spaces-perfect-lemma-ext-perfect

Let $S$ be a scheme. Let $B$ be a Noetherian algebraic space over $S$. Let $f : X \to B$ be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let $E \in D(\mathcal{O}_X)$ be perfect. Let $\mathcal{G}^\bullet$ be a bounded complex of coherent $\mathcal{O}_X$-modules flat over $B$ with support proper over $B$. Then $K = Rf_*R\SheafHom(E, \mathcal{G})$ is a perfect object of $D(\mathcal{O}_B)$.


spaces-perfect-lemma-compute-tensor-perfect

Assumptions and notation as in Lemma \ref{lemma-tensor-perfect}. Then there are functorial isomorphisms $$ H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}) \longrightarrow H^i(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})) $$ for $\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps (see proof).


spaces-perfect-lemma-compute-ext-perfect

Assumption and notation as in Lemma \ref{lemma-ext-perfect}. Then there are functorial isomorphisms $$ H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) $$ for $\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps (see proof).


spaces-perfect-lemma-compute-ext

Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$, $E \in D(\mathcal{O}_X)$, and $\mathcal{F}^\bullet$ a complex of $\mathcal{O}_X$-modules. Assume \begin{enumerate} \item $B$ is Noetherian, \item $f$ is locally of finite type and quasi-separated, \item $E \in D^-_{\textit{Coh}}(\mathcal{O}_X)$, \item $\mathcal{G}^\bullet$ is a bounded complex of coherent $\mathcal{O}_X$-module flat over $B$ with support proper over $B$. \end{enumerate} Then the following two statements are true \begin{enumerate} \item[(A)] for every $m \in \mathbf{Z}$ there exists a perfect object $K$ of $D(\mathcal{O}_B)$ and functorial maps $$ \alpha^i_\mathcal{F} : \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \longrightarrow H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}) $$ for $\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps (see proof) such that $\alpha^i_\mathcal{F}$ is an isomorphism for $i \leq m$, and \item[(B)] there exists a pseudo-coherent $L \in D(\mathcal{O}_B)$ and functorial isomorphisms $$ \Ext^i_{\mathcal{O}_B}(L, \mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) $$ for $\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps. \end{enumerate}


spaces-perfect-lemma-descend-homomorphisms

In Situation \ref{situation-descent}. Let $E_0$ and $K_0$ be objects of $D(\mathcal{O}_{X_0})$. Set $E_i = Lf_{i0}^*E_0$ and $K_i = Lf_{i0}^*K_0$ for $i \geq 0$ and set $E = Lf_0^*E_0$ and $K = Lf_0^*K_0$. Then the map $$ \colim_{i \geq 0} \Hom_{D(\mathcal{O}_{X_i})}(E_i, K_i) \longrightarrow \Hom_{D(\mathcal{O}_X)}(E, K) $$ is an isomorphism if either \begin{enumerate} \item $E_0$ is perfect and $K_0 \in D_\QCoh(\mathcal{O}_{X_0})$, or \item $E_0$ is pseudo-coherent and $K_0 \in D_\QCoh(\mathcal{O}_{X_0})$ has finite tor dimension. \end{enumerate}


spaces-perfect-lemma-perfect-on-limit

In Situation \ref{situation-descent} the category of perfect objects of $D(\mathcal{O}_X)$ is the colimit of the categories of perfect objects of $D(\mathcal{O}_{X_i})$.


spaces-perfect-lemma-base-change-tensor-perfect

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of finite presentation between algebraic spaces over $S$. Let $E \in D(\mathcal{O}_X)$ be a perfect object. Let $\mathcal{G}^\bullet$ be a bounded complex of finitely presented $\mathcal{O}_X$-modules, flat over $Y$, with support proper over $Y$. Then $$ K = Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet) $$ is a perfect object of $D(\mathcal{O}_Y)$ and its formation commutes with arbitrary base change.


spaces-perfect-lemma-base-change-tensor-pseudo-coherent

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of finite presentation between algebraic spaces over $S$. Let $E \in D(\mathcal{O}_X)$ be a pseudo-coherent object. Let $\mathcal{G}^\bullet$ be a bounded above complex of finitely presented $\mathcal{O}_X$-modules, flat over $Y$, with support proper over $Y$. Then $$ K = Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet) $$ is a pseudo-coherent object of $D(\mathcal{O}_Y)$ and its formation commutes with arbitrary base change.


spaces-perfect-lemma-flat-proper-perfect-direct-image-general

Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of finite presentation of algebraic spaces over $S$. \begin{enumerate} \item Let $E \in D(\mathcal{O}_X)$ be perfect and $f$ flat. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_Y)$ and its formation commutes with arbitrary base change. \item Let $\mathcal{G}$ be an $\mathcal{O}_X$-module of finite presentation, flat over $S$. Then $Rf_*\mathcal{G}$ is a perfect object of $D(\mathcal{O}_Y)$ and its formation commutes with arbitrary base change. \end{enumerate}


spaces-perfect-lemma-flat-proper-pseudo-coherent-direct-image-general

Let $S$ be a scheme. Let $f : X \to Y$ be a flat proper morphism of finite presentation of algebraic spaces over $S$. Let $E \in D(\mathcal{O}_X)$ be pseudo-coherent. Then $Rf_*E$ is a pseudo-coherent object of $D(\mathcal{O}_Y)$ and its formation commutes with arbitrary base change.


spaces-perfect-lemma-pullback-and-limits

Let $R$ be a ring. Let $X$ be an algebraic space and let $f : X \to \Spec(R)$ be proper, flat, and of finite presentation. Let $(M_n)$ be an inverse system of $R$-modules with surjective transition maps. Then the canonical map $$ \mathcal{O}_X \otimes_R (\lim M_n) \longrightarrow \lim \mathcal{O}_X \otimes_R M_n $$ induces an isomorphism from the source to $DQ_X$ applied to the target.


spaces-perfect-lemma-perfect-enough

Let $A$ be a ring. Let $X$ be an algebraic space over $A$ which is quasi-compact and quasi-separated. Let $K \in D^-_\QCoh(\mathcal{O}_X)$. If $R\Gamma(X, E \otimes^\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every perfect $E$ in $D(\mathcal{O}_X)$, then $R\Gamma(X, E \otimes^\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\mathcal{O}_X)$.


spaces-perfect-lemma-base-change-RHom-perfect

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of finite presentation between algebraic spaces over $S$. Let $E \in D(\mathcal{O}_X)$ be a perfect object. Let $\mathcal{G}^\bullet$ be a bounded complex of finitely presented $\mathcal{O}_X$-modules, flat over $Y$, with support proper over $Y$. Then $$ K = Rf_*R\SheafHom(E, \mathcal{G}^\bullet) $$ is a perfect object of $D(\mathcal{O}_Y)$ and its formation commutes with arbitrary base change.


spaces-perfect-lemma-jump-loci

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \in D(\mathcal{O}_X)$ be pseudo-coherent (for example perfect). For any $i \in \mathbf{Z}$ consider the function $$ \beta_i : |X| \longrightarrow \{0, 1, 2, \ldots\} $$ defined above. Then we have \begin{enumerate} \item formation of $\beta_i$ commutes with arbitrary base change, \item the functions $\beta_i$ are upper semi-continuous, and \item the level sets of $\beta_i$ are \'etale locally constructible. \end{enumerate}


spaces-perfect-lemma-jump-loci-geometric

Let $Y$ be a scheme and let $X$ be an algebraic space over $Y$ such that the structure morphism $f : X \to Y$ is flat, proper, and of finite presentation. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module of finite presentation, flat over $Y$. For fixed $i \in \mathbf{Z}$ consider the function $$ \beta_i : |Y| \to \{0, 1, 2, \ldots\},\quad y \longmapsto \dim_{\kappa(y)} H^i(X_y, \mathcal{F}_y) $$ Then we have \begin{enumerate} \item formation of $\beta_i$ commutes with arbitrary base change, \item the functions $\beta_i$ are upper semi-continuous, and \item the level sets of $\beta_i$ are locally constructible in $Y$. \end{enumerate}


spaces-perfect-lemma-open-where-cohomology-in-degree-i-rank-r

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \in D(\mathcal{O}_X)$ be perfect. Given $i, r \in \mathbf{Z}$, there exists an open subspace $U \subset X$ characterized by the following \begin{enumerate} \item $E|_U \cong H^i(E|_U)[-i]$ and $H^i(E|_U)$ is a locally free $\mathcal{O}_U$-module of rank $r$, \item a morphism $f : Y \to X$ factors through $U$ if and only if $Lf^*E$ is isomorphic to a locally free module of rank $r$ placed in degree $i$. \end{enumerate}


spaces-perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is proper, flat, and of finite presentation. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module of finite presentation, flat over $Y$. Fix $i, r \in \mathbf{Z}$. Then there exists an open subspace $V \subset Y$ with the following property: A morphism $T \to Y$ factors through $V$ if and only if $Rf_{T, *}\mathcal{F}_T$ is isomorphic to a finite locally free module of rank $r$ placed in degree $i$.


spaces-perfect-lemma-locally-closed-where-H0-locally-free

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \in D(\mathcal{O}_X)$ be perfect of tor-amplitude in $[a, b]$ for some $a, b \in \mathbf{Z}$. Let $r \geq 0$. Then there exists a locally closed subspace $j : Z \to X$ characterized by the following \begin{enumerate} \item $H^a(Lj^*E)$ is a locally free $\mathcal{O}_Z$-module of rank $r$, and \item a morphism $f : Y \to X$ factors through $Z$ if and only if for all morphisms $g : Y' \to Y$ the $\mathcal{O}_{Y'}$-module $H^a(L(f \circ g)^*E)$ is locally free of rank $r$. \end{enumerate} Moreover, $j : Z \to X$ is of finite presentation and we have \begin{enumerate} \item[(3)] if $f : Y \to X$ factors as $Y \xrightarrow{g} Z \to X$, then $H^a(Lf^*E) = g^*H^a(Lj^*E)$, \item[(4)] if $\beta_a(x) \leq r$ for all $x \in |X|$, then $j$ is a closed immersion and given $f : Y \to X$ the following are equivalent \begin{enumerate} \item $f : Y \to X$ factors through $Z$, \item $H^0(Lf^*E)$ is a locally free $\mathcal{O}_Y$-module of rank $r$, \end{enumerate} and if $r = 1$ these are also equivalent to \begin{enumerate} \item[(c)] $\mathcal{O}_Y \to \SheafHom_{\mathcal{O}_Y}(H^0(Lf^*E), H^0(Lf^*E))$ is injective. \end{enumerate} \end{enumerate}


spaces-perfect-lemma-proper-flat-h0

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $f$ is proper, flat, and of finite presentation, and \item for a morphism $\Spec(k) \to Y$ where $k$ is a field, we have $k = H^0(X_k, \mathcal{O}_{X_k})$. \end{enumerate} Then we have \begin{enumerate} \item[(a)] $f_*\mathcal{O}_X = \mathcal{O}_S$ and this holds after any base change, \item[(b)] \'etale locally on $Y$ we have $$ Rf_*\mathcal{O}_X = \mathcal{O}_Y \oplus P $$ in $D(\mathcal{O}_Y)$ where $P$ is perfect of tor amplitude in $[1, \infty)$. \end{enumerate}


spaces-perfect-lemma-proper-flat-geom-red-connected

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $f$ is proper, flat, and of finite presentation, and \item the geometric fibres of $f$ are reduced and connected. \end{enumerate} Then $f_*\mathcal{O}_X = \mathcal{O}_Y$ and this holds after any base change.


