begin{enumerate} \item $\mathcal{P}$ is smooth local on the source, \item $\mathcal{P}$ is smooth local on the target, and \item $\mathcal{P}$ is stable under postcomposing with smooth morphisms: if $f : X \to Y$ has $\mathcal{P}$ and $Y \to Z$ is a smooth morphism then $X \to Z$ has $\mathcal{P}$. \end{enumerate} Then $\mathcal{P}$ is smooth local on the source-and-target.


spaces-descent-lemma-etale-smooth-local-source-target-implies

Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is \'etale-smooth local on source-and-target. Then \begin{enumerate} \item $\mathcal{P}$ is \'etale local on the source, \item $\mathcal{P}$ is smooth local on the target, \item $\mathcal{P}$ is stable under postcomposing with \'etale morphisms: if $f : X \to Y$ has $\mathcal{P}$ and $g : Y \to Z$ is \'etale, then $g \circ f$ has $\mathcal{P}$, and \item $\mathcal{P}$ has a permanence property: given $f : X \to Y$ and $g : Y \to Z$ \'etale such that $g \circ f$ has $\mathcal{P}$, then $f$ has $\mathcal{P}$. \end{enumerate}


spaces-descent-lemma-etale-smooth-local-source-target-characterize

Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is etale-smooth local on source-and-target. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item[(a)] $f$ has property $\mathcal{P}$, \item[(b)] for every $x \in |X|$ there exists a smooth morphism $b : V \to Y$, an \'etale morphism $a : U \to V \times_Y X$, and a point $u \in |U|$ mapping to $x$ such that $U \to V$ has $\mathcal{P}$, \item[(c)] for some commutative diagram $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y } $$ with $b$ smooth, $U \to V \times_Y X$ \'etale, and $a$ surjective the morphism $h$ has $\mathcal{P}$, \item[(d)] for any commutative diagram $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y } $$ with $b$ smooth and $U \to X \times_Y V$ \'etale, the morphism $h$ has $\mathcal{P}$, \item[(e)] there exists a smooth covering $\{Y_i \to Y\}_{i \in I}$ such that each base change $Y_i \times_Y X \to Y_i$ has $\mathcal{P}$, \item[(f)] there exists an \'etale covering $\{X_i \to X\}_{i \in I}$ such that each composition $X_i \to Y$ has $\mathcal{P}$, \item[(g)] there exists a smooth covering $\{Y_i \to Y\}_{i \in I}$ and for each $i \in I$ an \'etale covering $\{X_{ij} \to Y_i \times_Y X\}_{j \in J_i}$ such that each morphism $X_{ij} \to Y_i$ has $\mathcal{P}$. \end{enumerate}


spaces-descent-lemma-etale-smooth-local-source-target

Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Assume \begin{enumerate} \item $\mathcal{P}$ is \'etale local on the source, \item $\mathcal{P}$ is smooth local on the target, and \item $\mathcal{P}$ is stable under postcomposing with open immersions: if $f : X \to Y$ has $\mathcal{P}$ and $Y \subset Z$ is an open embedding then $X \to Z$ has $\mathcal{P}$. \end{enumerate} Then $\mathcal{P}$ is \'etale-smooth local on the source-and-target.


spaces-descent-lemma-family-is-one

Let $S$ be a scheme. Let $\{X_i \to X\}_{i \in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$. Set $Y = \coprod_{i \in I} X_i$. There is a canonical equivalence of categories $$ \begin{matrix} \text{category of descent data } \\ \text{relative to the family } \{X_i \to X\}_{i \in I} \end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data} \\ \text{ relative to } Y/X \end{matrix} $$ which maps $(V_i, \varphi_{ij})$ to $(V, \varphi)$ with $V = \coprod_{i\in I} V_i$ and $\varphi = \coprod \varphi_{ij}$.


spaces-descent-lemma-pullback

Pullback of descent data. Let $S$ be a scheme. \begin{enumerate} \item Let $$ \xymatrix{ Y' \ar[r]_f \ar[d]_{a'} & Y \ar[d]^a \\ X' \ar[r]^h & X } $$ be a commutative diagram of algebraic spaces over $S$. The construction $$ (V \to Y, \varphi) \longmapsto f^*(V \to Y, \varphi) = (V' \to Y', \varphi') $$ where $V' = Y' \times_Y V$ and where $\varphi'$ is defined as the composition $$ \xymatrix{ V' \times_{X'} Y' \ar@{=}[r] & (Y' \times_Y V) \times_{X'} Y' \ar@{=}[r] & (Y' \times_{X'} Y') \times_{Y \times_X Y} (V \times_X Y) \ar[d]^{\text{id} \times \varphi} \\ Y' \times_{X'} V' \ar@{=}[r] & Y' \times_{X'} (Y' \times_Y V) & (Y' \times_X Y') \times_{Y \times_X Y} (Y \times_X V) \ar@{=}[l] } $$ defines a functor from the category of descent data relative to $Y \to X$ to the category of descent data relative to $Y' \to X'$. \item Given two morphisms $f_i : Y' \to Y$, $i = 0, 1$ making the diagram commute the functors $f_0^*$ and $f_1^*$ are canonically isomorphic. \end{enumerate}


spaces-descent-lemma-pullback-family

Let $S$ be a scheme. Let $\mathcal{U}' = \{X'_i \to X'\}_{i \in I'}$ and $\mathcal{U} = \{X_j \to X\}_{i \in I}$ be families of morphisms with fixed target. Let $\alpha : I' \to I$, $g : X' \to X$ and $g_i : X'_i \to X_{\alpha(i)}$ be a morphism of families of maps with fixed target, see Sites, Definition \ref{sites-definition-morphism-coverings}. \begin{enumerate} \item Let $(V_i, \varphi_{ij})$ be a descent datum relative to the family $\mathcal{U}$. The system $$ \left( g_i^*V_{\alpha(i)}, (g_i \times g_j)^*\varphi_{\alpha(i) \alpha(j)} \right) $$ (with notation as in Remark \ref{remark-easier-family}) is a descent datum relative to $\mathcal{U}'$. \item This construction defines a functor between the category of descent data relative to $\mathcal{U}$ and the category of descent data relative to $\mathcal{U}'$. \item Given a second $\beta : I' \to I$, $h : X' \to X$ and $h'_i : X'_i \to X_{\beta(i)}$ morphism of families of maps with fixed target, then if $g = h$ the two resulting functors between descent data are canonically isomorphic. \item These functors agree, via Lemma \ref{lemma-family-is-one}, with the pullback functors constructed in Lemma \ref{lemma-pullback}. \end{enumerate}


