divisors-lemma-div-additive

Let $X$ be a locally Noetherian integral scheme. Let $f, g \in R(X)^*$. Then $$ \text{div}_X(fg) = \text{div}_X(f) + \text{div}_X(g) $$ as Weil divisors on $X$.


divisors-lemma-divisor-meromorphic-locally-finite

Let $X$ be a locally Noetherian integral scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \mathcal{K}_X(\mathcal{L})$ be a regular (i.e., nonzero) meromorphic section of $\mathcal{L}$. Then the sets $$ \{Z \subset X \mid Z \text{ a prime divisor with generic point }\xi \text{ and }s\text{ not in }\mathcal{L}_\xi\} $$ and $$ \{Z \subset X \mid Z \text{ is a prime divisor and } \text{ord}_{Z, \mathcal{L}}(s) \not = 0\} $$ are locally finite in $X$.


divisors-lemma-divisor-meromorphic-well-defined

Let $X$ be a locally Noetherian integral scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s, s' \in \mathcal{K}_X(\mathcal{L})$ be nonzero meromorphic sections of $\mathcal{L}$. Then $f = s/s'$ is an element of $R(X)^*$ and we have $$ \sum \text{ord}_{Z, \mathcal{L}}(s)[Z] = \sum \text{ord}_{Z, \mathcal{L}}(s')[Z] + \text{div}(f) $$ as Weil divisors.


divisors-lemma-c1-additive

Let $X$ be a locally Noetherian integral scheme. Let $\mathcal{L}$, $\mathcal{N}$ be invertible $\mathcal{O}_X$-modules. Let $s$, resp.\ $t$ be a nonzero meromorphic section of $\mathcal{L}$, resp.\ $\mathcal{N}$. Then $st$ is a nonzero meromorphic section of $\mathcal{L} \otimes \mathcal{N}$, and $$ \text{div}_{\mathcal{L} \otimes \mathcal{N}}(st) = \text{div}_\mathcal{L}(s) + \text{div}_\mathcal{N}(t) $$ in $\text{Div}(X)$. In particular, the Weil divisor class of $\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{N}$ is the sum of the Weil divisor classes of $\mathcal{L}$ and $\mathcal{N}$.


divisors-lemma-normal-c1-injective

Let $X$ be a locally Noetherian integral scheme. If $X$ is normal, then the map (\ref{equation-c1}) $\Pic(X) \to \text{Cl}(X)$ is injective.


divisors-lemma-local-rings-UFD-c1-bijective

Let $X$ be a locally Noetherian integral scheme. Consider the map (\ref{equation-c1}) $\Pic(X) \to \text{Cl}(X)$. The following are equivalent \begin{enumerate} \item the local rings of $X$ are UFDs, and \item $X$ is normal and $\Pic(X) \to \text{Cl}(X)$ is surjective. \end{enumerate} In this case $\Pic(X) \to \text{Cl}(X)$ is an isomorphism.


divisors-lemma-in-image-pullback

Let $\varphi : X \to Y$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume that \begin{enumerate} \item $X$ is locally Noetherian, \item $Y$ is locally Noetherian, integral, and normal, \item $\varphi$ is flat with integral (hence nonempty) fibres, \item $\varphi$ is either quasi-compact or locally of finite type, \item $\mathcal{L}$ is trivial when restricted to the generic fibre of $\varphi$. \end{enumerate} Then $\mathcal{L} \cong \varphi^*\mathcal{N}$ for some invertible $\mathcal{O}_Y$-module $\mathcal{N}$.


divisors-lemma-closure-effective-cartier-divisor

Let $X$ be a locally Noetherian scheme. Let $U \subset X$ be an open and let $D \subset U$ be an effective Cartier divisor. If $\mathcal{O}_{X, x}$ is a UFD for all $x \in X \setminus U$, then there exists an effective Cartier divisor $D' \subset X$ with $D = U \cap D'$.


divisors-lemma-extend-invertible-module

Let $X$ be a locally Noetherian scheme. Let $U \subset X$ be an open and let $\mathcal{L}$ be an invertible $\mathcal{O}_U$-module. If $\mathcal{O}_{X, x}$ is a UFD for all $x \in X \setminus U$, then there exists an invertible $\mathcal{O}_X$-module $\mathcal{L}'$ with $\mathcal{L} \cong \mathcal{L}'|_U$.


divisors-lemma-open-subscheme-UFD

Let $R$ be a UFD. The Picard groups of the following are trivial. \begin{enumerate} \item $\Spec(R)$ and any open subscheme of it. \item $\mathbf{A}^n_R = \Spec(R[x_1, \ldots, x_n])$ and any open subscheme of it. \end{enumerate} In particular, the Picard group of any open subscheme of affine $n$-space $\mathbf{A}^n_k$ over a field $k$ is trivial.


divisors-lemma-Pic-projective-space-UFD

Let $R$ be a UFD. The Picard group of $\mathbf{P}^n_R$ is $\mathbf{Z}$. More precisely, there is an isomorphism $$ \mathbf{Z} \longrightarrow \Pic(\mathbf{P}^n_R),\quad m \longmapsto \mathcal{O}_{\mathbf{P}^n_R}(m) $$ In particular, the Picard group of $\mathbf{P}^n_k$ of projective space over a field $k$ is $\mathbf{Z}$.


divisors-lemma-structure-sheaf-Xs

Let $X$ be an integral locally Noetherian normal scheme. Let $\mathcal{F}$ be a rank 1 coherent reflexive $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{F})$. Let $$ U = \{x \in X \mid s : \mathcal{O}_{X, x} \to \mathcal{F}_x \text{ is an isomorphism}\} $$ Then $j : U \to X$ is an open subscheme of $X$ and $$ j_*\mathcal{O}_U = \colim (\mathcal{O}_X \xrightarrow{s} \mathcal{F} \xrightarrow{s} \mathcal{F}^{[2]} \xrightarrow{s} \mathcal{F}^{[3]} \xrightarrow{s} \ldots) $$ where $\mathcal{F}^{[1]} = \mathcal{F}$ and inductively $\mathcal{F}^{[n + 1]} = (\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{F}^{[n]})^{**}$.


divisors-lemma-Xs-codim-complement

Assumptions and notation as in Lemma \ref{lemma-structure-sheaf-Xs}. If $s$ is nonzero, then every irreducible component of $X \setminus U$ has codimension $1$ in $X$.


divisors-lemma-relative-proj-quasi-compact

Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_S$-algebra. Let $p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. If one of the following holds \begin{enumerate} \item $\mathcal{A}$ is of finite type as a sheaf of $\mathcal{A}_0$-algebras, \item $\mathcal{A}$ is generated by $\mathcal{A}_1$ as an $\mathcal{A}_0$-algebra and $\mathcal{A}_1$ is a finite type $\mathcal{A}_0$-module, \item there exists a finite type quasi-coherent $\mathcal{A}_0$-submodule $\mathcal{F} \subset \mathcal{A}_{+}$ such that $\mathcal{A}_{+}/\mathcal{F}\mathcal{A}$ is a locally nilpotent sheaf of ideals of $\mathcal{A}/\mathcal{F}\mathcal{A}$, \end{enumerate} then $p$ is quasi-compact.


divisors-lemma-relative-proj-finite-type

Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_S$-algebra. Let $p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. If $\mathcal{A}$ is of finite type as a sheaf of $\mathcal{O}_S$-algebras, then $p$ is of finite type and $\mathcal{O}_X(d)$ is a finite type $\mathcal{O}_X$-module.


divisors-lemma-relative-proj-universally-closed

Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_S$-algebra. Let $p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. If $\mathcal{O}_S \to \mathcal{A}_0$ is an integral algebra map\footnote{In other words, the integral closure of $\mathcal{O}_S$ in $\mathcal{A}_0$, see Morphisms, Definition \ref{morphisms-definition-integral-closure}, equals $\mathcal{A}_0$.} and $\mathcal{A}$ is of finite type as an $\mathcal{A}_0$-algebra, then $p$ is universally closed.


divisors-lemma-relative-proj-proper

Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_S$-algebra. Let $p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. The following conditions are equivalent \begin{enumerate} \item $\mathcal{A}_0$ is a finite type $\mathcal{O}_S$-module and $\mathcal{A}$ is of finite type as an $\mathcal{A}_0$-algebra, \item $\mathcal{A}_0$ is a finite type $\mathcal{O}_S$-module and $\mathcal{A}$ is of finite type as an $\mathcal{O}_S$-algebra \end{enumerate} If these conditions hold, then $p$ is locally projective and in particular proper.


divisors-lemma-relative-proj-projective

Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_S$-algebra. Let $p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. If $\mathcal{A}$ is generated by $\mathcal{A}_1$ over $\mathcal{A}_0$ and $\mathcal{A}_1$ is a finite type $\mathcal{O}_S$-module, then $p$ is projective.


divisors-lemma-relative-proj-flat

Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_S$-algebra. Let $p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. If $\mathcal{A}_d$ is a flat $\mathcal{O}_S$-module for $d \gg 0$, then $p$ is flat and $\mathcal{O}_X(d)$ is flat over $S$.


divisors-lemma-relative-proj-finite-presentation

Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_S$-algebra. Let $p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. If $\mathcal{A}$ is a finitely presented $\mathcal{O}_S$-algebra, then $p$ is of finite presentation and $\mathcal{O}_X(d)$ is an $\mathcal{O}_X$-module of finite presentation.


divisors-lemma-closed-subscheme-proj

Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_S$-algebra. Let $p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. Let $i : Z \to X$ be a closed subscheme. Denote $\mathcal{I} \subset \mathcal{A}$ the kernel of the canonical map $$ \mathcal{A} \longrightarrow \bigoplus\nolimits_{d \geq 0} p_*\left((i_*\mathcal{O}_Z)(d)\right) $$ If $p$ is quasi-compact, then there is an isomorphism $Z = \underline{\text{Proj}}_S(\mathcal{A}/\mathcal{I})$.


divisors-lemma-closed-subscheme-proj-finite

Let $S$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_S$-algebra. Let $p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. Let $i : Z \to X$ be a closed subscheme. If $p$ is quasi-compact and $i$ of finite presentation, then there exists a $d > 0$ and a quasi-coherent finite type $\mathcal{O}_S$-submodule $\mathcal{F} \subset \mathcal{A}_d$ such that $Z = \underline{\text{Proj}}_S(\mathcal{A}/\mathcal{F}\mathcal{A})$.


divisors-lemma-closed-subscheme-proj-finite-type

Let $S$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_S$-algebra. Let $p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. Let $i : Z \to X$ be a closed subscheme. Let $U \subset X$ be an open. Assume that \begin{enumerate} \item $p$ is quasi-compact, \item $i$ of finite presentation, \item $U \cap p(i(Z)) = \emptyset$, \item $U$ is quasi-compact, \item $\mathcal{A}_n$ is a finite type $\mathcal{O}_S$-module for all $n$. \end{enumerate} Then there exists a $d > 0$ and a quasi-coherent finite type $\mathcal{O}_S$-submodule $\mathcal{F} \subset \mathcal{A}_d$ with (a) $Z = \underline{\text{Proj}}_S(\mathcal{A}/\mathcal{F}\mathcal{A})$ and (b) the support of $\mathcal{A}_d/\mathcal{F}$ is disjoint from $U$.


divisors-lemma-blowing-up-affine

Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $U = \Spec(A)$ be an affine open subscheme of $X$ and let $I \subset A$ be the ideal corresponding to $\mathcal{I}|_U$. If $b : X' \to X$ is the blowup of $X$ in $\mathcal{I}$, then there is a canonical isomorphism $$ b^{-1}(U) = \text{Proj}(\bigoplus\nolimits_{d \geq 0} I^d) $$ of $b^{-1}(U)$ with the homogeneous spectrum of the Rees algebra of $I$ in $A$. Moreover, $b^{-1}(U)$ has an affine open covering by spectra of the affine blowup algebras $A[\frac{I}{a}]$.


divisors-lemma-flat-base-change-blowing-up

\begin{slogan} Blowing up commutes with flat base change. \end{slogan} Let $X_1 \to X_2$ be a flat morphism of schemes. Let $Z_2 \subset X_2$ be a closed subscheme. Let $Z_1$ be the inverse image of $Z_2$ in $X_1$. Let $X'_i$ be the blowup of $Z_i$ in $X_i$. Then there exists a cartesian diagram $$ \xymatrix{ X_1' \ar[r] \ar[d] & X_2' \ar[d] \\ X_1 \ar[r] & X_2 } $$ of schemes.


divisors-lemma-blowing-up-gives-effective-Cartier-divisor

Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. The blowing up $b : X' \to X$ of $Z$ in $X$ has the following properties: \begin{enumerate} \item $b|_{b^{-1}(X \setminus Z)} : b^{-1}(X \setminus Z) \to X \setminus Z$ is an isomorphism, \item the exceptional divisor $E = b^{-1}(Z)$ is an effective Cartier divisor on $X'$, \item there is a canonical isomorphism $\mathcal{O}_{X'}(-1) = \mathcal{O}_{X'}(E)$ \end{enumerate}


divisors-lemma-universal-property-blowing-up

Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. Let $\mathcal{C}$ be the full subcategory of $(\Sch/X)$ consisting of $Y \to X$ such that the inverse image of $Z$ is an effective Cartier divisor on $Y$. Then the blowing up $b : X' \to X$ of $Z$ in $X$ is a final object of $\mathcal{C}$.


