schemes-lemma-scheme-finite-discrete-affine

Let $X$ be a scheme whose underlying topological space is a finite discrete set. Then $X$ is affine.


schemes-lemma-reduced

A scheme $X$ is reduced if and only if $\mathcal{O}_X(U)$ is a reduced ring for all $U \subset X$ open.


schemes-lemma-affine-reduced

An affine scheme $\Spec(R)$ is reduced if and only if $R$ is reduced.


schemes-lemma-reduced-closed-subscheme

Let $X$ be a scheme. Let $T \subset X$ be a closed subset. There exists a unique closed subscheme $Z \subset X$ with the following properties: (a) the underlying topological space of $Z$ is equal to $T$, and (b) $Z$ is reduced.


schemes-lemma-map-into-reduction

Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. Let $Y$ be a reduced scheme. A morphism $f : Y \to X$ factors through $Z$ if and only if $f(Y) \subset Z$ (set theoretically). In particular, any morphism $Y \to X$ factors as $Y \to X_{red} \to X$.


schemes-lemma-morphism-from-spec-local-ring

Let $X$ be a scheme. Let $R$ be a local ring. The construction above gives a bijective correspondence between morphisms $\Spec(R) \to X$ and pairs $(x, \varphi)$ consisting of a point $x \in X$ and a local homomorphism of local rings $\varphi : \mathcal{O}_{X, x} \to R$.


schemes-lemma-specialize-points

Let $X$ be a scheme. Let $x, x' \in X$ be points of $X$. Then $x' \in X$ is a generalization of $x$ if and only if $x'$ is in the image of the canonical morphism $\Spec(\mathcal{O}_{X, x}) \to X$.


schemes-lemma-characterize-points

Let $X$ be a scheme. Points of $X$ correspond bijectively to equivalence classes of morphisms from spectra of fields into $X$. Moreover, each equivalence class contains a (unique up to unique isomorphism) smallest element $\Spec(\kappa(x)) \to X$.


schemes-lemma-glue

\begin{slogan} If you have two locally ringed spaces, and a subspace of the first one is isomorphic to a subspace of the other, then you can glue them together into one big locally ringed space. \end{slogan} Given any glueing data of locally ringed spaces there exists a locally ringed space $X$ and open subspaces $U_i \subset X$ together with isomorphisms $\varphi_i : X_i \to U_i$ of locally ringed spaces such that \begin{enumerate} \item $\varphi_i(U_{ij}) = U_i \cap U_j$, and \item $\varphi_{ij} = \varphi_j^{-1}|_{U_i \cap U_j} \circ \varphi_i|_{U_{ij}}$. \end{enumerate} The locally ringed space $X$ is characterized by the following mapping properties: Given a locally ringed space $Y$ we have \begin{eqnarray*} \Mor(X, Y) & = & \{ (f_i)_{i\in I} \mid f_i : X_i \to Y, \ f_j \circ \varphi_{ij} = f_i|_{U_{ij}}\} \\ f & \mapsto & (f|_{U_i} \circ \varphi_i)_{i \in I} \\ \Mor(Y, X) & = & \left\{ \begin{matrix} \text{open covering }Y = \bigcup\nolimits_{i \in I} V_i\text{ and } (g_i : V_i \to X_i)_{i \in I} \text{ such that}\\ g_i^{-1}(U_{ij}) = V_i \cap V_j \text{ and } g_j|_{V_i \cap V_j} = \varphi_{ij} \circ g_i|_{V_i \cap V_j} \end{matrix} \right\} \\ g & \mapsto & V_i = g^{-1}(U_i), \ g_i = \varphi_i^{-1} \circ g|_{V_i} \end{eqnarray*}


schemes-lemma-glue-schemes

\begin{slogan} Schemes can be glued to give new schemes. \end{slogan} In Lemma \ref{lemma-glue} above, assume that all $X_i$ are schemes. Then the resulting locally ringed space $X$ is a scheme.


schemes-lemma-glue-functors

Let $F$ be a contravariant functor on the category of schemes with values in the category of sets. Suppose that \begin{enumerate} \item $F$ satisfies the sheaf property for the Zariski topology, \item there exists a set $I$ and a collection of subfunctors $F_i \subset F$ such that \begin{enumerate} \item each $F_i$ is representable, \item each $F_i \subset F$ is representable by open immersions, and \item the collection $(F_i)_{i \in I}$ covers $F$. \end{enumerate} \end{enumerate} Then $F$ is representable.


schemes-lemma-fibre-product-affines

Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. If $X, Y, S$ are all affine then $X \times_S Y$ is affine.


schemes-lemma-open-fibre-product

Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. Let $X \times_S Y$, $p$, $q$ be the fibre product. Suppose that $U \subset S$, $V \subset X$, $W \subset Y$ are open subschemes such that $f(V) \subset U$ and $g(W) \subset U$. Then the canonical morphism $V \times_U W \to X \times_S Y$ is an open immersion which identifies $V \times_U W$ with $p^{-1}(V) \cap q^{-1}(W)$.


schemes-lemma-affine-covering-fibre-product

\begin{slogan} Bare-hands construction of fiber products: an affine open cover of a fiber product of schemes can be assembled from compatible affine open covers of the pieces. \end{slogan} Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. Let $S = \bigcup U_i$ be any affine open covering of $S$. For each $i \in I$, let $f^{-1}(U_i) = \bigcup_{j \in J_i} V_j$ be an affine open covering of $f^{-1}(U_i)$ and let $g^{-1}(U_i) = \bigcup_{k \in K_i} W_k$ be an affine open covering of $g^{-1}(U_i)$. Then $$ X \times_S Y = \bigcup\nolimits_{i \in I} \bigcup\nolimits_{j \in J_i, \ k \in K_i} V_j \times_{U_i} W_k $$ is an affine open covering of $X \times_S Y$.


schemes-lemma-points-fibre-product

Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. Points $z$ of $X \times_S Y$ are in bijective correspondence to quadruples $$ (x, y, s, \mathfrak p) $$ where $x \in X$, $y \in Y$, $s \in S$ are points with $f(x) = s$, $g(y) = s$ and $\mathfrak p$ is a prime ideal of the ring $\kappa(x) \otimes_{\kappa(s)} \kappa(y)$. The residue field of $z$ corresponds to the residue field of the prime $\mathfrak p$.


schemes-lemma-fibre-product-immersion

Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. \begin{enumerate} \item If $f : X \to S$ is a closed immersion, then $X \times_S Y \to Y$ is a closed immersion. Moreover, if $X \to S$ corresponds to the quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_S$, then $X \times_S Y \to Y$ corresponds to the sheaf of ideals $\Im(g^*\mathcal{I} \to \mathcal{O}_Y)$. \item If $f : X \to S$ is an open immersion, then $X \times_S Y \to Y$ is an open immersion. \item If $f : X \to S$ is an immersion, then $X \times_S Y \to Y$ is an immersion. \end{enumerate}


schemes-lemma-base-change-immersion

Let $S$ be a scheme. Let $f : X \to Y$ be an immersion (resp.\ closed immersion, resp. open immersion) of schemes over $S$. Then any base change of $f$ is an immersion (resp.\ closed immersion, resp. open immersion).


schemes-lemma-fibre-topological

Let $f : X \to S$ be a morphism of schemes. Consider the diagrams $$ \xymatrix{ X_s \ar[r] \ar[d] & X \ar[d] & \Spec(\mathcal{O}_{S, s}) \times_S X \ar[r] \ar[d] & X \ar[d] \\ \Spec(\kappa(s)) \ar[r] & S & \Spec(\mathcal{O}_{S, s}) \ar[r] & S } $$ In both cases the top horizontal arrow is a homeomorphism onto its image.


schemes-lemma-quasi-compact-affine

Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $f : X \to S$ is quasi-compact, \item the inverse image of every affine open is quasi-compact, and \item there exists some affine open covering $S = \bigcup_{i \in I} U_i$ such that $f^{-1}(U_i)$ is quasi-compact for all $i$. \end{enumerate}


schemes-lemma-quasi-compact-preserved-base-change

Being quasi-compact is a property of morphisms of schemes over a base which is preserved under arbitrary base change.


schemes-lemma-composition-quasi-compact

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


schemes-lemma-closed-immersion-quasi-compact

A closed immersion is quasi-compact.


schemes-lemma-image-quasi-compact-closed

Let $f : X \to S$ be a quasi-compact morphism of schemes. The following are equivalent \begin{enumerate} \item $f(X) \subset S$ is closed, and \item $f(X) \subset S$ is stable under specialization. \end{enumerate}


schemes-lemma-quasi-compact-closed

Let $f : X \to S$ be a quasi-compact morphism of schemes. Then $f$ is closed if and only if specializations lift along $f$, see Topology, Definition \ref{topology-definition-lift-specializations}.


schemes-lemma-specializations-lift

Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item If $f$ is universally closed then specializations lift along any base change of $f$, see Topology, Definition \ref{topology-definition-lift-specializations}. \item If $f$ is quasi-compact and specializations lift along any base change of $f$, then $f$ is universally closed. \end{enumerate}


schemes-lemma-points-specialize

Let $S$ be a scheme. Let $s' \leadsto s$ be a specialization of points of $S$. Then \begin{enumerate} \item there exists a valuation ring $A$ and a morphism $f : \Spec(A) \to S$ such that the generic point $\eta$ of $\Spec(A)$ maps to $s'$ and the special point maps to $s$, and \item given a field extension $\kappa(s') \subset K$ we may arrange it so that the extension $\kappa(s') \subset \kappa(\eta)$ induced by $f$ is isomorphic to the given extension. \end{enumerate}


schemes-lemma-lift-specializations-valuative

Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item Specializations lift along any base change of $f$ \item The morphism $f$ satisfies the existence part of the valuative criterion. \end{enumerate}


schemes-lemma-diagonal-affines-closed

The diagonal morphism of a morphism between affines is closed.


schemes-lemma-diagonal-immersion

\begin{slogan} The diagonal morphism for relative schemes is an immersion. \end{slogan} Let $X$ be a scheme over $S$. The diagonal morphism $\Delta_{X/S}$ is an immersion.


schemes-lemma-where-are-they-equal

Let $X$, $Y$ be schemes over $S$. Let $a, b : X \to Y$ be morphisms of schemes over $S$. There exists a largest locally closed subscheme $Z \subset X$ such that $a|_Z = b|_Z$. In fact $Z$ is the equalizer of $(a, b)$. Moreover, if $Y$ is separated over $S$, then $Z$ is a closed subscheme.


schemes-lemma-characterize-quasi-separated

Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item The morphism $f$ is quasi-separated. \item For every pair of affine opens $U, V \subset X$ which map into a common affine open of $S$ the intersection $U \cap V$ is a finite union of affine opens of $X$. \item There exists an affine open covering $S = \bigcup_{i \in I} U_i$ and for each $i$ an affine open covering $f^{-1}U_i = \bigcup_{j \in I_i} V_j$ such that for each $i$ and each pair $j, j' \in I_i$ the intersection $V_j \cap V_{j'}$ is a finite union of affine opens of $X$. \end{enumerate}


schemes-lemma-characterize-separated

Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item If $f$ is separated then for every pair of affine opens $(U, V)$ of $X$ which map into a common affine open of $S$ we have \begin{enumerate} \item the intersection $U \cap V$ is affine. \item the ring map $\mathcal{O}_X(U) \otimes_{\mathbf{Z}} \mathcal{O}_X(V) \to \mathcal{O}_X(U \cap V)$ is surjective. \end{enumerate} \item If any pair of points $x_1, x_2 \in X$ lying over a common point $s \in S$ are contained in affine opens $x_1 \in U$, $x_2 \in V$ which map into a common affine open of $S$ such that (a), (b) hold, then $f$ is separated. \end{enumerate}


schemes-lemma-fibre-product-after-map

Let $f : X \to T$ and $g : Y \to T$ be morphisms of schemes with the same target. Let $h : T \to S$ be a morphism of schemes. Then the induced morphism $i : X \times_T Y \to X \times_S Y$ is an immersion. If $T \to S$ is separated, then $i$ is a closed immersion. If $T \to S$ is quasi-separated, then $i$ is a quasi-compact morphism.


schemes-lemma-semi-diagonal

Let $g : X \to Y$ be a morphism of schemes over $S$. The morphism $i : X \to X \times_S Y$ is an immersion. If $Y$ is separated over $S$ it is a closed immersion. If $Y$ is quasi-separated over $S$ it is quasi-compact.


schemes-lemma-separated-permanence

Permanence properties. \begin{enumerate} \item A composition of separated morphisms is separated. \item A composition of quasi-separated morphisms is quasi-separated. \item The base change of a separated morphism is separated. \item The base change of a quasi-separated morphism is quasi-separated. \item A (fibre) product of separated morphisms is separated. \item A (fibre) product of quasi-separated morphisms is quasi-separated. \end{enumerate}


schemes-lemma-compose-after-separated

\begin{slogan} Separated and quasi-separated morphisms satisfy cancellation. \end{slogan} Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes. If $g \circ f$ is separated then so is $f$. If $g \circ f$ is quasi-separated then so is $f$.


schemes-lemma-quasi-compact-permanence

Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes. If $g \circ f$ is quasi-compact and $g$ is quasi-separated then $f$ is quasi-compact.


schemes-lemma-affine-separated

An affine scheme is separated. A morphism from an affine scheme to another scheme is separated.


schemes-lemma-separated-implies-valuative

Let $f : X \to S$ be a morphism of schemes. If $f$ is separated, then $f$ satisfies the uniqueness part of the valuative criterion.