spaces-descent-lemma-descent-data-sheaves

Let $S$ be a scheme. Let $\{X_i \to X\}_{i \in I}$ be an fppf covering of algebraic spaces over $S$ (Topologies on Spaces, Definition \ref{spaces-topologies-definition-fppf-covering}). There is an equivalence of categories $$ \left\{ \begin{matrix} \text{descent data }(V_i, \varphi_{ij})\\ \text{relative to }\{X_i \to X\} \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{sheaves }F\text{ on }(\Sch/S)_{fppf}\text{ endowed}\\ \text{with a map }F \to X\text{ such that each}\\ X_i \times_X F\text{ is an algebraic space} \end{matrix} \right\}. $$ Moreover, \begin{enumerate} \item the algebraic space $X_i \times_X F$ on the right hand side corresponds to $V_i$ on the left hand side, and \item the sheaf $F$ is an algebraic space\footnote{We will see later that this is always the case if $I$ is not too large, see Bootstrap, Lemma \ref{bootstrap-lemma-descend-algebraic-space}.} if and only if the corresponding descent datum $(X_i, \varphi_{ij})$ is effective. \end{enumerate}


spaces-descent-proposition-fpqc-descent-quasi-coherent

Let $S$ be a scheme. Let $\{X_i \to X\}$ be an fpqc covering of algebraic spaces over $S$, see Topologies on Spaces, Definition \ref{spaces-topologies-definition-fpqc-covering}. Any descent datum on quasi-coherent sheaves for $\{X_i \to X\}$ is effective. Moreover, the functor from the category of quasi-coherent $\mathcal{O}_X$-modules to the category of descent data with respect to $\{X_i \to X\}$ is fully faithful.


decent-spaces-theorem-decent-open-dense-scheme

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


decent-spaces-lemma-composition-universally-bounded

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $V \to U$ be a morphism of schemes over $S$, and let $U \to X$ be a morphism from $U$ to $X$. If the fibres of $V \to U$ and $U \to X$ are universally bounded, then so are the fibres of $V \to X$.


decent-spaces-lemma-base-change-universally-bounded

Let $S$ be a scheme. Let $Y \to X$ be a representable morphism of algebraic spaces over $S$. Let $U \to X$ be a morphism from a scheme to $X$. If the fibres of $U \to X$ are universally bounded, then the fibres of $U \times_X Y \to Y$ are universally bounded.


decent-spaces-lemma-descent-universally-bounded

Let $S$ be a scheme. Let $g : Y \to X$ be a representable morphism of algebraic spaces over $S$. Let $f : U \to X$ be a morphism from a scheme towards $X$. Let $f' : U \times_X Y \to Y$ be the base change of $f$. If $$ \Im(|f| : |U| \to |X|) \subset \Im(|g| : |Y| \to |X|) $$ and $f'$ has universally bounded fibres, then $f$ has universally bounded fibres.


decent-spaces-lemma-universally-bounded-permanence

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider a commutative diagram $$ \xymatrix{ U \ar[rd]_g \ar[rr]_f & & V \ar[ld]^h \\ & X & } $$ where $U$ and $V$ are schemes. If $g$ has universally bounded fibres, and $f$ is surjective and flat, then also $h$ has universally bounded fibres.


decent-spaces-lemma-universally-bounded-finite-fibres

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$, and let $U$ be a scheme over $S$. Let $\varphi : U \to X$ be a morphism over $S$. If the fibres of $\varphi$ are universally bounded, then there exists an integer $n$ such that each fibre of $|U| \to |X|$ has at most $n$ elements.


decent-spaces-lemma-U-finite-above-x

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 family of schemes $U_i$ and \'etale morphisms $\varphi_i : U_i \to X$ such that $\coprod \varphi_i : \coprod U_i \to X$ is surjective, and such that for each $i$ the fibre of $|U_i| \to |X|$ over $x$ is finite, and \item for every affine scheme $U$ and \'etale morphism $\varphi : U \to X$ the fibre of $|U| \to |X|$ over $x$ is finite. \end{enumerate}


decent-spaces-lemma-R-finite-above-x

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 scheme $U$, an \'etale morphism $\varphi : U \to X$, and points $u, u' \in U$ mapping to $x$ such that setting $R = U \times_X U$ the fibre of $$ |R| \to |U| \times_{|X|} |U| $$ over $(u, u')$ is finite, \item for every scheme $U$, \'etale morphism $\varphi : U \to X$ and any points $u, u' \in U$ mapping to $x$ setting $R = U \times_X U$ the fibre of $$ |R| \to |U| \times_{|X|} |U| $$ over $(u, u')$ is finite, \item there exists a morphism $\Spec(k) \to X$ with $k$ a field in the equivalence class of $x$ such that the projections $\Spec(k) \times_X \Spec(k) \to \Spec(k)$ are \'etale and quasi-compact, and \item there exists a monomorphism $\Spec(k) \to X$ with $k$ a field in the equivalence class of $x$. \end{enumerate}


decent-spaces-lemma-weak-UR-finite-above-x

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. Let $U$ be a scheme and let $\varphi : U \to X$ be an \'etale morphism. The following are equivalent: \begin{enumerate} \item $x$ is in the image of $|U| \to |X|$, and setting $R = U \times_X U$ the fibres of both $$ |U| \longrightarrow |X| \quad\text{and}\quad |R| \longrightarrow |X| $$ over $x$ are finite, \item there exists a monomorphism $\Spec(k) \to X$ with $k$ a field in the equivalence class of $x$, and the fibre product $\Spec(k) \times_X U$ is a finite nonempty scheme over $k$. \end{enumerate}