spaces-perfect-lemma-countable-cohomology

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $K$ be an object of $D_\QCoh(\mathcal{O}_X)$ such that the cohomology sheaves $H^i(K)$ have countable sets of sections over affine schemes \'etale over $X$. Then for any quasi-compact and quasi-separated \'etale morphism $U \to X$ and any perfect object $E$ in $D(\mathcal{O}_X)$ the sets $$ H^i(U, K \otimes^\mathbf{L} E),\quad \Ext^i(E|_U, K|_U) $$ are countable.


spaces-perfect-lemma-countable

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Assume the sets of sections of $\mathcal{O}_X$ over affines \'etale over $X$ are countable. Let $K$ be an object of $D_\QCoh(\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item $K = \text{hocolim} E_n$ with $E_n$ a perfect object of $D(\mathcal{O}_X)$, and \item the cohomology sheaves $H^i(K)$ have countable sets of sections over affines \'etale over $X$. \end{enumerate}


spaces-perfect-lemma-computing-sections-as-colim

Let $A$ be a ring. Let $f : U \to X$ be a flat morphism of algebraic spaces of finite presentation over $A$. Then \begin{enumerate} \item there exists an inverse system of perfect objects $L_n$ of $D(\mathcal{O}_X)$ such that $$ R\Gamma(U, Lf^*K) = \text{hocolim}\ R\Hom_X(L_n, K) $$ in $D(A)$ functorially in $K$ in $D_\QCoh(\mathcal{O}_X)$, and \item there exists a system of perfect objects $E_n$ of $D(\mathcal{O}_X)$ such that $$ R\Gamma(U, Lf^*K) = \text{hocolim}\ R\Gamma(X, E_n \otimes^\mathbf{L} K) $$ in $D(A)$ functorially in $K$ in $D_\QCoh(\mathcal{O}_X)$. \end{enumerate}


spaces-perfect-lemma-bounded-truncation

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $P \in D_{perf}(\mathcal{O}_X)$ and $E \in D_{\QCoh}(\mathcal{O}_X)$. Let $a \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item $\Hom_{D(\mathcal{O}_X)}(P[-i], E) = 0$ for $i \gg 0$, and \item $\Hom_{D(\mathcal{O}_X)}(P[-i], \tau_{\geq a} E) = 0$ for $i \gg 0$. \end{enumerate}


spaces-perfect-lemma-bounded-below-truncation

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $P \in D_{perf}(\mathcal{O}_X)$ and $E \in D_{\QCoh}(\mathcal{O}_X)$. Let $a \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item $\Hom_{D(\mathcal{O}_X)}(P[-i], E) = 0$ for $i \ll 0$, and \item $\Hom_{D(\mathcal{O}_X)}(P[-i], \tau_{\leq a} E) = 0$ for $i \ll 0$. \end{enumerate}


spaces-perfect-proposition-quasi-compact-affine-diagonal

Let $S$ be a scheme. Let $X$ be a quasi-compact algebraic space over $S$ with affine diagonal. Then the functor (\ref{equation-compare}) $$ D(\QCoh(\mathcal{O}_X)) \longrightarrow D_\QCoh(\mathcal{O}_X) $$ is an equivalence with quasi-inverse given by $RQ_X$.


spaces-perfect-proposition-compact-is-perfect

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. An object of $D_\QCoh(\mathcal{O}_X)$ is compact if and only if it is perfect.


spaces-perfect-proposition-detecting-bounded-above

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $G \in D_{perf}(\mathcal{O}_X)$ be a perfect complex which generates $D_\QCoh (\mathcal{O}_X)$. Let $E \in D_\QCoh (\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item $E \in D^-_\QCoh (\mathcal{O}_X)$, \item $\Hom_{D(\mathcal{O}_X)}(G[-i], E) = 0$ for $i \gg 0$, \item $\Ext^i_X(G, E) = 0$ for $i \gg 0$, \item $R\Hom_X(G, E)$ is in $D^-(\mathbf{Z})$, \item $H^i(X, G^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E) = 0$ for $i \gg 0$, \item $R\Gamma(X, G^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E)$ is in $D^-(\mathbf{Z})$, \item for every perfect object $P$ of $D(\mathcal{O}_X)$ \begin{enumerate} \item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and \item $H^i(X, P \otimes_{\mathcal{O}_X}^\mathbf{L} E) = 0$ for $i \gg 0$, \item $R\Gamma(X, P \otimes_{\mathcal{O}_X}^\mathbf{L} E)$ is in $D^-(\mathbf{Z})$. \end{enumerate} \end{enumerate}


spaces-perfect-proposition-detecting-bounded-below

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $G \in D_{perf}(\mathcal{O}_X)$ be a perfect complex which generates $D_\QCoh (\mathcal{O}_X)$. Let $E \in D_\QCoh (\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item $E \in D^+_\QCoh (\mathcal{O}_X)$, \item $\Hom_{D(\mathcal{O}_X)}(G[-i], E) = 0$ for $i \ll 0$, \item $\Ext^i_X(G, E) = 0$ for $i \ll 0$, \item $R\Hom_X(G, E)$ is in $D^+(\mathbf{Z})$, \item $H^i(X, G^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E) = 0$ for $i \ll 0$, \item $R\Gamma(X, G^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E)$ is in $D^+(\mathbf{Z})$, \item for every perfect object $P$ of $D(\mathcal{O}_X)$ \begin{enumerate} \item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and \item $H^i(X, P \otimes_{\mathcal{O}_X}^\mathbf{L} E) = 0$ for $i \ll 0$, \item $R\Gamma(X, P \otimes_{\mathcal{O}_X}^\mathbf{L} E)$ is in $D^+(\mathbf{Z})$. \end{enumerate} \end{enumerate}


dualizing-theorem-local-duality

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $\omega_A^\bullet$ be a normalized dualizing complex. Let $E$ be an injective hull of the residue field. Let $Z = V(\mathfrak m) \subset \Spec(A)$. Denote ${}^\wedge$ derived completion with respect to $\mathfrak m$. Then $$ R\Hom_A(K, \omega_A^\bullet)^\wedge \cong R\Hom_A(R\Gamma_Z(K), E[0]) $$ for $K$ in $D(A)$.


dualizing-lemma-essential

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item If $A \subset B$ and $B \subset C$ are essential extensions, then $A \subset C$ is an essential extension. \item If $A \subset B$ is an essential extension and $C \subset B$ is a subobject, then $A \cap C \subset C$ is an essential extension. \item If $A \to B$ and $B \to C$ are essential surjections, then $A \to C$ is an essential surjection. \item Given an essential surjection $f : A \to B$ and a surjection $A \to C$ with kernel $K$, the morphism $C \to B/f(K)$ is an essential surjection. \end{enumerate}


dualizing-lemma-union-essential-extensions

Let $R$ be a ring. Let $M$ be an $R$-module. Let $E = \colim E_i$ be a filtered colimit of $R$-modules. Suppose given a compatible system of essential injections $M \to E_i$ of $R$-modules. Then $M \to E$ is an essential injection.


dualizing-lemma-essential-extension

Let $R$ be a ring. Let $M \subset N$ be $R$-modules. The following are equivalent \begin{enumerate} \item $M \subset N$ is an essential extension, \item for all $x \in N$ nonzero there exists an $f \in R$ such that $fx \in M$ and $fx \not = 0$. \end{enumerate}


dualizing-lemma-injective-flat

Let $R \to S$ be a flat ring map. If $E$ is an injective $S$-module, then $E$ is injective as an $R$-module.


dualizing-lemma-injective-epimorphism

Let $R \to S$ be an epimorphism of rings. Let $E$ be an $S$-module. If $E$ is injective as an $R$-module, then $E$ is an injective $S$-module.


dualizing-lemma-hom-injective

Let $R \to S$ be a ring map. If $E$ is an injective $R$-module, then $\Hom_R(S, E)$ is an injective $S$-module.


dualizing-lemma-essential-extensions-in-injective

Let $R$ be a ring. Let $I$ be an injective $R$-module. Let $E \subset I$ be a submodule. The following are equivalent \begin{enumerate} \item $E$ is injective, and \item for all $E \subset E' \subset I$ with $E \subset E'$ essential we have $E = E'$. \end{enumerate} In particular, an $R$-module is injective if and only if every essential extension is trivial.


dualizing-lemma-sum-injective-modules

Let $R$ be a Noetherian ring. A direct sum of injective modules is injective.


dualizing-lemma-localization-injective-modules

Let $R$ be a Noetherian ring. Let $S \subset R$ be a multiplicative subset. If $E$ is an injective $R$-module, then $S^{-1}E$ is an injective $S^{-1}R$-module.


dualizing-lemma-injective-module-divide

Let $R$ be a Noetherian ring. Let $I$ be an injective $R$-module. \begin{enumerate} \item Let $f \in R$. Then $E = \bigcup I[f^n] = I[f^\infty]$ is an injective submodule of $I$. \item Let $J \subset R$ be an ideal. Then the $J$-power torsion submodule $I[J^\infty]$ is an injective submodule of $I$. \end{enumerate}


dualizing-lemma-injective-dimension-over-polynomial-ring

Let $A$ be a Noetherian ring. Let $E$ be an injective $A$-module. Then $E \otimes_A A[x]$ has injective-amplitude $[0, 1]$ as an object of $D(A[x])$. In particular, $E \otimes_A A[x]$ has finite injective dimension as an $A[x]$-module.


dualizing-lemma-projective-cover-unique

Let $R$ be a ring and let $M$ be an $R$-module. If a projective cover of $M$ exists, then it is unique up to isomorphism.