schemes-lemma-valuative-criterion-separatedness

\begin{reference} \cite[II Proposition 7.2.3]{EGA} \end{reference} Let $f : X \to S$ be a morphism. Assume \begin{enumerate} \item the morphism $f$ is quasi-separated, and \item the morphism $f$ satisfies the uniqueness part of the valuative criterion. \end{enumerate} Then $f$ is separated.


schemes-lemma-monomorphism

\begin{slogan} A scheme morphism is a monomorphism iff its diagonal is an isomorphism. \end{slogan} Let $j : X \to Y$ be a morphism of schemes. Then $j$ is a monomorphism if and only if the diagonal morphism $\Delta_{X/Y} : X \to X \times_Y X$ is an isomorphism.


schemes-lemma-monomorphism-separated

A monomorphism of schemes is separated.


schemes-lemma-composition-monomorphism

A composition of monomorphisms is a monomorphism.


schemes-lemma-base-change-monomorphism

The base change of a monomorphism is a monomorphism.


schemes-lemma-injective-points-surjective-stalks

Let $j : X \to Y$ be a morphism of schemes. If \begin{enumerate} \item $j$ is injective on points, and \item for any $x \in X$ the ring map $j^\sharp_x : \mathcal{O}_{Y, j(x)} \to \mathcal{O}_{X, x}$ is surjective, \end{enumerate} then $j$ is a monomorphism.


schemes-lemma-immersions-monomorphisms

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


schemes-lemma-subscheme-of-separated-scheme

Let $f : X \to S$ be a separated morphism. Any locally closed subscheme $Z \subset X$ is separated over $S$.


schemes-lemma-mono-towards-spec-field

Let $k_1, \ldots, k_n$ be fields. For any monomorphism of schemes $X \to \Spec(k_1 \times \ldots \times k_n)$ there exists a subset $I \subset \{1, \ldots, n\}$ such that $X \cong \Spec(\prod_{i \in I} k_i)$ as schemes over $\Spec(k_1 \times \ldots \times k_n)$. More generally, if $X = \coprod_{i \in I} \Spec(k_i)$ is a disjoint union of spectra of fields and $Y \to X$ is a monomorphism, then there exists a subset $J \subset I$ such that $Y = \coprod_{i \in J} \Spec(k_i)$.


schemes-lemma-push-forward-quasi-coherent

Let $f : X \to S$ be a morphism of schemes. If $f$ is quasi-compact and quasi-separated then $f_*$ transforms quasi-coherent $\mathcal{O}_X$-modules into quasi-coherent $\mathcal{O}_S$-modules.


schemes-lemma-characterize-closed-immersions

Let $f : X \to Y$ be a morphism of schemes. Suppose that \begin{enumerate} \item $f$ induces a homeomorphism of $X$ with a closed subset of $Y$, and \item $f^\sharp : \mathcal{O}_Y \to f_*\mathcal{O}_X$ is surjective. \end{enumerate} Then $f$ is a closed immersion of schemes.


schemes-lemma-composition-immersion

A composition of immersions of schemes is an immersion, a composition of closed immersions of schemes is a closed immersion, and a composition of open immersions of schemes is an open immersion.


schemes-proposition-characterize-universally-closed

Let $f$ be a quasi-compact morphism of schemes. Then $f$ is universally closed if and only if $f$ satisfies the existence part of the valuative criterion.


injectives-theorem-baer-grothendieck

Let $\kappa$ be the cardinality of the set of ideals in $R$, and let $\alpha$ be an ordinal whose cofinality is greater than $\kappa$. Then $\mathbf{M}_\alpha(N)$ is an injective $R$-module, and $N \to \mathbf{M}_\alpha(N)$ is a functorial injective embedding.


injectives-theorem-sheaves-injectives

The category of sheaves of abelian groups on a site has enough injectives. In fact there exists a functorial injective embedding, see Homology, Definition \ref{homology-definition-functorial-injective-embedding}.


injectives-theorem-sheaves-modules-injectives

Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. The category of sheaves of $\mathcal{O}$-modules on a site has enough injectives. In fact there exists a functorial injective embedding, see Homology, Definition \ref{homology-definition-functorial-injective-embedding}.


injectives-theorem-injective-embedding-grothendieck

Let $\mathcal{A}$ be a Grothendieck abelian category. Then $\mathcal{A}$ has functorial injective embeddings.


injectives-theorem-K-injective-embedding-grothendieck

\begin{slogan} Existence of K-injective complexes for Grothendieck abelian categories. \end{slogan} Let $\mathcal{A}$ be a Grothendieck abelian category. For every complex $M^\bullet$ there exists a quasi-isomorphism $M^\bullet \to I^\bullet$ such that $M^n \to I^n$ is injective and $I^n$ is an injective object of $\mathcal{A}$ for all $n$ and $I^\bullet$ is a K-injective complex. Moreover, the construction is functorial in $M^\bullet$.


injectives-theorem-gabriel-popescu

Let $\mathcal{A}$ be a Grothendieck abelian category. Then there exists a (noncommutative) ring $R$ and functors $G : \mathcal{A} \to \text{Mod}_R$ and $F : \text{Mod}_R \to \mathcal{A}$ such that \begin{enumerate} \item $F$ is the left adjoint to $G$, \item $G$ is fully faithful, and \item $F$ is exact. \end{enumerate} Moreover, the functors are the ones constructed above.


injectives-lemma-out-of-finite

Suppose that, in (\ref{equation-compare}), $\mathcal{C}$ is the category of sets and $A$ is a {\it finite set}, then the map is a bijection.


injectives-lemma-criterion-baer

\begin{reference} \cite[Theorem 1]{Baer} \end{reference} Let $R$ be a ring. An $R$-module $Q$ is injective if and only if in every commutative diagram $$ \xymatrix{ \mathfrak{a} \ar[d] \ar[r] &  Q \\ R \ar@{-->}[ru] } $$ for $\mathfrak{a} \subset R$ an ideal, the dotted arrow exists.


injectives-lemma-construction

Let $R$ be a ring. \begin{enumerate} \item The construction $M \mapsto (M \to \mathbf{M}(M))$ is functorial in $M$. \item The map $M \to \mathbf{M}(M)$ is injective. \item For any ideal $\mathfrak{a}$ and any $R$-module map $\varphi : \mathfrak a \to M$ there is an $R$-module map $\varphi' : R \to \mathbf{M}(M)$ such that $$ \xymatrix{ \mathfrak{a} \ar[d] \ar[r]_\varphi &  M \ar[d] \\ R \ar[r]^{\varphi'} & \mathbf{M}(M) } $$ commutes. \end{enumerate}


injectives-lemma-abelian-sheaves-space

Let $X$ be a topological space. The category of abelian sheaves on $X$ has enough injectives. In fact it has functorial injective embeddings.


injectives-lemma-sheaves-modules-space

Let $(X, \mathcal{O}_X)$ be a ringed space, see Sheaves, Section \ref{sheaves-section-ringed-spaces}. The category of sheaves of $\mathcal{O}_X$-modules on $X$ has enough injectives. In fact it has functorial injective embeddings.


injectives-lemma-map-into-next-one

With notation as above. Suppose that $\mathcal{G}_1 \to \mathcal{G}_2$ is an injective map of abelian sheaves on $\mathcal{C}$. Let $\alpha$ be an ordinal and let $\mathcal{G}_1 \to J_\alpha(\mathcal{F})$ be a morphism of sheaves. There exists a morphism $\mathcal{G}_2 \to J_{\alpha + 1}(\mathcal{F})$ such that the following diagram commutes $$ \xymatrix{ \mathcal{G}_1 \ar[d] \ar[r] & \mathcal{G}_2 \ar[d] \\ J_{\alpha}(\mathcal{F}) \ar[r] & J_{\alpha + 1}(\mathcal{F}) } $$


injectives-lemma-map-into-smaller

Suppose that $\mathcal{G}_i$, $i\in I$ is set of abelian sheaves on $\mathcal{C}$. There exists an ordinal $\beta$ such that for any sheaf $\mathcal{F}$, any $i\in I$, and any map $\varphi : \mathcal{G}_i \to J_\beta(\mathcal{F})$ there exists an $\alpha < \beta$ such that $ \varphi $ factors through $J_\alpha(\mathcal{F})$.


injectives-lemma-characterize-injectives

Suppose $\mathcal{J}$ is a sheaf of abelian groups with the following property: For all $X\in \Ob(\mathcal{C})$, for any abelian subsheaf $\mathcal{S} \subset \mathbf{Z}_X^\#$ and any morphism $\varphi : \mathcal{S} \to \mathcal{J}$, there exists a morphism $\mathbf{Z}_X^\# \to \mathcal{J}$ extending $\varphi$. Then $\mathcal{J}$ is an injective sheaf of abelian groups.


injectives-lemma-vee-exact-sheaves

The functor $\mathcal{F} \mapsto \mathcal{F}^\vee$ is exact.


injectives-lemma-ev-injective-sheaves

For any $\mathcal{O}$-module $\mathcal{F}$ the evaluation map $ev : \mathcal{F} \to (\mathcal{F}^\vee)^\vee$ is injective.


injectives-lemma-JM-injective-sheaves

Let $\mathcal{O}$ be a sheaf of rings. For every $\mathcal{O}$-module $\mathcal{F}$ the $\mathcal{O}$-module $J(\mathcal{F})$ is injective.


injectives-lemma-site-abelian-category

Let $\mathcal{A}$ be an abelian category. Let $$ \text{Cov} = \{\{f : V \to U\} \mid f\text{ is surjective}\}. $$ Then $(\mathcal{A}, \text{Cov})$ is a site, see Sites, Definition \ref{sites-definition-site}.


injectives-lemma-embedding

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{C} = (\mathcal{A}, \text{Cov})$ be the site defined in Lemma \ref{lemma-site-abelian-category}. Then $X \mapsto h_X$ defines a fully faithful, exact functor $$ \mathcal{A} \longrightarrow \textit{Ab}(\mathcal{C}). $$ Moreover, the site $\mathcal{C}$ has enough points.


injectives-lemma-set-of-subobjects

Let $\mathcal{A}$ be an abelian category with a generator $U$ and $X$ and object of $\mathcal{A}$. If $\kappa$ is the cardinality of $\Mor(U, X)$ then \begin{enumerate} \item There does not exist a strictly increasing (or strictly decreasing) chain of subobjects of $X$ indexed by a cardinal bigger than $\kappa$. \item If $\alpha$ is an ordinal of cofinality $> \kappa$ then any increasing (or decreasing) sequence of subobjects of $X$ indexed by $\alpha$ is eventually constant. \item The cardinality of the set of subobjects of $X$ is $\leq 2^\kappa$. \end{enumerate}


injectives-lemma-size-goes-down

Let $\mathcal{A}$ be a Grothendieck abelian category. If $0 \to M' \to M \to M'' \to 0$ is a short exact sequence of $\mathcal{A}$, then $|M'|, |M''| \leq |M|$.


injectives-lemma-set-iso-classes-bounded-size

Let $\mathcal{A}$ be a Grothendieck abelian category with generator $U$. \begin{enumerate} \item If $|M| \leq \kappa$, then $M$ is the quotient of a direct sum of at most $\kappa$ copies of $U$. \item For every cardinal $\kappa$ there exists a set of isomorphism classes of objects $M$ with $|M| \leq \kappa$. \end{enumerate}


injectives-lemma-characterize-injective

\begin{slogan} To check that an object is injective, one only needs to check that lifting holds for subobjects of a generator. \end{slogan} Let $\mathcal{A}$ be a Grothendieck abelian category with generator $U$. An object $I$ of $\mathcal{A}$ is injective if and only if in every commutative diagram $$ \xymatrix{ M \ar[d] \ar[r] &  I \\ U \ar@{-->}[ru] } $$ for $M \subset U$ a subobject, the dotted arrow exists.


injectives-lemma-surjection-bounded-size

Let $\mathcal{A}$ be a Grothendieck abelian category with generator $U$. Let $c$ be the function on cardinals defined by $c(\kappa) = |\bigoplus_{\alpha \in \kappa} U|$. If $\pi : M \to N$ is a surjection then there exists a subobject $M' \subset M$ which surjects onto $N$ with $|N'| \leq c(|N|)$.


injectives-lemma-acyclic-quotient-complexes-bounded-size

Let $\mathcal{A}$ be a Grothendieck abelian category. There exists a cardinal $\kappa$ such that given any acyclic complex $M^\bullet$ we have \begin{enumerate} \item if $M^\bullet$ is nonzero, there is a nonzero subcomplex $N^\bullet$ which is bounded above, acyclic, and $|N^n| \leq \kappa$, \item there exists a surjection of complexes $$ \bigoplus\nolimits_{i \in I} M_i^\bullet \longrightarrow M^\bullet $$ where $M_i^\bullet$ is bounded above, acyclic, and $|M_i^n| \leq \kappa$. \end{enumerate}


injectives-lemma-characterize-K-injective

Let $\mathcal{A}$ be a Grothendieck abelian category. Let $\kappa$ be a cardinal as in Lemma \ref{lemma-acyclic-quotient-complexes-bounded-size}. Suppose that $I^\bullet$ is a complex such that \begin{enumerate} \item each $I^j$ is injective, and \item for every bounded above acyclic complex $M^\bullet$ such that $|M^n| \leq \kappa$ we have $\Hom_{K(\mathcal{A})}(M^\bullet, I^\bullet) = 0$. \end{enumerate} Then $I^\bullet$ is an $K$-injective complex.