decent-spaces-lemma-UR-finite-above-x

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. The following are equivalent: \begin{enumerate} \item for every affine scheme $U$, any \'etale morphism $\varphi : U \to X$ setting $R = U \times_X U$ the fibres of both $$ |U| \longrightarrow |X| \quad\text{and}\quad |R| \longrightarrow |X| $$ over $x$ are finite, \item there exist schemes $U_i$ and \'etale morphisms $U_i \to X$ such that $\coprod U_i \to X$ is surjective and for each $i$, setting $R_i = U_i \times_X U_i$ the fibres of both $$ |U_i| \longrightarrow |X| \quad\text{and}\quad |R_i| \longrightarrow |X| $$ over $x$ are finite, \item there exists a monomorphism $\Spec(k) \to X$ with $k$ a field in the equivalence class of $x$, and for any affine scheme $U$ and \'etale morphism $U \to X$ the fibre product $\Spec(k) \times_X U$ is a finite scheme over $k$, \item there exists a quasi-compact monomorphism $\Spec(k) \to X$ with $k$ a field in the equivalence class of $x$, and \item there exists a quasi-compact morphism $\Spec(k) \to X$ with $k$ a field in the equivalence class of $x$. \end{enumerate}


decent-spaces-lemma-U-universally-bounded

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent: \begin{enumerate} \item there exist schemes $U_i$ and \'etale morphisms $U_i \to X$ such that $\coprod U_i \to X$ is surjective and each $U_i \to X$ has universally bounded fibres, and \item for every affine scheme $U$ and \'etale morphism $\varphi : U \to X$ the fibres of $U \to X$ are universally bounded. \end{enumerate}


decent-spaces-lemma-characterize-very-reasonable

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent: \begin{enumerate} \item there exists a Zariski covering $X = \bigcup X_i$ and for each $i$ a scheme $U_i$ and a quasi-compact surjective \'etale morphism $U_i \to X_i$, and \item there exist schemes $U_i$ and \'etale morphisms $U_i \to X$ such that the projections $U_i \times_X U_i \to U_i$ are quasi-compact and $\coprod U_i \to X$ is surjective. \end{enumerate}


decent-spaces-lemma-bounded-fibres

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider the following conditions on $X$: \begin{itemize} \item[] $(\alpha)$ For every $x \in |X|$, the equivalent conditions of Lemma \ref{lemma-U-finite-above-x} hold. \item[] $(\beta)$ For every $x \in |X|$, the equivalent conditions of Lemma \ref{lemma-R-finite-above-x} hold. \item[] $(\gamma)$ For every $x \in |X|$, the equivalent conditions of Lemma \ref{lemma-UR-finite-above-x} hold. \item[] $(\delta)$ The equivalent conditions of Lemma \ref{lemma-U-universally-bounded} hold. \item[] $(\epsilon)$ The equivalent conditions of Lemma \ref{lemma-characterize-very-reasonable} hold. \item[] $(\zeta)$ The space $X$ is Zariski locally quasi-separated. \item[] $(\eta)$ The space $X$ is quasi-separated \item[] $(\theta)$ The space $X$ is representable, i.e., $X$ is a scheme. \item[] $(\iota)$ The space $X$ is a quasi-separated scheme. \end{itemize} We have $$ \xymatrix{ & (\theta) \ar@{=>}[rd] & & & &  \\ (\iota) \ar@{=>}[ru] \ar@{=>}[rd] & & (\zeta) \ar@{=>}[r] & (\epsilon) \ar@{=>}[r] & (\delta) \ar@{=>}[r] & (\gamma) \ar@{<=>}[r] & (\alpha) + (\beta) \\ & (\eta) \ar@{=>}[ru] & & & & } $$


decent-spaces-lemma-properties-local

Let $S$ be a scheme. Let $\mathcal{P}$ be one of the properties $(\alpha)$, $(\beta)$, $(\gamma)$, $(\delta)$, $(\epsilon)$, $(\zeta)$, or $(\theta)$ of algebraic spaces listed in Lemma \ref{lemma-bounded-fibres}. Then if $X$ is an algebraic space over $S$, and $X = \bigcup X_i$ is a Zariski open covering such that each $X_i$ has $\mathcal{P}$, then $X$ has $\mathcal{P}$.


decent-spaces-lemma-representable-properties

Let $S$ be a scheme. Let $\mathcal{P}$ be one of the properties $(\beta)$, $(\gamma)$, $(\delta)$, $(\epsilon)$, or $(\theta)$ of algebraic spaces listed in Lemma \ref{lemma-bounded-fibres}. Let $X$, $Y$ be algebraic spaces over $S$. Let $X \to Y$ be a representable morphism. If $Y$ has property $\mathcal{P}$, so does $X$.


decent-spaces-lemma-representable-named-properties

Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $X \to Y$ be a representable morphism. If $Y$ is decent (resp.\ reasonable), then so is $X$.


decent-spaces-lemma-etale-named-properties

Let $S$ be a scheme. Let $X \to Y$ be an \'etale morphism of algebraic spaces over $S$. If $Y$ is decent, resp.\ reasonable, then so is $X$.


decent-spaces-lemma-no-specializations-map-to-same-point

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U \to X$ be an \'etale morphism from a scheme to $X$. Assume $u, u' \in |U|$ map to the same point $x$ of $|X|$, and $u' \leadsto u$. If the pair $(X, x)$ satisfies the equivalent conditions of Lemma \ref{lemma-U-finite-above-x} then $u = u'$.


decent-spaces-lemma-specialization

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x, x' \in |X|$ and assume $x' \leadsto x$, i.e., $x$ is a specialization of $x'$. Assume the pair $(X, x')$ satisfies the equivalent conditions of Lemma \ref{lemma-UR-finite-above-x}. Then for every \'etale morphism $\varphi : U \to X$ from a scheme $U$ and any $u \in U$ with $\varphi(u) = x$, exists a point $u'\in U$, $u' \leadsto u$ with $\varphi(u') = x'$.