dualizing-lemma-injective-hull-unique

Let $R$ be a ring. Let $M$, $N$ be $R$-modules and let $M \to E$ and $N \to E'$ be injective hulls. Then \begin{enumerate} \item for any $R$-module map $\varphi : M \to N$ there exists an $R$-module map $\psi : E \to E'$ such that $$ \xymatrix{ M \ar[r] \ar[d]_\varphi & E \ar[d]^\psi \\ N \ar[r] & E' } $$ commutes, \item if $\varphi$ is injective, then $\psi$ is injective, \item if $\varphi$ is an essential injection, then $\psi$ is an isomorphism, \item if $\varphi$ is an isomorphism, then $\psi$ is an isomorphism, \item if $M \to I$ is an embedding of $M$ into an injective $R$-module, then there is an isomorphism $I \cong E \oplus I'$ compatible with the embeddings of $M$, \end{enumerate} In particular, the injective hull $E$ of $M$ is unique up to isomorphism.


dualizing-lemma-indecomposable-injective

Let $R$ be a ring. Let $E$ be an indecomposable injective $R$-module. Then \begin{enumerate} \item $E$ is the injective hull of any nonzero submodule of $E$, \item the intersection of any two nonzero submodules of $E$ is nonzero, \item $\text{End}_R(E, E)$ is a noncommutative local ring with maximal ideal those $\varphi : E \to E$ whose kernel is nonzero, and \item the set of zerodivisors on $E$ is a prime ideal $\mathfrak p$ of $R$ and $E$ is an injective $R_\mathfrak p$-module. \end{enumerate}


dualizing-lemma-injective-hull-indecomposable

Let $\mathfrak p \subset R$ be a prime of a ring $R$. Let $E$ be the injective hull of $R/\mathfrak p$. Then \begin{enumerate} \item $E$ is indecomposable, \item $E$ is the injective hull of $\kappa(\mathfrak p)$, \item $E$ is the injective hull of $\kappa(\mathfrak p)$ over the ring $R_\mathfrak p$. \end{enumerate}


dualizing-lemma-indecomposable-injective-noetherian

Let $R$ be a Noetherian ring. Let $E$ be an indecomposable injective $R$-module. Then there exists a prime ideal $\mathfrak p$ of $R$ such that $E$ is the injective hull of $\kappa(\mathfrak p)$.


dualizing-lemma-finite

Let $(R, \mathfrak m, \kappa)$ be an artinian local ring. Let $E$ be an injective hull of $\kappa$. For every finite $R$-module $M$ we have $$ \text{length}_R(M) = \text{length}_R(\Hom_R(M, E)) $$ In particular, the injective hull $E$ of $\kappa$ is a finite $R$-module.


dualizing-lemma-evaluate

Let $(R, \mathfrak m, \kappa)$ be an artinian local ring. Let $E$ be an injective hull of $\kappa$. For any finite $R$-module $M$ the evaluation map $$ M \longrightarrow \Hom_R(\Hom_R(M, E), E) $$ is an isomorphism. In particular $R = \Hom_R(E, E)$.


dualizing-lemma-duality

Let $(R, \mathfrak m, \kappa)$ be an artinian local ring. Let $E$ be an injective hull of $\kappa$. The functor $D(-) = \Hom_R(-, E)$ induces an exact anti-equivalence $\text{Mod}^{fg}_R \to \text{Mod}^{fg}_R$ and $D \circ D \cong \text{id}$.


dualizing-lemma-duality-torsion-cotorsion

Assumptions and notation as in Lemma \ref{lemma-duality}. Let $I \subset R$ be an ideal and $M$ a finite $R$-module. Then $$ D(M[I]) = D(M)/ID(M) \quad\text{and}\quad D(M/IM) = D(M)[I] $$


dualizing-lemma-quotient

Let $R \to S$ be a surjective map of local rings with kernel $I$. Let $E$ be the injective hull of the residue field of $R$ over $R$. Then $E[I]$ is the injective hull of the residue field of $S$ over $S$.


dualizing-lemma-torsion-submodule-sum-injective-hulls

Let $(R, \mathfrak m, \kappa)$ be a local ring. Let $E$ be the injective hull of $\kappa$. Let $M$ be a $\mathfrak m$-power torsion $R$-module with $n = \dim_\kappa(M[\mathfrak m]) < \infty$. Then $M$ is isomorphic to a submodule of $E^{\oplus n}$.


dualizing-lemma-union-artinian

Let $(R, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $E$ be an injective hull of $\kappa$ over $R$. Let $E_n$ be an injective hull of $\kappa$ over $R/\mathfrak m^n$. Then $E = \bigcup E_n$ and $E_n = E[\mathfrak m^n]$.


dualizing-lemma-compare

Let $R \to S$ be a flat local homomorphism of local Noetherian rings such that $R/\mathfrak m_R \cong S/\mathfrak m_R S$. Then the injective hull of the residue field of $R$ is the injective hull of the residue field of $S$.


dualizing-lemma-endos

Let $(R, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $E$ be an injective hull of $\kappa$ over $R$. Then $\Hom_R(E, E)$ is canonically isomorphic to the completion of $R$.


dualizing-lemma-injective-hull-has-dcc

Let $(R, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $E$ be an injective hull of $\kappa$ over $R$. Then $E$ satisfies the descending chain condition.


dualizing-lemma-describe-categories

Let $(R, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $E$ be an injective hull of $\kappa$. \begin{enumerate} \item For an $R$-module $M$ the following are equivalent: \begin{enumerate} \item $M$ satisfies the ascending chain condition, \item $M$ is a finite $R$-module, and \item there exist $n, m$ and an exact sequence $R^{\oplus m} \to R^{\oplus n} \to M \to 0$. \end{enumerate} \item For an $R$-module $M$ the following are equivalent: \begin{enumerate} \item $M$ satisfies the descending chain condition, \item $M$ is $\mathfrak m$-power torsion and $\dim_\kappa(M[\mathfrak m]) < \infty$, and \item there exist $n, m$ and an exact sequence $0 \to M \to E^{\oplus n} \to E^{\oplus m}$. \end{enumerate} \end{enumerate}


dualizing-lemma-adjoint

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. The functor $R\Gamma_I$ is right adjoint to the functor $D(I^\infty\text{-torsion}) \to D(A)$.


dualizing-lemma-local-cohomology-ext

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. For any object $K$ of $D(A)$ we have $$ R\Gamma_I(K) = \text{hocolim}\ R\Hom_A(A/I^n, K) $$ in $D(A)$ and $$ R^q\Gamma_I(K) = \colim_n \Ext_A^q(A/I^n, K) $$ as modules for all $q \in \mathbf{Z}$.


dualizing-lemma-bad-local-cohomology-vanishes

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Let $K^\bullet$ be a complex of $A$-modules such that $f : K^\bullet \to K^\bullet$ is an isomorphism for some $f \in I$, i.e., $K^\bullet$ is a complex of $A_f$-modules. Then $R\Gamma_I(K^\bullet) = 0$.


dualizing-lemma-not-equal

Let $A$ be a ring and let $I$ be a finitely generated ideal. Let $M$ and $N$ be $I$-power torsion modules. \begin{enumerate} \item $\Hom_{D(A)}(M, N) = \Hom_{D({I^\infty\text{-torsion}})}(M, N)$, \item $\Ext^1_{D(A)}(M, N) = \Ext^1_{D({I^\infty\text{-torsion}})}(M, N)$, \item $\Ext^2_{D({I^\infty\text{-torsion}})}(M, N) \to \Ext^2_{D(A)}(M, N)$ is not surjective in general, \item (\ref{equation-compare-torsion}) is not an equivalence in general. \end{enumerate}


dualizing-lemma-local-cohomology-adjoint

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. There exists a right adjoint $R\Gamma_Z$ (\ref{equation-local-cohomology}) to the inclusion functor $D_{I^\infty\text{-torsion}}(A) \to D(A)$. In fact, if $I$ is generated by $f_1, \ldots, f_r \in A$, then we have $$ R\Gamma_Z(K) = (A \to \prod\nolimits_{i_0} A_{f_{i_0}} \to \prod\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_r}) \otimes_A^\mathbf{L} K $$ functorially in $K \in D(A)$.


dualizing-lemma-local-cohomology-and-restriction

Let $A \to B$ be a ring homomorphism and let $I \subset A$ be a finitely generated ideal. Set $J = IB$. Set $Z = V(I)$ and $Y = V(J)$. Then $$ R\Gamma_Z(M_A) = R\Gamma_Y(M)_A $$ functorially in $M \in D(B)$. Here $(-)_A$ denotes the restriction functors $D(B) \to D(A)$ and $D_{J^\infty\text{-torsion}}(B) \to D_{I^\infty\text{-torsion}}(A)$.


dualizing-lemma-torsion-change-rings

Let $A \to B$ be a ring homomorphism and let $I \subset A$ be a finitely generated ideal. Set $J = IB$. Let $Z = V(I)$ and $Y = V(J)$. Then $$ R\Gamma_Z(K) \otimes_A^\mathbf{L} B = R\Gamma_Y(K \otimes_A^\mathbf{L} B) $$ functorially in $K \in D(A)$.


dualizing-lemma-local-cohomology-vanishes

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Let $K^\bullet$ be a complex of $A$-modules such that $f : K^\bullet \to K^\bullet$ is an isomorphism for some $f \in I$, i.e., $K^\bullet$ is a complex of $A_f$-modules. Then $R\Gamma_Z(K^\bullet) = 0$.


dualizing-lemma-torsion-tensor-product

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. For $K, L \in D(A)$ we have $$ R\Gamma_Z(K \otimes_A^\mathbf{L} L) = K \otimes_A^\mathbf{L} R\Gamma_Z(L) = R\Gamma_Z(K) \otimes_A^\mathbf{L} L = R\Gamma_Z(K) \otimes_A^\mathbf{L} R\Gamma_Z(L) $$ If $K$ or $L$ is in $D_{I^\infty\text{-torsion}}(A)$ then so is $K \otimes_A^\mathbf{L} L$.


dualizing-lemma-local-cohomology-ss

Let $A$ be a ring and let $I, J \subset A$ be finitely generated ideals. Set $Z = V(I)$ and $Y = V(J)$. Then $Z \cap Y = V(I + J)$ and $R\Gamma_Y \circ R\Gamma_Z = R\Gamma_{Y \cap Z}$ as functors $D(A) \to D_{(I + J)^\infty\text{-torsion}}(A)$. For $K \in D^+(A)$ there is a spectral sequence $$ E_2^{p, q} = H^p_Y(H^q_Z(K)) \Rightarrow H^{p + q}_{Y \cap Z}(K) $$ as in Derived Categories, Lemma \ref{derived-lemma-grothendieck-spectral-sequence}.