injectives-lemma-functorial-homotopies

Let $\mathcal{A}$ be a Grothendieck abelian category. Let $(K_i^\bullet)_{i \in I}$ be a set of acyclic complexes. There exists a functor $M^\bullet \mapsto \mathbf{M}^\bullet(M^\bullet)$ and a natural transformation $j_{M^\bullet} : M^\bullet \to \mathbf{M}^\bullet(M^\bullet)$ such \begin{enumerate} \item $j_{M^\bullet}$ is a (termwise) injective quasi-isomorphism, and \item for every $i \in I$ and $w : K_i^\bullet \to M^\bullet$ the morphism $j_{M^\bullet} \circ w$ is homotopic to zero. \end{enumerate}


injectives-lemma-functorial-injective

Let $\mathcal{A}$ be a Grothendieck abelian category. There exists a functor $M^\bullet \mapsto \mathbf{N}^\bullet(M^\bullet)$ and a natural transformation $j_{M^\bullet} : M^\bullet \to \mathbf{N}^\bullet(M^\bullet)$ such \begin{enumerate} \item $j_{M^\bullet}$ is a (termwise) injective quasi-isomorphism, and \item for every $n \in \mathbf{Z}$ the map $M^n \to \mathbf{N}^n(M^\bullet)$ factors through a subobject $I^n \subset \mathbf{N}^n(M^\bullet)$ where $I^n$ is an injective object of $\mathcal{A}$. \end{enumerate}


injectives-lemma-grothendieck-brown

Let $\mathcal{A}$ be a Grothendieck abelian category. Let $F : \mathcal{A}^{opp} \to \textit{Sets}$ be a functor. Then $F$ is representable if and only if $F$ commutes with colimits, i.e., $$ F(\colim_i N_i) = \lim F(N_i) $$ for any diagram $\mathcal{I} \to \mathcal{A}$, $i \in \mathcal{I}$.


injectives-lemma-grothendieck-products

A Grothendieck abelian category has Ab3*.


injectives-lemma-derived-products

Let $\mathcal{A}$ be a Grothendieck abelian category. Then \begin{enumerate} \item $D(\mathcal{A})$ has both direct sums and products, \item direct sums are obtained by taking termwise direct sums of any complexes, \item products are obtained by taking termwise products of K-injective complexes. \end{enumerate}


injectives-lemma-RF-commutes-with-Rlim

Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor of abelian categories. Assume \begin{enumerate} \item $\mathcal{A}$ is a Grothendieck abelian category, \item $\mathcal{B}$ has exact countable products, and \item $F$ commutes with countable products. \end{enumerate} Then $RF : D(\mathcal{A}) \to D(\mathcal{B})$ commutes with derived limits.


injectives-lemma-K-injective-embedding-filtration

Let $\mathcal{A}$ be a Grothendieck abelian category. Let $K^\bullet$ be a filtered complex of $\mathcal{A}$, see Homology, Definition \ref{homology-definition-filtered-complex}. Then there exists a morphism $j : K^\bullet \to J^\bullet$ of filtered complexes of $\mathcal{A}$ such that \begin{enumerate} \item $J^n$, $F^pJ^n$, $J^n/F^pJ^n$ and $F^pJ^n/F^{p'}J^n$ are injective objects of $\mathcal{A}$, \item $J^\bullet$, $F^pJ^\bullet$, $J^\bullet/F^pJ^\bullet$, and $F^pJ^\bullet/F^{p'}J^\bullet$ are K-injective complexes, \item $j$ induces quasi-isomorphisms $K^\bullet \to J^\bullet$, $F^pK^\bullet \to F^pJ^\bullet$, $K^\bullet/F^pK^\bullet \to J^\bullet/F^pJ^\bullet$, and $F^pK^\bullet/F^{p'}K^\bullet \to F^pJ^\bullet/F^{p'}J^\bullet$. \end{enumerate}


injectives-lemma-represent-by-filtered-complex

Let $\mathcal{A}$ be a Grothendieck abelian category. Suppose given an object $E \in D(\mathcal{A})$ and an inverse system $\{E^i\}_{i \in \mathbf{Z}}$ of objects of $D(\mathcal{A})$ over $\mathbf{Z}$ together with a compatible system of maps $E^i \to E$. Picture: $$ \ldots \to E^{i + 1} \to E^i \to E^{i - 1} \to \ldots \to E $$ Then there exists a filtered complex $K^\bullet$ of $\mathcal{A}$ (Homology, Definition \ref{homology-definition-filtered-complex}) such that $K^\bullet$ represents $E$ and $F^iK^\bullet$ represents $E^i$ compatibly with the given maps.


injectives-lemma-gabriel-popescu-left-adjoint

The functor $G$ above has a left adjoint $F : \text{Mod}_R \to \mathcal{A}$.


injectives-lemma-F-G-monos

Let $f : M \to G(A)$ be an injective map in $\text{Mod}_R$. Then the adjoint map $f' : F(M) \to A$ is injective too.


injectives-lemma-gabriel-popescu

\begin{reference} \cite[Corollary 4.1]{serpe} \end{reference} Let $\mathcal{A}$ be a Grothendieck abelian category. Let $R$, $F$, $G$ be as in the Gabriel-Popescu theorem (Theorem \ref{theorem-gabriel-popescu}). Then we obtain derived functors $$ RG : D(\mathcal{A}) \to D(\text{Mod}_R) \quad\text{and}\quad F : D(\text{Mod}_R) \to D(\mathcal{A}) $$ such that $F$ is left adjoint to $RG$, $RG$ is fully faithful, and $F \circ RG = \text{id}$.


injectives-lemma-brown

Let $\mathcal{A}$ be a Grothendieck abelian category. Let $H : D(\mathcal{A}) \to \textit{Ab}$ be a contravariant cohomological functor which transforms direct sums into products. Then $H$ is representable.


injectives-proposition-modules-are-small

Let $R$ be a ring. Let $M$ be an $R$-module. Let $\kappa$ the cardinality of the set of submodules of $M$. If $\alpha$ is an ordinal whose cofinality is bigger than $\kappa$, then $M$ is $\alpha$-small with respect to injections.


injectives-proposition-presheaves-injectives

For abelian presheaves on a category there is a functorial injective embedding.


injectives-proposition-presheaves-modules

Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$. The category $\textit{PMod}(\mathcal{O})$ of presheaves of $\mathcal{O}$-modules has functorial injective embeddings.


injectives-proposition-objects-are-small

Let $\mathcal{A}$ be a Grothendieck abelian category. Let $M$ be an object of $\mathcal{A}$. Let $\kappa = |M|$. If $\alpha$ is an ordinal whose cofinality is bigger than $\kappa$, then $M$ is $\alpha$-small with respect to injections.


injectives-proposition-brown

Let $\mathcal{A}$ be a Grothendieck abelian category. Let $\mathcal{D}$ be a triangulated category. Let $F : D(\mathcal{A}) \to \mathcal{D}$ be an exact functor of triangulated categories which transforms direct sums into direct sums. Then $F$ has an exact right adjoint.


brauer-theorem-wedderburn

\begin{slogan} Simple finite algebras over a field are matrix algebras over a skew field. \end{slogan} Let $A$ be a simple finite $k$-algebra. Then $A$ is a matrix algebra over a finite $k$-algebra $K$ which is a skew field.


brauer-theorem-skolem-noether

Let $A$ be a finite central simple $k$-algebra. Let $B$ be a simple $k$-algebra. Let $f, g : B \to A$ be two $k$-algebra homomorphisms. Then there exists an invertible element $x \in A$ such that $f(b) = xg(b)x^{-1}$ for all $b \in B$.


brauer-theorem-centralizer

Let $A$ be a finite central simple algebra over $k$, and let $B$ be a simple subalgebra of $A$. Then \begin{enumerate} \item the centralizer $C$ of $B$ in $A$ is simple, \item $[A : k] = [B : k][C : k]$, and \item the centralizer of $C$ in $A$ is $B$. \end{enumerate}


brauer-theorem-splitting

Let $A$ be a finite central simple $k$-algebra. Let $k \subset k'$ be a finite field extension. The following are equivalent \begin{enumerate} \item $k'$ splits $A$, and \item there exists a finite central simple algebra $B$ similar to $A$ such that $k' \subset B$ and $[B : k] = [k' : k]^2$. \end{enumerate}


brauer-lemma-rieffel

Let $A$ be a possibly noncommutative ring with $1$ which contains no nontrivial two-sided ideal. Let $M$ be a nonzero right ideal in $A$, and view $M$ as a right $A$-module. Then $A$ coincides with the bicommutant of $M$.


brauer-lemma-simple-module

Let $A$ be a $k$-algebra. If $A$ is finite, then \begin{enumerate} \item $A$ has a simple module, \item any nonzero module contains a simple submodule, \item a simple module over $A$ has finite dimension over $k$, and \item if $M$ is a simple $A$-module, then $\text{End}_A(M)$ is a skew field. \end{enumerate}


brauer-lemma-centralizer

Let $A$, $A'$ be $k$-algebras. Let $B \subset A$, $B' \subset A'$ be subalgebras with centralizers $C$, $C'$. Then the centralizer of $B \otimes_k B'$ in $A \otimes_k A'$ is $C \otimes_k C'$.


brauer-lemma-center-csa

Let $A$ be a finite simple $k$-algebra. Then the center $k'$ of $A$ is a finite field extension of $k$.


brauer-lemma-generate-two-sided-sub

Let $V$ be a $k$ vector space. Let $K$ be a central $k$-algebra which is a skew field. Let $W \subset V \otimes_k K$ be a two-sided $K$-sub vector space. Then $W$ is generated as a left $K$-vector space by $W \cap (V \otimes 1)$.


brauer-lemma-generate-two-sided-ideal

Let $A$ be a $k$-algebra. Let $K$ be a central $k$-algebra which is a skew field. Then any two-sided ideal $I \subset A \otimes_k K$ is of the form $J \otimes_k K$ for some two-sided ideal $J \subset A$. In particular, if $A$ is simple, then so is $A \otimes_k K$.


brauer-lemma-matrix-algebras

Let $R$ be a possibly noncommutative ring. Let $n \geq 1$ be an integer. Let $R_n = \text{Mat}(n \times n, R)$. \begin{enumerate} \item The functors $M \mapsto M^{\oplus n}$ and $N \mapsto Ne_{11}$ define quasi-inverse equivalences of categories $\text{Mod}_R \leftrightarrow \text{Mod}_{R_n}$. \item A two-sided ideal of $R_n$ is of the form $IR_n$ for some two-sided ideal $I$ of $R$. \item The center of $R_n$ is equal to the center of $R$. \end{enumerate}


brauer-lemma-simple-module-unique

Let $A$ be a finite simple $k$-algebra. \begin{enumerate} \item There exists exactly one simple $A$-module $M$ up to isomorphism. \item Any finite $A$-module is a direct sum of copies of a simple module. \item Two finite $A$-modules are isomorphic if and only if they have the same dimension over $k$. \item If $A = \text{Mat}(n \times n, K)$ with $K$ a finite skew field extension of $k$, then $M = K^{\oplus n}$ is a simple $A$-module and $\text{End}_A(M) = K^{op}$. \item If $M$ is a simple $A$-module, then $L = \text{End}_A(M)$ is a skew field finite over $k$ acting on the left on $M$, we have $A = \text{End}_L(M)$, and the centers of $A$ and $L$ agree. Also $[A : k] [L : k] = \dim_k(M)^2$. \item For a finite $A$-module $N$ the algebra $B = \text{End}_A(N)$ is a matrix algebra over the skew field $L$ of (5). Moreover $\text{End}_B(N) = A$. \end{enumerate}


brauer-lemma-tensor-simple

Let $A$, $A'$ be two simple $k$-algebras one of which is finite and central over $k$. Then $A \otimes_k A'$ is simple.


brauer-lemma-tensor-central-simple

The tensor product of finite central simple algebras over $k$ is finite, central, and simple.


brauer-lemma-inverse

Let $A$ be a finite central simple algebra over $k$. Then $A \otimes_k A^{op} \cong \text{Mat}(n \times n, k)$ where $n = [A : k]$.


brauer-lemma-similar

Similarity. \begin{enumerate} \item Similarity defines an equivalence relation on the set of isomorphism classes of finite central simple algebras over $k$. \item Every similarity class contains a unique (up to isomorphism) finite central skew field extension of $k$. \item If $A = \text{Mat}(n \times n, K)$ and $B = \text{Mat}(m \times m, K')$ for some finite central skew fields $K$, $K'$ over $k$ then $A$ and $B$ are similar if and only if $K \cong K'$ as $k$-algebras. \end{enumerate}


brauer-lemma-brauer-algebraically-closed

The Brauer group of an algebraically closed field is zero.


brauer-lemma-when-tensor-is-equal

Let $A$ be a finite central simple algebra over $k$, and let $B$ be a simple subalgebra of $A$. If $B$ is a central $k$-algebra, then $A = B \otimes_k C$ where $C$ is the (central simple) centralizer of $B$ in $A$.


brauer-lemma-self-centralizing-subfield

Let $A$ be a finite central simple algebra over $k$. If $K \subset A$ is a subfield, then the following are equivalent \begin{enumerate} \item $[A : k] = [K : k]^2$, \item $K$ is its own centralizer, and \item $K$ is a maximal commutative subring. \end{enumerate}


brauer-lemma-maximal-subfield

\begin{slogan} The dimension of a finite central skew field is the square of the dimension of any maximal subfield. \end{slogan} Let $A$ be a finite central skew field over $k$. Then every maximal subfield $K \subset A$ satisfies $[A : k] = [K : k]^2$.


brauer-lemma-maximal-subfield-splits

A maximal subfield of a finite central skew field $K$ over $k$ is a splitting field for $K$.