divisors-lemma-characterize-affine-blowup

Let $b : X' \to X$ be the blowing up of the scheme $X$ along a closed subscheme $Z$. Let $U = \Spec(A)$ be an affine open of $X$ and let $I \subset A$ be the ideal corresponding to $Z \cap U$. Let $a \in I$ and let $x' \in X'$ be a point mapping to a point of $U$. Then $x'$ is a point of the affine open $U' = \Spec(A[\frac{I}{a}])$ if and only if the image of $a$ in $\mathcal{O}_{X', x'}$ cuts out the exceptional divisor.


divisors-lemma-blow-up-effective-Cartier-divisor

Let $X$ be a scheme. Let $Z \subset X$ be an effective Cartier divisor. The blowup of $X$ in $Z$ is the identity morphism of $X$.


divisors-lemma-blow-up-reduced-scheme

Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. If $X$ is reduced, then the blowup $X'$ of $X$ in $\mathcal{I}$ is reduced.


divisors-lemma-blow-up-integral-scheme

Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a nonzero quasi-coherent sheaf of ideals. If $X$ is integral, then the blowup $X'$ of $X$ in $\mathcal{I}$ is integral.


divisors-lemma-blow-up-and-irreducible-components

Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. Let $b : X' \to X$ be the blowing up of $X$ along $Z$. Then $b$ induces an bijective map from the set of generic points of irreducible components of $X'$ to the set of generic points of irreducible components of $X$ which are not in $Z$.


divisors-lemma-blow-up-pullback-effective-Cartier

Let $X$ be a scheme. Let $b : X' \to X$ be a blowup of $X$ in a closed subscheme. The pullback $b^{-1}D$ is defined for all effective Cartier divisors $D \subset X$ and pullbacks of meromorphic functions are defined for $b$ (Definitions \ref{definition-pullback-effective-Cartier-divisor} and \ref{definition-pullback-meromorphic-sections}).


divisors-lemma-blowing-up-two-ideals

Let $X$ be a scheme. Let $\mathcal{I}, \mathcal{J} \subset \mathcal{O}_X$ be quasi-coherent sheaves of ideals. Let $b : X' \to X$ be the blowing up of $X$ in $\mathcal{I}$. Let $b' : X'' \to X'$ be the blowing up of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$. Then $X'' \to X$ is canonically isomorphic to the blowing up of $X$ in $\mathcal{I}\mathcal{J}$.


divisors-lemma-blowing-up-projective

Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $b : X' \to X$ be the blowing up of $X$ in the ideal sheaf $\mathcal{I}$. If $\mathcal{I}$ is of finite type, then \begin{enumerate} \item $b : X' \to X$ is a projective morphism, and \item $\mathcal{O}_{X'}(1)$ is a $b$-relatively ample invertible sheaf. \end{enumerate}


divisors-lemma-composition-finite-type-blowups

\begin{slogan} Composition of blowing ups is a blowing up \end{slogan} Let $X$ be a quasi-compact and quasi-separated scheme. Let $Z \subset X$ be a closed subscheme of finite presentation. Let $b : X' \to X$ be the blowing up with center $Z$. Let $Z' \subset X'$ be a closed subscheme of finite presentation. Let $X'' \to X'$ be the blowing up with center $Z'$. There exists a closed subscheme $Y \subset X$ of finite presentation, such that \begin{enumerate} \item $Y = Z \cup b(Z')$ set theoretically, and \item the composition $X'' \to X$ is isomorphic to the blowing up of $X$ in $Y$. \end{enumerate}


divisors-lemma-strict-transform

In the situation of Definition \ref{definition-strict-transform}. \begin{enumerate} \item The strict transform $X'$ of $X$ is the blowup of $X$ in the closed subscheme $f^{-1}Z$ of $X$. \item For a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the strict transform $\mathcal{F}'$ is canonically isomorphic to the pushforward along $X' \to X \times_S S'$ of the strict transform of $\mathcal{F}$ relative to the blowing up $X' \to X$. \end{enumerate}


divisors-lemma-strict-transform-flat

In the situation of Definition \ref{definition-strict-transform}. \begin{enumerate} \item If $X$ is flat over $S$ at all points lying over $Z$, then the strict transform of $X$ is equal to the base change $X \times_S S'$. \item Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $\mathcal{F}$ is flat over $S$ at all points lying over $Z$, then the strict transform $\mathcal{F}'$ of $\mathcal{F}$ is equal to the pullback $\text{pr}_X^*\mathcal{F}$. \end{enumerate}


divisors-lemma-strict-transform-affine

Let $S$ be a scheme. Let $Z \subset S$ be a closed subscheme. Let $b : S' \to S$ be the blowing up of $Z$ in $S$. Let $g : X \to Y$ be an affine morphism of schemes over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $g' : X \times_S S' \to Y \times_S S'$ be the base change of $g$. Let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$ relative to $b$. Then $g'_*\mathcal{F}'$ is the strict transform of $g_*\mathcal{F}$.


divisors-lemma-strict-transform-different-centers

Let $S$ be a scheme. Let $Z \subset S$ be a closed subscheme. Let $D \subset S$ be an effective Cartier divisor. Let $Z' \subset S$ be the closed subscheme cut out by the product of the ideal sheaves of $Z$ and $D$. Let $S' \to S$ be the blowup of $S$ in $Z$. \begin{enumerate} \item The blowup of $S$ in $Z'$ is isomorphic to $S' \to S$. \item Let $f : X \to S$ be a morphism of schemes and let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $\mathcal{F}$ has no nonzero local sections supported in $f^{-1}D$, then the strict transform of $\mathcal{F}$ relative to the blowing up in $Z$ agrees with the strict transform of $\mathcal{F}$ relative to the blowing up of $S$ in $Z'$. \end{enumerate}


divisors-lemma-strict-transform-composition-blowups

Let $S$ be a scheme. Let $Z \subset S$ be a closed subscheme. Let $b : S' \to S$ be the blowing up with center $Z$. Let $Z' \subset S'$ be a closed subscheme. Let $S'' \to S'$ be the blowing up with center $Z'$. Let $Y \subset S$ be a closed subscheme such that $Y = Z \cup b(Z')$ set theoretically and the composition $S'' \to S$ is isomorphic to the blowing up of $S$ in $Y$. In this situation, given any scheme $X$ over $S$ and $\mathcal{F} \in \QCoh(\mathcal{O}_X)$ we have \begin{enumerate} \item the strict transform of $\mathcal{F}$ with respect to the blowing up of $S$ in $Y$ is equal to the strict transform with respect to the blowup $S'' \to S'$ in $Z'$ of the strict transform of $\mathcal{F}$ with respect to the blowup $S' \to S$ of $S$ in $Z$, and \item the strict transform of $X$ with respect to the blowing up of $S$ in $Y$ is equal to the strict transform with respect to the blowup $S'' \to S'$ in $Z'$ of the strict transform of $X$ with respect to the blowup $S' \to S$ of $S$ in $Z$. \end{enumerate}


divisors-lemma-strict-transform-universally-injective

In the situation of Definition \ref{definition-strict-transform}. Suppose that $$ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 $$ is an exact sequence of quasi-coherent sheaves on $X$ which remains exact after any base change $T \to S$. Then the strict transforms of $\mathcal{F}_i'$ relative to any blowup $S' \to S$ form a short exact sequence $0 \to \mathcal{F}'_1 \to \mathcal{F}'_2 \to \mathcal{F}'_3 \to 0$ too.


divisors-lemma-composition-admissible-blowups

\begin{slogan} Admissible blowups are stable under composition. \end{slogan} Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be a quasi-compact open subscheme. Let $b : X' \to X$ be a $U$-admissible blowup. Let $X'' \to X'$ be a $U$-admissible blowup. Then the composition $X'' \to X$ is a $U$-admissible blowup.


divisors-lemma-extend-admissible-blowups

Let $X$ be a quasi-compact and quasi-separated scheme. Let $U, V \subset X$ be quasi-compact open subschemes. Let $b : V' \to V$ be a $U \cap V$-admissible blowup. Then there exists a $U$-admissible blowup $X' \to X$ whose restriction to $V$ is $V'$.


divisors-lemma-dominate-admissible-blowups

Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be a quasi-compact open subscheme. Let $b_i : X_i \to X$, $i = 1, \ldots, n$ be $U$-admissible blowups. There exists a $U$-admissible blowup $b : X' \to X$ such that (a) $b$ factors as $X' \to X_i \to X$ for $i = 1, \ldots, n$ and (b) each of the morphisms $X' \to X_i$ is a $U$-admissible blowup.


divisors-lemma-separate-disjoint-opens-by-blowing-up

\begin{slogan} Separate irreducible components by blowing up. \end{slogan} Let $X$ be a quasi-compact and quasi-separated scheme. Let $U, V$ be quasi-compact disjoint open subschemes of $X$. Then there exist a $U \cup V$-admissible blowup $b : X' \to X$ such that $X'$ is a disjoint union of open subschemes $X' = X'_1 \amalg X'_2$ with $b^{-1}(U) \subset X'_1$ and $b^{-1}(V) \subset X'_2$.


divisors-lemma-blowing-up-denominators

Let $X$ be a locally Noetherian scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s$ be a regular meromorphic section of $\mathcal{L}$. Let $U \subset X$ be the maximal open subscheme such that $s$ corresponds to a section of $\mathcal{L}$ over $U$. The blowup $b : X' \to X$ in the ideal of denominators of $s$ is $U$-admissible. There exists an effective Cartier divisor $D \subset X'$ and an isomorphism $$ b^*\mathcal{L} = \mathcal{O}_{X'}(D - E), $$ where $E \subset X'$ is the exceptional divisor such that the meromorphic section $b^*s$ corresponds, via the isomorphism, to the meromorphic section $1_D \otimes (1_E)^{-1}$.


divisors-lemma-strict-transform-blowup-fitting-ideal

Let $S$ be a scheme. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_S$-module. Let $Z_k \subset S$ be the closed subscheme cut out by $\text{Fit}_k(\mathcal{F})$, see Section \ref{section-fitting-ideals}. Let $S' \to S$ be the blowup of $S$ in $Z_k$ and let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$. Then $\mathcal{F}'$ can locally be generated by $\leq k$ sections.


divisors-lemma-strict-transform-blowup-fitting-ideal-locally-free

Let $S$ be a scheme. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_S$-module. Let $Z_k \subset S$ be the closed subscheme cut out by $\text{Fit}_k(\mathcal{F})$, see Section \ref{section-fitting-ideals}. Assume that $\mathcal{F}$ is locally free of rank $k$ on $S \setminus Z_k$. Let $S' \to S$ be the blowup of $S$ in $Z_k$ and let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$. Then $\mathcal{F}'$ is locally free of rank $k$.


divisors-lemma-blowup-fitting-ideal

Let $X$ be a scheme. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_X$-module. Let $U \subset X$ be a scheme theoretically dense open such that $\mathcal{F}|_U$ is finite locally free of constant rank $r$. Then \begin{enumerate} \item the blowup $b : X' \to X$ of $X$ in the $r$th Fitting ideal of $\mathcal{F}$ is $U$-admissible, \item the strict transform $\mathcal{F}'$ of $\mathcal{F}$ with respect to $b$ is locally free of rank $r$, \item the kernel $\mathcal{K}$ of the surjection $b^*\mathcal{F} \to \mathcal{F}'$ is finitely presented and $\mathcal{K}|_U = 0$, \item $b^*\mathcal{F}$ and $\mathcal{K}$ are perfect $\mathcal{O}_{X'}$-modules of tor dimension $\leq 1$. \end{enumerate}


divisors-lemma-filter-after-modification

Let $X$ be an integral scheme. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module. There exists a modification $f : X' \to X$ such that $f^*\mathcal{E}$ has a filtration whose successive quotients are invertible $\mathcal{O}_{X'}$-modules.


divisors-lemma-extend-rational-map-after-modification

Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Assume $X$ is Noetherian and $Y$ is proper over $S$. Given an $S$-rational map $f : U \to Y$ from $X$ to $Y$ there exists a morphism $p : X' \to X$ and an $S$-morphism $f' : X' \to Y$ such that \begin{enumerate} \item $p$ is proper and $p^{-1}(U) \to U$ is an isomorphism, \item $f'|_{p^{-1}(U)}$ is equal to $f \circ p|_{p^{-1}(U)}$. \end{enumerate}


divisors-proposition-push-down-ample

Let $\pi : X \to Y$ be a finite surjective morphism of schemes. Assume that $X$ has an ample invertible $\mathcal{O}_X$-module. If \begin{enumerate} \item $\pi$ is finite locally free, or \item $Y$ is an integral normal scheme, or \item $Y$ is Noetherian, $p\mathcal{O}_Y = 0$, and $X = Y_{red}$, \end{enumerate} then $Y$ has an ample invertible $\mathcal{O}_Y$-module.


spaces-theorem-presentation

Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j = (s, t) : R \to U \times_S U$ be an \'etale equivalence relation on $U$ over $S$. Then the quotient $U/R$ is an algebraic space, and $U \to U/R$ is \'etale and surjective, in other words $(U, R, U \to U/R)$ is a presentation of $U/R$.