decent-spaces-lemma-generalizations-lift-flat

Let $S$ be a scheme. Let $f : Y \to X$ be a  flat morphism of algebraic spaces over $S$. Let $x, x' \in |X|$ and assume $x' \leadsto x$, i.e., $x$ is a specialization of $x'$. Assume the pair $(X, x')$ satisfies the equivalent conditions of Lemma \ref{lemma-UR-finite-above-x} (for example if $X$ is decent, $X$ is quasi-separated, or $X$ is representable). Then for every $y \in |Y|$ with $f(y) = x$, there exists a point $y' \in |Y|$, $y' \leadsto y$ with $f(y') = x'$.


decent-spaces-lemma-quasi-compact-reasonable-stratification

Let $S$ be a scheme. Let $W \to X$ be a morphism of a scheme $W$ to an algebraic space $X$ which is flat, locally of finite presentation, separated, locally quasi-finite with universally bounded fibres. There exist reduced closed subspaces $$ \emptyset = Z_{-1} \subset Z_0 \subset Z_1 \subset Z_2 \subset \ldots \subset Z_n = X $$ such that with $X_r = Z_r \setminus Z_{r - 1}$ the stratification $X = \coprod_{r = 0, \ldots, n} X_r$ is characterized by the following universal property: Given $g : T \to X$ the projection $W \times_X T \to T$ is finite locally free of degree $r$ if and only if $g(|T|) \subset |X_r|$.


decent-spaces-lemma-stratify-flat-fp-lqf

Let $S$ be a scheme. Let $W \to X$ be a morphism of a scheme $W$ to an algebraic space $X$ which is flat, locally of finite presentation, separated, and locally quasi-finite. Then there exist open subspaces $$ X = X_0 \supset X_1 \supset X_2 \supset \ldots $$ such that a morphism $\Spec(k) \to X$ factors through $X_d$ if and only if $W \times_X \Spec(k)$ has degree $\geq d$ over $k$.


decent-spaces-lemma-filter-quasi-compact

Let $S$ be a scheme. Let $X$ be a quasi-compact algebraic space over $S$. There exist open subspaces $$ \ldots \subset U_4 \subset U_3 \subset U_2 \subset U_1 = X $$ with the following properties: \begin{enumerate} \item setting $T_p = U_p \setminus U_{p + 1}$ (with reduced induced subspace structure) there exists a separated scheme $V_p$ and a surjective \'etale morphism $f_p : V_p \to U_p$ such that $f_p^{-1}(T_p) \to T_p$ is an isomorphism, \item if $x \in |X|$ can be represented by a quasi-compact morphism $\Spec(k) \to X$ from a field, then $x \in T_p$ for some $p$. \end{enumerate}


decent-spaces-lemma-filter-reasonable

Let $S$ be a scheme. Let $X$ be a quasi-compact, reasonable algebraic space over $S$. There exist an integer $n$ and open subspaces $$ \emptyset = U_{n + 1} \subset U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X $$ with the following property: setting $T_p = U_p \setminus U_{p + 1}$ (with reduced induced subspace structure) there exists a separated scheme $V_p$ and a surjective \'etale morphism $f_p : V_p \to U_p$ such that $f_p^{-1}(T_p) \to T_p$ is an isomorphism.


decent-spaces-lemma-filter-quasi-compact-quasi-separated

\begin{reference} This result is almost identical to \cite[Proposition 5.7.8]{GruRay}. \end{reference} Let $X$ be a quasi-compact and quasi-separated algebraic space over $\Spec(\mathbf{Z})$. There exist an integer $n$ and open subspaces $$ \emptyset = U_{n + 1} \subset U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X $$ with the following property: setting $T_p = U_p \setminus U_{p + 1}$ (with reduced induced subspace structure) there exists a quasi-compact separated scheme $V_p$ and a surjective \'etale morphism $f_p : V_p \to U_p$ such that $f_p^{-1}(T_p) \to T_p$ is an isomorphism.


decent-spaces-lemma-extend-integral-morphism

Let $S$ be a scheme. Let $j : V \to Y$ be a quasi-compact open immersion of algebraic spaces over $S$. Let $\pi : Z \to V$ be an integral morphism. Then there exists an integral morphism $\nu : Y' \to Y$ such that $Z$ is $V$-isomorphic to the inverse image of $V$ in $Y'$.


decent-spaces-lemma-there-is-a-scheme-integral-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 integral morphism $Y \to X$ 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}


decent-spaces-lemma-when-quotient-scheme-at-point

Let $S$ be a scheme. Let $X \to Y$ be a surjective finite locally free morphism of algebraic spaces over $S$. For $y \in |Y|$ the following are equivalent \begin{enumerate} \item $y$ is in the schematic locus of $Y$, and \item there exists an affine open $U \subset X$ containing the preimage of $y$. \end{enumerate}


decent-spaces-lemma-decent-points-monomorphism

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider the map $$ \{\Spec(k) \to X \text{ monomorphism where }k\text{ is a field}\} \longrightarrow |X| $$ This map is always injective. If $X$ is decent then this map is a bijection.


decent-spaces-lemma-identifies-residue-fields

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of decent algebraic spaces over $S$. Let $x \in |X|$ be a point with image $y = f(x) \in |Y|$. The following are equivalent \begin{enumerate} \item $f$ induces an isomorphism $\kappa(y) \to \kappa(x)$, and \item the induced morphism $\Spec(\kappa(x)) \to Y$ is a monomorphism. \end{enumerate}


decent-spaces-lemma-decent-space-elementary-etale-neighbourhood

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. For every point $x \in |X|$ there exists an \'etale morphism $$ (U, u) \longrightarrow (X, x) $$ where $U$ is an affine scheme, $u$ is the only point of $U$ lying over $x$, and the induced homomorphism $\kappa(x) \to \kappa(u)$ is an isomorphism.


decent-spaces-lemma-elementary-etale-neighbourhoods

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x$ be a point of $X$. The category of elementary \'etale neighborhoods of $(X, x)$ is cofiltered (see Categories, Definition \ref{categories-definition-codirected}).


decent-spaces-lemma-describe-henselian-local-ring

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. Let $(U, u) \to (X, x)$ be an elementary \'etale neighbourhood. Then $$ \mathcal{O}_{X, x}^h = \mathcal{O}_{U, u}^h $$ In words: the henselian local ring of $X$ at $x$ is equal to the henselization $\mathcal{O}_{U, u}^h$ of the local ring $\mathcal{O}_{U, u}$ of $U$ at $u$.