brauer-lemma-finite-central-simple-algebra

Let $k$ be a field. For a $k$-algebra $A$ the following are equivalent \begin{enumerate} \item $A$ is finite central simple $k$-algebra, \item $A$ is a finite dimensional $k$-vector space, $k$ is the center of $A$, and $A$ has no nontrivial two-sided ideal, \item there exists $d \geq 1$ such that $A \otimes_k \bar k \cong \text{Mat}(d \times d, \bar k)$, \item there exists $d \geq 1$ such that $A \otimes_k k^{sep} \cong \text{Mat}(d \times d, k^{sep})$, \item there exist $d \geq 1$ and a finite Galois extension $k \subset k'$ such that $A \otimes_k k' \cong \text{Mat}(d \times d, k')$, \item there exist $n \geq 1$ and a finite central skew field $K$ over $k$ such that $A \cong \text{Mat}(n \times n, K)$. \end{enumerate} The integer $d$ is called the {\it degree} of $A$.


brauer-proposition-separable-splitting-field

Consider a finite central skew field $K$ over $k$. There exists a maximal subfield $k \subset k' \subset K$ which is separable over $k$. In particular, every Brauer class has a finite separable spitting field.


divisors-lemma-associated-affine-open

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\Spec(A) = U \subset X$ be an affine open, and set $M = \Gamma(U, \mathcal{F})$. Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime. \begin{enumerate} \item If $\mathfrak p$ is associated to $M$, then $x$ is associated to $\mathcal{F}$. \item If $\mathfrak p$ is finitely generated, then the converse holds as well. \end{enumerate} In particular, if $X$ is locally Noetherian, then the equivalence $$ \mathfrak p \in \text{Ass}(M) \Leftrightarrow x \in \text{Ass}(\mathcal{F}) $$ holds for all pairs $(\mathfrak p, x)$ as above.


divisors-lemma-ass-support

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\text{Ass}(\mathcal{F}) \subset \text{Supp}(\mathcal{F})$.


divisors-lemma-ses-ass

Let $X$ be a scheme. Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ be a short exact sequence of quasi-coherent sheaves on $X$. Then $\text{Ass}(\mathcal{F}_2) \subset \text{Ass}(\mathcal{F}_1) \cup \text{Ass}(\mathcal{F}_3)$ and $\text{Ass}(\mathcal{F}_1) \subset \text{Ass}(\mathcal{F}_2)$.


divisors-lemma-finite-ass

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $\text{Ass}(\mathcal{F}) \cap U$ is finite for every quasi-compact open $U \subset X$.


divisors-lemma-ass-zero

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $$ \mathcal{F} = 0 \Leftrightarrow \text{Ass}(\mathcal{F}) = \emptyset. $$


divisors-lemma-restriction-injective-open-contains-ass

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $U \subset X$ is open and $\text{Ass}(\mathcal{F}) \subset U$, then $\Gamma(X, \mathcal{F}) \to \Gamma(U, \mathcal{F})$ is injective.


divisors-lemma-minimal-support-in-ass

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in \text{Supp}(\mathcal{F})$ be a point in the support of $\mathcal{F}$ which is not a specialization of another point of $\text{Supp}(\mathcal{F})$. Then $x \in \text{Ass}(\mathcal{F})$. In particular, any generic point of an irreducible component of $X$ is an associated point of $X$.


divisors-lemma-check-injective-on-ass

Let $X$ be a locally Noetherian scheme. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of quasi-coherent $\mathcal{O}_X$-modules. Assume that for every $x \in X$ at least one of the following happens \begin{enumerate} \item $\mathcal{F}_x \to \mathcal{G}_x$ is injective, or \item $x \not \in \text{Ass}(\mathcal{F})$. \end{enumerate} Then $\varphi$ is injective.


divisors-lemma-check-isomorphism-via-depth-and-ass

Let $X$ be a locally Noetherian scheme. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of quasi-coherent $\mathcal{O}_X$-modules. Assume $\mathcal{F}$ is coherent and that for every $x \in X$ one of the following happens \begin{enumerate} \item $\mathcal{F}_x \to \mathcal{G}_x$ is an isomorphism, or \item $\text{depth}(\mathcal{F}_x) \geq 2$ and $x \not \in \text{Ass}(\mathcal{G})$. \end{enumerate} Then $\varphi$ is an isomorphism.


divisors-lemma-bourbaki

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$ which is flat over $S$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $S$. Then we have $$ \text{Ass}_X(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}) \supset \bigcup\nolimits_{s \in \text{Ass}_S(\mathcal{G})} \text{Ass}_{X_s}(\mathcal{F}_s) $$ and equality holds if $S$ is locally Noetherian (for the notation $\mathcal{F}_s$ see above).


divisors-lemma-S1-no-embedded

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf on $X$. Then the following are equivalent: \begin{enumerate} \item $\mathcal{F}$ has no embedded associated points, and \item $\mathcal{F}$ has property $(S_1)$. \end{enumerate}


divisors-lemma-noetherian-dim-1-CM-no-embedded-points

Let $X$ be a locally Noetherian scheme of dimension $\leq 1$. The following are equivalent \begin{enumerate} \item $X$ is Cohen-Macaulay, and \item $X$ has no embedded points. \end{enumerate}


divisors-lemma-remove-embedded-points

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf on $X$. The set of coherent subsheaves $$ \{ \mathcal{K} \subset \mathcal{F} \mid \text{Supp}(\mathcal{K})\text{ is nowhere dense in }\text{Supp}(\mathcal{F}) \} $$ has a maximal element $\mathcal{K}$. Setting $\mathcal{F}' = \mathcal{F}/\mathcal{K}$ we have the following \begin{enumerate} \item $\text{Supp}(\mathcal{F}') = \text{Supp}(\mathcal{F})$, \item $\mathcal{F}'$ has no embedded associated points, and \item there exists a dense open $U \subset X$ such that $U \cap \text{Supp}(\mathcal{F})$ is dense in $\text{Supp}(\mathcal{F})$ and $\mathcal{F}'|_U \cong \mathcal{F}|_U$. \end{enumerate}


divisors-lemma-no-embedded-points-endos

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module without embedded associated points. Set $$ \mathcal{I} = \Ker(\mathcal{O}_X \longrightarrow \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{F})). $$ This is a coherent sheaf of ideals which defines a closed subscheme $Z \subset X$ without embedded points. Moreover there exists a coherent sheaf $\mathcal{G}$ on $Z$ such that (a) $\mathcal{F} = (Z \to X)_*\mathcal{G}$, (b) $\mathcal{G}$ has no associated embedded points, and (c) $\text{Supp}(\mathcal{G}) = Z$ (as sets).


divisors-lemma-weakly-associated-affine-open

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\Spec(A) = U \subset X$ be an affine open, and set $M = \Gamma(U, \mathcal{F})$. Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime. The following are equivalent \begin{enumerate} \item $\mathfrak p$ is weakly associated to $M$, and \item $x$ is weakly associated to $\mathcal{F}$. \end{enumerate}


divisors-lemma-weakly-ass-support

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $$ \text{Ass}(\mathcal{F}) \subset \text{WeakAss}(\mathcal{F}) \subset \text{Supp}(\mathcal{F}). $$


divisors-lemma-ses-weakly-ass

Let $X$ be a scheme. Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ be a short exact sequence of quasi-coherent sheaves on $X$. Then $\text{WeakAss}(\mathcal{F}_2) \subset \text{WeakAss}(\mathcal{F}_1) \cup \text{WeakAss}(\mathcal{F}_3)$ and $\text{WeakAss}(\mathcal{F}_1) \subset \text{WeakAss}(\mathcal{F}_2)$.


divisors-lemma-weakly-ass-zero

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $$ \mathcal{F} = (0) \Leftrightarrow \text{WeakAss}(\mathcal{F}) = \emptyset $$


divisors-lemma-restriction-injective-open-contains-weakly-ass

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $\text{WeakAss}(\mathcal{F}) \subset U \subset X$ is open, then $\Gamma(X, \mathcal{F}) \to \Gamma(U, \mathcal{F})$ is injective.


divisors-lemma-ass-weakly-ass

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $\mathfrak m_x$ is a finitely generated ideal of $\mathcal{O}_{X, x}$, then $$ x \in \text{Ass}(\mathcal{F}) \Leftrightarrow x \in \text{WeakAss}(\mathcal{F}). $$ In particular, if $X$ is locally Noetherian, then $\text{Ass}(\mathcal{F}) = \text{WeakAss}(\mathcal{F})$.


divisors-lemma-weakass-pushforward

Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $s \in S$ be a point which is not in the image of $f$. Then $s$ is not weakly associated to $f_*\mathcal{F}$.


divisors-lemma-depth-2-hartog

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Let $j : U \to X$ be an open subscheme such that for $x \in X \setminus U$ we have $\text{depth}(\mathcal{F}_x) \geq 2$. Then $$ \mathcal{F} \longrightarrow j_*(\mathcal{F}|_U) $$ is an isomorphism and consequently $\Gamma(X, \mathcal{F}) \to \Gamma(U, \mathcal{F})$ is an isomorphism too.


divisors-lemma-weakass-reduced

Let $X$ be a reduced scheme. Then the weakly associated points of $X$ are exactly the generic points of the irreducible components of $X$.


divisors-lemma-weakly-ass-reverse-functorial

Let $f : X \to S$ be an affine morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then we have $$ \text{WeakAss}_S(f_*\mathcal{F}) \subset f(\text{WeakAss}_X(\mathcal{F})) $$


divisors-lemma-ass-functorial-equal

Let $f : X \to S$ be an affine morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $X$ is locally Noetherian, then we have $$ f(\text{Ass}_X(\mathcal{F})) = \text{Ass}_S(f_*\mathcal{F}) = \text{WeakAss}_S(f_*\mathcal{F}) = f(\text{WeakAss}_X(\mathcal{F})) $$


divisors-lemma-weakly-associated-finite

Let $f : X \to S$ be a finite morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\text{WeakAss}(f_*\mathcal{F}) = f(\text{WeakAss}(\mathcal{F}))$.


divisors-lemma-weakly-ass-pullback

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_S$-module. Let $x \in X$ with $s = f(x)$. If $f$ is flat at $x$, the point $x$ is a generic point of the fibre $X_s$, and $s \in \text{WeakAss}_S(\mathcal{G})$, then $x \in \text{WeakAss}(f^*\mathcal{G})$.


divisors-lemma-weakly-ass-change-fields

Let $K/k$ be a field extension. Let $X$ be a scheme over $k$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $y \in X_K$ with image $x \in X$. If $y$ is a weakly associated point of the pullback $\mathcal{F}_K$, then $x$ is a weakly associated point of $\mathcal{F}$.


divisors-lemma-depth-pushforward

Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $s \in S$. \begin{enumerate} \item If $s \not \in f(X)$, then $s$ is not weakly associated to $f_*\mathcal{F}$. \item If $s \not \in f(X)$ and $\mathcal{O}_{S, s}$ is Noetherian, then $s$ is not associated to $f_*\mathcal{F}$. \item If $s \not \in f(X)$, $(f_*\mathcal{F})_s$ is a finite $\mathcal{O}_{S, s}$-module, and $\mathcal{O}_{S, s}$ is Noetherian, then $\text{depth}((f_*\mathcal{F})_s) \geq 2$. \item If $\mathcal{F}$ is flat over $S$ and $a \in \mathfrak m_s$ is a nonzerodivisor, then $a$ is a nonzerodivisor on $(f_*\mathcal{F})_s$. \item If $\mathcal{F}$ is flat over $S$ and $a, b \in \mathfrak m_s$ is a regular sequence, then $a$ is a nonzerodivisor on $(f_*\mathcal{F})_s$ and $b$ is a nonzerodivisor on $(f_*\mathcal{F})_s/a(f_*\mathcal{F})_s$. \item If $\mathcal{F}$ is flat over $S$ and $(f_*\mathcal{F})_s$ is a finite $\mathcal{O}_{S, s}$-module, then $\text{depth}((f_*\mathcal{F})_s) \geq \min(2, \text{depth}(\mathcal{O}_{S, s}))$. \end{enumerate}


divisors-lemma-base-change-relative-assassin

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $g : S' \to S$ be a morphism of schemes. Consider the base change diagram $$ \xymatrix{ X' \ar[d] \ar[r]_{g'} & X \ar[d] \\ S' \ar[r]^g & S } $$ and set $\mathcal{F}' = (g')^*\mathcal{F}$. Let $x' \in X'$ be a point with images $x \in X$, $s' \in S'$ and $s \in S$. Assume $f$ locally of finite type. Then $x' \in \text{Ass}_{X'/S'}(\mathcal{F}')$ if and only if $x \in \text{Ass}_{X/S}(\mathcal{F})$ and $x'$ corresponds to a generic point of an irreducible component of $\Spec(\kappa(s') \otimes_{\kappa(s)} \kappa(x))$.


divisors-lemma-relative-weak-assassin-finite

Let $f : X \to S$ be a morphism of schemes. Let $i : Z \to X$ be a finite morphism. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_Z$-module. Then $\text{WeakAss}_{X/S}(i_*\mathcal{F}) = i(\text{WeakAss}_{Z/S}(\mathcal{F}))$.


divisors-lemma-base-change-fitting-ideal

Let $f : T \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_S$-module. Then $f^{-1}\text{Fit}_i(\mathcal{F}) \cdot \mathcal{O}_T = \text{Fit}_i(f^*\mathcal{F})$.


divisors-lemma-fitting-ideal-of-finitely-presented

Let $S$ be a scheme. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_S$-module. Then $\text{Fit}_r(\mathcal{F})$ is a quasi-coherent ideal of finite type.