spaces-lemma-morphism-schemes-gives-representable-transformation

Let $S$ be a scheme contained in $\Sch_{fppf}$ and let $X$, $Y$ be objects of $(\Sch/S)_{fppf}$. Let $f : X \to Y$ be a morphism of schemes. Then $$ h_f : h_X \longrightarrow h_Y $$ is a representable transformation of functors.


spaces-lemma-composition-representable-transformations

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : G \to H$ be representable transformations of functors. Then $$ b \circ a : F \longrightarrow H $$ is a representable transformation of functors.


spaces-lemma-base-change-representable-transformations

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$ be a representable transformation of functors. Let $b : H \to G$ be any transformation of functors. Consider the fibre product diagram $$ \xymatrix{ H \times_{b, G, a} F \ar[r]_-{b'} \ar[d]_{a'} & F \ar[d]^a \\ H \ar[r]^b & G } $$ Then the base change $a'$ is a representable transformation of functors.


spaces-lemma-product-representable-transformations

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F_i, G_i : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$, $i = 1, 2$. Let $a_i : F_i \to G_i$, $i = 1, 2$ be representable transformations of functors. Then $$ a_1 \times a_2 : F_1 \times F_2 \longrightarrow G_1 \times G_2 $$ is a representable transformation of functors.


spaces-lemma-representable-transformation-to-sheaf

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$ be a representable transformation of functors. If $G$ is a sheaf, then so is $F$.


spaces-lemma-representable-transformation-diagonal

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$ be a representable transformation of functors. Then $\Delta_{F/G} : F \to F \times_G F$ is representable.


spaces-lemma-morphism-schemes-gives-representable-transformation-property

Let $S$, $X$, $Y$ be objects of $\Sch_{fppf}$. Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{P}$ be as in Definition \ref{definition-relative-representable-property}. Then $h_X \longrightarrow h_Y$ has property $\mathcal{P}$ if and only if $f$ has property $\mathcal{P}$.


spaces-lemma-composition-representable-transformations-property

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $\mathcal{P}$ be a property as in Definition \ref{definition-relative-representable-property} which is stable under composition. Let $a : F \to G$, $b : G \to H$ be representable transformations of functors. If $a$ and $b$ have property $\mathcal{P}$ so does $b \circ a : F \longrightarrow H$.


spaces-lemma-base-change-representable-transformations-property

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $\mathcal{P}$ be a property as in Definition \ref{definition-relative-representable-property}. Let $a : F \to G$ be a representable transformations of functors. Let $b : H \to G$ be any transformation of functors. Consider the fibre product diagram $$ \xymatrix{ H \times_{b, G, a} F \ar[r]_-{b'} \ar[d]_{a'} & F \ar[d]^a \\ H \ar[r]^b & G } $$ If $a$ has property $\mathcal{P}$ then also the base change $a'$ has property $\mathcal{P}$.


spaces-lemma-descent-representable-transformations-property

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $\mathcal{P}$ be a property as in Definition \ref{definition-relative-representable-property}. Let $a : F \to G$ be a representable transformations of functors. Let $b : H \to G$ be any transformation of functors. Consider the fibre product diagram $$ \xymatrix{ H \times_{b, G, a} F \ar[r]_-{b'} \ar[d]_{a'} & F \ar[d]^a \\ H \ar[r]^b & G } $$ Assume that $b$ induces a surjective map of fppf sheaves $H^\# \to G^\#$. In this case, if $a'$ has property $\mathcal{P}$, then also $a$ has property $\mathcal{P}$.


spaces-lemma-product-representable-transformations-property

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F_i, G_i : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$, $i = 1, 2$. Let $a_i : F_i \to G_i$, $i = 1, 2$ be representable transformations of functors. Let $\mathcal{P}$ be a property as in Definition \ref{definition-relative-representable-property} which is stable under composition. If $a_1$ and $a_2$ have property $\mathcal{P}$ so does $a_1 \times a_2 : F_1 \times F_2 \longrightarrow G_1 \times G_2$.


spaces-lemma-representable-transformations-property-implication

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$ be a representable transformation of functors. Let $\mathcal{P}$, $\mathcal{P}'$ be properties as in Definition \ref{definition-relative-representable-property}. Suppose that for any morphism of schemes $f : X \to Y$ we have $\mathcal{P}(f) \Rightarrow \mathcal{P}'(f)$. If $a$ has property $\mathcal{P}$ then $a$ has property $\mathcal{P}'$.


spaces-lemma-surjective-flat-locally-finite-presentation

Let $S$ be a scheme. Let $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ be sheaves. Let $a : F \to G$ be representable, flat, locally of finite presentation, and surjective. Then $a : F \to G$ is surjective as a map of sheaves.


spaces-lemma-representable-diagonal

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F$ be a presheaf of sets on $(\Sch/S)_{fppf}$. The following are equivalent: \begin{enumerate} \item the diagonal $F \to F \times F$ is representable, \item for $U \in \Ob((\Sch/S)_{fppf})$ and any $a \in F(U)$ the map $a : h_U \to F$ is representable, \item for every pair $U, V \in \Ob((\Sch/S)_{fppf})$ and any $a \in F(U)$, $b \in F(V)$ the fibre product $h_U \times_{a, F, b} h_V$ is representable. \end{enumerate}


spaces-lemma-transformation-diagonal-properties

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F$ be a presheaf of sets on $(\Sch/S)_{fppf}$. Let $\mathcal{P}$ be a property as in Definition \ref{definition-relative-representable-property}. If for every $U, V \in \Ob((\Sch/S)_{fppf})$ and $a \in F(U)$, $b \in F(V)$ we have \begin{enumerate} \item $h_U \times_{a, F, b} h_V$ is representable, say by the scheme $W$, and \item the morphism $W \to U \times_S V$ corresponding to $h_U \times_{a, F, b} h_V \to h_U \times h_V$ has property $\mathcal{P}$, \end{enumerate} then $\Delta : F \to F \times F$ is representable and has property $\mathcal{P}$.


spaces-lemma-scheme-is-space

A scheme is an algebraic space. More precisely, given a scheme $T \in \Ob((\Sch/S)_{fppf})$ the representable functor $h_T$ is an algebraic space.


spaces-lemma-fibre-product-spaces-over-sheaf-with-representable-diagonal

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $H$ be a sheaf on $(\Sch/S)_{fppf}$ whose diagonal is representable. Let $F, G$ be algebraic spaces over $S$. Let $F \to H$, $G \to H$ be maps of sheaves. Then $F \times_H G$ is an algebraic space.


spaces-lemma-fibre-product-spaces

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F \to H$, $G \to H$ be morphisms of algebraic spaces over $S$. Then $F \times_H G$ is an algebraic space, and is a fibre product in the category of algebraic spaces over $S$.


spaces-lemma-coproduct-sheaves-open-and-closed

Let $S \in \Ob(\Sch_{fppf})$. Let $F$ and $G$ be sheaves on $(\Sch/S)_{fppf}^{opp}$ and denote $F \amalg G$ the coproduct in the category of sheaves. The map $F \to F \amalg G$ is representable by open and closed immersions.


spaces-lemma-representable-sheaf-coproduct-sheaves

Let $S \in \Ob(\Sch_{fppf})$. Let $U \in \Ob((\Sch/S)_{fppf})$. Given a set $I$ and sheaves $F_i$ on $\Ob((\Sch/S)_{fppf})$, if $U \cong \coprod_{i\in I} F_i$ as sheaves, then each $F_i$ is representable by an open and closed subscheme $U_i$ and $U \cong \coprod U_i$ as schemes.


spaces-lemma-coproduct-algebraic-spaces

Let $S \in \Ob(\Sch_{fppf})$. Suppose given a set $I$ and algebraic spaces $F_i$, $i \in I$. Then $F = \coprod_{i \in I} F_i$ is an algebraic space provided $I$, and the $F_i$ are not too ``large'': for example if we can choose surjective \'etale morphisms $U_i \to F_i$ such that $\coprod_{i \in I} U_i$ is isomorphic to an object of $(\Sch/S)_{fppf}$, then $F$ is an algebraic space.


spaces-lemma-glueing-algebraic-spaces

Let $S \in \Ob(\Sch_{fppf})$. Let $F$ be a presheaf of sets on $(\Sch/S)_{fppf}$. Assume \begin{enumerate} \item $F$ is a sheaf, \item there exists an index set $I$ and subfunctors $F_i \subset F$ such that \begin{enumerate} \item each $F_i$ is an algebraic space, \item each $F_i \to F$ is representable, \item each $F_i \to F$ is an open immersion (see Definition \ref{definition-relative-representable-property}), \item the map $\coprod F_i \to F$ is surjective as a map of sheaves, and \item $\coprod F_i$ is an algebraic space (set theoretic condition, see Lemma \ref{lemma-coproduct-algebraic-spaces}). \end{enumerate} \end{enumerate} Then $F$ is an algebraic space.


spaces-lemma-space-presentation

Let $F$ be an algebraic space over $S$. Let $f : U \to F$ be a surjective \'etale morphism from a scheme to $F$. Set $R = U \times_F U$. Then \begin{enumerate} \item $j : R \to U \times_S U$ defines an equivalence relation on $U$ over $S$ (see Groupoids, Definition \ref{groupoids-definition-equivalence-relation}). \item the morphisms $s, t : R \to U$ are \'etale, and \item the diagram $$ \xymatrix{ R \ar@<1ex>[r] \ar@<-1ex>[r] & U \ar[r] & F } $$ is a coequalizer diagram in $\Sh((\Sch/S)_{fppf})$. \end{enumerate}


spaces-lemma-pullback-etale-equivalence-relation

Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j = (s, t) : R \to U \times_S U$ be an \'etale equivalence relation on $U$ over $S$. Let $U' \to U$ be an \'etale morphism. Let $R'$ be the restriction of $R$ to $U'$, see Groupoids, Definition \ref{groupoids-definition-restrict-relation}. Then $j' : R' \to U' \times_S U'$ is an \'etale equivalence relation also.


spaces-lemma-finding-opens

Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j = (s, t) : R \to U \times_S U$ be a pre-relation. Let $g : U' \to U$ be a morphism. Assume \begin{enumerate} \item $j$ is an equivalence relation, \item $s, t : R \to U$ are surjective, flat and locally of finite presentation, \item $g$ is flat and locally of finite presentation. \end{enumerate} Let $R' = R|_{U'}$ be the restriction of $R$ to $U'$. Then $U'/R' \to U/R$ is representable, and is an open immersion.


spaces-lemma-when-it-works-it-works

Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j = (s, t) : R \to U \times_S U$ be an \'etale equivalence relation on $U$ over $S$. If the quotient $U/R$ is an algebraic space, then $U \to U/R$ is \'etale and surjective. Hence $(U, R, U \to U/R)$ is a presentation of the algebraic space $U/R$.


spaces-lemma-presentation-quasi-compact

Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j = (s, t) : R \to U \times_S U$ be an \'etale equivalence relation on $U$ over $S$. Assume that $U$ is affine. Then the quotient $F = U/R$ is an algebraic space, and $U \to F$ is \'etale and surjective.


spaces-lemma-etale-locally-representable-gives-space

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F$ be a sheaf on $(\Sch/S)_{fppf}$ such that there exists $U \in \Ob((\Sch/S)_{fppf})$ and a map $U \to F$ which is representable, surjective, and \'etale. Then $F$ is an algebraic space.


spaces-lemma-etale-locally-representable-by-space-gives-space

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $G$ be an algebraic space over $S$, let $F$ be a sheaf on $(\Sch/S)_{fppf}$, and let $G \to F$ be a representable transformation of functors which is surjective and \'etale. Then $F$ is an algebraic space.


spaces-lemma-representable-over-space

\begin{slogan} A functor that admits a representable morphism to an algebraic space is an algebraic space. \end{slogan} Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F$ be an algebraic space over $S$. Let $G \to F$ be a representable transformation of functors. Then $G$ is an algebraic space.


spaces-lemma-representable-morphisms-spaces-property

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F$, $G$ be algebraic spaces over $S$. Let $G \to F$ be a representable morphism. Let $U \in \Ob((\Sch/S)_{fppf})$, and $q : U \to F$ surjective and \'etale. Set $V = G \times_F U$. Finally, let $\mathcal{P}$ be a property of morphisms of schemes as in Definition \ref{definition-relative-representable-property}. Then $G \to F$ has property $\mathcal{P}$ if and only if $V \to U$ has property $\mathcal{P}$.


spaces-lemma-morphism-sheaves-with-P-effective-descent-etale

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $G \to F$ be a transformation of presheaves on $(\Sch/S)_{fppf}$. Let $\mathcal{P}$ be a property of morphisms of schemes. Assume \begin{enumerate} \item $\mathcal{P}$ is preserved under any base change, fppf local on the base, and morphisms of type $\mathcal{P}$ satisfy descent for fppf coverings, see Descent, Definition \ref{descent-definition-descending-types-morphisms}, \item $G$ is a sheaf, \item $F$ is an algebraic space, \item there exists a $U \in \Ob((\Sch/S)_{fppf})$ and a surjective \'etale morphism $U \to F$ such that $V = G \times_F U$ is representable, and \item $V \to U$ has $\mathcal{P}$. \end{enumerate} Then $G$ is an algebraic space, $G \to F$ is representable and has property $\mathcal{P}$.