decent-spaces-lemma-henselian-local-ring-strict

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$ lying over $x \in |X|$.  The \'etale local ring $\mathcal{O}_{X, \overline{x}}$ of $X$ at $\overline{x}$ (Properties of Spaces, Definition \ref{spaces-properties-definition-etale-local-rings}) is the strict henselization of the henselian local ring $\mathcal{O}_{X, x}^h$ of $X$ at $x$.


decent-spaces-lemma-decent-no-specializations-map-to-same-point

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $U \to X$ be an \'etale morphism from a scheme to $X$. If $u, u' \in |U|$ map to the same point of $|X|$, and $u' \leadsto u$, then $u = u'$.


decent-spaces-lemma-decent-specialization

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x, x' \in |X|$ and assume $x' \leadsto x$, i.e., $x$ is a specialization of $x'$. Then for every \'etale morphism $\varphi : U \to X$ from a scheme $U$ and any $u \in U$ with $\varphi(u) = x$, exists a point $u'\in U$, $u' \leadsto u$ with $\varphi(u') = x'$.


decent-spaces-lemma-kolmogorov

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


decent-spaces-lemma-dimension-decent-space

Let $S$ be a scheme. Dimension as defined in Properties of Spaces, Section \ref{spaces-properties-section-dimension} behaves well on decent algebraic spaces $X$ over $S$. \begin{enumerate} \item If $x \in |X|$, then $\dim_x(|X|) = \dim_x(X)$, and \item $\dim(|X|) = \dim(X)$. \end{enumerate}


decent-spaces-lemma-dimension-local-ring-quasi-finite

Let $S$ be a scheme. Let $X \to Y$ be a locally quasi-finite morphism of algebraic spaces over $S$. Let $x \in |X|$ with image $y \in |Y|$. Then the dimension of the local ring of $Y$ at $y$ is $\geq$ to the dimension of the local ring of $X$ at $x$.


decent-spaces-lemma-dimension-quasi-finite

Let $S$ be a scheme. Let $X \to Y$ be a locally quasi-finite morphism of algebraic spaces over $S$. Then $\dim(X) \leq \dim(Y)$.


decent-spaces-lemma-decent-point-like-spaces

Let $S$ be a scheme. Let $k$ be a field. Let $X$ be an algebraic space over $S$ and assume that there exists a surjective \'etale morphism $\Spec(k) \to X$. If $X$ is decent, then $X \cong \Spec(k')$ where $k' \subset k$ is a finite separable extension.


decent-spaces-lemma-flat-cover-by-field

Let $S$ be a scheme. Let $Z$ be an algebraic space over $S$. Let $k$ be a field and let $\Spec(k) \to Z$ be surjective and flat. Then any morphism $\Spec(k') \to Z$ where $k'$ is a field is surjective and flat.


decent-spaces-lemma-unique-point

Let $S$ be a scheme. Let $Z$ be an algebraic space over $S$. The following are equivalent \begin{enumerate} \item $Z$ is reduced and $|Z|$ is a singleton, \item there exists a surjective flat morphism $\Spec(k) \to Z$ where $k$ is a field, and \item there exists a locally of finite type, surjective, flat morphism $\Spec(k) \to Z$ where $k$ is a field. \end{enumerate}


decent-spaces-lemma-unique-point-better

Let $S$ be a scheme. Let $Z$ be an algebraic space over $S$. The following are equivalent \begin{enumerate} \item $Z$ is reduced, locally Noetherian, and $|Z|$ is a singleton, and \item there exists a locally finitely presented, surjective, flat morphism $\Spec(k) \to Z$ where $k$ is a field. \end{enumerate}


decent-spaces-lemma-monomorphism-into-point

Let $S$ be a scheme. Let $Z' \to Z$ be a monomorphism of algebraic spaces over $S$. Assume there exists a field $k$ and a locally finitely presented, surjective, flat morphism $\Spec(k) \to Z$. Then either $Z'$ is empty or $Z' = Z$.


decent-spaces-lemma-find-singleton-from-point

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. Then there exists a unique monomorphism $Z \to X$ of algebraic spaces over $S$ such that $Z$ is an algebraic space which satisfies the equivalent conditions of Lemma \ref{lemma-unique-point-better} and such that the image of $|Z| \to |X|$ is $\{x\}$.


decent-spaces-lemma-locally-Noetherian-decent-quasi-separated

Any locally Noetherian decent algebraic space is quasi-separated.


decent-spaces-lemma-when-field

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. \begin{enumerate} \item If $|X|$ is a singleton then $X$ is a scheme. \item If $|X|$ is a singleton and $X$ is reduced, then $X \cong \Spec(k)$ for some field $k$. \end{enumerate}


decent-spaces-lemma-algebraic-residue-field-extension-closed-point

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Consider a commutative diagram $$ \xymatrix{ \Spec(k) \ar[rr] \ar[rd] & & X \ar[ld] \\ & S } $$ Assume that the image point $s \in S$ of $\Spec(k) \to S$ is a closed point and that $\kappa(s) \subset k$ is algebraic. Then the image $x$ of $\Spec(k) \to X$ is a closed point of $|X|$.


decent-spaces-lemma-finite-residue-field-extension-finite

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Consider a commutative diagram $$ \xymatrix{ \Spec(k) \ar[rr] \ar[rd] & & X \ar[ld] \\ & S } $$ Assume that the image point $s \in S$ of $\Spec(k) \to S$ is a closed point and that $\kappa(s) \subset k$ is finite. Then $\Spec(k) \to X$ is finite morphism. If $\kappa(s) = k$ then $\Spec(k) \to X$ is closed immersion.


decent-spaces-lemma-decent-space-closed-point

Let $S$ be a scheme. Suppose $X$ is a decent algebraic space over $S$. Let $x \in |X|$ be a closed point. Then $x$ can be represented by a closed immersion $i : \Spec(k) \to X$ from the spectrum of a field.