divisors-lemma-on-subscheme-cut-out-by-Fit-0

Let $S$ be a scheme. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_S$-module. Let $Z_0 \subset S$ be the closed subscheme cut out by $\text{Fit}_0(\mathcal{F})$. Let $Z \subset S$ be the scheme theoretic support of $\mathcal{F}$. Then \begin{enumerate} \item $Z \subset Z_0 \subset S$ as closed subschemes, \item $Z = Z_0 = \text{Supp}(\mathcal{F})$ as closed subsets, \item there exists a finite type, quasi-coherent $\mathcal{O}_{Z_0}$-module $\mathcal{G}_0$ with $$ (Z_0 \to X)_*\mathcal{G}_0 = \mathcal{F}. $$ \end{enumerate}


divisors-lemma-fitting-ideal-generate-locally

Let $S$ be a scheme. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_S$-module. Let $s \in S$. Then $\mathcal{F}$ can be generated by $r$ elements in a neighbourhood of $s$ if and only if $\text{Fit}_r(\mathcal{F})_s = \mathcal{O}_{S, s}$.


divisors-lemma-fitting-ideal-finite-locally-free

Let $S$ be a scheme. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_S$-module. Let $r \geq 0$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is finite locally free of rank $r$ \item $\text{Fit}_{r - 1}(\mathcal{F}) = 0$ and $\text{Fit}_r(\mathcal{F}) = \mathcal{O}_S$, and \item $\text{Fit}_k(\mathcal{F}) = 0$ for $k < r$ and $\text{Fit}_k(\mathcal{F}) = \mathcal{O}_S$ for $k \geq r$. \end{enumerate}


divisors-lemma-locally-free-rank-r-pullback

Let $S$ be a scheme. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_S$-module. The closed subschemes $$ S = Z_{-1} \supset Z_0 \supset Z_1 \supset Z_2 \ldots $$ defined by the Fitting ideals of $\mathcal{F}$ have the following properties \begin{enumerate} \item The intersection $\bigcap Z_r$ is empty. \item The functor $(\Sch/S)^{opp} \to \textit{Sets}$ defined by the rule $$ T \longmapsto \left\{ \begin{matrix} \{*\} & \text{if }\mathcal{F}_T\text{ is locally generated by } \leq r\text{ sections} \\ \emptyset & \text{otherwise} \end{matrix} \right. $$ is representable by the open subscheme $S \setminus Z_r$. \item The functor $F_r : (\Sch/S)^{opp} \to \textit{Sets}$ defined by the rule $$ T \longmapsto \left\{ \begin{matrix} \{*\} & \text{if }\mathcal{F}_T\text{ locally free rank }r\\ \emptyset & \text{otherwise} \end{matrix} \right. $$ is representable by the locally closed subscheme $Z_{r - 1} \setminus Z_r$ of $S$. \end{enumerate} If $\mathcal{F}$ is of finite presentation, then $Z_r \to S$, $S \setminus Z_r \to S$, and $Z_{r - 1} \setminus Z_r \to S$ are of finite presentation.


divisors-lemma-finite-presentation-module

Let $S$ be a scheme. Let $\mathcal{F}$ be an $\mathcal{O}_S$-module of finite presentation. Let $S = Z_{-1} \supset Z_0 \supset Z_1 \supset \ldots$ be as in Lemma \ref{lemma-locally-free-rank-r-pullback}. Set $S_r = Z_{r - 1} \setminus Z_r$. Then $S' = \coprod_{r \geq 0} S_r$ represents the functor $$ F_{flat} : \Sch/S \longrightarrow \textit{Sets},\quad\quad T \longmapsto \left\{ \begin{matrix} \{*\} & \text{if }\mathcal{F}_T\text{ flat over }T\\ \emptyset & \text{otherwise} \end{matrix} \right. $$ Moreover, $\mathcal{F}|_{S_r}$ is locally free of rank $r$ and the morphisms $S_r \to S$ and $S' \to S$ are of finite presentation.


divisors-lemma-base-change-and-fitting-ideal-omega

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $X = Z_{-1} \supset Z_0 \supset Z_1 \supset \ldots$ be the closed subschemes defined by the fitting ideals of $\Omega_{X/S}$. Then the formation of $Z_i$ commutes with arbitrary base change.


divisors-lemma-d-fitting-ideal-omega-smooth

Let $f : X \to S$ be a morphism of schemes. Let $d \geq 0$ be an integer. Assume \begin{enumerate} \item $f$ is flat, \item $f$ is locally of finite presentation, and \item every nonempty fibre of $f$ is equidimensional of dimension $d$. \end{enumerate} Let $Z \subset X$ be the closed subscheme cut out by the $d$th fitting ideal of $\Omega_{X/S}$. Then $Z$ is exactly the set of points where $f$ is not smooth.


divisors-lemma-torsion-sections

Let $X$ be an integral scheme with generic point $\eta$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $U \subset X$ be nonempty open and $s \in \mathcal{F}(U)$. The following are equivalent \begin{enumerate} \item for some $x \in U$ the image of $s$ in $\mathcal{F}_x$ is torsion, \item for all $x \in U$ the image of $s$ in $\mathcal{F}_x$ is torsion, \item the image of $s$ in $\mathcal{F}_\eta$ is zero, \item the image of $s$ in $j_*\mathcal{F}_\eta$ is zero, where $j : \eta \to X$ is the inclusion morphism. \end{enumerate}


divisors-lemma-check-torsion-on-affines

Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is torsion free, \item for $U \subset X$ affine open $\mathcal{F}(U)$ is a torsion free $\mathcal{O}(U)$-module. \end{enumerate}


divisors-lemma-flat-over-integral-integral-fibre

Let $f : X \to Y$ be a flat morphism of schemes. If $Y$ is integral and the generic fibre of $f$ is integral, then $X$ is integral.


divisors-lemma-torsion-free-finite-noetherian-domain

Let $X$ be a locally Noetherian integral scheme with generic point $\eta$. Let $\mathcal{F}$ be a nonzero coherent $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is torsion free, \item $\eta$ is the only associated prime of $\mathcal{F}$, \item $\eta$ is in the support of $\mathcal{F}$ and $\mathcal{F}$ has property $(S_1)$, and \item $\eta$ is in the support of $\mathcal{F}$ and $\mathcal{F}$ has no embedded associated prime. \end{enumerate}


divisors-lemma-torsion-free-over-regular-dim-1

Let $X$ be an integral regular scheme of dimension $\leq 1$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is torsion free, \item $\mathcal{F}$ is finite locally free. \end{enumerate}


divisors-lemma-hom-into-torsion-free

Let $X$ be an integral scheme. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent $\mathcal{O}_X$-modules. If $\mathcal{G}$ is torsion free and $\mathcal{F}$ is of finite presentation, then $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is torsion free.


divisors-lemma-check-reflexive-on-affines

Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is reflexive, \item for $U \subset X$ affine open $\mathcal{F}(U)$ is a reflexive $\mathcal{O}(U)$-module. \end{enumerate}


divisors-lemma-sequence-reflexive

Let $X$ be an integral locally Noetherian scheme. Let $0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}''$ an exact sequence of coherent $\mathcal{O}_X$-modules. If $\mathcal{F}'$ is reflexive and $\mathcal{F}''$ is torsion free, then $\mathcal{F}$ is reflexive.


divisors-lemma-dual-reflexive

Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. If $\mathcal{G}$ is reflexive, then $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is reflexive.


divisors-lemma-reflexive-S2

Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$ be a coherent reflexive $\mathcal{O}_X$-module. Let $x \in X$. \begin{enumerate} \item If $\text{depth}(\mathcal{O}_{X, x}) \geq 2$, then $\text{depth}(\mathcal{F}_x) \geq 2$. \item If $X$ is $(S_2)$, then $\mathcal{F}$ is $(S_2)$. \end{enumerate}


divisors-lemma-reflexive-S2-extend

Let $X$ be an integral locally Noetherian scheme. Let $j : U \to X$ be an open subscheme with complement $Z$. Assume $\mathcal{O}_{X, z}$ has depth $\geq 2$ for all $z \in Z$. Then $j^*$ and $j_*$ define an equivalence of categories between the category of coherent reflexive $\mathcal{O}_X$-modules and the category of coherent reflexive $\mathcal{O}_U$-modules.


divisors-lemma-reflexive-over-normal

Let $X$ be an integral locally Noetherian normal scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is reflexive, \item $\mathcal{F}$ is torsion free and has property $(S_2)$, and \item there exists an open subscheme $j : U \to X$ such that \begin{enumerate} \item every irreducible component of $X \setminus U$ has codimension $\geq 2$ in $X$, \item $j^*\mathcal{F}$ is finite locally free, and \item $\mathcal{F} = j_*j^*\mathcal{F}$. \end{enumerate} \end{enumerate}


divisors-lemma-describe-reflexive-hull

Let $X$ be an integral locally Noetherian normal scheme with generic point $\eta$. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. Let $T : \mathcal{G}_\eta \to \mathcal{F}_\eta$ be a linear map. Then $T$ extends to a map $\mathcal{G} \to \mathcal{F}^{**}$ of $\mathcal{O}_X$-modules if and only if \begin{itemize} \item[(*)] for every $x \in X$ with $\dim(\mathcal{O}_{X, x}) = 1$ we have $$ T\left(\Im(\mathcal{G}_x \to \mathcal{G}_\eta)\right) \subset \Im(\mathcal{F}_x \to \mathcal{F}_\eta). $$ \end{itemize}


divisors-lemma-reflexive-over-regular-dim-2

Let $X$ be a regular scheme of dimension $\leq 2$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is reflexive, \item $\mathcal{F}$ is finite locally free. \end{enumerate}


divisors-lemma-characterize-effective-Cartier-divisor

Let $S$ be a scheme. Let $D \subset S$ be a closed subscheme. The following are equivalent: \begin{enumerate} \item The subscheme $D$ is an effective Cartier divisor on $S$. \item For every $x \in D$ there exists an affine open neighbourhood $\Spec(A) = U \subset S$ of $x$ such that $U \cap D = \Spec(A/(f))$ with $f \in A$ a nonzerodivisor. \end{enumerate}


divisors-lemma-complement-locally-principal-closed-subscheme

Let $S$ be a scheme. Let $Z \subset S$ be a locally principal closed subscheme. Let $U = S \setminus Z$. Then $U \to S$ is an affine morphism.


divisors-lemma-complement-effective-Cartier-divisor

Let $S$ be a scheme. Let $D \subset S$ be an effective Cartier divisor. Let $U = S \setminus D$. Then $U \to S$ is an affine morphism and $U$ is scheme theoretically dense in $S$.


divisors-lemma-effective-Cartier-makes-dimension-drop

Let $S$ be a scheme. Let $D \subset S$ be an effective Cartier divisor. Let $s \in D$. If $\dim_s(S) < \infty$, then $\dim_s(D) < \dim_s(S)$.


divisors-lemma-sum-effective-Cartier-divisors

The sum of two effective Cartier divisors is an effective Cartier divisor.


divisors-lemma-difference-effective-Cartier-divisors

Let $X$ be a scheme. Let $D, D'$ be two effective Cartier divisors on $X$. If $D \subset D'$ (as closed subschemes of $X$), then there exists an effective Cartier divisor $D''$ such that $D' = D + D''$.


divisors-lemma-sum-closed-subschemes-effective-Cartier

Let $X$ be a scheme. Let $Z, Y$ be two closed subschemes of $X$ with ideal sheaves $\mathcal{I}$ and $\mathcal{J}$. If $\mathcal{I}\mathcal{J}$ defines an effective Cartier divisor $D \subset X$, then $Z$ and $Y$ are effective Cartier divisors and $D = Z + Y$.


divisors-lemma-sum-effective-Cartier-divisors-union

Let $X$ be a scheme. Let $D, D' \subset X$ be effective Cartier divisors such that the scheme theoretic intersection $D \cap D'$ is an effective Cartier divisor on $D'$. Then $D + D'$ is the scheme theoretic union of $D$ and $D'$.


divisors-lemma-pullback-locally-principal

Let $f : S' \to S$ be a morphism of schemes. Let $Z \subset S$ be a locally principal closed subscheme. Then the inverse image $f^{-1}(Z)$ is a locally principal closed subscheme of $S'$.


divisors-lemma-pullback-effective-Cartier-defined

Let $f : X \to Y$ be a morphism of schemes. Let $D \subset Y$ be an effective Cartier divisor. The pullback of $D$ by $f$ is defined in each of the following cases: \begin{enumerate} \item $f(x) \not \in D$ for any weakly associated point $x$ of $X$, \item $X$, $Y$ integral and $f$ dominant, \item $X$ reduced and $f(\xi) \not \in D$ for any generic point $\xi$ of any irreducible component of $X$, \item $X$ is locally Noetherian and $f(x) \not \in D$ for any associated point $x$ of $X$, \item $X$ is locally Noetherian, has no embedded points, and $f(\xi) \not \in D$ for any generic point $\xi$ of an irreducible component of $X$, \item $f$ is flat, and \item add more here as needed. \end{enumerate}


divisors-lemma-pullback-effective-Cartier-divisors-additive

Let $f : S' \to S$ be a morphism of schemes. Let $D_1$, $D_2$ be effective Cartier divisors on $S$. If the pullbacks of $D_1$ and $D_2$ are defined then the pullback of $D = D_1 + D_2$ is defined and $f^*D = f^*D_1 + f^*D_2$.


divisors-lemma-conormal-effective-Cartier-divisor

Let $S$ be a scheme and let $D \subset S$ be an effective Cartier divisor. Then the conormal sheaf is $\mathcal{C}_{D/S} = \mathcal{I}_D|D = \mathcal{O}_S(-D)|_D$ and the normal sheaf is $\mathcal{N}_{D/S} = \mathcal{O}_S(D)|_D$.