spaces-lemma-lift-morphism-presentations

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G$ be algebraic spaces over $S$. Let $a : F \to G$ be a morphism. Given any $V \in \Ob((\Sch/S)_{fppf})$ and a surjective \'etale morphism $q : V \to G$ there exists a $U \in \Ob((\Sch/S)_{fppf})$ and a commutative diagram $$ \xymatrix{ U \ar[d]_p \ar[r]_\alpha & V \ar[d]^q \\ F \ar[r]^a & G } $$ with $p$ surjective and \'etale.


spaces-lemma-composition-immersions

Let $S \in \Ob(\Sch_{fppf})$ be a scheme. A composition of (closed, resp.\ open) immersions of algebraic spaces over $S$ is a (closed, resp.\ open) immersion of algebraic spaces over $S$.


spaces-lemma-base-change-immersions

Let $S \in \Ob(\Sch_{fppf})$ be a scheme. A base change of a (closed, resp.\ open) immersion of algebraic spaces over $S$ is a (closed, resp.\ open) immersion of algebraic spaces over $S$.


spaces-lemma-sub-subspaces

Let $S \in \Ob(\Sch_{fppf})$ be a scheme. Let $F$ be an algebraic space over $S$. Let $F_1$, $F_2$ be locally closed subspaces of $F$. If $F_1 \subset F_2$ as subfunctors of $F$, then $F_1$ is a locally closed subspace of $F_2$. Similarly for closed and open subspaces.


spaces-lemma-properties-diagonal

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F$ be an algebraic space over $S$. Let $\Delta : F \to F \times F$ be the diagonal morphism. Then \begin{enumerate} \item $\Delta$ is locally of finite type, \item $\Delta$ is a monomorphism, \item $\Delta$ is separated, and \item $\Delta$ is locally quasi-finite. \end{enumerate}


spaces-lemma-quotient

Let $U \to S$ be a morphism of $\Sch_{fppf}$. Let $G$ be an abstract group. Let $G \to \text{Aut}_S(U)$ be a group homomorphism. Assume \begin{itemize} \item[(*)] if $u \in U$ is a point, and $g(u) = u$ for some non-identity element $g \in G$, then $g$ induces a nontrivial automorphism of $\kappa(u)$. \end{itemize} Then $$ j : R = \coprod\nolimits_{g \in G} U \longrightarrow U \times_S U, \quad (g, x) \longmapsto (g(x), x) $$ is an \'etale equivalence relation and hence $$ F = U/R $$ is an algebraic space by Theorem \ref{theorem-presentation}.


spaces-lemma-quotient-finite-separated

Notation and assumptions as in Lemma \ref{lemma-quotient}. Assume $G$ is finite. Then \begin{enumerate} \item if $U \to S$ is quasi-separated, then $U/G$ is quasi-separated over $S$, and \item if $U \to S$ is separated, then $U/G$ is separated over $S$. \end{enumerate}


spaces-lemma-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 $S$ be an object of $\Sch_{fppf}$. Let \begin{align*} g : \Sh((\Sch/S)_{fppf}) \longrightarrow \Sh((\Sch'/S)_{fppf}), \\ f : \Sh((\Sch'/S)_{fppf}) \longrightarrow \Sh((\Sch/S)_{fppf}) \end{align*} be the morphisms of topoi of Topologies, Lemma \ref{topologies-lemma-change-alpha}. Let $F$ be a sheaf of sets on $(\Sch/S)_{fppf}$. Then \begin{enumerate} \item if $F$ is representable by a scheme $X \in \Ob((\Sch/S)_{fppf})$ over $S$, then $f^{-1}F$ is representable too, in fact it is representable by the same scheme $X$, now viewed as an object of $(\Sch'/S)_{fppf}$, and \item if $F$ is an algebraic space over $S$, then $f^{-1}F$ is an algebraic space over $S$ also. \end{enumerate}


spaces-lemma-fully-faithful

Suppose $\Sch_{fppf}$ is contained in $\Sch'_{fppf}$. Let $S$ be an object of $\Sch_{fppf}$. Denote $\textit{Spaces}/S$ the category of algebraic spaces over $S$ defined using $\Sch_{fppf}$. Similarly, denote $\textit{Spaces}'/S$ the category of algebraic spaces over $S$ defined using $\Sch'_{fppf}$. The construction of Lemma \ref{lemma-change-big-site} defines a fully faithful functor $$ \textit{Spaces}/S \longrightarrow \textit{Spaces}'/S $$ whose essential image consists of those $X' \in \Ob(\textit{Spaces}'/S)$ such that there exist $U, R \in \Ob((\Sch/S)_{fppf})$\footnote{Requiring the existence of $R$ is necessary because of our choice of the function $Bound$ in Sets, Equation (\ref{sets-equation-bound}). The size of the fibre product $U \times_{X'} U$ can grow faster than $Bound$ in terms of the size of $U$. We can illustrate this by setting $S = \Spec(A)$, $U = \Spec(A[x_i, i \in I])$ and $R = \coprod_{(\lambda_i) \in A^I} \Spec(A[x_i, y_i]/(x_i - \lambda_i y_i))$. In this case the size of $R$ grows like $\kappa^\kappa$ where $\kappa$ is the size of $U$.} and morphisms $$ U \longrightarrow X' \quad\text{and}\quad R \longrightarrow U \times_{X'} U $$ in $\Sh((\Sch'/S)_{fppf})$ which are surjective as maps of sheaves (for example if the displayed morphisms are surjective and \'etale).


spaces-lemma-change-base-scheme

Suppose given a big site $\Sch_{fppf}$. Let $g : S \to S'$ be morphism of $\Sch_{fppf}$. Let $j : (\Sch/S)_{fppf} \to (\Sch/S')_{fppf}$ be the corresponding localization functor. Let $F$ be a sheaf of sets on $(\Sch/S)_{fppf}$. Then \begin{enumerate} \item for a scheme $T'$ over $S'$ we have $j_!F(T'/S') = \coprod\nolimits_{\varphi : T' \to S} F(T' \xrightarrow{\varphi} S),$ \item if $F$ is representable by a scheme $X \in \Ob((\Sch/S)_{fppf})$, then $j_!F$ is representable by $j(X)$ which is $X$ viewed as a scheme over $S'$, and \item if $F$ is an algebraic space over $S$, then $j_!F$ is an algebraic space over $S'$, and if $F = U/R$ is a presentation, then $j_!F = j(U)/j(R)$ is a presentation. \end{enumerate} Let $F'$ be a sheaf of sets on $(\Sch/S')_{fppf}$. Then \begin{enumerate} \item[(4)] for a scheme $T$ over $S$ we have $j^{-1}F'(T/S) = F'(T/S')$, \item[(5)] if $F'$ is representable by a scheme $X' \in \Ob((\Sch/S')_{fppf})$, then $j^{-1}F'$ is representable, namely by $X'_S = S \times_{S'} X'$, and \item[(6)] if $F'$ is an algebraic space, then $j^{-1}F'$ is an algebraic space, and if $F' = U'/R'$ is a presentation, then $j^{-1}F' = U'_S/R'_S$ is a presentation. \end{enumerate}


spaces-lemma-category-of-spaces-over-smaller-base-scheme

Let $\Sch_{fppf}$ be a big fppf site. Let $S \to S'$ be a morphism of this site. The construction above give an equivalence of categories $$ \left\{ \begin{matrix} \text{category of algebraic}\\ \text{spaces over }S \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{category of pairs }(F', F' \to S)\text{ consisting}\\ \text{of an algebraic space }F'\text{ over }S'\text{ and a}\\ \text{morphism }F' \to S\text{ of algebraic spaces over }S' \end{matrix} \right\} $$


spaces-lemma-rephrase

Let $\Sch_{fppf}$ be a big fppf site. Let $S \to S'$ be a morphism of this site. Let $F'$ be a sheaf on $(\Sch/S')_{fppf}$. The following are equivalent: \begin{enumerate} \item The restriction $F'|_{(\Sch/S)_{fppf}}$ is an algebraic space over $S$, and \item the sheaf $h_S \times F'$ is an algebraic space over $S'$. \end{enumerate}


topology-theorem-tychonov

A product of quasi-compact spaces is quasi-compact.


topology-theorem-characterize-proper

\begin{reference} In \cite[I, p. 75, Theorem 1]{Bourbaki} you can find: (2) $\Leftrightarrow$ (4). In \cite[I, p. 77, Proposition 6]{Bourbaki} you can find: (2) $\Rightarrow$ (1). \end{reference} Let $f: X\to Y$ be a continuous map between topological spaces. The following conditions are equivalent: \begin{enumerate} \item The map $f$ is quasi-proper and closed. \item The map $f$ is proper. \item The map $f$ is universally closed. \item The map $f$ is closed and $f^{-1}(y)$ is quasi-compact for any $y\in Y$. \end{enumerate}


topology-lemma-Hausdorff

Let $X$ be a topological space. The following are equivalent: \begin{enumerate} \item $X$ is Hausdorff, \item the diagonal $\Delta(X) \subset X \times X$ is closed. \end{enumerate}


topology-lemma-graph-closed

\begin{slogan} Graphs of maps to Hausdorff spaces are closed. \end{slogan} Let $f : X \to Y$ be a continuous map of topological spaces. If $Y$ is Hausdorff, then the graph of $f$ is closed in $X \times Y$.


topology-lemma-fibre-product-closed

Let $X \to Z$ and $Y \to Z$ be continuous maps of topological spaces. If $Z$ is Hausdorff, then  $X \times_Z Y$ is closed in $X \times Y$.


topology-lemma-separated

Let $f : X \to Y$ be continuous map of topological spaces. The following are equivalent: \begin{enumerate} \item $f$ is separated, \item $\Delta(X) \subset X \times_Y X$ is a closed subset, \item given distinct points $x, x' \in X$ mapping to the same point of $Y$, there exist disjoint open neighbourhoods of $x$ and $x'$. \end{enumerate}


topology-lemma-base-change-separated

Let $f : X \to Y$ and $Z \to Y$ be continuous maps of topological spaces. If $f$ is separated, then $f' : Z \times_Y X \to Z$ is separated.


topology-lemma-make-base

Let $X$ be a set and let $\mathcal{B}$ be a collection of subsets. Assume that $X = \bigcup_{B \in \mathcal{B}} B$ and that given $x \in B_1 \cap B_2$ with $B_1, B_2 \in \mathcal{B}$ there is a $B_3 \in \mathcal{B}$ with $x \in B_3 \subset B_1 \cap B_2$. Then there is a unique topology on $X$ such that $\mathcal{B}$ is a basis for this topology.


topology-lemma-refine-covering-basis

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. Let $\mathcal{U} : U = \bigcup_i U_i$ be an open covering of $U \subset X$. There exists an open covering $U = \bigcup V_j$ which is a refinement of $\mathcal{U}$ such that each $V_j$ is an element of the basis $\mathcal{B}$.


topology-lemma-subbase

Let $X$ be a set. Given any collection $\mathcal{B}$ of subsets of $X$ there is a unique topology on $X$ such that $\mathcal{B}$ is a subbase for this topology.


topology-lemma-create-map-from-subcollection

Let $X$ be a topological space. Let $\mathcal{B}$ be a collection of opens of $X$. Assume $X = \bigcup_{U \in \mathcal{B}} U$ and for $U, V \in \mathcal{B}$ we have $U \cap V = \bigcup_{W \in \mathcal{B}, W \subset U \cap V} W$. Then there is a continuous map $f : X \to Y$ of topological spaces such that \begin{enumerate} \item for $U \in \mathcal{B}$ the image $f(U)$ is open, \item for $U \in \mathcal{B}$ we have $f^{-1}(f(U)) = U$, and \item the opens $f(U)$, $U \in \mathcal{B}$ form a basis for the topology on $Y$. \end{enumerate}


topology-lemma-induced

Let $X$ be a topological space. Let $Y$ be a set and let $f : Y \to X$ be an injective map of sets. The induced topology on $Y$ is the topology characterized by each of the following statements: \begin{enumerate} \item it is the weakest topology on $Y$ such that $f$ is continuous, \item the open subsets of $Y$ are $f^{-1}(U)$ for $U \subset X$ open, \item the closed subsets of $Y$ are the sets $f^{-1}(Z)$ for $Z \subset X$ closed. \end{enumerate}


topology-lemma-quotient

Let $X$ be a topological space. Let $Y$ be a set and let $f : X \to Y$ be a surjective map of sets. The quotient topology on $Y$ is the topology characterized by each of the following statements: \begin{enumerate} \item it is the strongest topology on $Y$ such that $f$ is continuous, \item a subset $V$ of $Y$ is open if and only if $f^{-1}(V)$ is open, \item a subset $Z$ of $Y$ is closed if and only if $f^{-1}(Z)$ is closed. \end{enumerate}


topology-lemma-open-morphism-quotient-topology

Let $f : X \to Y$ be surjective, open, continuous map of topological spaces. Let $T \subset Y$ be a subset. Then \begin{enumerate} \item $f^{-1}(\overline{T}) = \overline{f^{-1}(T)}$, \item $T \subset Y$ is closed if and only if $f^{-1}(T)$ is closed, \item $T \subset Y$ is open if and only if $f^{-1}(T)$ is open, and \item $T \subset Y$ is locally closed if and only if $f^{-1}(T)$ is locally closed. \end{enumerate} In particular we see that $f$ is submersive.