dualizing-lemma-torsion-flat-change-rings

Let $A \to B$ be a flat ring map and let $I \subset A$ be a finitely generated ideal such that $A/I = B/IB$. Then base change and restriction induce quasi-inverse equivalences $D_{I^\infty\text{-torsion}}(A) = D_{(IB)^\infty\text{-torsion}}(B)$.


dualizing-lemma-local-cohomology-noetherian

Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. \begin{enumerate} \item the adjunction $R\Gamma_I(K) \to K$ is an isomorphism for $K \in D_{I^\infty\text{-torsion}}(A)$, \item the functor (\ref{equation-compare-torsion}) $D(I^\infty\text{-torsion}) \to D_{I^\infty\text{-torsion}}(A)$ is an equivalence, \item the transformation of functors (\ref{equation-compare-torsion-functors}) is an isomorphism, in other words $R\Gamma_I(K) = R\Gamma_Z(K)$ for $K \in D(A)$. \end{enumerate}


dualizing-lemma-compute-local-cohomology-noetherian

Let $A$ be a Noetherian ring and let $I = (f_1, \ldots, f_r)$ be an ideal of $A$. Set $Z = V(I) \subset \Spec(A)$. There are canonical isomorphisms $$ R\Gamma_I(A) \to (A \to \prod\nolimits_{i_0} A_{f_{i_0}} \to \prod\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_r}) \to R\Gamma_Z(A) $$ in $D(A)$. If $M$ is an $A$-module, then we have similarly $$ R\Gamma_I(M) \cong (M \to \prod\nolimits_{i_0} M_{f_{i_0}} \to \prod\nolimits_{i_0 < i_1} M_{f_{i_0}f_{i_1}} \to \ldots \to M_{f_1\ldots f_r}) \cong R\Gamma_Z(M) $$ in $D(A)$.


dualizing-lemma-local-cohomology-change-rings

If $A \to B$ is a homomorphism of Noetherian rings and $I \subset A$ is an ideal, then in $D(B)$ we have $$ R\Gamma_I(A) \otimes_A^\mathbf{L} B = R\Gamma_Z(A) \otimes_A^\mathbf{L} B = R\Gamma_Y(B) = R\Gamma_{IB}(B) $$ where $Y = V(IB) \subset \Spec(B)$.


dualizing-lemma-depth

Let $A$ be a Noetherian ring, let $I \subset A$ be an ideal, and let $M$ be a finite $A$-module such that $IM \not = M$. Then the following integers are equal: \begin{enumerate} \item $\text{depth}_I(M)$, \item the smallest integer $i$ such that $\Ext_A^i(A/I, M)$ is nonzero, and \item the smallest integer $i$ such that $H^i_I(M)$ is nonzero. \end{enumerate} Moreover, we have $\Ext^i_A(N, M) = 0$ for $i < \text{depth}_I(M)$ for any finite $A$-module $N$ annihilated by a power of $I$.


dualizing-lemma-depth-in-ses

Let $A$ be a Noetherian ring. Let $0 \to N' \to N \to N'' \to 0$ be a short exact sequence of finite $A$-modules. Let $I \subset A$ be an ideal. \begin{enumerate} \item $\text{depth}_I(N) \geq \min\{\text{depth}_I(N'), \text{depth}_I(N'')\}$ \item $\text{depth}_I(N'') \geq \min\{\text{depth}_I(N), \text{depth}_I(N') - 1\}$ \item $\text{depth}_I(N') \geq \min\{\text{depth}_I(N), \text{depth}_I(N'') + 1\}$ \end{enumerate}


dualizing-lemma-depth-drops-by-one

Let $A$ be a Noetherian ring, let $I \subset A$ be an ideal, and let $M$ a finite $A$-module with $IM \not = M$. \begin{enumerate} \item If $x \in I$ is a nonzerodivisor on $M$, then $\text{depth}_I(M/xM) = \text{depth}_I(M) - 1$. \item Any $M$-regular sequence $x_1, \ldots, x_r$ in $I$ can be extended to an $M$-regular sequence in $I$ of length $\text{depth}_I(M)$. \end{enumerate}


dualizing-lemma-depth-CM

Let $R$ be a Noetherian local ring. If $M$ is a finite Cohen-Macaulay $R$-module and $I \subset R$ a nontrivial ideal. Then $$ \text{depth}_I(M) = \dim(\text{Supp}(M)) - \dim(\text{Supp}(M/IM)). $$


dualizing-lemma-depth-flat-CM

Let $R \to S$ be a flat local ring homomorphism of Noetherian local rings. Denote $\mathfrak m \subset R$ the maximal ideal. Let $I \subset S$ be an ideal. If $S/\mathfrak mS$ is Cohen-Macaulay, then $$ \text{depth}_I(S) \geq \dim(S/\mathfrak mS) - \dim(S/\mathfrak mS + I) $$


dualizing-lemma-divide-by-torsion

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Let $M$ be an $A$-module. Let $Z = V(I)$. Then $H^0_I(M) = H^0_Z(M)$. Let $N$ be the common value and set $M' = M/N$. Then \begin{enumerate} \item $H^0_I(M') = 0$ and $H^p_I(M) = H^p_I(M')$ and $H^p_I(N) = 0$ for all $p > 0$, \item $H^0_Z(M') = 0$ and $H^p_Z(M) = H^p_Z(M')$ and $H^p_Z(N) = 0$ for all $p > 0$. \end{enumerate}


dualizing-lemma-complete-and-local

\begin{slogan} Results of this nature are sometimes referred to as Greenlees-May duality. \end{slogan} Let $A$ be a ring and let $I$ be a finitely generated ideal. Let $R\Gamma_Z$ be as in Lemma \ref{lemma-local-cohomology-adjoint}. Let ${\ }^\wedge$ denote derived completion as in More on Algebra, Lemma \ref{more-algebra-lemma-derived-completion}. For an object $K$ in $D(A)$ we have $$ R\Gamma_Z(K^\wedge) = R\Gamma_Z(K) \quad\text{and}\quad (R\Gamma_Z(K))^\wedge = K^\wedge $$ in $D(A)$.


dualizing-lemma-compare-RHom

With notation as in Lemma \ref{lemma-complete-and-local}. For objects $K, L$ in $D(A)$ there is a canonical isomorphism $$ R\Hom_A(K^\wedge, L^\wedge) \longrightarrow R\Hom_A(R\Gamma_Z(K), R\Gamma_Z(L)) $$ in $D(A)$.


dualizing-lemma-completion-local

Let $I$ and $J$ be ideals in a Noetherian ring $A$. Let $M$ be a finite $A$-module. Set $Z =V(J)$. Consider the derived $I$-adic completion $R\Gamma_Z(M)^\wedge$ of local cohomology. Then \begin{enumerate} \item we have $R\Gamma_Z(M)^\wedge = R\lim R\Gamma_Z(M/I^nM)$, and \item there are short exact sequences $$ 0 \to R^1\lim H^{i - 1}_Z(M/I^nM) \to H^i(R\Gamma_Z(M)^\wedge) \to \lim H^i_Z(M/I^nM) \to 0 $$ \end{enumerate} In particular $R\Gamma_Z(M)^\wedge$ has vanishing cohomology in negative degrees.


dualizing-lemma-completion-local-H0

With notation and hypotheses as in Lemma \ref{lemma-completion-local} assume $A$ is $I$-adically complete. Then $$ H^0(R\Gamma_Z(M)^\wedge) = \colim H^0_{V(J')}(M) $$ where the filtered colimit is over $J' \subset J$ such that $V(J') \cap V(I) = V(J) \cap V(I)$.


dualizing-lemma-right-adjoint

Let $A \to B$ be a ring homomorphism. The functor $R\Hom(B, -)$ constructed above is right adjoint to the restriction functor $D(B) \to D(A)$.


dualizing-lemma-composition-right-adjoints

Let $A \to B \to C$ be ring maps. Then $R\Hom(C, -) \circ R\Hom(B, -) : D(A) \to D(C)$ is the functor $R\Hom(C, -) : D(A) \to D(C)$.


dualizing-lemma-RHom-ext

Let $\varphi : A \to B$ be a ring homomorphism. For $K$ in $D(A)$ we have $$ \varphi_*R\Hom(B, K) = R\Hom_A(B, K) $$ where $\varphi_* : D(B) \to D(A)$ is restriction. In particular $R^q\Hom(B, K) = \Ext_A^q(B, K)$.


dualizing-lemma-exact-support-coherent

With notation as above, assume $A \to B$ is a finite ring map of Noetherian rings. Then $R\Hom(B, -)$ maps $D^+_{\textit{Coh}}(A)$ into $D^+_{\textit{Coh}}(B)$.


dualizing-lemma-RHom-dga

In Situation \ref{situation-resolution} the functor $R\Hom(A, -)$ is equal to the composition of $R\Hom(E, -) : D(R) \to D(E, \text{d})$ and the equivalence $- \otimes^\mathbf{L}_E A : D(E, \text{d}) \to D(A)$.


dualizing-lemma-RHom-is-tensor

In Situation \ref{situation-resolution} assume that \begin{enumerate} \item $E$ viewed as an object of $D(R)$ is compact, and \item $N = \Hom^\bullet_R(E^\bullet, R)$ computes $R\Hom(E, R)$. \end{enumerate} Then $R\Hom(E, -) : D(R) \to D(E)$ is isomorphic to $K \mapsto K \otimes_R^\mathbf{L} N$.


dualizing-lemma-RHom-is-tensor-special

In Situation \ref{situation-resolution} assume $A$ is a perfect $R$-module. Then $$ R\Hom(A, -) : D(R) \to D(A) $$ is given by $K \mapsto K \otimes_R^\mathbf{L} M$ where $M = R\Hom(A, R) \in D(A)$.


dualizing-lemma-compute-for-effective-Cartier-algebraic

Let $R \to A$ be a surjective ring map whose kernel $I$ is an invertible $R$-module. The functor $R\Hom(A, -) : D(R) \to D(A)$ is isomorphic to $K \mapsto K \otimes_R^\mathbf{L} N[-1]$ where $N$ is inverse of the invertible $A$-module $I \otimes_R A$.