divisors-lemma-ses-add-divisor

Let $X$ be a scheme. Let $D, C \subset X$ be effective Cartier divisors with $C \subset D$ and let $D' = D + C$. Then there is a short exact sequence $$ 0 \to \mathcal{O}_X(-D)|_C \to \mathcal{O}_{D'} \to \mathcal{O}_D \to 0 $$ of $\mathcal{O}_X$-modules.


divisors-lemma-invertible-sheaf-sum-effective-Cartier-divisors

Let $S$ be a scheme. Let $D_1$, $D_2$ be effective Cartier divisors on $S$. Let $D = D_1 + D_2$. Then there is a unique isomorphism $$ \mathcal{O}_S(D_1) \otimes_{\mathcal{O}_S} \mathcal{O}_S(D_2) \longrightarrow \mathcal{O}_S(D) $$ which maps $1_{D_1} \otimes 1_{D_2}$ to $1_D$.


divisors-lemma-pullback-effective-Cartier-divisors

Let $f : S' \to S$ be a morphism of schemes. Let $D$ be a effective Cartier divisors on $S$. If the pullback of $D$ is defined then $f^*\mathcal{O}_S(D) = \mathcal{O}_{S'}(f^*D)$ and the canonical section $1_D$ pulls back to the canonical section $1_{f^*D}$.


divisors-lemma-regular-section-structure-sheaf

Let $X$ be a locally ringed space. Let $f \in \Gamma(X, \mathcal{O}_X)$. The following are equivalent: \begin{enumerate} \item $f$ is a regular section, and \item for any $x \in X$ the image $f \in \mathcal{O}_{X, x}$ is a nonzerodivisor. \end{enumerate} If $X$ is a scheme these are also equivalent to \begin{enumerate} \item[(3)] for any affine open $\Spec(A) = U \subset X$ the image $f \in A$ is a nonzerodivisor, \item[(4)] there exists an affine open covering $X = \bigcup \Spec(A_i)$ such that the image of $f$ in $A_i$ is a nonzerodivisor for all $i$. \end{enumerate}


divisors-lemma-zero-scheme

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible sheaf. Let $s \in \Gamma(X, \mathcal{L})$. \begin{enumerate} \item Consider closed immersions $i : Z \to X$ such that $i^*s \in \Gamma(Z, i^*\mathcal{L})$ is zero ordered by inclusion. The zero scheme $Z(s)$ is the maximal element of this ordered set. \item For any morphism of schemes $f : Y \to X$ we have $f^*s = 0$ in $\Gamma(Y, f^*\mathcal{L})$ if and only if $f$ factors through $Z(s)$. \item The zero scheme $Z(s)$ is a locally principal closed subscheme. \item The zero scheme $Z(s)$ is an effective Cartier divisor if and only if $s$ is a regular section of $\mathcal{L}$. \end{enumerate}


divisors-lemma-characterize-OD

\begin{slogan} Effective Cartier divisors on a scheme are the same as invertible sheaves with fixed regular global section. \end{slogan} Let $X$ be a scheme. \begin{enumerate} \item If $D \subset X$ is an effective Cartier divisor, then the canonical section $1_D$ of $\mathcal{O}_X(D)$ is regular. \item Conversely, if $s$ is a regular section of the invertible sheaf $\mathcal{L}$, then there exists a unique effective Cartier divisor $D = Z(s) \subset X$ and a unique isomorphism $\mathcal{O}_X(D) \to \mathcal{L}$ which maps $1_D$ to $s$. \end{enumerate} The constructions $D \mapsto (\mathcal{O}_X(D), 1_D)$ and $(\mathcal{L}, s) \mapsto Z(s)$ give mutually inverse maps $$ \left\{ \begin{matrix} \text{effective Cartier divisors on }X \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{isomorphism classes of pairs }(\mathcal{L}, s)\\ \text{consisting of an invertible } \mathcal{O}_X\text{-module}\\ \mathcal{L}\text{ and a regular global section }s \end{matrix} \right\} $$


divisors-lemma-effective-Cartier-in-points

Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be a closed subscheme corresponding to the quasi-coherent ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. \begin{enumerate} \item If for every $x \in D$ the ideal $\mathcal{I}_x \subset \mathcal{O}_{X, x}$ can be generated by one element, then $D$ is locally principal. \item If for every $x \in D$ the ideal $\mathcal{I}_x \subset \mathcal{O}_{X, x}$ can be generated by a single nonzerodivisor, then $D$ is an effective Cartier divisor. \end{enumerate}


divisors-lemma-effective-Cartier-codimension-1

Let $X$ be a locally Noetherian scheme. \begin{enumerate} \item Let $D \subset X$ be a locally principal closed subscheme. Let $\xi \in D$ be a generic point of an irreducible component of $D$. Then $\dim(\mathcal{O}_{X, \xi}) \leq 1$. \item Let $D \subset X$ be an effective Cartier divisor. Let $\xi \in D$ be a generic point of an irreducible component of $D$. Then $\dim(\mathcal{O}_{X, \xi}) = 1$. \end{enumerate}


divisors-lemma-integral-effective-Cartier-divisor-dvr

Let $X$ be a Noetherian scheme. Let $D \subset X$ be an integral closed subscheme which is also an effective Cartier divisor. Then the local ring of $X$ at the generic point of $D$ is a discrete valuation ring.


divisors-lemma-effective-Cartier-divisor-Sk

Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be an effective Cartier divisor. If $X$ is $(S_k)$, then $D$ is $(S_{k - 1})$.


divisors-lemma-normal-effective-Cartier-divisor-S1

Let $X$ be a locally Noetherian normal scheme. Let $D \subset X$ be an effective Cartier divisor. Then $D$ is $(S_1)$.


divisors-lemma-weil-divisor-is-cartier-UFD

Let $X$ be a Noetherian scheme. Let $D \subset X$ be a integral closed subscheme. Assume that \begin{enumerate} \item $D$ has codimension $1$ in $X$, and \item $\mathcal{O}_{X, x}$ is a UFD for all $x \in D$. \end{enumerate} Then $D$ is an effective Cartier divisor.


divisors-lemma-codim-1-part

Let $X$ be a Noetherian scheme. Let $Z \subset X$ be a closed subscheme. Assume there exist integral effective Cartier divisors $D_i \subset X$ and a closed subset $Z' \subset X$ of codimension $\geq 2$ such that $Z \subset Z' \cup \bigcup D_i$ set-theoretically. Then there exists an effective Cartier divisor of the form $$ D = \sum a_i D_i \subset Z $$ such that $D \to Z$ is an isomorphism away from codimension $2$ in $X$. The existence of the $D_i$ is guaranteed if $\mathcal{O}_{X, x}$ is a UFD for all $x \in Z$ or if $X$ is regular.


divisors-lemma-codimension-1-is-effective-Cartier

Let $Z \subset X$ be a closed subscheme of a Noetherian scheme. Assume \begin{enumerate} \item $Z$ has no embedded points, \item every irreducible component of $Z$ has codimension $1$ in $X$, \item every local ring $\mathcal{O}_{X, x}$, $x \in Z$ is a UFD or $X$ is regular. \end{enumerate} Then $Z$ is an effective Cartier divisor.


divisors-lemma-effective-Cartier-divisor-is-a-sum

Let $X$ be a Noetherian scheme. Let $D \subset X$ be an effective Cartier divisor. Assume that there exist integral effective Cartier divisors $D_i \subset X$ such that $D \subset \bigcup D_i$ set theoretically. Then $D = \sum a_i D_i$ for some $a_i \geq 0$. The existence of the $D_i$ is guaranteed if $\mathcal{O}_{X, x}$ is a UFD for all $x \in D$ or if $X$ is regular.


divisors-lemma-quasi-projective-Noetherian-pic-effective-Cartier

\begin{slogan} On a projective scheme, every line bundle has a regular meromorphic section. \end{slogan} Let $X$ be a Noetherian scheme which has an ample invertible sheaf. Then every invertible $\mathcal{O}_X$-module is isomorphic to $$ \mathcal{O}_X(D - D') = \mathcal{O}_X(D) \otimes_{\mathcal{O}_X} \mathcal{O}_X(D')^{\otimes -1} $$ for some effective Cartier divisors $D, D'$ in $X$. Moreover, given a finite subset $E \subset X$ we may choose $D, D'$ such that $E \cap D = \emptyset$ and $E \cap D' = \emptyset$. If $X$ is quasi-affine, then we may choose $D' = \emptyset$.


divisors-lemma-wedge-product-ses

Let $X$ be an integral regular scheme of dimension $2$. Let $i : D \to X$ be the immersion of an effective Cartier divisor. Let $\mathcal{F} \to \mathcal{F}' \to i_*\mathcal{G} \to 0$ be an exact sequence of coherent $\mathcal{O}_X$-modules. Assume \begin{enumerate} \item $\mathcal{F}, \mathcal{F}'$ are locally free of rank $r$ on a nonempty open of $X$, \item $D$ is an integral scheme, \item $\mathcal{G}$ is a finite locally free $\mathcal{O}_D$-module of rank $s$. \end{enumerate} Then $\mathcal{L} = (\wedge^r\mathcal{F})^{**}$ and $\mathcal{L}' = (\wedge^r \mathcal{F}')^{**}$ are invertible $\mathcal{O}_X$-modules and $\mathcal{L}' \cong \mathcal{L}(k D)$ for some $k \in \{0, \ldots, \min(s, r)\}$.


divisors-lemma-affine-punctured-spec

Let $(A, \mathfrak m)$ be a Noetherian local ring. The punctured spectrum $U = \Spec(A) \setminus \{\mathfrak m\}$ of $A$ is affine if and only if $\dim(A) \leq 1$.


divisors-lemma-complement-affine-open-immersion

\begin{reference} \cite[EGA IV, Corollaire 21.12.7]{EGA4} \end{reference} Let $X$ be a locally Noetherian scheme. Let $U \subset X$ be an open subscheme such that the inclusion morphism $U \to X$ is affine. For every generic point $\xi$ of an irreducible component of $X \setminus U$ the local ring $\mathcal{O}_{X, \xi}$ has dimension $\leq 1$. If $U$ is dense or if $\xi$ is in the closure of $U$, then $\dim(\mathcal{O}_{X, \xi}) = 1$.


divisors-lemma-complement-affine-open

Let $X$ be a separated locally Noetherian scheme. Let $U \subset X$ be an affine open. For every generic point $\xi$ of an irreducible component of $X \setminus U$ the local ring $\mathcal{O}_{X, \xi}$ has dimension $\leq 1$. If $U$ is dense or if $\xi$ is in the closure of $U$, then $\dim(\mathcal{O}_{X, \xi}) = 1$.


divisors-lemma-complement-open-affine-effective-cartier-divisor

Let $X$ be a Noetherian separated scheme. Let $U \subset X$ be a dense affine open. 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 $U = X \setminus D$.


divisors-lemma-complement-open-affine-effective-cartier-divisor-bis

Let $X$ be a Noetherian scheme with affine diagonal. Let $U \subset X$ be a dense affine open. 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 $U = X \setminus D$.


divisors-lemma-finite-trivialize-invertible-upstairs

Let $\pi : X \to Y$ be a finite morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $y \in Y$. There exists an open neighbourhood $V \subset Y$ of $y$ such that $\mathcal{L}|_{\pi^{-1}(V)}$ is trivial.


divisors-lemma-norm-invertible

Let $\pi : X \to Y$ be a finite morphism of schemes. If there exists a norm of degree $d$ for $\pi$, then there exists a homomorphism of abelian groups $$ \text{Norm}_\pi : \Pic(X) \to \Pic(Y) $$ such that $\text{Norm}_\pi(\pi^*\mathcal{N}) \cong \mathcal{N}^{\otimes d}$ for all invertible $\mathcal{O}_Y$-modules $\mathcal{N}$.


divisors-lemma-norm-map-invertible

Let $\pi : X \to Y$ be a finite morphism of schemes. Assume there exists a norm of degree $d$ for $\pi$. For any $\mathcal{O}_X$-linear map $\varphi : \mathcal{L} \to \mathcal{L}'$ of invertible $\mathcal{O}_X$-modules there is an $\mathcal{O}_Y$-linear map $$ \text{Norm}_\pi(\varphi) : \text{Norm}_\pi(\mathcal{L}) \longrightarrow \text{Norm}_\pi(\mathcal{L}') $$ with $\text{Norm}_\pi(\mathcal{L})$, $\text{Norm}_\pi(\mathcal{L}')$ as in Lemma \ref{lemma-norm-invertible}. Moreover, for $y \in Y$ the following are equivalent \begin{enumerate} \item $\varphi$ is zero at a point of $x \in X$ with $\pi(x) = y$, and \item $\text{Norm}_\pi(\varphi)$ is zero at $y$. \end{enumerate}


divisors-lemma-norm-ample

Let $\pi : X \to Y$ be a finite morphism of schemes. Assume $X$ has an ample invertible sheaf and there exists a norm of degree $d$ for $\pi$. Then $Y$ has an ample invertible sheaf.


divisors-lemma-norm-quasi-affine

Let $\pi : X \to Y$ be a finite morphism of schemes. Assume $X$ is quasi-affine and there exists a norm of degree $d$ for $\pi$. Then $Y$ is quasi-affine.


divisors-lemma-finite-locally-free-has-norm

Let $\pi : X \to Y$ be a finite locally free morphism of degree $d \geq 1$. Then there exists a canonical norm of degree $d$ whose formation commutes with arbitrary base change.