topology-lemma-closed-morphism-quotient-topology

Let $f : X \to Y$ be surjective, closed, continuous map of topological spaces. Let $T \subset Y$ be a subset. Then \begin{enumerate} \item $\overline{T} = f(\overline{f^{-1}(T)})$, \item $T \subset Y$ is closed if and only if $f^{-1}(T)$ is closed, \item $T \subset Y$ is open if and only if $f^{-1}(T)$ is open, and \item $T \subset Y$ is locally closed if and only if $f^{-1}(T)$ is locally closed. \end{enumerate} In particular we see that $f$ is submersive.


topology-lemma-image-connected-space

Let $f : X \to Y$ be a continuous map of topological spaces. If $E \subset X$ is a connected subset, then $f(E) \subset Y$ is connected as well.


topology-lemma-connected-components

Let $X$ be a topological space. \begin{enumerate} \item If $T \subset X$ is connected, then so is its closure. \item Any connected component of $X$ is closed (but not necessarily open). \item Every connected subset of $X$ is contained in a unique connected component of $X$. \item Every point of $X$ is contained in a unique connected component, in other words, $X$ is the union of its connected components. \end{enumerate}


topology-lemma-connected-fibres-quotient-topology-connected-components

Let $f : X \to Y$ be a continuous map of topological spaces. Assume that \begin{enumerate} \item all fibres of $f$ are connected, and \item a set $T \subset Y$ is closed if and only if $f^{-1}(T)$ is closed. \end{enumerate} Then $f$ induces a bijection between the sets of connected components of $X$ and $Y$.


topology-lemma-connected-fibres-connected-components

Let $f : X \to Y$ be a continuous map of topological spaces. Assume that (a) $f$ is open, (b) all fibres of $f$ are connected. Then $f$ induces a bijection between the sets of connected components of $X$ and $Y$.


topology-lemma-finite-fibre-connected-components

Let $f : X \to Y$ be a continuous map of nonempty topological spaces. Assume that (a) $Y$ is connected, (b) $f$ is open and closed, and (c) there is a point $y\in Y$ such that the fiber $f^{-1}(y)$ is a finite set. Then $X$ has at most $|f^{-1}(y)|$ connected components. Hence any connected  component $T$ of $X$ is open and closed, and $f(T)$ is a nonempty open and  closed subset of $Y$, which is therefore equal to $Y$.


topology-lemma-space-connected-components

Let $X$ be a topological space. Let $\pi_0(X)$ be the set of connected components of $X$. Let $X \to \pi_0(X)$ be the map which sends $x \in X$ to the connected component of $X$ passing through $x$. Endow $\pi_0(X)$ with the quotient topology. Then $\pi_0(X)$ is a totally disconnected space and any continuous map $X \to Y$ from $X$ to a totally disconnected space $Y$ factors through $\pi_0(X)$.


topology-lemma-locally-connected

Let $X$ be a topological space. If $X$ is locally connected, then \begin{enumerate} \item any open subset of $X$ is locally connected, and \item the connected components of $X$ are open. \end{enumerate} So also the connected components of open subsets of $X$ are open. In particular, every point has a fundamental system of open connected neighbourhoods.


topology-lemma-image-irreducible-space

Let $f : X \to Y$ be a continuous map of topological spaces. If $E \subset X$ is an irreducible subset, then $f(E) \subset Y$ is irreducible as well.


topology-lemma-irreducible

Let $X$ be a topological space. \begin{enumerate} \item If $T \subset X$ is irreducible so is its closure in $X$. \item Any irreducible component of $X$ is closed. \item Any irreducible subset of $X$ is contained in an irreducible component of $X$. \item Every point of $X$ is contained in some irreducible component of $X$, in other words, $X$ is the union of its irreducible components. \end{enumerate}


topology-lemma-pick-irreducible-components

Let $X$ be a topological space and suppose $X = \bigcup_{i = 1, \ldots, n} X_i$ where each $X_i$ is an irreducible closed subset of $X$ and no $X_i$ is contained in the union of the other members.  Then each $X_i$ is an irreducible component of $X$ and each irreducible component of $X$ is one of the $X_i$.


topology-lemma-sober-subspace

Let $X$ be a topological space and let $Y\subset X$. \begin{enumerate} \item If $X$ is Kolmogorov then so is $Y$. \item Suppose $Y$ is locally closed in $X$. If $X$ is quasi-sober then so is $Y$. \item Suppose $Y$ is locally closed in $X$. If $X$ is sober then so is $Y$. \end{enumerate}


topology-lemma-sober-local

Let $X$ be a topological space and let $(X_i)_{i\in I}$ be a covering of $X$. \begin{enumerate} \item Suppose $X_i$ is locally closed in $X$ for every $i\in I$. Then, $X$ is Kolmogorov if and only if $X_i$ is Kolmogorov for every $i\in I$. \item Suppose $X_i$ is open in $X$ for every $i\in I$. Then, $X$ is quasi-sober if and only if $X_i$ is quasi-sober for every $i\in I$. \item Suppose $X_i$ is open in $X$ for every $i\in I$. Then, $X$ is sober if and  only if $X_i$ is sober for every $i\in I$. \end{enumerate}


topology-lemma-irreducible-on-top

Let $f : X \to Y$ be a continuous map of topological spaces. Assume that (a) $Y$ is irreducible, (b) $f$ is open, and (c) there exists a dense collection of points $y \in Y$ such that $f^{-1}(y)$ is irreducible. Then $X$ is irreducible.


topology-lemma-irreducible-fibres-irreducible-components

Let $f : X \to Y$ be a continuous map of topological spaces. Assume that (a) $f$ is open, and (b) for every $y \in Y$ the fibre $f^{-1}(y)$ is irreducible. Then $f$ induces a bijection between irreducible components.


topology-lemma-make-sober

Let $X$ be a topological space. There is a canonical continuous map $$ c : X \longrightarrow X' $$ from $X$ to a sober topological space $X'$ which is universal among continuous maps from $X$ to sober topological spaces. Moreover, the assignment $U' \mapsto c^{-1}(U')$ is a bijection between opens of $X'$ and $X$ which commutes with finite intersections and arbitrary unions. The image $c(X)$ is a Kolmogorov topological space and the map $c : X \to c(X)$ is universal for maps of $X$ into Kolmogorov spaces.


topology-lemma-Noetherian

Let $X$ be a Noetherian topological space. \begin{enumerate} \item Any subset of $X$ with the induced topology is Noetherian. \item The space $X$ has finitely many irreducible components. \item Each irreducible component of $X$ contains a nonempty open of $X$. \end{enumerate}


topology-lemma-image-Noetherian

Let $f : X \to Y$ be a continuous map of topological spaces. \begin{enumerate} \item If $X$ is Noetherian, then $f(X)$ is Noetherian. \item If $X$ is locally Noetherian and $f$ open, then $f(X)$ is locally Noetherian. \end{enumerate}


topology-lemma-finite-union-Noetherian

Let $X$ be a topological space. Let $X_i \subset X$, $i = 1, \ldots, n$ be a finite collection of subsets. If each $X_i$ is Noetherian (with the induced topology), then $\bigcup_{i = 1, \ldots, n}  X_i$ is Noetherian (with the induced topology).


topology-lemma-locally-Noetherian-locally-connected

Let $X$ be a locally Noetherian topological space. Then $X$ is locally connected.


topology-lemma-dimension-supremum-local-dimensions

Let $X$ be a topological space. Then $\dim(X) = \sup \dim_x(X)$ where the supremum runs over the points $x$ of $X$.


topology-lemma-codimension-at-generic-point

Let $X$ be a topological space. Let $Y \subset X$ be an irreducible closed subset. Let $U \subset X$ be an open subset such that $Y \cap U$ is nonempty. Then $$ \text{codim}(Y, X) = \text{codim}(Y \cap U, U) $$


topology-lemma-catenary

Let $X$ be a topological space. The following are equivalent: \begin{enumerate} \item $X$ is catenary, \item $X$ has an open covering by catenary spaces. \end{enumerate} Moreover, in this case any locally closed subspace of $X$ is catenary.


topology-lemma-catenary-in-codimension

Let $X$ be a topological space. The following are equivalent: \begin{enumerate} \item $X$ is catenary, and \item for every pair of irreducible closed subsets $Y \subset Y'$ we have $\text{codim}(Y, Y') < \infty$ and for every triple $Y \subset Y' \subset Y''$ of irreducible closed subsets we have $$ \text{codim}(Y, Y'') = \text{codim}(Y, Y') + \text{codim}(Y', Y''). $$ \end{enumerate}


topology-lemma-composition-quasi-compact

A composition of quasi-compact maps is quasi-compact.


topology-lemma-closed-in-quasi-compact

A closed subset of a quasi-compact topological space is quasi-compact.


topology-lemma-quasi-compact-in-Hausdorff

Let $X$ be a Hausdorff topological space. \begin{enumerate} \item If $E \subset X$ is quasi-compact, then it is closed. \item If $E_1, E_2 \subset X$ are disjoint quasi-compact subsets then there exists opens $E_i \subset U_i$ with $U_1 \cap U_2 = \emptyset$. \end{enumerate}


topology-lemma-closed-in-compact

Let $X$ be a quasi-compact Hausdorff space. Let $E \subset X$. The following are equivalent: (a) $E$ is closed in $X$, (b) $E$ is quasi-compact.


topology-lemma-intersection-closed-in-quasi-compact

Let $X$ be a quasi-compact topological space. If $\{Z_\alpha\}_{\alpha \in A}$ is a collection of closed subsets such that the intersection of each finite subcollection is nonempty, then $\bigcap_{\alpha \in A} Z_\alpha$ is nonempty.


topology-lemma-image-quasi-compact

Let $f : X \to Y$ be a continuous map of topological spaces. \begin{enumerate} \item If $X$ is quasi-compact, then $f(X)$ is quasi-compact. \item If $f$ is quasi-compact, then $f(X)$ is retrocompact. \end{enumerate}


topology-lemma-quasi-compact-closed-point

Let $X$ be a topological space. Assume that \begin{enumerate} \item $X$ is nonempty, \item $X$ is quasi-compact, and \item $X$ is Kolmogorov. \end{enumerate} Then $X$ has a closed point.


topology-lemma-closed-points-quasi-compact

Let $X$ be a quasi-compact Kolmogorov space. Then the set $X_0$ of closed points of $X$ is quasi-compact.


topology-lemma-connected-component-intersection

Let $X$ be a topological space. Assume \begin{enumerate} \item $X$ is quasi-compact, \item $X$ has a basis for the topology consisting of quasi-compact opens, and \item the intersection of two quasi-compact opens is quasi-compact. \end{enumerate} For any $x \in X$ the connected component of $X$ containing $x$ is the intersection of all open and closed subsets of $X$ containing $x$.


topology-lemma-connected-component-intersection-compact-Hausdorff

Let $X$ be a topological space. Assume $X$ is quasi-compact and Hausdorff. For any $x \in X$ the connected component of $X$ containing $x$ is the intersection of all open and closed subsets of $X$ containing $x$.


topology-lemma-closed-union-connected-components

Let $X$ be a topological space. Assume \begin{enumerate} \item $X$ is quasi-compact, \item $X$ has a basis for the topology consisting of quasi-compact opens, and \item the intersection of two quasi-compact opens is quasi-compact. \end{enumerate} For a subset $T \subset X$ the following are equivalent: \begin{enumerate} \item[(a)] $T$ is an intersection of open and closed subsets of $X$, and \item[(b)] $T$ is closed in $X$ and is a union of connected components of $X$. \end{enumerate}


topology-lemma-Noetherian-quasi-compact

Let $X$ be a Noetherian topological space. \begin{enumerate} \item The space $X$ is quasi-compact. \item Any subset of $X$ is retrocompact. \end{enumerate}


topology-lemma-quasi-compact-locally-Noetherian-Noetherian

A quasi-compact locally Noetherian space is Noetherian.


topology-lemma-subbase-theorem

Let $X$ be a topological space. Let $\mathcal{B}$ be a subbase for $X$. If every covering of $X$ by elements of $\mathcal{B}$ has a finite refinement, then $X$ is quasi-compact.


topology-lemma-locally-quasi-compact-Hausdorff

A Hausdorff space is locally quasi-compact if and only if every point has a quasi-compact neighbourhood.


topology-lemma-relatively-compact-refinement

Let $X$ be a Hausdorff and quasi-compact space. Let $X = \bigcup_{i \in I} U_i$ be an open covering. Then there exists an open covering $X = \bigcup_{i \in I} V_i$ such that $\overline{V_i} \subset U_i$ for all $i$.


topology-lemma-refine-covering

Let $X$ be a Hausdorff and quasi-compact space. Let $X = \bigcup_{i \in I} U_i$ be an open covering. Suppose given an integer $p \geq 0$ and for every $(p + 1)$-tuple $i_0, \ldots, i_p$ of $I$ an open covering $U_{i_0} \cap \ldots \cap U_{i_p} = \bigcup W_{i_0 \ldots i_p, k}$. Then there exists an open covering $X = \bigcup_{j \in J} V_j$ and a map $\alpha : J \to I$ such that $\overline{V_j} \subset U_{\alpha(j)}$ and such that each $V_{j_0} \cap \ldots \cap V_{j_p}$ is contained in $W_{\alpha(j_0) \ldots \alpha(j_p), k}$ for some $k$.


topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset

Let $X$ be a topological space. Let $Z \subset X$ be a quasi-compact subset such that any two points of $Z$ have disjoint open neighbourhoods in $X$. Suppose given an integer $p \geq 0$, a set $I$, for every $i \in I$ an open $U_i \subset X$, and for every $(p + 1)$-tuple $i_0, \ldots, i_p$ of $I$ an open $W_{i_0 \ldots i_p} \subset U_{i_0} \cap \ldots \cap U_{i_p}$ such that \begin{enumerate} \item $Z \subset \bigcup U_i$, and \item for every $i_0, \ldots, i_p$ we have $W_{i_0 \ldots i_p} \cap Z = U_{i_0} \cap \ldots \cap U_{i_p} \cap Z$. \end{enumerate} Then there exist opens $V_i$ of $X$ such that \begin{enumerate} \item $Z \subset \bigcup V_i$, \item $V_i \subset U_i$ for all $i$, \item $\overline{V_i} \cap Z \subset U_i$ for all $i$, and \item $V_{i_0} \cap \ldots \cap V_{i_p} \subset W_{i_0 \ldots i_p}$ for all $(p + 1)$-tuples $i_0, \ldots, i_p$. \end{enumerate}


topology-lemma-limits

The category of topological spaces has limits and the forgetful functor to sets commutes with them.


topology-lemma-describe-limits

Let $\mathcal{I}$ be a cofiltered category. Let $i \mapsto X_i$ be a diagram of topological spaces over $\mathcal{I}$. Let $X = \lim X_i$ be the limit with projection maps $f_i : X \to X_i$. \begin{enumerate} \item Any open of $X$ is of the form $\bigcup_{j \in J} f_j^{-1}(U_j)$ for some subset $J \subset I$ and opens $U_j \subset X_j$. \item Any quasi-compact open of $X$ is of the form $f_i^{-1}(U_i)$ for some $i$ and some $U_i \subset X_i$ open. \end{enumerate}


topology-lemma-characterize-limit

Let $\mathcal{I}$ be a cofiltered category. Let $i \mapsto X_i$ be a diagram of topological spaces over $\mathcal{I}$. Let $X$ be a topological space such that \begin{enumerate} \item $X = \lim X_i$ as a set (denote $f_i$ the projection maps), \item the sets $f_i^{-1}(U_i)$ for $i \in \Ob(\mathcal{I})$ and $U_i \subset X_i$ open form a basis for the topology of $X$. \end{enumerate} Then $X$ is the limit of the $X_i$ as a topological space.


topology-lemma-inverse-limit-quasi-compact

Let $\mathcal{I}$ be a category and let $i \mapsto X_i$ be a diagram over $\mathcal{I}$ in the category of topological spaces. If each $X_i$ is quasi-compact and Hausdorff, then $\lim X_i$ is quasi-compact.


topology-lemma-nonempty-limit

Let $\mathcal{I}$ be a cofiltered category and let $i \mapsto X_i$ be a diagram over $\mathcal{I}$ in the category of topological spaces. If each $X_i$ is quasi-compact, Hausdorff, and nonempty, then $\lim X_i$ is nonempty.


topology-lemma-constructible

The collection of constructible sets is closed under finite intersections, finite unions and complements.


topology-lemma-inverse-images-constructibles

Let $f : X \to Y$ be a continuous map of topological spaces. If the inverse image of every retrocompact open subset of $Y$ is retrocompact in $X$, then inverse images of constructible sets are constructible.


topology-lemma-open-immersion-constructible-inverse-image

Let $U \subset X$ be open. For a constructible set $E \subset X$ the intersection $E \cap U$ is constructible in $U$.


topology-lemma-quasi-compact-open-immersion-constructible-image

Let $U \subset X$ be a retrocompact open. Let $E \subset U$. If $E$ is constructible in $U$, then $E$ is constructible in $X$.


topology-lemma-collate-constructible

Let $X$ be a topological space. Let $E \subset X$ be a subset. Let $X = V_1 \cup \ldots \cup V_m$ be a finite covering by retrocompact opens. Then $E$ is constructible in $X$ if and only if $E \cap V_j$ is constructible in $V_j$ for each $j = 1, \ldots, m$.


topology-lemma-intersect-constructible-with-closed

Let $X$ be a topological space. Let $Z \subset X$ be a closed subset such that $X \setminus Z$ is quasi-compact. Then for a constructible set $E \subset X$ the intersection $E \cap Z$ is constructible in $Z$.


topology-lemma-intersect-constructible-with-retrocompact

Let $X$ be a topological space. Let $T \subset X$ be a subset. Suppose \begin{enumerate} \item $T$ is retrocompact in $X$, \item quasi-compact opens form a basis for the topology on $X$. \end{enumerate} Then for a constructible set $E \subset X$ the intersection $E \cap T$ is constructible in $T$.


topology-lemma-closed-constructible-image

Let $Z \subset X$ be a closed subset whose complement is retrocompact open. Let $E \subset Z$. If $E$ is constructible in $Z$, then $E$ is constructible in $X$.


topology-lemma-constructible-is-retrocompact

Let $X$ be a topological space. Every constructible subset of $X$ is retrocompact.


topology-lemma-intersect-constructible-with-constructible

Let $X$ be a topological space. Assume $X$ has a basis consisting of quasi-compact opens. For $E, E'$ constructible in $X$, the intersection $E \cap E'$ is constructible in $E$.


topology-lemma-constructible-in-constructible

Let $X$ be a topological space. Assume $X$ has a basis consisting of quasi-compact opens. Let $E$ be constructible in $X$ and $F \subset E$ constructible in $E$. Then $F$ is constructible in $X$.


topology-lemma-locally-closed-constructible-image

Let $X$ be a quasi-compact topological space having a basis consisting of quasi-compact opens such that the intersection of any two quasi-compact opens is quasi-compact. Let $T \subset X$ be a locally closed subset such that $T$ is quasi-compact and $T^c$ is retrocompact in $X$. Then $T$ is constructible in $X$.


topology-lemma-collate-constructible-from-constructible

Let $X$ be a topological space which has a basis for the topology consisting of quasi-compact opens. Let $E \subset X$ be a subset. Let $X = E_1 \cup \ldots \cup E_m$ be a finite covering by constructible subsets. Then $E$ is constructible in $X$ if and only if $E \cap E_j$ is constructible in $E_j$ for each $j = 1, \ldots, m$.


topology-lemma-generic-point-in-constructible

Let $X$ be a topological space. Suppose that $Z \subset X$ is irreducible. Let $E \subset X$ be a finite union of locally closed subsets (e.g.\ $E$ is constructible). The following are equivalent \begin{enumerate} \item The intersection $E \cap Z$ contains an open dense subset of $Z$. \item The intersection $E \cap Z$ is dense in $Z$. \end{enumerate} If $Z$ has a generic point $\xi$, then this is also equivalent to \begin{enumerate} \item[(3)] We have $\xi \in E$. \end{enumerate}


topology-lemma-constructible-Noetherian-space

Let $X$ be a Noetherian topological space. The constructible sets in $X$ are precisely the finite unions of locally closed subsets of $X$.


topology-lemma-constructible-map-Noetherian

Let $f : X \to Y$ be a continuous map of Noetherian topological spaces. If $E \subset Y$ is constructible in $Y$, then $f^{-1}(E)$ is constructible in $X$.


topology-lemma-characterize-constructible-Noetherian

Let $X$ be a Noetherian topological space. Let $E \subset X$ be a subset. The following are equivalent: \begin{enumerate} \item $E$ is constructible in $X$, and \item for every irreducible closed $Z \subset X$ the intersection $E \cap Z$ either contains a nonempty open of $Z$ or is not dense in $Z$. \end{enumerate}


topology-lemma-constructible-neighbourhood-Noetherian

Let $X$ be a Noetherian topological space. Let $x \in X$. Let $E \subset X$ be constructible in $X$. The following are equivalent: \begin{enumerate} \item $E$ is a neighbourhood of $x$, and \item for every irreducible closed subset $Y$ of $X$ which contains $x$ the intersection $E \cap Y$ is dense in $Y$. \end{enumerate}


topology-lemma-characterize-open-Noetherian

Let $X$ be a Noetherian topological space. Let $E \subset X$ be a subset. The following are equivalent: \begin{enumerate} \item $E$ is open in $X$, and \item for every irreducible closed subset $Y$ of $X$ the intersection $E \cap Y$ is either empty or contains a nonempty open of $Y$. \end{enumerate}


topology-lemma-tube

Let $X$ and $Y$ be topological spaces. Let $A \subset X$ and $B \subset Y$ be quasi-compact subsets. Let $A \times B \subset W \subset X \times Y$ with $W$ open in $X \times Y$. Then there exists opens $A \subset U \subset X$ and $B \subset V \subset Y$ such that $U \times V \subset W$.


topology-lemma-characterize-quasi-compact

\begin{reference} Combination of \cite[I, p. 75, Lemme 1]{Bourbaki} and \cite[I, p. 76, Corrolaire 1]{Bourbaki}. \end{reference} A topological space $X$ is quasi-compact if and only if the projection map $Z \times X \to Z$ is closed for any topological space $Z$.


topology-lemma-closed-map

\begin{slogan} A map from a compact space to a Hausdorff space is a proper. \end{slogan} Let $f : X \to Y$ be a continuous map of topological spaces. If $X$ is quasi-compact and $Y$ is Hausdorff, then $f$ is proper.


topology-lemma-bijective-map

Let $f : X \to Y$ be a continuous map of topological spaces. If $f$ is bijective, $X$ is quasi-compact, and $Y$ is Hausdorff, then $f$ is a homeomorphism.


topology-lemma-jacobson-check-irreducible-closed

Let $X$ be a topological space. Let $X_0$ be the set of closed points of $X$. Suppose that for every point $x\in X$ the intersection $X_0 \cap \overline{\{x\}}$ is dense in $\overline{\{x\}}$. Then $X$ is Jacobson.


topology-lemma-non-jacobson-Noetherian-characterize

Let $X$ be a Kolmogorov topological space with a basis of quasi-compact open sets. If $X$ is not Jacobson, then there exists a non-closed point $x \in X$ such that $\{x\}$ is locally closed.


topology-lemma-jacobson-local

Let $X$ be a topological space. Let $X = \bigcup U_i$ be an open covering. Then $X$ is Jacobson if and only if each $U_i$ is Jacobson. Moreover, in this case $X_0 = \bigcup U_{i, 0}$.


topology-lemma-jacobson-inherited

Let $X$ be Jacobson. The following types of subsets $T \subset X$ are Jacobson: \begin{enumerate} \item Open subspaces. \item Closed subspaces. \item Locally closed subspaces. \item Unions of locally closed subspaces. \item Constructible sets. \item Any subset $T \subset X$ which locally on $X$ is a union of locally closed subsets. \end{enumerate} In each of these cases closed points of $T$ are closed in $X$.


topology-lemma-finite-jacobson

A finite Jacobson space is discrete.


topology-lemma-jacobson-equivalent-locally-closed

\begin{slogan} For Jacobson spaces, closed points see everything about the topology. \end{slogan} Suppose $X$ is a Jacobson topological space. Let $X_0$ be the set of closed points of $X$. There is a bijective, inclusion preserving correspondence $$ \{\text{finite unions loc.\ closed subsets of } X\} \leftrightarrow \{\text{finite unions loc.\ closed subsets of } X_0\} $$ given by $E \mapsto E \cap X_0$. This correspondence preserves the subsets of locally closed, of open and of closed subsets.


topology-lemma-jacobson-equivalent-constructible

Suppose $X$ is a Jacobson topological space. Let $X_0$ be the set of closed points of $X$. There is a bijective, inclusion preserving correspondence $$ \{\text{constructible subsets of } X\} \leftrightarrow \{\text{constructible subsets of } X_0\} $$ given by $E \mapsto E \cap X_0$. This correspondence preserves the subset of retrocompact open subsets, as well as complements of these.


topology-lemma-open-closed-specialization

Let $X$ be a topological space. \begin{enumerate} \item Any closed subset of $X$ is stable under specialization. \item Any open subset of $X$ is stable under generalization. \item A subset $T \subset X$ is stable under specialization if and only if the complement $T^c$ is stable under generalization. \end{enumerate}


topology-lemma-stable-specialization

Let $T \subset X$ be a subset of a topological space $X$. The following are equivalent \begin{enumerate} \item $T$ is stable under specialization, and \item $T$ is a (directed) union of closed subsets of $X$. \end{enumerate}


topology-lemma-lift-specialization-composition

Suppose $f : X \to Y$ and $g : Y \to Z$ are continuous maps of topological spaces. If specializations lift along both $f$ and $g$ then specializations lift along $g \circ f$. Similarly for ``generalizations lift along''.