dualizing-lemma-check-base-change-is-iso

In the situation above, the map (\ref{equation-base-change}) is an isomorphism if and only if the map $$ R\Hom_R(A, K) \otimes_R^\mathbf{L} R' \longrightarrow R\Hom_R(A, K \otimes_R^\mathbf{L} R') $$ of More on Algebra, Lemma \ref{more-algebra-lemma-internal-hom-diagonal-better} is an isomorphism.


dualizing-lemma-flat-bc-surjection

Let $R \to A$ and $R \to R'$ be ring maps and $A' = A \otimes_R R'$. Assume \begin{enumerate} \item $A$ is pseudo-coherent as an $R$-module, \item $R'$ has finite tor dimension as an $R$-module (for example $R \to R'$ is flat), \item $A$ and $R'$ are tor independent over $R$. \end{enumerate} Then (\ref{equation-base-change}) is an isomorphism for $K \in D^+(R)$.


dualizing-lemma-bc-surjection

Let $R \to A$ and $R \to R'$ be ring maps and $A' = A \otimes_R R'$. Assume \begin{enumerate} \item $A$ is perfect as an $R$-module, \item $A$ and $R'$ are tor independent over $R$. \end{enumerate} Then (\ref{equation-base-change}) is an isomorphism for all $K \in D(R)$.


dualizing-lemma-finite-ext-into-bounded-injective

Let $A$ be a Noetherian ring. Let $K, L \in D_{\textit{Coh}}(A)$ and assume $L$ has finite injective dimension. Then $R\Hom_A(K, L)$ is in $D_{\textit{Coh}}(A)$.


dualizing-lemma-dualizing

Let $A$ be a Noetherian ring. If $\omega_A^\bullet$ is a dualizing complex, then the functor $$ D : K \longmapsto R\Hom_A(K, \omega_A^\bullet) $$ is an anti-equivalence $D_{\textit{Coh}}(A) \to D_{\textit{Coh}}(A)$ which exchanges $D^+_{\textit{Coh}}(A)$ and $D^-_{\textit{Coh}}(A)$ and induces an anti-equivalence $D^b_{\textit{Coh}}(A) \to D^b_{\textit{Coh}}(A)$. Moreover $D \circ D$ is isomorphic to the identity functor.


dualizing-lemma-equivalence-comes-from-invertible

Let $A$ be a Noetherian ring. Let $F : D^b_{\textit{Coh}}(A) \to D^b_{\textit{Coh}}(A)$ be an $A$-linear equivalence of categories. Then $F(A)$ is an invertible object of $D(A)$.


dualizing-lemma-dualizing-unique

Let $A$ be a Noetherian ring. If $\omega_A^\bullet$ and $(\omega'_A)^\bullet$ are dualizing complexes, then $(\omega'_A)^\bullet$ is quasi-isomorphic to $\omega_A^\bullet \otimes_A^\mathbf{L} L$ for some invertible object $L$ of $D(A)$.


dualizing-lemma-dualizing-localize

Let $A$ be a Noetherian ring. Let $B = S^{-1}A$ be a localization. If $\omega_A^\bullet$ is a dualizing complex, then $\omega_A^\bullet \otimes_A B$ is a dualizing complex for $B$.


dualizing-lemma-dualizing-glue

Let $A$ be a Noetherian ring. Let $f_1, \ldots, f_n \in A$ generate the unit ideal. If $\omega_A^\bullet$ is a complex of $A$-modules such that $(\omega_A^\bullet)_{f_i}$ is a dualizing complex for $A_{f_i}$ for all $i$, then $\omega_A^\bullet$ is a dualizing complex for $A$.


dualizing-lemma-dualizing-finite

Let $A \to B$ be a finite ring map of Noetherian rings. Let $\omega_A^\bullet$ be a dualizing complex. Then $R\Hom(B, \omega_A^\bullet)$ is a dualizing complex for $B$.


dualizing-lemma-dualizing-quotient

Let $A \to B$ be a surjective homomorphism of Noetherian rings. Let $\omega_A^\bullet$ be a dualizing complex. Then $R\Hom(B, \omega_A^\bullet)$ is a dualizing complex for $B$.


dualizing-lemma-dualizing-polynomial-ring

Let $A$ be a Noetherian ring. If $\omega_A^\bullet$ is a dualizing complex, then $\omega_A^\bullet \otimes_A A[x]$ is a dualizing complex for $A[x]$.


dualizing-lemma-find-function

Let $A$ be a Noetherian ring. Let $\omega_A^\bullet$ be a dualizing complex. Let $\mathfrak m \subset A$ be a maximal ideal and set $\kappa = A/\mathfrak m$. Then $R\Hom_A(\kappa, \omega_A^\bullet) \cong \kappa[n]$ for some $n \in \mathbf{Z}$.


dualizing-lemma-normalized-finite

Let $(A, \mathfrak m, \kappa) \to (B, \mathfrak m', \kappa')$ be a finite local map of Noetherian local rings. Let $\omega_A^\bullet$ be a normalized dualizing complex. Then $\omega_B^\bullet = R\Hom(B, \omega_A^\bullet)$ is a normalized dualizing complex for $B$.


dualizing-lemma-normalized-quotient

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring with normalized dualizing complex $\omega_A^\bullet$. Let $A \to B$ be surjective. Then $\omega_B^\bullet = R\Hom_A(B, \omega_A^\bullet)$ is a normalized dualizing complex for $B$.


dualizing-lemma-equivalence-finite-length

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $F$ be an $A$-linear self-equivalence of the category of finite length $A$-modules. Then $F$ is isomorphic to the identity functor.


dualizing-lemma-dualizing-finite-length

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring with normalized dualizing complex $\omega_A^\bullet$. Let $E$ be an injective hull of $\kappa$. Then there exists a functorial isomorphism $$ R\Hom_A(N, \omega_A^\bullet) = \Hom_A(N, E)[0] $$ for $N$ running through the finite length $A$-modules.


dualizing-lemma-sitting-in-degrees

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring with normalized dualizing complex $\omega_A^\bullet$. Let $M$ be a finite $A$-module and let $d = \dim(\text{Supp}(M))$. Then \begin{enumerate} \item if $\Ext^i_A(M, \omega_A^\bullet)$ is nonzero, then $i \in \{-d, \ldots, 0\}$, \item the dimension of the support of $\Ext^i_A(M, \omega_A^\bullet)$ is at most $-i$, \item $\text{depth}(M)$ is the smallest integer $\delta \geq 0$ such that $\Ext^{-\delta}_A(M, \omega_A^\bullet) \not = 0$. \end{enumerate}


dualizing-lemma-local-CM

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring with normalized dualizing complex $\omega_A^\bullet$. Let $M$ be a finite $A$-module. The following are equivalent \begin{enumerate} \item $M$ is Cohen-Macaulay, \item $\Ext^i_A(M, \omega_A^\bullet)$ is nonzero for a single $i$, \item $\Ext^{-i}_A(M, \omega_A^\bullet)$ is zero for $i \not = \dim(\text{Supp}(M))$. \end{enumerate} Denote $CM_d$ the category of finite Cohen-Macaulay $A$-modules of depth $d$. Then $M \mapsto \Ext^{-d}_A(M, \omega_A^\bullet)$ defines an anti-auto-equivalence of $CM_d$.


dualizing-lemma-dualizing-artinian

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring with normalized dualizing complex $\omega_A^\bullet$. If $\dim(A) = 0$, then $\omega_A^\bullet \cong E[0]$ where $E$ is an injective hull of the residue field.


dualizing-lemma-divide-by-finite-length-ideal

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring with normalized dualizing complex. Let $I \subset \mathfrak m$ be an ideal of finite length. Set $B = A/I$. Then there is a distinguished triangle $$ \omega_B^\bullet \to \omega_A^\bullet \to \Hom_A(I, E)[0] \to \omega_B^\bullet[1] $$ in $D(A)$ where $E$ is an injective hull of $\kappa$ and $\omega_B^\bullet$ is a normalized dualizing complex for $B$.


dualizing-lemma-divide-by-nonzerodivisor

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring with normalized dualizing complex $\omega_A^\bullet$. Let $f \in \mathfrak m$ be a nonzerodivisor. Set $B = A/(f)$. Then there is a distinguished triangle $$ \omega_B^\bullet \to \omega_A^\bullet \to \omega_A^\bullet \to \omega_B^\bullet[1] $$ in $D(A)$ where $\omega_B^\bullet$ is a normalized dualizing complex for $B$.


dualizing-lemma-nonvanishing-generically-local

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring with normalized dualizing complex $\omega_A^\bullet$. Let $\mathfrak p$ be a minimal prime of $A$ with $\dim(A/\mathfrak p) = e$. Then $H^i(\omega_A^\bullet)_\mathfrak p$ is nonzero if and only if $i = -e$.


dualizing-lemma-nonvanishing-generically

Let $A$ be a Noetherian ring. Let $\mathfrak p$ be a minimal prime of $A$. Then $H^i(\omega_A^\bullet)_\mathfrak p$ is nonzero for exactly one $i$.


dualizing-lemma-quotient-function

Let $A$ be a Noetherian ring and let $\omega_A^\bullet$ be a dualizing complex. Let $A \to B$ be a surjective ring map and let $\omega_B^\bullet = R\Hom(B, \omega_A^\bullet)$ be the dualizing complex for $B$ of Lemma \ref{lemma-dualizing-quotient}. Then we have $$ \delta_{\omega_B^\bullet} = \delta_{\omega_A^\bullet}|_{\Spec(B)} $$


dualizing-lemma-dimension-function

Let $A$ be a Noetherian ring and let $\omega_A^\bullet$ be a dualizing complex. The function $\delta = \delta_{\omega_A^\bullet}$ defined above is a dimension function (Topology, Definition \ref{topology-definition-dimension-function}).


dualizing-lemma-universally-catenary

Let $A$ be a Noetherian ring which has a dualizing complex. Then $A$ is universally catenary of finite dimension.


dualizing-lemma-depth-dualizing-module

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring with normalized dualizing complex $\omega_A^\bullet$. Let $d = \dim(A)$ and $\omega_A = H^{-d}(\omega_A^\bullet)$. Then \begin{enumerate} \item the support of $\omega_A$ is the union of the irreducible components of $\Spec(A)$ of dimension $d$, \item $\omega_A$ satisfies $(S_2)$, see Algebra, Definition \ref{algebra-definition-conditions}. \end{enumerate}


dualizing-lemma-local-cohomology-of-dualizing

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $\omega_A^\bullet$ be a normalized dualizing complex. Let $Z = V(\mathfrak m) \subset \Spec(A)$. Then $E = R^0\Gamma_Z(\omega_A^\bullet)$ is an injective hull of $\kappa$ and $R\Gamma_Z(\omega_A^\bullet) = E[0]$.