decent-spaces-lemma-infinite-number

Let $A$ be a ring. Let $k$ be a field. Let $\mathfrak p_n$, $n \geq 1$ be a sequence of pairwise distinct primes of $A$. Moreover, for each $n$ let $k \to \kappa(\mathfrak p_n)$ be an embedding. Then the closure of the image of $$ \coprod\nolimits_{n \not = m} \Spec(\kappa(\mathfrak p_n) \otimes_k \kappa(\mathfrak p_m)) \longrightarrow \Spec(A \otimes A) $$ meets the diagonal.


decent-spaces-lemma-locally-separated-decent

A locally separated algebraic space is decent.


decent-spaces-lemma-properties-trivial-implications

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We have the following implications among the conditions on $f$: $$ \xymatrix{ \text{representable} \ar@{=>}[rd] & & & & \\ & \text{very reasonable} \ar@{=>}[r] & \text{reasonable} \ar@{=>}[r] & \text{decent} \ar@{=>}[r] & (\beta) \\ \text{quasi-separated} \ar@{=>}[ru] & & & & } $$


decent-spaces-lemma-property-for-morphism-out-of-property

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $X$ is decent (resp.\ is reasonable, resp.\ has property $(\beta)$ of Lemma \ref{lemma-bounded-fibres}), then $f$ is decent (resp.\ reasonable, resp.\ has property $(\beta)$).


decent-spaces-lemma-base-change-relative-conditions

Having property $(\beta)$, being decent, or being reasonable is preserved under arbitrary base change.


decent-spaces-lemma-property-over-property

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\omega \in \{\beta, decent, reasonable\}$. Suppose that $Y$ has property $(\omega)$ and $f : X \to Y$ has $(\omega)$. Then $X$ has $(\omega)$.


decent-spaces-lemma-composition-relative-conditions

Having property $(\beta)$, being decent, or being reasonable is preserved under compositions.


decent-spaces-lemma-descent-conditions

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{P} \in \{(\beta), decent, reasonable\}$. Assume \begin{enumerate} \item $f$ is quasi-compact, \item $f$ is \'etale, \item $|f| : |X| \to |Y|$ is surjective, and \item the algebraic space $X$ has property $\mathcal{P}$. \end{enumerate} Then $Y$ has property $\mathcal{P}$.


decent-spaces-lemma-surjective-on-fibres

In the situation of (\ref{equation-points-fibres}) if $Z' \to Z$ is a morphism and $z' \in |Z'|$ maps to $z$, then the induced map $F_{x, z'} \to F_{x, z}$ is surjective.


decent-spaces-lemma-qf-and-qc-finite-fibre

In diagram (\ref{equation-points-fibres}) the set (\ref{equation-fibre}) is finite if $f$ is of finite type and $f$ is quasi-finite at $x$.


decent-spaces-lemma-decent-finite-fibre

In diagram (\ref{equation-points-fibres}) the set (\ref{equation-fibre}) is finite if $y$ can be represented by a monomorphism $\Spec(k) \to Y$ where $k$ is a field and $g$ is quasi-finite at $z$. (Special case: $Y$ is decent and $g$ is \'etale.)


decent-spaces-lemma-topology-fibre

\begin{slogan} Fibers of field points of algebraic spaces have the expected Zariski topologies. \end{slogan} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $y \in |Y|$ and assume that $y$ is represented by a quasi-compact monomorphism $\Spec(k) \to Y$. Then $|X_k| \to |X|$ is a homeomorphism onto $f^{-1}(\{y\}) \subset |X|$ with induced topology.


decent-spaces-lemma-conditions-on-point-on-space-over-field

Let $X$ be an algebraic space locally of finite type over a field $k$. Let $x \in |X|$. Consider the conditions \begin{enumerate} \item $\dim_x(|X|) = 0$, \item $x$ is closed in $|X|$ and if $x' \leadsto x$  in $|X|$ then $x' = x$, \item $x$ is an isolated point of $|X|$, \item $\dim_x(X) = 0$, \item $X \to \Spec(k)$ is quasi-finite at $x$. \end{enumerate} Then (2), (3), (4), and (5) are equivalent. If $X$ is decent, then (1) is equivalent to the others.


decent-spaces-lemma-conditions-on-space-over-field

Let $X$ be an algebraic space locally of finite type over a field $k$. Consider the conditions \begin{enumerate} \item $|X|$ is a finite set, \item $|X|$ is a discrete space, \item $\dim(|X|) = 0$, \item $\dim(X) = 0$, \item $X \to \Spec(k)$ is locally quasi-finite, \end{enumerate} Then (2), (3), (4), and (5) are equivalent. If $X$ is decent, then (1) implies the others.


decent-spaces-lemma-conditions-on-point-in-fibre-and-qf

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $x \in |X|$ with image $y \in |Y|$. Let $F = f^{-1}(\{y\})$ with induced topology from $|X|$. Let $k$ be a field and let $\Spec(k) \to Y$ be in the equivalence class defining $y$. Set $X_k = \Spec(k) \times_Y X$. Let $\tilde x \in |X_k|$ map to $x \in |X|$. Consider the following conditions \begin{enumerate} \item \label{item-fibre-at-x-dim-0} $\dim_x(F) = 0$, \item \label{item-isolated-in-fibre} $x$ is isolated in $F$, \item \label{item-no-specializations-in-fibre} $x$ is closed in $F$ and if $x' \leadsto x$ in $F$, then $x = x'$, \item \label{item-dimension-top-k-fibre} $\dim_{\tilde x}(|X_k|) = 0$, \item \label{item-isolated-in-k-fibre} $\tilde x$ is isolated in $|X_k|$, \item \label{item-no-specializations-in-k-fibre} $\tilde x$ is closed in $|X_k|$ and if $\tilde x' \leadsto \tilde x$ in $|X_k|$, then $\tilde x = \tilde x'$, \item \label{item-k-fibre-at-x-dim-0} $\dim_{\tilde x}(X_k) = 0$, \item \label{item-quasi-finite-at-x} $f$ is quasi-finite at $x$. \end{enumerate} Then we have $$ \xymatrix{ (\ref{item-dimension-top-k-fibre}) \ar@{=>}[r]_{f\text{ decent}} & (\ref{item-isolated-in-k-fibre}) \ar@{<=>}[r] & (\ref{item-no-specializations-in-k-fibre}) \ar@{<=>}[r] & (\ref{item-k-fibre-at-x-dim-0}) \ar@{<=>}[r] & (\ref{item-quasi-finite-at-x}) } $$ If $Y$ is decent, then conditions (\ref{item-isolated-in-fibre}) and (\ref{item-no-specializations-in-fibre}) are equivalent to each other and to conditions (\ref{item-isolated-in-k-fibre}), (\ref{item-no-specializations-in-k-fibre}), (\ref{item-k-fibre-at-x-dim-0}), and (\ref{item-quasi-finite-at-x}). If $Y$ and $X$ are decent, then all conditions are equivalent.