topology-lemma-lift-specializations-images

Let $f : X \to Y$ be a continuous map of topological spaces. \begin{enumerate} \item If specializations lift along $f$, and if $T \subset X$ is stable under specialization, then $f(T) \subset Y$ is stable under specialization. \item If generalizations lift along $f$, and if $T \subset X$ is stable under generalization, then $f(T) \subset Y$ is stable under generalization. \end{enumerate}


topology-lemma-closed-open-map-specialization

Let $f : X \to Y$ be a continuous map of topological spaces. \begin{enumerate} \item If $f$ is closed then specializations lift along $f$. \item If $f$ is open, $X$ is a Noetherian topological space, each irreducible closed subset of $X$ has a generic point, and $Y$ is Kolmogorov then generalizations lift along $f$. \end{enumerate}


topology-lemma-quotient-kolmogorov

Suppose that $s, t : R \to U$ and $\pi : U \to X$ are continuous maps of topological spaces such that \begin{enumerate} \item $\pi$ is open, \item $U$ is sober, \item $s, t$ have finite fibres, \item generalizations lift along $s, t$, \item $(t, s)(R) \subset U \times U$ is an equivalence relation on $U$ and $X$ is the quotient of $U$ by this equivalence relation (as a set). \end{enumerate} Then $X$ is Kolmogorov.


topology-lemma-dimension-specializations-lift

Let $f : X \to Y$ be a morphism of topological spaces. Suppose that $Y$ is a sober topological space, and $f$ is surjective. If either specializations or generalizations lift along $f$, then $\dim(X) \geq \dim(Y)$.


topology-lemma-characterize-closed-Noetherian

Let $X$ be a Noetherian sober topological space. Let $E \subset X$ be a subset of $X$. \begin{enumerate} \item If $E$ is constructible and stable under specialization, then $E$ is closed. \item If $E$ is constructible and stable under generalization, then $E$ is open. \end{enumerate}


topology-lemma-dimension-function-catenary

Let $X$ be a topological space. If $X$ is sober and has a dimension function, then $X$ is catenary. Moreover, for any $x \leadsto y$ we have $$ \delta(x) - \delta(y) = \text{codim}\left(\overline{\{y\}}, \ \overline{\{x\}}\right). $$


topology-lemma-dimension-function-unique

Let $X$ be a topological space. Let $\delta$, $\delta'$ be two dimension functions on $X$. If $X$ is locally Noetherian and sober then $\delta - \delta'$ is locally constant on $X$.


topology-lemma-locally-dimension-function

Let $X$ be locally Noetherian, sober and catenary. Then any point has an open neighbourhood $U \subset X$ which has a dimension function.


topology-lemma-nowhere-dense

Let $X$ be a topological space. The union of a finite number of nowhere dense sets is a nowhere dense set.


topology-lemma-image-nowhere-dense-open

Let $X$ be a topological space. Let $U \subset X$ be an open. Let $T \subset U$ be a subset. If $T$ is nowhere dense in $U$, then $T$ is nowhere dense in $X$.


topology-lemma-nowhere-dense-local

Let $X$ be a topological space. Let $X = \bigcup U_i$ be an open covering. Let $T \subset X$ be a subset. If $T \cap U_i$ is nowhere dense in $U_i$ for all $i$, then $T$ is nowhere dense in $X$.


topology-lemma-closed-image-nowhere-dense

Let $f : X \to Y$ be a continuous map of topological spaces. Let $T \subset X$ be a subset. If $f$ is a homeomorphism of $X$ onto a closed subset of $Y$ and $T$ is nowhere dense in $X$, then also $f(T)$ is nowhere dense in $Y$.


topology-lemma-open-inverse-image-closed-nowhere-dense

Let $f : X \to Y$ be a continuous map of topological spaces. Let $T \subset Y$ be a subset. If $f$ is open and $T$ is a closed nowhere dense subset of $Y$, then also $f^{-1}(T)$ is a closed nowhere dense subset of $X$. If $f$ is surjective and open, then $T$ is closed nowhere dense if and only if $f^{-1}(T)$ is closed nowhere dense.


topology-lemma-profinite

Let $X$ be a topological space. The following are equivalent \begin{enumerate} \item $X$ is a profinite space, and \item $X$ is Hausdorff, quasi-compact, and totally disconnected. \end{enumerate} If this is true, then $X$ is a cofiltered limit of finite discrete spaces.


topology-lemma-directed-inverse-limit-profinite

A limit of profinite spaces is profinite.


topology-lemma-profinite-refine-open-covering

Let $X$ be a profinite space. Every open covering of $X$ has a refinement by a finite covering $X = \coprod U_i$ with $U_i$ open and closed.


topology-lemma-pi0-profinite

Let $X$ be a topological space. If $X$ is quasi-compact and every connected component of $X$ is the intersection of the open and closed subsets containing it, then $\pi_0(X)$ is a profinite space.


topology-lemma-constructible-hausdorff-quasi-compact

Let $X$ be a spectral space. The constructible topology is Hausdorff, totally disconnected, and quasi-compact.


topology-lemma-fibres-spectral-map-quasi-compact

Let $f : X \to Y$ be a spectral map of spectral spaces. Then \begin{enumerate} \item $f$ is continuous in the constructible topology, \item the fibres of $f$ are quasi-compact, and \item the image is closed in the constructible topology. \end{enumerate}


topology-lemma-spectral-sub

Let $X$ be a spectral space. Let $E \subset X$ be closed in the constructible topology (for example constructible or closed). Then $E$ with the induced topology is a spectral space.


topology-lemma-constructible-stable-specialization-closed

Let $X$ be a spectral space. Let $E \subset X$ be a subset closed in the constructible topology (for example constructible). \begin{enumerate} \item If $x \in \overline{E}$, then $x$ is the specialization of a point of $E$. \item If $E$ is stable under specialization, then $E$ is closed. \item If $E' \subset X$ is open in the constructible topology (for example constructible) and stable under generalization, then $E'$ is open. \end{enumerate}


topology-lemma-characterize-profinite-spectral

Let $X$ be a spectral space. The following are equivalent: \begin{enumerate} \item $X$ is profinite, \item $X$ is Hausdorff, \item $X$ is totally disconnected, \item every quasi-compact open is closed, \item there are no nontrivial specializations between points, \item every point of $X$ is closed, \item every point of $X$ is the generic point of an irreducible component of $X$, \item the constructible topology equals the given topology on $X$, and \item add more here. \end{enumerate}


topology-lemma-spectral-pi0

If $X$ is a spectral space, then $\pi_0(X)$ is a profinite space.


topology-lemma-product-spectral-spaces

The product of two spectral spaces is spectral.


topology-lemma-spectral-bijective

Let $f : X \to Y$ be a continuous map of topological spaces. If \begin{enumerate} \item $X$ and $Y$ are spectral, \item $f$ is spectral and bijective, and \item generalizations (resp.\ specializations) lift along $f$. \end{enumerate} Then $f$ is a homeomorphism.


topology-lemma-directed-inverse-limit-finite-sober-spectral-spaces

The inverse limit of a directed inverse system of finite sober topological spaces is a spectral topological space.


topology-lemma-spectral-closed-in-product-two-point-space

Let $W$ be the topological space with two points, one closed, the other not. A topological space is spectral if and only if it is homeomorphic to a subspace of a product of copies of $W$ which is closed in the constructible topology.


topology-lemma-spectral-inverse-limit-finite-sober-spaces

A topological space is spectral if and only if it is a directed inverse limit of finite sober topological spaces.


topology-lemma-Noetherian-goes-to-spectral

Let $X$ be a topological space and let $c : X \to X'$ be the universal map from $X$ to a sober topological space, see Lemma \ref{lemma-make-sober}. \begin{enumerate} \item If $X$ is quasi-compact, so is $X'$. \item If $X$ is quasi-compact, has a basis of quasi-compact opens, and the intersection of two quasi-compact opens is quasi-compact, then  $X'$ is spectral. \item If $X$ is Noetherian, then $X'$ is a Noetherian spectral space. \end{enumerate}


topology-lemma-inverse-limit-spectral-spaces-quasi-compact

Let $\mathcal{I}$ be a category. Let $i \mapsto X_i$ be a diagram of spectral spaces such that for $a : j \to i$ in $\mathcal{I}$ the corresponding map $f_a : X_j \to X_i$ is spectral. \begin{enumerate} \item Given subsets $Z_i \subset X_i$ closed in the constructible topology with $f_a(Z_j) \subset Z_i$ for all $a : j \to i$ in $\mathcal{I}$, then $\lim Z_i$ is quasi-compact. \item The space $X = \lim X_i$ is quasi-compact. \end{enumerate}


topology-lemma-inverse-limit-spectral-spaces-nonempty

Let $\mathcal{I}$ be a cofiltered category. Let $i \mapsto X_i$ be a diagram of spectral spaces such that for $a : j \to i$ in $\mathcal{I}$ the corresponding map $f_a : X_j \to X_i$ is spectral. \begin{enumerate} \item Given nonempty subsets $Z_i \subset X_i$ closed in the constructible topology with $f_a(Z_j) \subset Z_i$ for all $a : j \to i$ in $\mathcal{I}$, then $\lim Z_i$ is nonempty. \item If each $X_i$ is nonempty, then $X = \lim X_i$ is nonempty. \end{enumerate}


topology-lemma-inverse-limit-spectral-spaces-equal

Let $\mathcal{I}$ be a cofiltered category. Let $i \mapsto X_i$ be a diagram of spectral spaces such that for $a : j \to i$ in $\mathcal{I}$ the corresponding map $f_a : X_j \to X_i$ is spectral. Let $X = \lim X_i$ with projections $p_i : X \to X_i$. Let $i \in \Ob(\mathcal{I})$ and let $E, F \subset X_i$ be subsets with $E$ closed in the constructible topology and $F$ open in the constructible topology. Then $p_i^{-1}(E) \subset p_i^{-1}(F)$ if and only if there is a morphism $a : j \to i$ in $\mathcal{I}$ such that $f_a^{-1}(E) \subset f_a^{-1}(F)$.


topology-lemma-inverse-limit-spectral-spaces-constructible

Let $\mathcal{I}$ be a cofiltered category. Let $i \mapsto X_i$ be a diagram of spectral spaces such that for $a : j \to i$ in $\mathcal{I}$ the corresponding map $f_a : X_j \to X_i$ is spectral. Let $X = \lim X_i$ with projections $p_i : X \to X_i$. Let $E \subset X$ be a constructible subset. Then there exists an $i \in \Ob(\mathcal{I})$ and a constructible subset $E_i \subset X_i$ such that $p_i^{-1}(E_i) = E$. If $E$ is open, resp.\ closed, we may choose $E_i$ open, resp.\ closed.


topology-lemma-directed-inverse-limit-spectral-spaces

Let $\mathcal{I}$ be a cofiltered index category. Let $i \mapsto X_i$ be a diagram of spectral spaces such that for $a : j \to i$ in $\mathcal{I}$ the corresponding map $f_a : X_j \to X_i$ is spectral. Then the inverse limit $X = \lim X_i$ is a spectral topological space and the projection maps $p_i : X \to X_i$ are spectral.


topology-lemma-descend-opens

Let $\mathcal{I}$ be a cofiltered index category. Let $i \mapsto X_i$ be a diagram of spectral spaces such that for $a : j \to i$ in $\mathcal{I}$ the corresponding map $f_a : X_j \to X_i$ is spectral. Set $X = \lim X_i$ and denote $p_i : X \to X_i$ the projection. \begin{enumerate} \item Given any quasi-compact open $U \subset X$ there exists an $i \in \Ob(\mathcal{I})$ and a quasi-compact open $U_i \subset X_i$ such that $p_i^{-1}(U_i) = U$. \item Given $U_i \subset X_i$ and $U_j \subset X_j$ quasi-compact opens such that $p_i^{-1}(U_i) \subset p_j^{-1}(U_j)$ there exist $k \in \Ob(\mathcal{I})$ and morphisms $a : k \to i$ and $b : k \to j$ such that $f_a^{-1}(U_i) \subset f_b^{-1}(U_j)$. \item If $U_i, U_{1, i}, \ldots, U_{n, i} \subset X_i$ are quasi-compact opens and $p_i^{-1}(U_i) = p_i^{-1}(U_{1, i}) \cup \ldots \cup p_i^{-1}(U_{n, i})$ then $f_a^{-1}(U_i) = f_a^{-1}(U_{1, i}) \cup \ldots \cup f_a^{-1}(U_{n, i})$ for some morphism $a : j \to i$ in $\mathcal{I}$. \item Same statement as in (3) but for intersections. \end{enumerate}


topology-lemma-make-spectral-space

Let $W$ be a subset of a spectral space $X$. The following are equivalent: \begin{enumerate} \item $W$ is an intersection of constructible sets and closed under generalizations, \item $W$ is quasi-compact and closed under generalizations, \item there exists a quasi-compact subset $E \subset X$ such that $W$ is the set of points specializing to $E$, \item $W$ is an intersection of quasi-compact open subsets, \item \label{item-intersection-quasi-compact-open} there exists a nonempty set $I$ and quasi-compact opens $U_i \subset X$, $i \in I$ such that $W = \bigcap U_i$ and for all $i, j \in I$ there exists a $k \in I$ with $U_k \subset U_i \cap U_j$. \end{enumerate} In this case we have (a) $W$ is a spectral space, (b) $W = \lim U_i$ as topological spaces, and (c) for any open $U$ containing $W$ there exists an $i$ with $U_i \subset U$.