dualizing-lemma-special-case-local-duality

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $\omega_A^\bullet$ be a normalized dualizing complex. Let $E$ be an injective hull of the residue field. Let $K \in D_{\textit{Coh}}(A)$. Then $$ \Ext^{-i}_A(K, \omega_A^\bullet)^\wedge = \Hom_A(H^i_{\mathfrak m}(K), E) $$ where ${}^\wedge$ denotes $\mathfrak m$-adic completion.


dualizing-lemma-depth-in-terms-dualizing-complex

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring with normalized dualizing complex $\omega_A^\bullet$. Then $\text{depth}(A)$ is equal to the smallest integer $\delta \geq 0$ such that $H^{-\delta}(\omega_A^\bullet) \not = 0$.


dualizing-lemma-apply-CM

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring with normalized dualizing complex $\omega_A^\bullet$ and dualizing module $\omega_A = H^{-\dim(A)}(\omega_A^\bullet)$. The following are equivalent \begin{enumerate} \item $A$ is Cohen-Macaulay, \item $\omega_A^\bullet$ is concentrated in a single degree, and \item $\omega_A^\bullet = \omega_A[\dim(A)]$. \end{enumerate} In this case $\omega_A$ is a maximal Cohen-Macaulay module.


dualizing-lemma-has-dualizing-module-CM

Let $A$ be a Noetherian ring. If there exists a finite $A$-module $\omega_A$ such that $\omega_A[0]$ is a dualizing complex, then $A$ is Cohen-Macaulay.


dualizing-lemma-CM-open

Let $A$ be a Noetherian ring with dualizing complex $\omega_A^\bullet$. Let $M$ be a finite $A$-module. Then $$ U = \{\mathfrak p \in \Spec(A) \mid M_\mathfrak p\text{ is Cohen-Macaulay}\} $$ is an open subset of $\Spec(A)$ whose intersection with $\text{Supp}(M)$ is dense.


dualizing-lemma-gorenstein-CM

A Gorenstein ring is Cohen-Macaulay.


dualizing-lemma-regular-gorenstein

A regular local ring is Gorenstein. A regular ring is Gorenstein.


dualizing-lemma-gorenstein

Let $A$ be a Noetherian ring. \begin{enumerate} \item If $A$ has a dualizing complex $\omega_A^\bullet$, then \begin{enumerate} \item $A$ is Gorenstein $\Leftrightarrow$ $\omega_A^\bullet$ is an invertible object of $D(A)$, \item $A_\mathfrak p$ is Gorenstein $\Leftrightarrow$ $(\omega_A^\bullet)_\mathfrak p$ is an invertible object of $D(A_\mathfrak p)$, \item $\{\mathfrak p \in \Spec(A) \mid A_\mathfrak p\text{ is Gorenstein}\}$ is an open subset. \end{enumerate} \item If $A$ is Gorenstein, then $A$ has a dualizing complex if and only if $A[0]$ is a dualizing complex. \end{enumerate}


dualizing-lemma-gorenstein-ext

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring. Then $A$ is Gorenstein if and only if $\Ext^i_A(\kappa, A)$ is zero for $i \gg 0$.


dualizing-lemma-gorenstein-divide-by-nonzerodivisor

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $f \in \mathfrak m$ be a nonzerodivisor. Set $B = A/(f)$. Then $A$ is Gorenstein if and only if $B$ is Gorenstein.


dualizing-lemma-gorenstein-lci

If $A \to B$ is a local complete intersection homomorphism of rings and $A$ is a Noetherian Gorenstein ring, then $B$ is a Gorenstein ring.


dualizing-lemma-flat-under-gorenstein

Let $A \to B$ be a flat local homomorphism of Noetherian local rings. The following are equivalent \begin{enumerate} \item $B$ is Gorenstein, and \item $A$ and $B/\mathfrak m_A B$ are Gorenstein. \end{enumerate}


dualizing-lemma-tor-injective-hull

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local Gorenstein ring of dimension $d$. Let $E$ be the injective hull of $\kappa$. Then $\text{Tor}_i^A(E, \kappa)$ is zero for $i \not = d$ and $\text{Tor}_d^A(E, \kappa) = \kappa$.


dualizing-lemma-flat-unramified

Let $A \to B$ be a local homomorphism of Noetherian local rings. Let $\omega_A^\bullet$ be a normalized dualizing complex. If $A \to B$ is flat and $\mathfrak m_A B = \mathfrak m_B$, then $\omega_A^\bullet \otimes_A B$ is a normalized dualizing complex for $B$.


dualizing-lemma-flat-iso-mod-I

Let $A \to B$ be a flat map of Noetherian rings. Let $I \subset A$ be an ideal such that $A/I = B/IB$ and such that $IB$ is contained in the Jacobson radical of $B$. Let $\omega_A^\bullet$ be a dualizing complex. Then $\omega_A^\bullet \otimes_A B$ is a dualizing complex for $B$.


dualizing-lemma-completion-henselization-dualizing

Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Let $\omega_A^\bullet$ be a dualizing complex. \begin{enumerate} \item $\omega_A^\bullet \otimes_A A^h$ is a dualizing complex on the henselization $(A^h, I^h)$ of the pair $(A, I)$, \item $\omega_A^\bullet \otimes_A A^\wedge$ is a dualizing complex on the $I$-adic completion $A^\wedge$, and \item if $A$ is local, then $\omega_A^\bullet \otimes_A A^h$, resp.\ $\omega_A^\bullet \otimes_A A^{sh}$ is a dualzing complex on the henselization, resp.\ strict henselization of $A$. \end{enumerate}


dualizing-lemma-ubiquity-dualizing

The following types of rings have a dualizing complex: \begin{enumerate} \item fields, \item Noetherian complete local rings, \item $\mathbf{Z}$, \item Dedekind domains, \item any ring which is obtained from one of the rings above by taking an algebra essentially of finite type, or by taking an ideal-adic completion, or by taking a henselization,  or by taking a strict henselization. \end{enumerate}


dualizing-lemma-formal-fibres-gorenstein

Properties (A), (B), (C), (D), and (E) of More on Algebra, Section \ref{more-algebra-section-properties-formal-fibres} hold for $P(k \to R) =$``$R$ is a Gorenstein ring''.


dualizing-lemma-dualizing-gorenstein-formal-fibres

Let $A$ be a Noetherian local ring. If $A$ has a dualizing complex, then the formal fibres of $A$ are Gorenstein.


dualizing-lemma-shriek-boundedness

Let $\varphi : R \to A$ be a finite type homomorphism of Noetherian rings. \begin{enumerate} \item $\varphi^!$ maps $D^+(R)$ into $D^+(A)$ and $D^+_{\textit{Coh}}(R)$ into $D^+_{\textit{Coh}}(A)$. \item if $\varphi$ is perfect, then $\varphi^!$ maps $D^-(R)$ into $D^-(A)$, $D^-_{\textit{Coh}}(R)$ into $D^-_{\textit{Coh}}(A)$, and $D^b_{\textit{Coh}}(R)$ into $D^b_{\textit{Coh}}(A)$. \end{enumerate}


dualizing-lemma-shriek-dualizing-algebraic

Let $\varphi$ be a finite type homomorphism of Noetherian rings. If $\omega_R^\bullet$ is a dualizing complex for $R$, then $\varphi^!(\omega_R^\bullet)$ is a dualizing complex for $A$.


dualizing-lemma-flat-bc

Let $R \to R'$ be a flat homomorphism of Noetherian rings. Let $\varphi : R \to A$ be a finite type ring map. Let $\varphi' : R' \to A' = A \otimes_R R'$ be the map induced by $\varphi$. Then we have a functorial maps $$ \varphi^!(K) \otimes_A^\mathbf{L} A' \longrightarrow (\varphi')^!(K \otimes_R^\mathbf{L} R') $$ for $K$ in $D(R)$ which are isomorphisms for $K \in D^+(R)$.


dualizing-lemma-bc

Let $R \to R'$ be a homomorphism of Noetherian rings. Let $\varphi : R \to A$ be a perfect ring map (More on Algebra, Definition \ref{more-algebra-definition-pseudo-coherent-perfect}) such that $R'$ and $A$ are tor independent over $R$. Let $\varphi' : R' \to A' = A \otimes_R R'$ be the map induced by $\varphi$. Then we have a functorial isomorphism $$ \varphi^!(K) \otimes_A^\mathbf{L} A' = (\varphi')^!(K \otimes_R^\mathbf{L} R') $$ for $K$ in $D(R)$.


dualizing-lemma-bc-flat

Let $R \to R'$ be a homomorphism of Noetherian rings. Let $\varphi : R \to A$ be flat of finite type. Let $\varphi' : R' \to A' = A \otimes_R R'$ be the map induced by $\varphi$. Then we have a functorial isomorphism $$ \varphi^!(K) \otimes_A^\mathbf{L} A' = (\varphi')^!(K \otimes_R^\mathbf{L} R') $$ for $K$ in $D(R)$.


dualizing-lemma-composition-shriek-algebraic

Let $A \xrightarrow{a} B \xrightarrow{b} C$ be finite type homomorphisms of Noetherian rings. Then there is a transformation of functors $b^! \circ a^! \to (b \circ a)^!$ which is an isomorphism on $D^+(A)$.


dualizing-lemma-upper-shriek-finite

Let $\varphi : R \to A$ be a finite map of Noetherian rings. Then $\varphi^!$ is isomorphic to the functor $R\Hom(A, -) : D(R) \to D(A)$ from Section \ref{section-trivial}.


dualizing-lemma-upper-shriek-localize

Let $R$ be a Noetherian ring and let $f \in R$. If $\varphi$ denotes the map $R \to R_f$, then $\varphi^!$ is isomorphic to $- \otimes_R^\mathbf{L} R_f$. More generally, if $\varphi : R \to R'$ is a map such that $\Spec(R') \to \Spec(R)$ is an open immersion, then $\varphi^!$ is isomorphic to $- \otimes_R^\mathbf{L} R'$.


dualizing-lemma-upper-shriek-is-tensor-functor

Let $\varphi : R \to A$ be a perfect homomorphism of Noetherian rings (for example $\varphi$ is flat of finite type). Then $\varphi^!(K) = K \otimes_R^\mathbf{L} \varphi^!(R)$ for $K \in D(R)$.