divisors-lemma-norm-in-normal-case

Let $\pi : X \to Y$ be a finite surjective morphism with $X$ and $Y$ integral and $Y$ normal. Then there exists a norm of degree $[R(X) : R(Y)]$ for $\pi$.


divisors-lemma-Frobenius-gives-norm-for-reduction

Let $X$ be a Noetherian scheme. Let $p$ be a prime number such that $p\mathcal{O}_X = 0$. Then for some $e > 0$ there exists a norm of degree $p^e$ for $X_{red} \to X$ where $X_{red}$ is the reduction of $X$.


divisors-lemma-push-down-quasi-affine

Let $\pi : X \to Y$ be a finite surjective morphism of schemes. Assume that $X$ is quasi-affine. If either \begin{enumerate} \item $\pi$ is finite locally free, or \item $Y$ is an integral normal scheme \end{enumerate} then $Y$ is quasi-affine.


divisors-lemma-relative-Cartier

Let $f : X \to S$ be a morphism of schemes. Let $D \subset X$ be a closed subscheme. Assume \begin{enumerate} \item $D$ is an effective Cartier divisor, and \item $D \to S$ is a flat morphism. \end{enumerate} Then for every morphism of schemes $g : S' \to S$ the pullback $(g')^{-1}D$ is an effective Cartier divisor on $X' = S' \times_S X$ where $g' : X' \to X$ is the projection.


divisors-lemma-sum-relative-effective-Cartier-divisor

Let $f : X \to S$ be a morphism of schemes. If $D_1, D_2 \subset X$ are relative effective Cartier divisor on $X/S$ then so is $D_1 + D_2$ (Definition \ref{definition-sum-effective-Cartier-divisors}).


divisors-lemma-difference-relative-effective-Cartier-divisor

Let $f : X \to S$ be a morphism of schemes. If $D_1, D_2 \subset X$ are relative effective Cartier divisor on $X/S$ and $D_1 \subset D_2$ as closed subschemes, then the effective Cartier divisor $D$ such that $D_2 = D_1 + D$ (Lemma \ref{lemma-difference-effective-Cartier-divisors}) is a relative effective Cartier divisor on $X/S$.


divisors-lemma-flat-at-x

Let $f : X \to S$ be a morphism of schemes. Let $D \subset X$ be a relative effective Cartier divisor on $X/S$. If $x \in D$ and $\mathcal{O}_{X, x}$ is Noetherian, then $f$ is flat at $x$.


divisors-lemma-fibre-Cartier

Let $\varphi : X \to S$ be a flat morphism which is locally of finite presentation. Let $Z \subset X$ be a closed subscheme. Let $x \in Z$ with image $s \in S$. \begin{enumerate} \item If $Z_s \subset X_s$ is a Cartier divisor in a neighbourhood of $x$, then there exists an open $U \subset X$ and a relative effective Cartier divisor $D \subset U$ such that $Z \cap U \subset D$ and $Z_s \cap U = D_s$. \item If $Z_s \subset X_s$ is a Cartier divisor in a neighbourhood of $x$, the morphism $Z \to X$ is of finite presentation, and $Z \to S$ is flat at $x$, then we can choose $U$ and $D$ such that $Z \cap U = D$. \item If $Z_s \subset X_s$ is a Cartier divisor in a neighbourhood of $x$ and $Z$ is a locally principal closed subscheme of $X$ in a neighbourhood of $x$, then we can choose $U$ and $D$ such that $Z \cap U = D$. \end{enumerate} In particular, if $Z \to S$ is locally of finite presentation and flat and all fibres $Z_s \subset X_s$ are effective Cartier divisors, then $Z$ is a relative effective Cartier divisor. Similarly, if $Z$ is a locally principal closed subscheme of $X$ such that all fibres $Z_s \subset X_s$ are effective Cartier divisors, then $Z$ is a relative effective Cartier divisor.


divisors-lemma-affine-conormal-sheaf

Let $i : Z \to X$ be an immersion. The conormal algebra of $i$ has the following properties: \begin{enumerate} \item Let $U \subset X$ be any open such that $i(Z)$ is a closed subset of $U$. Let $\mathcal{I} \subset \mathcal{O}_U$ be the sheaf of ideals corresponding to the closed subscheme $i(Z) \subset U$. Then $$ \mathcal{C}_{Z/X, *} = i^*\left(\bigoplus\nolimits_{n \geq 0} \mathcal{I}^n\right) = i^{-1}\left( \bigoplus\nolimits_{n \geq 0} \mathcal{I}^n/\mathcal{I}^{n + 1} \right) $$ \item For any affine open $\Spec(R) = U \subset X$ such that $Z \cap U = \Spec(R/I)$ there is a canonical isomorphism $\Gamma(Z \cap U, \mathcal{C}_{Z/X, *}) = \bigoplus_{n \geq 0} I^n/I^{n + 1}$. \end{enumerate}


divisors-lemma-conormal-algebra-functorial

Let $$ \xymatrix{ Z \ar[r]_i \ar[d]_f & X \ar[d]^g \\ Z' \ar[r]^{i'} & X' } $$ be a commutative diagram in the category of schemes. Assume $i$, $i'$ immersions. There is a canonical map of graded $\mathcal{O}_Z$-algebras $$ f^*\mathcal{C}_{Z'/X', *} \longrightarrow \mathcal{C}_{Z/X, *} $$ characterized by the following property: For every pair of affine opens $(\Spec(R) = U \subset X, \Spec(R') = U' \subset X')$ with $f(U) \subset U'$ such that $Z \cap U = \Spec(R/I)$ and $Z' \cap U' = \Spec(R'/I')$ the induced map $$ \Gamma(Z' \cap U', \mathcal{C}_{Z'/X', *}) = \bigoplus\nolimits (I')^n/(I')^{n + 1} \longrightarrow \bigoplus\nolimits_{n \geq 0} I^n/I^{n + 1} = \Gamma(Z \cap U, \mathcal{C}_{Z/X, *}) $$ is the one induced by the ring map $f^\sharp : R' \to R$ which has the property $f^\sharp(I') \subset I$.


divisors-lemma-conormal-algebra-functorial-flat

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


divisors-lemma-types-regular-sequences-implications

Let $X$ be a ringed space. Let $f_1, \ldots, f_r \in \Gamma(X, \mathcal{O}_X)$. We have the following implications $f_1, \ldots, f_r$ is a regular sequence $\Rightarrow$ $f_1, \ldots, f_r$ is a Koszul-regular sequence $\Rightarrow$ $f_1, \ldots, f_r$ is an $H_1$-regular sequence $\Rightarrow$ $f_1, \ldots, f_r$ is a quasi-regular sequence.


divisors-lemma-regular-quasi-regular-scheme

Let $X$ be a ringed space. Let $\mathcal{J}$ be a sheaf of ideals. We have the following implications: $\mathcal{J}$ is regular $\Rightarrow$ $\mathcal{J}$ is Koszul-regular $\Rightarrow$ $\mathcal{J}$ is $H_1$-regular $\Rightarrow$ $\mathcal{J}$ is quasi-regular.


divisors-lemma-quasi-regular-ideal

Let $X$ be a locally ringed space. Let $\mathcal{J} \subset \mathcal{O}_X$ be a sheaf of ideals. Then $\mathcal{J}$ is quasi-regular if and only if the following conditions are satisfied: \begin{enumerate} \item $\mathcal{J}$ is an $\mathcal{O}_X$-module of finite type, \item $\mathcal{J}/\mathcal{J}^2$ is a finite locally free $\mathcal{O}_X/\mathcal{J}$-module, and \item the canonical maps $$ \text{Sym}^n_{\mathcal{O}_X/\mathcal{J}}(\mathcal{J}/\mathcal{J}^2) \longrightarrow \mathcal{J}^n/\mathcal{J}^{n + 1} $$ are isomorphisms for all $n \geq 0$. \end{enumerate}


divisors-lemma-generate-regular-ideal

Let $(X, \mathcal{O}_X)$ be a locally ringed space. Let $\mathcal{J} \subset \mathcal{O}_X$ be a sheaf of ideals. Let $x \in X$ and $f_1, \ldots, f_r \in \mathcal{J}_x$ whose images give a basis for the $\kappa(x)$-vector space $\mathcal{J}_x/\mathfrak m_x\mathcal{J}_x$. \begin{enumerate} \item If $\mathcal{J}$ is quasi-regular, then there exists an open neighbourhood such that $f_1, \ldots, f_r \in \mathcal{O}_X(U)$ form a quasi-regular sequence generating $\mathcal{J}|_U$. \item If $\mathcal{J}$ is $H_1$-regular, then there exists an open neighbourhood such that $f_1, \ldots, f_r \in \mathcal{O}_X(U)$ form an $H_1$-regular sequence generating $\mathcal{J}|_U$. \item If $\mathcal{J}$ is Koszul-regular, then there exists an open neighbourhood such that $f_1, \ldots, f_r \in \mathcal{O}_X(U)$ form an Koszul-regular sequence generating $\mathcal{J}|_U$. \end{enumerate}


divisors-lemma-regular-ideal-sheaf-quasi-coherent

Any regular, Koszul-regular, $H_1$-regular, or quasi-regular sheaf of ideals on a scheme is a finite type quasi-coherent sheaf of ideals.


divisors-lemma-regular-ideal-sheaf-scheme

Let $X$ be a scheme. Let $\mathcal{J}$ be a sheaf of ideals. Then $\mathcal{J}$ is regular (resp.\ Koszul-regular, $H_1$-regular, quasi-regular) if and only if for every $x \in \text{Supp}(\mathcal{O}_X/\mathcal{J})$ there exists an affine open neighbourhood $x \in U \subset X$, $U = \Spec(A)$ such that $\mathcal{J}|_U = \widetilde{I}$ and such that $I$ is generated by a regular (resp.\ Koszul-regular, $H_1$-regular, quasi-regular) sequence $f_1, \ldots, f_r \in A$.


divisors-lemma-Noetherian-scheme-regular-ideal

Let $X$ be a locally Noetherian scheme. Let $\mathcal{J} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $x$ be a point of the support of $\mathcal{O}_X/\mathcal{J}$. The following are equivalent \begin{enumerate} \item $\mathcal{J}_x$ is generated by a regular sequence in $\mathcal{O}_{X, x}$, \item $\mathcal{J}_x$ is generated by a Koszul-regular sequence in $\mathcal{O}_{X, x}$, \item $\mathcal{J}_x$ is generated by an $H_1$-regular sequence in $\mathcal{O}_{X, x}$, \item $\mathcal{J}_x$ is generated by a quasi-regular sequence in $\mathcal{O}_{X, x}$, \item there exists an affine neighbourhood $U = \Spec(A)$ of $x$ such that $\mathcal{J}|_U = \widetilde{I}$ and $I$ is generated by a regular sequence in $A$, and \item there exists an affine neighbourhood $U = \Spec(A)$ of $x$ such that $\mathcal{J}|_U = \widetilde{I}$ and $I$ is generated by a Koszul-regular sequence in $A$, and \item there exists an affine neighbourhood $U = \Spec(A)$ of $x$ such that $\mathcal{J}|_U = \widetilde{I}$ and $I$ is generated by an $H_1$-regular sequence in $A$, and \item there exists an affine neighbourhood $U = \Spec(A)$ of $x$ such that $\mathcal{J}|_U = \widetilde{I}$ and $I$ is generated by a quasi-regular sequence in $A$, \item there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_U$ is regular, and \item there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_U$ is Koszul-regular, and \item there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_U$ is $H_1$-regular, and \item there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_U$ is quasi-regular. \end{enumerate} In particular, on a locally Noetherian scheme the notions of regular, Koszul-regular, $H_1$-regular, or quasi-regular ideal sheaf all agree.


divisors-lemma-regular-quasi-regular-immersion

Let $i : Z \to X$ be an immersion of schemes. We have the following implications: $i$ is regular $\Rightarrow$ $i$ is Koszul-regular $\Rightarrow$ $i$ is $H_1$-regular $\Rightarrow$ $i$ is quasi-regular.


divisors-lemma-regular-immersion-noetherian

Let $i : Z \to X$ be an immersion of schemes. Assume $X$ is locally Noetherian. Then $i$ is regular $\Leftrightarrow$ $i$ is Koszul-regular $\Leftrightarrow$ $i$ is $H_1$-regular $\Leftrightarrow$ $i$ is quasi-regular.


divisors-lemma-flat-base-change-regular-immersion

Let $i : Z \to X$ be a regular (resp.\ Koszul-regular, $H_1$-regular, quasi-regular) immersion. Let $X' \to X$ be a flat morphism. Then the base change $i' : Z \times_X X' \to X'$ is a regular (resp.\ Koszul-regular, $H_1$-regular, quasi-regular) immersion.


divisors-lemma-quasi-regular-immersion

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


divisors-lemma-transitivity-conormal-quasi-regular

Let $Z \to Y \to X$ be immersions of schemes. Assume that $Z \to Y$ is $H_1$-regular. Then the canonical sequence of Morphisms, Lemma \ref{morphisms-lemma-transitivity-conormal} $$ 0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0 $$ is exact and locally split.