topology-lemma-make-spectral-space-minus

Let $X$ be a spectral space. Let $E \subset X$ be a constructible subset. Let $W \subset X$ be the set of points of $X$ which specialize to a point of $E$. Then $W \setminus E$ is a spectral space. If $W = \bigcap U_i$ with $U_i$ as in Lemma \ref{lemma-make-spectral-space} (\ref{item-intersection-quasi-compact-open}) then $W \setminus E = \lim (U_i \setminus E)$.


topology-lemma-dense-image

Let $f : X \to Y$ be a continuous map of topological spaces. Assume that $f(X)$ is dense in $Y$ and that $Y$ is Hausdorff. Then the cardinality of $Y$ is at most the cardinality of $P(P(X))$ where $P$ is the power set operation.


topology-lemma-image-open-technical

Let $f : X \to Y$ be a continuous map of topological spaces. Assume $f$ is surjective and $f(E) \not = Y$ for all proper closed subsets $E \subset X$. Then for $U \subset X$ open the subset $f(U)$ is contained in the closure of $Y \setminus f(X \setminus U)$.


topology-lemma-intersection-empty

Let $X$ be an extremally disconnected space. If $U, V \subset X$ are disjoint open subsets, then $\overline{U}$ and $\overline{V}$ are disjoint too.


topology-lemma-isomorphism

Let $f : X \to Y$ be a continuous map of Hausdorff quasi-compact topological spaces. If $Y$ is extremally disconnected, $f$ is surjective, and $f(Z) \not = Y$ for every proper closed subset $Z$ of $X$, then $f$ is a homeomorphism.


topology-lemma-find-compact-subset

Let $f : X \to Y$ be a continuous surjective map of Hausdorff quasi-compact topological spaces. There exists a quasi-compact subset $E \subset X$ such that $f(E) = Y$ but $f(E') \not = Y$ for all proper closed subsets $E' \subset E$.


topology-lemma-rainwater

Let $f : X \to X$ be a surjective continuous selfmap of a Hausdorff topological space. If $f$ is not $\text{id}_X$, then there exists a proper closed subset $E \subset X$ such that $X = E \cup f(E)$.


topology-lemma-existence-projective-cover

\begin{slogan} Every quasi-compact Hausdorff space has a canonical extremally disconnected cover \end{slogan} Let $X$ be a quasi-compact Hausdorff space. There exists a continuous surjection $X' \to X$ with $X'$ quasi-compact, Hausdorff, and extremally disconnected. If we require that every proper closed subset of $X'$ does not map onto $X$, then $X'$ is unique up to isomorphism.


topology-lemma-topology-quasi-separated-scheme

Let $X$ be a topological space which \begin{enumerate} \item has a basis of the topology consisting of quasi-compact opens, and \item has the property that the intersection of any two quasi-compact opens is quasi-compact. \end{enumerate} Then \begin{enumerate} \item $X$ is locally quasi-compact, \item a quasi-compact open $U \subset X$ is retrocompact, \item any quasi-compact open $U \subset X$ has a cofinal system of open coverings $\mathcal{U} : U = \bigcup_{j\in J} U_j$ with $J$ finite and all $U_j$ and $U_j \cap U_{j'}$ quasi-compact, \item add more here. \end{enumerate}


topology-lemma-partition-refined-by-stratification

Let $X$ be a topological space. Let $X = \coprod X_i$ be a finite partition of $X$. Then there exists a finite stratification of $X$ refining it.


topology-lemma-constructible-partition-refined-by-stratification

Let $X$ be a topological space. Suppose $X = T_1 \cup \ldots \cup T_n$ is written as a union of constructible subsets. There exists a finite stratification $X = \coprod X_i$ with each $X_i$ constructible such that each $T_k$ is a union of strata.


topology-lemma-colimits

The category of topological spaces has colimits and the forgetful functor to sets commutes with them.


topology-lemma-topological-group-limits

The category of topological groups has limits and limits commute with the forgetful functors to (a) the category of topological spaces and (b) the category of groups.


topology-lemma-profinite-group

Let $G$ be a topological group. The following are equivalent \begin{enumerate} \item $G$ as a topological space is profinite, \item $G$ is a limit of a diagram of finite discrete topological groups, \item $G$ is a cofiltered limit of finite discrete topological groups. \end{enumerate}


topology-lemma-topological-group-colimits

The category of topological groups has colimits and colimits commute with the forgetful functor to the category of groups.


topology-proposition-projective-in-category-hausdorff-qc

Let $X$ be a Hausdorff, quasi-compact topological space. The following are equivalent \begin{enumerate} \item $X$ is extremally disconnected, \item for any surjective continuous map $f : Y \to X$ with $Y$ Hausdorff quasi-compact there exists a continuous section, and \item for any solid commutative diagram $$ \xymatrix{ & Y \ar[d] \\ X \ar@{..>}[ru] \ar[r] & Z } $$ of continuous maps of quasi-compact Hausdorff spaces with $Y \to Z$ surjective, there is a dotted arrow in the category of topological spaces making the diagram commute. \end{enumerate}


hypercovering-theorem-cohomology-hypercoverings

Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $i \geq 0$. The functors \begin{eqnarray*} \textit{Ab}(\mathcal{C}) & \longrightarrow & \textit{Ab} \\ \mathcal{F} & \longmapsto & H^i(X, \mathcal{F}) \\ \mathcal{F} & \longmapsto & \check{H}^i_{\text{HC}}(X, \mathcal{F}) \end{eqnarray*} are canonically isomorphic.


hypercovering-lemma-coprod-prod-SR

Let $\mathcal{C}$ be a category. \begin{enumerate} \item the category $\text{SR}(\mathcal{C})$ has coproducts and $F$ commutes with them, \item the functor $F : \text{SR}(\mathcal{C}) \to \textit{PSh}(\mathcal{C})$ commutes with limits, \item if $\mathcal{C}$ has fibre products, then $\text{SR}(\mathcal{C})$ has fibre products, \item if $\mathcal{C}$ has products of pairs, then $\text{SR}(\mathcal{C})$ has products of pairs, \item if $\mathcal{C}$ has equalizers, so does $\text{SR}(\mathcal{C})$, and \item if $\mathcal{C}$ has a final object, so does $\text{SR}(\mathcal{C})$. \end{enumerate} Let $X \in \Ob(\mathcal{C})$. \begin{enumerate} \item the category $\text{SR}(\mathcal{C}, X)$ has coproducts and $F$ commutes with them, \item if $\mathcal{C}$ has fibre products, then $\text{SR}(\mathcal{C}, X)$ has finite limits and $F : \text{SR}(\mathcal{C}, X) \to \textit{PSh}(\mathcal{C})/h_X$ commutes with them. \end{enumerate}


hypercovering-lemma-covering-permanence

Let $\mathcal{C}$ be a site. \begin{enumerate} \item A composition of coverings in $\text{SR}(\mathcal{C})$ is a covering. \item If $K \to L$ is a covering in $\text{SR}(\mathcal{C})$ and $L' \to L$ is a morphism, then $L' \times_L K$ exists and $L' \times_L K \to L'$ is a covering. \item If $\mathcal{C}$ has products of pairs, and $A \to B$ and $K \to L$ are coverings in $\text{SR}(\mathcal{C})$, then $A \times K \to B \times L$ is a covering. \end{enumerate} Let $X \in \Ob(\mathcal{C})$. Then (1) and (2) holds for $\text{SR}(\mathcal{C}, X)$ and (3) holds if $\mathcal{C}$ has fibre products.


hypercovering-lemma-hypercoverings-set

Let $\mathcal{C}$ be a site with fibre products. Let $X \in \Ob(\mathcal{C})$ be an object of $\mathcal{C}$. The collection of all hypercoverings of $X$ forms a set.


hypercovering-lemma-hypercovering-F

Let $\mathcal{C}$ be a site with fibre products. Let $X \in \Ob(\mathcal{C})$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Consider the simplicial object $F(K)$ of $\textit{PSh}(\mathcal{C})$, endowed with its augmentation to the constant simplicial presheaf $h_X$. \begin{enumerate} \item The morphism of presheaves $F(K)_0 \to h_X$ becomes a surjection after sheafification. \item The morphism $$ (d^1_0, d^1_1) : F(K)_1 \longrightarrow F(K)_0 \times_{h_X} F(K)_0 $$ becomes a surjection after sheafification. \item For every $n \geq 1$ the morphism $$ F(K)_{n + 1} \longrightarrow (\text{cosk}_n \text{sk}_n F(K))_{n + 1} $$ turns into a surjection after sheafification. \end{enumerate}


hypercovering-lemma-compare-cosk0

Let $\mathcal{C}$ be a site. Let $\mathcal{F} \to \mathcal{G}$ be a morphism of presheaves of sets. Denote $K$ the simplicial object of $\textit{PSh}(\mathcal{C})$ whose $n$th term is the $(n + 1)$st fibre product of $\mathcal{F}$ over $\mathcal{G}$, see Simplicial, Example \ref{simplicial-example-fibre-products-simplicial-object}. Then, if $\mathcal{F} \to \mathcal{G}$ is surjective after sheafification, we have $$ H_i(K) = \left\{ \begin{matrix} 0 & \text{if} & i > 0\\ \mathbf{Z}_\mathcal{G}^\# & \text{if} & i = 0 \end{matrix} \right. $$ The isomorphism in degree $0$ is given by the morphism $H_0(K) \to \mathbf{Z}_\mathcal{G}^\#$ coming from the map $(\mathbf{Z}_K^\#)_0 = \mathbf{Z}_\mathcal{F}^\# \to \mathbf{Z}_\mathcal{G}^\#$.


hypercovering-lemma-acyclicity

Let $\mathcal{C}$ be a site. Let $f : L \to K$ be a morphism of simplicial objects of $\textit{PSh}(\mathcal{C})$. Let $n \geq 0$ be an integer. Assume that \begin{enumerate} \item For $i < n$ the morphism $L_i \to K_i$ is an isomorphism. \item The morphism $L_n \to K_n$ is surjective after sheafification. \item The canonical map $L \to \text{cosk}_n \text{sk}_n L$ is an isomorphism. \item The canonical map $K \to \text{cosk}_n \text{sk}_n K$ is an isomorphism. \end{enumerate} Then $H_i(f) : H_i(L) \to H_i(K)$ is an isomorphism.


hypercovering-lemma-acyclic-hypercover-sheaves

Let $\mathcal{C}$ be a site. Let $K$ be a simplicial presheaf. Let $\mathcal{G}$ be a presheaf. Let $K \to \mathcal{G}$ be an augmentation of $K$ towards $\mathcal{G}$. Assume that \begin{enumerate} \item The morphism of presheaves $K_0 \to \mathcal{G}$ becomes a surjection after sheafification. \item The morphism $$ (d^1_0, d^1_1) : K_1 \longrightarrow K_0 \times_\mathcal{G} K_0 $$ becomes a surjection after sheafification. \item For every $n \geq 1$ the morphism $$ K_{n + 1} \longrightarrow (\text{cosk}_n \text{sk}_n K)_{n + 1} $$ turns into a surjection after sheafification. \end{enumerate} Then $H_i(K) = 0$ for $i > 0$ and $H_0(K) = \mathbf{Z}_\mathcal{G}^\#$.


hypercovering-lemma-hypercovering-acyclic

Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. The homology of the simplicial presheaf $F(K)$ is $0$ in degrees $> 0$ and equal to $\mathbf{Z}_X^\#$ in degree $0$.


hypercovering-lemma-h0-cech

Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups on $\mathcal{C}$. Then $\check{H}^0(K, \mathcal{F}) = \mathcal{F}(X)$.


hypercovering-lemma-injective-trivial-cech

Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $\mathcal{I}$ be an injective sheaf of abelian groups on $\mathcal{C}$. Then $$ \check{H}^p(K, \mathcal{I}) = \left\{ \begin{matrix} \mathcal{I}(X) & \text{if} & p = 0 \\ 0 & \text{if} & p > 0 \end{matrix} \right. $$


hypercovering-lemma-cech-spectral-sequence

Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups on $\mathcal{C}$. There is a map $$ s(\mathcal{F}(K)) \longrightarrow R\Gamma(X, \mathcal{F}) $$ in $D^{+}(\textit{Ab})$ functorial in $\mathcal{F}$, which induces natural transformations $$ \check{H}^i(K, -) \longrightarrow H^i(X, -) $$ as functors $\textit{Ab}(\mathcal{C}) \to \textit{Ab}$. Moreover, there is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with $$ E_2^{p, q} = \check{H}^p(K, \underline{H}^q(\mathcal{F})) $$ converging to $H^{p + q}(X, \mathcal{F})$. This spectral sequence is functorial in $\mathcal{F}$ and in the hypercovering $K$.


hypercovering-lemma-h0-cech-variant

Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{G}$ be a presheaf on $\mathcal{C}$. Let $K$ be a hypercovering of $\mathcal{G}$. Let $\mathcal{F}$ be a sheaf of abelian groups on $\mathcal{C}$. Then $\check{H}^0(K, \mathcal{F}) = H^0(\mathcal{G}, \mathcal{F})$.