decent-spaces-lemma-conditions-on-fibre-and-qf

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $y \in |Y|$. Let $k$ be a field and let $\Spec(k) \to Y$ be in the equivalence class defining $y$. Set $X_k = \Spec(k) \times_Y X$ and let $F = f^{-1}(\{y\})$ with the induced topology from $|X|$. Consider the following conditions \begin{enumerate} \item \label{item-fibre-finite} $F$ is finite, \item \label{item-fibre-discrete} $F$ is a discrete topological space, \item \label{item-fibre-no-specializations} $\dim(F) = 0$, \item \label{item-k-fibre-finite} $|X_k|$ is a finite set, \item \label{item-k-fibre-discrete} $|X_k|$ is a discrete space, \item \label{item-k-fibre-no-specializations} $\dim(|X_k|) = 0$, \item \label{item-k-fibre-dim-0} $\dim(X_k) = 0$, \item \label{item-quasi-finite-at-points-fibre} $f$ is quasi-finite at all points of $|X|$ lying over $y$. \end{enumerate} Then we have $$ \xymatrix{ (\ref{item-fibre-finite}) & (\ref{item-k-fibre-finite}) \ar@{=>}[l] \ar@{=>}[r]_{f\text{ decent}} & (\ref{item-k-fibre-discrete}) \ar@{<=>}[r] & (\ref{item-k-fibre-no-specializations}) \ar@{<=>}[r] & (\ref{item-k-fibre-dim-0}) \ar@{<=>}[r] & (\ref{item-quasi-finite-at-points-fibre}) } $$ If $Y$ is decent, then conditions (\ref{item-fibre-discrete}) and (\ref{item-fibre-no-specializations}) are equivalent to each other and to conditions (\ref{item-k-fibre-discrete}), (\ref{item-k-fibre-no-specializations}), (\ref{item-k-fibre-dim-0}), and (\ref{item-quasi-finite-at-points-fibre}). If $Y$ and $X$ are decent, then (\ref{item-fibre-finite}) implies all the other conditions.


decent-spaces-lemma-monomorphism-toward-disjoint-union-dim-0-rings

Let $S$ be a scheme. Let $Y$ be a disjoint union of spectra of zero dimensional local rings over $S$. Let $f : X \to Y$ be a monomorphism of algebraic spaces over $S$. Then $f$ is representable, i.e., $X$ is a scheme.


decent-spaces-lemma-decent-generic-points

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. The following are equivalent \begin{enumerate} \item $x$ is a generic point of an irreducible component of $|X|$, \item for any \'etale morphism $(Y, y) \to (X, x)$ of pointed algebraic spaces, $y$ is a generic point of an irreducible component of $|Y|$, \item for some \'etale morphism $(Y, y) \to (X, x)$ of pointed algebraic spaces, $y$ is a generic point of an irreducible component of $|Y|$, \item the dimension of the local ring of $X$ at $x$ is zero, and \item $x$ is a point of codimension $0$ on $X$ \end{enumerate}


decent-spaces-lemma-codimension-local-ring

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $T \subset |X|$ be an irreducible closed subset. Let $\xi \in T$ be the generic point (Proposition \ref{proposition-reasonable-sober}). Then $\text{codim}(T, |X|)$ (Topology, Definition \ref{topology-definition-codimension}) is the dimension of the local ring of $X$ at $\xi$ (Properties of Spaces, Definition \ref{spaces-properties-definition-dimension-local-ring}).


decent-spaces-lemma-get-reasonable

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume \begin{enumerate} \item every quasi-compact scheme \'etale over $X$ has finitely many irreducible components, and \item every $x \in |X|$ of codimension $0$ on $X$ can be represented by a monomorphism $\Spec(k) \to X$. \end{enumerate} Then $X$ is a reasonable algebraic space.


decent-spaces-lemma-finitely-many-irreducible-components

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \begin{enumerate} \item $X$ is decent and $|X|$ has finitely many irreducible components, \item every quasi-compact scheme \'etale over $X$ has finitely many irreducible components, there are finitely many $x \in |X|$ of codimension $0$ on $X$, and each of these can be represented by a monomorphism $\Spec(k) \to X$, \item there exists a dense open $X' \subset X$ which is a scheme, $X'$ has finitely many irreducible components with generic points $\{x'_1, \ldots, x'_m\}$, and the morphism $x'_j \to X$ is quasi-compact for $j = 1, \ldots, m$. \end{enumerate} Moreover, if these conditions hold, then $X$ is reasonable and the points $x'_j \in |X|$ are the generic points of the irreducible components of $|X|$.