divisors-lemma-composition-regular-immersion

Let $i : Z \to Y$ and $j : Y \to X$ be immersions of schemes. \begin{enumerate} \item If $i$ and $j$ are regular immersions, so is $j \circ i$. \item If $i$ and $j$ are Koszul-regular immersions, so is $j \circ i$. \item If $i$ and $j$ are $H_1$-regular immersions, so is $j \circ i$. \item If $i$ is an $H_1$-regular immersion and $j$ is a quasi-regular immersion, then $j \circ i$ is a quasi-regular immersion. \end{enumerate}


divisors-lemma-permanence-regular-immersion

Let $i : Z \to Y$ and $j : Y \to X$ be immersions of schemes. Assume that the sequence $$ 0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0 $$ of Morphisms, Lemma \ref{morphisms-lemma-transitivity-conormal} is exact and locally split. \begin{enumerate} \item If $j \circ i$ is a quasi-regular immersion, so is $i$. \item If $j \circ i$ is a $H_1$-regular immersion, so is $i$. \item If both $j$ and $j \circ i$ are Koszul-regular immersions, so is $i$. \end{enumerate}


divisors-lemma-extra-permanence-regular-immersion-noetherian

Let $i : Z \to Y$ and $j : Y \to X$ be immersions of schemes. Pick $z \in Z$ and denote $y \in Y$, $x \in X$ the corresponding points. Assume $X$ is locally Noetherian. The following are equivalent \begin{enumerate} \item $i$ is a regular immersion in a neighbourhood of $z$ and $j$ is a regular immersion in a neighbourhood of $y$, \item $i$ and $j \circ i$ are regular immersions in a neighbourhood of $z$, \item $j \circ i$ is a regular immersion in a neighbourhood of $z$ and the conormal sequence $$ 0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0 $$ is split exact in a neighbourhood of $z$. \end{enumerate}


divisors-lemma-immersion-regular-regular-immersion

Let $i : Z \to X$ be an immersion. If $Z$ and $X$ are regular schemes, then $i$ is a regular immersion.


divisors-lemma-relative-regular-immersion

Let $f : X \to S$ be a morphism of schemes. Let $i : Z \subset X$ be an immersion. Assume \begin{enumerate} \item $i$ is an $H_1$-regular (resp.\ quasi-regular) immersion, and \item $Z \to S$ is a flat morphism. \end{enumerate} Then for every morphism of schemes $g : S' \to S$ the base change $Z' = S' \times_S Z \to X' = S' \times_S X$ is an $H_1$-regular (resp.\ quasi-regular) immersion.


divisors-lemma-quasi-regular-immersion-flat-at-x

Let $f : X \to S$ be a morphism of schemes. Let $Z \to X$ be a relative quasi-regular immersion. If $x \in Z$ and $\mathcal{O}_{X, x}$ is Noetherian, then $f$ is flat at $x$.


divisors-lemma-relative-regular-immersion-flat-in-neighbourhood

Let $X \to S$ be a morphism of schemes. Let $Z \to X$ be an immersion. Assume \begin{enumerate} \item $X \to S$ is flat and locally of finite presentation, \item $Z \to X$ is a relative quasi-regular immersion. \end{enumerate} Then $Z \to X$ is a regular immersion and the same remains true after any base change.


divisors-lemma-fibre-quasi-regular

Let $\varphi : X \to S$ be a flat morphism which is locally of finite presentation. Let $T \subset X$ be a closed subscheme. Let $x \in T$ with image $s \in S$. \begin{enumerate} \item If $T_s \subset X_s$ is a quasi-regular immersion in a neighbourhood of $x$, then there exists an open $U \subset X$ and a relative quasi-regular immersion $Z \subset U$ such that $Z_s = T_s \cap U_s$ and $T \cap U \subset Z$. \item If $T_s \subset X_s$ is a quasi-regular immersion in a neighbourhood of $x$, the morphism $T \to X$ is of finite presentation, and $T \to S$ is flat at $x$, then we can choose $U$ and $Z$ as in (1) such that $T \cap U = Z$. \item If $T_s \subset X_s$ is a quasi-regular immersion in a neighbourhood of $x$, and $T$ is cut out by $c$ equations in a neighbourhood of $x$, where $c = \dim_x(X_s) - \dim_x(T_s)$, then we can choose $U$ and $Z$ as in (1) such that $T \cap U = Z$. \end{enumerate} In each case $Z \to U$ is a regular immersion by Lemma \ref{lemma-relative-regular-immersion-flat-in-neighbourhood}. In particular, if $T \to S$ is locally of finite presentation and flat and all fibres $T_s \subset X_s$ are quasi-regular immersions, then $T \to X$ is a relative quasi-regular immersion.


divisors-lemma-lift-regular-immersion-to-smooth

Let $$ \xymatrix{ Y \ar[rd]_j \ar[rr]_i & & X \ar[ld] \\ & S } $$ be a commutative diagram of morphisms of schemes. Assume $X \to S$ smooth, and $i$, $j$ immersions. If $j$ is a regular (resp.\ Koszul-regular, $H_1$-regular, quasi-regular) immersion, then so is $i$.


divisors-lemma-immersion-lci-into-smooth-regular-immersion

Let $$ \xymatrix{ Y \ar[rd] \ar[rr]_i & & X \ar[ld] \\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that $Y \to S$ is syntomic, $X \to S$ smooth, and $i$ an immersion. Then $i$ is a regular immersion.


divisors-lemma-immersion-smooth-into-smooth-regular-immersion

Let $$ \xymatrix{ Y \ar[rd] \ar[rr]_i & & X \ar[ld] \\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that $Y \to S$ is smooth, $X \to S$ smooth, and $i$ an immersion. Then $i$ is a regular immersion.


divisors-lemma-push-regular-immersion-thru-smooth

Let $$ \xymatrix{ Y \ar[rd]_j \ar[rr]_i & & X \ar[ld] \\ & S } $$ be a commutative diagram of morphisms of schemes. Assume $X \to S$ smooth, and $i$, $j$ immersions. If $i$ is a Koszul-regular (resp.\ $H_1$-regular, quasi-regular) immersion, then so is $j$.


divisors-lemma-pullback-meromorphic-sections-defined

Let $f : X \to Y$ be a morphism of schemes. In each of the following cases pullbacks of meromorphic functions are defined. \begin{enumerate} \item every weakly associated point of $X$ maps to a generic point of an irreducible component of $Y$, \item $X$, $Y$ are integral and $f$ is dominant, \item $X$ is integral and the generic point of $X$ maps to a generic point of an irreducible component of $Y$, \item $X$ is reduced and every generic point of every irreducible component of $X$ maps to the generic point of an irreducible component of $Y$, \item $X$ is locally Noetherian, and any associated point of $X$ maps to a generic point of an irreducible component of $Y$, \item $X$ is locally Noetherian, has no embedded points and any generic point of an irreducible component of $X$ maps to the generic point of an irreducible component of $Y$, and \item $f$ is flat. \end{enumerate}


divisors-lemma-meromorphic-weakass-finite

Let $X$ be a scheme such that \begin{enumerate} \item[(a)] every weakly associated point of $X$ is a generic point of an irreducible component of $X$, and \item[(b)] any quasi-compact open has a finite number of irreducible components. \end{enumerate} Let $X^0$ be the set of generic points of irreducible components of $X$. Then we have $$ \mathcal{K}_X = \bigoplus\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{O}_{X, \eta} = \prod\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{O}_{X, \eta} $$ where $j_\eta : \Spec(\mathcal{O}_{X, \eta}) \to X$ is the canonical map of Schemes, Section \ref{schemes-section-points}. Moreover \begin{enumerate} \item $\mathcal{K}_X$ is a quasi-coherent sheaf of $\mathcal{O}_X$-algebras, \item for every quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the sheaf $$ \mathcal{K}_X(\mathcal{F}) = \bigoplus\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{F}_\eta = \prod\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{F}_\eta $$ of meromorphic sections of $\mathcal{F}$ is quasi-coherent, \item $\mathcal{S}_x \subset \mathcal{O}_{X, x}$ is the set of nonzerodivisors for any $x \in X$, \item $\mathcal{K}_{X, x}$ is the total quotient ring of $\mathcal{O}_{X, x}$ for any $x \in X$, \item $\mathcal{K}_X(U)$ equals the total quotient ring of $\mathcal{O}_X(U)$ for any affine open $U \subset X$, \item the ring of rational functions of $X$ (Morphisms, Definition \ref{morphisms-definition-rational-function}) is the ring of meromorphic functions on $X$, in a formula: $R(X) = \Gamma(X, \mathcal{K}_X)$. \end{enumerate}


divisors-lemma-meromorphic-sections-pullback

Let $f : X \to Y$ be a morphism of locally ringed spaces. Assume that pullbacks of meromorphic functions are defined for $f$ (see Definition \ref{definition-pullback-meromorphic-sections}). \begin{enumerate} \item Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_Y$-modules. There is a canonical pullback map $f^* : \Gamma(Y, \mathcal{K}_Y(\mathcal{F})) \to \Gamma(X, \mathcal{K}_X(f^*\mathcal{F}))$ for meromorphic sections of $\mathcal{F}$. \item Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. A regular meromorphic section $s$ of $\mathcal{L}$ pulls back to a regular meromorphic section $f^*s$ of $f^*\mathcal{L}$. \end{enumerate}


divisors-lemma-regular-meromorphic-ideal-denominators

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s$ be a regular meromorphic section of $\mathcal{L}$. Let us denote $\mathcal{I} \subset \mathcal{O}_X$ the sheaf of ideals defined by the rule $$ \mathcal{I}(V) = \{f \in \mathcal{O}_X(V) \mid fs \in \mathcal{L}(V)\}. $$ The formula makes sense since $\mathcal{L}(V) \subset \mathcal{K}_X(\mathcal{L})(V)$. Then $\mathcal{I}$ is a quasi-coherent sheaf of ideals and we have injective maps $$ 1 : \mathcal{I} \longrightarrow \mathcal{O}_X, \quad s : \mathcal{I} \longrightarrow \mathcal{L} $$ whose cokernels are supported on closed nowhere dense subsets of $X$.


divisors-lemma-meromorphic-section-restricts-to-zero

Let $X$ be a quasi-compact scheme. Let $h \in \Gamma(X, \mathcal{O}_X)$ and $f \in \Gamma(X, \mathcal{K}_X)$ such that $f$ restricts to zero on $X_h$. Then $h^n f = 0$ for some $n \gg 0$.


divisors-lemma-quasi-coherent-K

Let $X$ be a locally Noetherian scheme having no embedded points. Let $X^0$ be the set of generic points of irreducible components of $X$. Then we have $$ \mathcal{K}_X = \bigoplus\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{O}_{X, \eta} = \prod\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{O}_{X, \eta} $$ where $j_\eta : \Spec(\mathcal{O}_{X, \eta}) \to X$ is the canonical map of Schemes, Section \ref{schemes-section-points}. Moreover \begin{enumerate} \item $\mathcal{K}_X$ is a quasi-coherent sheaf of $\mathcal{O}_X$-algebras, \item for every quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the sheaf $$ \mathcal{K}_X(\mathcal{F}) = \bigoplus\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{F}_\eta = \prod\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{F}_\eta $$ of meromorphic sections of $\mathcal{F}$ is quasi-coherent, and \item the ring of rational functions of $X$ is the ring of meromorphic functions on $X$, in a formula: $R(X) = \Gamma(X, \mathcal{K}_X)$. \end{enumerate}


divisors-lemma-reduced-finite-irreducible

Let $X$ be a reduced scheme such that any quasi-compact open has a finite number of irreducible components. Let $X^0$ be the set of generic points of irreducible components of $X$. Then we have $$ \mathcal{K}_X = \bigoplus\nolimits_{\eta \in X^0} j_{\eta, *}\kappa(\eta) = \prod\nolimits_{\eta \in X^0} j_{\eta, *}\kappa(\eta) $$ where $j_\eta : \Spec(\kappa(\eta)) \to X$ is the canonical map of Schemes, Section \ref{schemes-section-points}. Moreover \begin{enumerate} \item $\mathcal{K}_X$ is a quasi-coherent sheaf of $\mathcal{O}_X$-algebras, \item for every quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the sheaf $$ \mathcal{K}_X(\mathcal{F}) = \bigoplus\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{F}_\eta = \prod\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{F}_\eta $$ of meromorphic sections of $\mathcal{F}$ is quasi-coherent, \item $\mathcal{S}_x \subset \mathcal{O}_{X, x}$ is the set of nonzerodivisors for any $x \in X$, \item $\mathcal{K}_{X, x}$ is the total quotient ring of $\mathcal{O}_{X, x}$ for any $x \in X$, \item $\mathcal{K}_X(U)$ equals the total quotient ring of $\mathcal{O}_X(U)$ for any affine open $U \subset X$, \item the ring of rational functions of $X$ is the ring of meromorphic functions on $X$, in a formula: $R(X) = \Gamma(X, \mathcal{K}_X)$. \end{enumerate}


divisors-lemma-meromorphic-functions-integral-scheme

Let $X$ be an integral scheme with generic point $\eta$. We have \begin{enumerate} \item the sheaf of meromorphic functions is isomorphic to the constant sheaf with value the function field (see Morphisms, Definition \ref{morphisms-definition-function-field}) of $X$. \item for any quasi-coherent sheaf $\mathcal{F}$ on $X$ the sheaf $\mathcal{K}_X(\mathcal{F})$ is isomorphic to the constant sheaf with value $\mathcal{F}_\eta$. \end{enumerate}


divisors-lemma-components-locally-finite

Let $X$ be a locally Noetherian scheme. Let $Z \subset X$ be a closed subscheme. The collection of irreducible components of $Z$ is locally finite in $X$.


divisors-lemma-divisor-locally-finite

Let $X$ be a locally Noetherian integral scheme. Let $f \in R(X)^*$. Then the collections $$ \{Z \subset X \mid Z\text{ a prime divisor with generic point }\xi \text{ and }f\text{ not in }\mathcal{O}_{X, \xi}\} $$ and $$ \{Z \subset X \mid Z \text{ a prime divisor and }\text{ord}_Z(f) \not = 0\} $$ are locally finite in $X$.