dualizing-lemma-relative-dualizing-if-have-omega

Let $\varphi : A \to B$ be a finite type homomorphism of Noetherian rings. Let $\omega_A^\bullet$ be a dualizing complex for $A$. Set $\omega_B^\bullet = \varphi^!(\omega_A^\bullet)$. Denote $D_A(K) = R\Hom_A(K, \omega_A^\bullet)$ for $K \in D_{\textit{Coh}}(A)$ and $D_B(L) = R\Hom_B(L, \omega_B^\bullet)$ for $L \in D_{\textit{Coh}}(B)$. Then there is a functorial isomorphism $$ \varphi^!(K) = D_B(D_A(K) \otimes_A^\mathbf{L} B) $$ for $K \in D_{\textit{Coh}}(A)$.


dualizing-lemma-base-change-relative-algebraic

Let $R \to R'$ be a homomorphism of Noetherian rings. Let $R \to A$ be of finite type. Set $A' = A \otimes_R R'$. If \begin{enumerate} \item $R \to R'$ is flat, or \item $R \to A$ is flat, or \item $R \to A$ is perfect and $R'$ and $A$ are tor independent over $R$, \end{enumerate} then there is an isomorphism $\omega_{A/R}^\bullet \otimes_A^\mathbf{L} A' \to \omega^\bullet_{A'/R'}$ in $D(A')$.


dualizing-lemma-relative-dualizing-algebraic

Let $\varphi : R \to A$ be a flat finite type map of Noetherian rings. Then \begin{enumerate} \item $\omega_{A/R}^\bullet$ is in $D^b_{\textit{Coh}}(A)$ and $R$-perfect (More on Algebra, Definition \ref{more-algebra-definition-relatively-perfect}), \item $A \to R\Hom_A(\omega_{A/R}^\bullet, \omega_{A/R}^\bullet)$ is an isomorphism, and \item for every map $R \to k$ to a field the base change $\omega_{A/R}^\bullet \otimes_A^\mathbf{L} (A \otimes_R k)$ is a dualizing complex for $A \otimes_R k$. \end{enumerate}


dualizing-lemma-base-change-dualizing-over-field

Let $K/k$ be an extension of fields. Let $A$ be a finite type $k$-algebra. Let $A_K = A \otimes_k K$. If $\omega_A^\bullet$ is a dualizing complex for $A$, then $\omega_A^\bullet \otimes_A A_K$ is a dualizing complex for $A_K$.


dualizing-lemma-lci-shriek

Let $\varphi : R \to A$ be a local complete intersection homomorphism of Noetherian rings. Then $\omega_{A/R}^\bullet$ is an invertible object of $D(A)$ and $\varphi^!(K) = K \otimes_R^\mathbf{L} \omega_{A/R}^\bullet$ for all $K \in D(R)$.


dualizing-lemma-gorenstein-shriek

Let $\varphi : R \to A$ be a flat finite type homomorphism of Noetherian rings. The following are equivalent \begin{enumerate} \item the fibres $A \otimes_R \kappa(\mathfrak p)$ are Gorenstein for all primes $\mathfrak p \subset R$, and \item $\omega_{A/R}^\bullet$ is an invertible object of $D(A)$, see More on Algebra, Lemma \ref{more-algebra-lemma-invertible-derived}. \end{enumerate}


dualizing-lemma-shriek-normalized

Let $\varphi : R \to A$ be a finite type homomorphism of Noetherian rings. Assume $R$ local and let $\mathfrak m \subset A$ be a maximal ideal lying over the maximal ideal of $R$. If $\omega_R^\bullet$ is a normalized dualizing complex for $R$, then $\varphi^!(\omega_R^\bullet)_\mathfrak m$ is a normalized dualizing complex for $A_\mathfrak m$.


dualizing-lemma-relative-dualizing-trivial-vanishing

Let $R \to A$ be a finite type homomorphism of Noetherian rings. Let $\mathfrak q \subset A$ be a prime ideal lying over $\mathfrak p \subset R$. Then $$ H^i(\omega_{A/R}^\bullet)_\mathfrak q \not = 0 \Rightarrow - d \leq i $$ where $d$ is the dimension of the fibre of $\Spec(A) \to \Spec(R)$ over $\mathfrak p$ at the point $\mathfrak q$.


dualizing-lemma-relative-dualizing-flat-vanishing

Let $R \to A$ be a flat finite type homomorphism of Noetherian rings. Let $\mathfrak q \subset A$ be a prime ideal lying over $\mathfrak p \subset R$. Then $$ H^i(\omega_{A/R}^\bullet)_\mathfrak q \not = 0 \Rightarrow - d \leq i \leq 0 $$ where $d$ is the dimension of the fibre of $\Spec(A) \to \Spec(R)$ over $\mathfrak p$ at the point $\mathfrak q$. If all fibres of $\Spec(A) \to \Spec(R)$ have dimension $\leq d$, then $\omega_{A/R}^\bullet$ has tor amplitude in $[-d, 0]$ as a complex of $R$-modules.


dualizing-lemma-relative-dualizing-CM-vanishing

Let $R \to A$ be a finite type homomorphism of Noetherian rings. Let $\mathfrak p \subset R$ be a prime ideal. Assume \begin{enumerate} \item $R_\mathfrak p$ is Cohen-Macaulay, and \item for any minimal prime $\mathfrak q \subset A$ we have $\text{trdeg}_{\kappa(R \cap \mathfrak q)} \kappa(\mathfrak q) \leq r$. \end{enumerate} Then $$ H^i(\omega_{A/R}^\bullet)_\mathfrak p \not = 0 \Rightarrow - r \leq i $$ and $H^{-r}(\omega_{A/R}^\bullet)_\mathfrak p$ is $(S_2)$ as an $A_\mathfrak p$-module.


dualizing-lemma-descent

Let $A \to B$ be a faithfully flat map of Noetherian rings. If $K \in D(A)$ and $K \otimes_A^\mathbf{L} B$ is a dualizing complex for $B$, then $K$ is a dualizing complex for $A$.


dualizing-lemma-descent-ascent

Let $\varphi : A \to B$ be a homomorphism of Noetherian rings. Assume \begin{enumerate} \item $A \to B$ is syntomic and induces a surjective map on spectra, or \item $A \to B$ is a faithfully flat local complete intersection, or \item $A \to B$ is faithfully flat of finite type with Gorenstein fibres. \end{enumerate} Then $K \in D(A)$ is a dualizing complex for $A$ if and only if $K \otimes_A^\mathbf{L} B$ is a dualizing complex for $B$.


dualizing-lemma-injective-hull-goes-up

Let $(A, \mathfrak m, \kappa) \to (B, \mathfrak n, l)$ be a flat local homorphism of Noetherian rings such that $\mathfrak n = \mathfrak m B$. If $E$ is the injective hull of $\kappa$, then $E \otimes_A B$ is the injective hull of $l$.


dualizing-lemma-injective-goes-up

Let $\varphi : A \to B$ be a flat homorphism of Noetherian rings such that for all primes $\mathfrak q \subset B$ we have $\mathfrak p B_\mathfrak q = \mathfrak qB_\mathfrak q$ where $\mathfrak p = \varphi^{-1}(\mathfrak q)$, for example if $\varphi$ is \'etale. If $I$ is an injective $A$-module, then $I \otimes_A B$ is an injective $B$-module.


dualizing-lemma-uniqueness-relative-dualizing

Let $R \to A$ be a flat ring map of finite presentation. Any two relative dualizing complexes for $R \to A$ are isomorphic.


dualizing-lemma-relative-dualizing-noetherian

Let $\varphi : R \to A$ be a flat finite type ring map of Noetherian rings. Then the relative dualizing complex $\omega_{A/R}^\bullet = \varphi^!(R)$ of Section \ref{section-relative-dualizing-complexes-Noetherian} is a relative dualizing complex in the sense of Definition \ref{definition-relative-dualizing-complex}.


dualizing-lemma-base-change-relative-dualizing

Let $R \to A$ be a flat ring map of finite presentation. Then \begin{enumerate} \item there exists a relative dualizing complex $K$ in $D(A)$, and \item for any ring map $R \to R'$ setting $A' = A \otimes_R R'$ and $K' = K \otimes_A^\mathbf{L} A'$, then $K'$ is a relative dualizing complex for $R' \to A'$. \end{enumerate} Moreover, if $$ \xi : A \longrightarrow K \otimes_A^\mathbf{L} (A \otimes_R A) $$ is a generator for the cyclic module $\Hom_{D(A \otimes_R A)}(A, K \otimes_A^\mathbf{L} (A \otimes_R A))$ then in (2) the derived base change of $\xi$ by $A \otimes_R A \to A' \otimes_{R'} A'$ is a generator for the cyclic module $\Hom_{D(A' \otimes_{R'} A')}(A', K' \otimes_{A'}^\mathbf{L} (A' \otimes_{R'} A'))$


dualizing-lemma-relative-dualizing-RHom

Let $R \to A$ be a flat ring map of finite presentation. Let $K$ be a relative dualizing complex. Then $A \to R\Hom_A(K, K)$ is an isomorphism.


dualizing-lemma-relative-dualizing-composition

Let $R \to A \to B$ be a ring maps which are flat and of finite presentation. Let $K_{A/R}$ and $K_{B/A}$ be relative dualizing complexes for $R \to A$ and $A \to B$. Then $K = K_{A/R} \otimes_A^\mathbf{L} K_{B/A}$ is a relative dualizing complex for $R \to B$.


dualizing-proposition-structure-injectives-noetherian

Let $R$ be a Noetherian ring. Every injective module is a direct sum of indecomposable injective modules. Every indecomposable injective module is the injective hull of the residue field at a prime.


dualizing-proposition-matlis

Let $(R, \mathfrak m, \kappa)$ be a complete local Noetherian ring. Let $E$ be an injective hull of $\kappa$ over $R$. The functor $D(-) = \Hom_R(-, E)$ induces an anti-equivalence $$ \left\{ \begin{matrix} R\text{-modules with the} \\ \text{descending chain condition} \end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} R\text{-modules with the} \\ \text{ascending chain condition} \end{matrix} \right\} $$ and we have $D \circ D = \text{id}$ on either side of the equivalence.


dualizing-proposition-torsion-complete

\begin{reference} This is a special case of \cite[Theorem 1.1]{Porta-Liran-Yekutieli}. \end{reference} Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. The functors $R\Gamma_Z$ and ${\ }^\wedge$ define quasi-inverse equivalences of categories $$ D_{I^\infty\text{-torsion}}(A) \leftrightarrow D_{comp}(A, I) $$


dualizing-proposition-dualizing-essentially-finite-type

Let $A$ be a Noetherian ring which has a dualizing complex. Then any $A$-algebra essentially of finite type over $A$ has a dualizing complex.