decent-spaces-lemma-generically-finite

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is quasi-separated of finite type. Let $y \in |Y|$ be a point of codimension $0$ on $Y$. The following are equivalent: \begin{enumerate} \item the space $|X_k|$ is finite where $\Spec(k) \to Y$ represents $y$, \item $X \to Y$ is quasi-finite at all points of $|X|$ over $y$, \item there exists an open subspace $Y' \subset Y$ with $y \in |Y'|$ such that $Y' \times_Y X \to Y'$ is finite. \end{enumerate} If $Y$ is decent these are also equivalent to \begin{enumerate} \item[(4)] the set $f^{-1}(\{y\})$ is finite. \end{enumerate}


decent-spaces-lemma-generically-finite-reprise

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is quasi-separated and locally of finite type and $Y$ quasi-separated. Let $y \in |Y|$ be a point of codimension $0$ on $Y$. The following are equivalent: \begin{enumerate} \item the set $f^{-1}(\{y\})$ is finite, \item the space $|X_k|$ is finite where $\Spec(k) \to Y$ represents $y$, \item there exist open subspaces $X' \subset X$ and $Y' \subset Y$ with $f(X') \subset Y'$, $y \in |Y'|$, and $f^{-1}(\{y\}) \subset |X'|$ such that $f|_{X'} : X' \to Y'$ is finite. \end{enumerate}


decent-spaces-lemma-quasi-finiteness-over-generic-point

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. Let $X^0 \subset |X|$, resp.\ $Y^0 \subset |Y|$ denote the set of codimension $0$ points of $X$, resp.\ $Y$. Let $y \in Y^0$. The following are equivalent \begin{enumerate} \item $f^{-1}(\{y\}) \subset X^0$, \item $f$ is quasi-finite at all points lying over $y$, \item $f$ is quasi-finite at all $x \in X^0$ lying over $y$. \end{enumerate}


decent-spaces-lemma-finite-over-dense-open

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. Let $X^0 \subset |X|$, resp.\ $Y^0 \subset |Y|$ denote the set of codimension $0$ points of $X$, resp.\ $Y$. Assume \begin{enumerate} \item $Y$ is decent, \item $X^0$ and $Y^0$ are finite and $f^{-1}(Y^0) = X^0$, \item either $f$ is quasi-compact or $f$ is separated. \end{enumerate} Then there exists a dense open $V \subset Y$ such that $f^{-1}(V) \to V$ is finite.


decent-spaces-lemma-birational-isomorphism-over-dense-open

Let $S$ be a scheme. Let $f : X \to Y$ be a birational morphism of algebraic spaces over $S$ which are decent and have finitely many irreducible components. Assume \begin{enumerate} \item either $f$ is quasi-compact or $f$ is separated, and \item either $f$ is locally of finite type and $Y$ is reduced or $f$ is locally of finite presentation. \end{enumerate} Then there exists a dense open $V \subset Y$ such that $f^{-1}(V) \to V$ is an isomorphism.


decent-spaces-lemma-birational-etale-localization

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which are decent and have finitely many irreducible components. If $f$ is birational and $V \to Y$ is an \'etale morphism with $V$ affine, then $X \times_Y V$ is decent with finitely many irreducible components and $X \times_Y V \to V$ is birational.


decent-spaces-lemma-birational-induced-morphism-normalizations

Let $S$ be a scheme. Let $f : X \to Y$ be a birational morphism between algebraic spaces over $S$ which are decent and have finitely many irreducible components. Then the normalizations $X^\nu \to X$ and $Y^\nu \to Y$ exist and there is a commutative diagram $$ \xymatrix{ X^\nu \ar[r] \ar[d] &  Y^\nu \ar[d] \\ X \ar[r] & Y } $$ of algebraic spaces over $S$. The morphism $X^\nu \to Y^\nu$ is birational.


decent-spaces-lemma-finite-birational-over-normal

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $X$ and $Y$ are decent and have finitely many irreducible components, \item $f$ is integral and birational, \item $Y$ is normal, and \item $X$ is reduced. \end{enumerate} Then $f$ is an isomorphism.


decent-spaces-lemma-Jacobson-universally-Jacobson

Let $S$ be a scheme. Let $X$ be a Jacobson algebraic space over $S$. Any algebraic space locally of finite type over $X$ is Jacobson.


decent-spaces-lemma-decent-Jacobson-ft-pts

Let $S$ be a scheme. Let $X$ be a decent Jacobson algebraic space over $S$. Then $X_{\text{ft-pts}} \subset |X|$ is the set of closed points.


decent-spaces-lemma-irreducible-local-ring

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$ be a point. The following are equivalent \begin{enumerate} \item for any elementary \'etale neighbourhood $(U, u) \to (X, x)$ the local ring $\mathcal{O}_{U, u}$ has a unique minimal prime, \item for any elementary \'etale neighbourhood $(U, u) \to (X, x)$ there is a unique irreducible component of $U$ through $u$, \item for any elementary \'etale neighbourhood $(U, u) \to (X, x)$ the local ring $\mathcal{O}_{U, u}$ is unibranch, \item the henselian local ring $\mathcal{O}_{X, x}^h$ has a unique minimal prime. \end{enumerate}


decent-spaces-lemma-nr-branches-local-ring

Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$ be a point. Let $n \in \{1, 2, \ldots\}$ be an integer. The following are equivalent \begin{enumerate} \item for any elementary \'etale neighbourhood $(U, u) \to (X, x)$ the number of minimal primes of the local ring $\mathcal{O}_{U, u}$ is $\leq n$ and for at least one choice of $(U, u)$ it is $n$, \item for any elementary \'etale neighbourhood $(U, u) \to (X, x)$ the number irreducible components of $U$ passing through $u$ is $\leq n$ and for at least one choice of $(U, u)$ it is $n$, \item for any elementary \'etale neighbourhood $(U, u) \to (X, x)$ the number of branches of $U$ at $u$ is $\leq n$ and for at least one choice of $(U, u)$ it is $n$, \item the number of minimal prime ideals of $\mathcal{O}_{X, x}^h$ is $n$. \end{enumerate}


decent-spaces-lemma-scheme-with-dimension-function

Let $S$ be a locally Noetherian and universally catenary scheme. Let $\delta : S \to \mathbf{Z}$ be a dimension function. Let $X$ be a decent algebraic space over $S$ such that the structure morphism $X \to S$ is locally of finite type. Let $\delta_X : |X| \to \mathbf{Z}$ be the map sending $x$ to $\delta(f(x))$ plus the transcendence degree of $x/f(x)$. Then $\delta_X$ is a dimension function on $|X|$.


decent-spaces-lemma-check-dimension-function-finite-cover

Let $S$ be a scheme. Let $f : Y \to X$ be a surjective finite morphism of decent and locally Noetherian algebraic spaces. Let $\delta : |X| \to \mathbf{Z}$ be a function. If $\delta \circ |f|$ is a dimension function, then $\delta$ is a dimension function.