properties-lemma-locally-constructible

Let $X$ be a scheme. A subset $E$ of $X$ is locally constructible in $X$ if and only if $E \cap U$ is constructible in $U$ for every affine open $U$ of $X$.


properties-lemma-generic-point-in-constructible

Let $X$ be a scheme and let $E \subset X$ be a locally constructible subset. Let $\xi \in X$ be a generic point of an irreducible component of $X$. \begin{enumerate} \item If $\xi \in E$, then an open neighbourhood of $\xi$ is contained in $E$. \item If $\xi \not \in E$, then an open neighbourhood of $\xi$ is disjoint from $E$. \end{enumerate}


properties-lemma-quasi-separated-quasi-compact-open-retrocompact

Let $X$ be a quasi-separated scheme. The intersection of any two quasi-compact opens of $X$ is a quasi-compact open of $X$. Every quasi-compact open of $X$ is retrocompact in $X$.


properties-lemma-quasi-compact-quasi-separated-spectral

Let $X$ be a quasi-compact and quasi-separated scheme. Then the underlying topological space of $X$ is a spectral space.


properties-lemma-constructible-quasi-compact-quasi-separated

Let $X$ be a quasi-compact and quasi-separated scheme. Any locally constructible subset of $X$ is constructible.


properties-lemma-retrocompact

Let $X$ be a scheme. A subset $E$ of $X$ is retrocompact in $X$ if and only if $E \cap U$ is quasi-compact for every affine open $U$ of $X$.


properties-lemma-stratification-locally-finite-constructible

A partition $X = \coprod_{i \in I} X_i$ of a scheme $X$ with retrocompact parts is locally finite if and only if the parts are locally constructible.


properties-lemma-characterize-reduced

Let $X$ be a scheme. The following are equivalent. \begin{enumerate} \item The scheme $X$ is reduced, see Schemes, Definition \ref{schemes-definition-reduced}. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\Gamma(U_i, \mathcal{O}_X)$ is reduced. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is reduced. \item For every open $U \subset X$ the ring $\mathcal{O}_X(U)$ is reduced. \end{enumerate}


properties-lemma-characterize-irreducible

Let $X$ be a scheme. The following are equivalent. \begin{enumerate} \item The scheme $X$ is irreducible. \item There exists an affine open covering $X = \bigcup_{i \in I} U_i$ such that $I$ is not empty, $U_i$ is irreducible for all $i \in I$, and $U_i \cap U_j \not = \emptyset$ for all $i, j \in I$. \item The scheme $X$ is nonempty and every nonempty affine open $U \subset X$ is irreducible. \end{enumerate}


properties-lemma-characterize-integral

A scheme $X$ is integral if and only if it is reduced and irreducible.


properties-lemma-locally-P

Let $X$ be a scheme. Let $P$ be a local property of rings. The following are equivalent: \begin{enumerate} \item The scheme $X$ is locally $P$. \item For every affine open $U \subset X$ the property $P(\mathcal{O}_X(U))$ holds. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ satisfies $P$. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is locally $P$. \end{enumerate} Moreover, if $X$ is locally $P$ then every open subscheme is locally $P$.


properties-lemma-properties-local

The following properties of a ring $R$ are local. \begin{enumerate} \item (Cohen-Macaulay.) The ring $R$ is Noetherian and CM, see Algebra, Definition \ref{algebra-definition-ring-CM}. \item (Regular.) The ring $R$ is Noetherian and regular, see Algebra, Definition \ref{algebra-definition-regular}. \item (Absolutely Noetherian.) The ring $R$ is of finite type over $Z$. \item Add more here as needed.\footnote{But we only list those properties here which we have not already dealt with separately somewhere else.} \end{enumerate}


properties-lemma-locally-Noetherian

Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is locally Noetherian. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is Noetherian. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is Noetherian. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is locally Noetherian. \end{enumerate} Moreover, if $X$ is locally Noetherian then every open subscheme is locally Noetherian.


properties-lemma-immersion-into-noetherian

Any immersion $Z \to X$ with $X$ locally Noetherian is quasi-compact.


properties-lemma-locally-Noetherian-quasi-separated

A locally Noetherian scheme is quasi-separated.


properties-lemma-Noetherian-topology

A (locally) Noetherian scheme has a (locally) Noetherian underlying topological space, see Topology, Definition \ref{topology-definition-noetherian}.


properties-lemma-locally-closed-in-Noetherian

Any locally closed subscheme of a (locally) Noetherian scheme is (locally) Noetherian.


properties-lemma-Noetherian-irreducible-components

A Noetherian scheme has a finite number of irreducible components.


properties-lemma-morphism-Noetherian-schemes-quasi-compact

Any morphism of schemes $f : X \to Y$ with $X$ Noetherian is quasi-compact.


properties-lemma-locally-Noetherian-closed-point

Any nonempty locally Noetherian scheme has a closed point. Any nonempty closed subset of a locally Noetherian scheme has a closed point. Equivalently, any point of a locally Noetherian scheme specializes to a closed point.


properties-lemma-locally-Noetherian-specialization-dvr

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


properties-lemma-thin-infinite-sequence

Let $S$ be a Noetherian scheme. Let $T \subset S$ be an infinite subset. Then there exists an infinite subset $T' \subset T$ such that there are no nontrivial specializations among the points $T'$.


properties-lemma-maximal-points

Let $S$ be a Noetherian scheme. Let $T \subset S$ be a subset. Let $T_0 \subset T$ be the set of $t \in T$ such that there is no nontrivial specialization $t' \leadsto t$ with $t' \in T'$. Then (a) there are no specializations among the points of $T_0$, (b) every point of $T$ is a specialization of a point of $T_0$, and (c) the closures of $T$ and $T_0$ are the same.


properties-lemma-countable-dense-subset

Let $S$ be a Noetherian scheme. Let $T \subset S$ be an infinite dense subset. Then there exist a countable subset $E \subset T$ which is dense in $S$.


properties-lemma-affine-jacobson

An affine scheme $\Spec(R)$ is Jacobson if and only if the ring $R$ is Jacobson.


properties-lemma-locally-jacobson

Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is Jacobson. \item The scheme $X$ is ``locally Jacobson'' in the sense of Definition \ref{definition-locally-P}. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is Jacobson. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is Jacobson. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is Jacobson. \end{enumerate} Moreover, if $X$ is Jacobson then every open subscheme is Jacobson.


properties-lemma-complement-closed-point-Jacobson

Examples of Noetherian Jacobson schemes. \begin{enumerate} \item If $(R, \mathfrak m)$ is a Noetherian local ring, then the punctured spectrum $\Spec(R) \setminus \{\mathfrak m\}$ is a Jacobson scheme. \item If $R$ is a Noetherian ring with Jacobson radical $\text{rad}(R)$ then $\Spec(R) \setminus V(\text{rad}(R))$ is a Jacobson scheme. \item If $(R, I)$ is a Zariski pair (More on Algebra, Definition \ref{more-algebra-definition-zariski-pair}) with $R$ Noetherian, then $\Spec(R) \setminus V(I)$ is a Jacobson scheme. \end{enumerate}


properties-lemma-locally-normal

Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is normal. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is normal. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is normal. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is normal. \end{enumerate} Moreover, if $X$ is normal then every open subscheme is normal.


properties-lemma-normal-reduced

A normal scheme is reduced.


properties-lemma-normal-locally-finite-nr-irreducibles

Let $X$ be a scheme such that any quasi-compact open has a finite number of irreducible components. The following are equivalent: \begin{enumerate} \item $X$ is normal, and \item $X$ is a disjoint union of normal integral schemes. \end{enumerate}


properties-lemma-normal-Noetherian

Let $X$ be a Noetherian scheme. The following are equivalent: \begin{enumerate} \item $X$ is normal, and \item $X$ is a finite disjoint union of normal integral schemes. \end{enumerate}


properties-lemma-normal-locally-Noetherian

Let $X$ be a locally Noetherian scheme. The following are equivalent: \begin{enumerate} \item $X$ is normal, and \item $X$ is a disjoint union of integral normal schemes. \end{enumerate}


properties-lemma-normal-integral-sections

\begin{slogan} The ring of functions on a normal scheme is normal. \end{slogan} Let $X$ be an integral normal scheme. Then $\Gamma(X, \mathcal{O}_X)$ is a normal domain.


properties-lemma-characterize-Cohen-Macaulay

Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item $X$ is Cohen-Macaulay, \item $X$ is locally Noetherian and all of its local rings are Cohen-Macaulay, and \item $X$ is locally Noetherian and for any closed point $x \in X$ the local ring $\mathcal{O}_{X, x}$ is Cohen-Macaulay. \end{enumerate}


properties-lemma-characterize-regular

Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item $X$ is regular, \item $X$ is locally Noetherian and all of its local rings are regular, and \item $X$ is locally Noetherian and for any closed point $x \in X$ the local ring $\mathcal{O}_{X, x}$ is regular. \end{enumerate}


properties-lemma-regular-normal

A regular scheme is normal.


properties-lemma-dimension

Let $X$ be a scheme. The following are equal \begin{enumerate} \item The dimension of $X$. \item The supremum of the dimensions of the local rings of $X$. \item The supremum of $\dim_x(X)$ for $x \in X$. \end{enumerate}


properties-lemma-codimension-local-ring

Let $X$ be a scheme. Let $Y \subset X$ be an irreducible closed subset. Let $\xi \in Y$ be the generic point. Then $$ \text{codim}(Y, X) = \dim(\mathcal{O}_{X, \xi}) $$ where the codimension is as defined in Topology, Definition \ref{topology-definition-codimension}.


properties-lemma-generic-point

Let $X$ be a scheme. Let $x \in X$. Then $x$ is a generic point of an irreducible component of $X$ if and only if $\dim(\mathcal{O}_{X, x}) = 0$.


properties-lemma-locally-Noetherian-dimension-0

A locally Noetherian scheme of dimension $0$ is a disjoint union of spectra of Artinian local rings.


properties-lemma-catenary-local

Let $S$ be a scheme. The following are equivalent \begin{enumerate} \item $S$ is catenary, \item there exists an open covering of $S$ all of whose members are catenary schemes, \item for every affine open $\Spec(R) = U \subset S$ the ring $R$ is catenary, and \item there exists an affine open covering $S = \bigcup U_i$ such that each $U_i$ is the spectrum of a catenary ring. \end{enumerate} Moreover, in this case any locally closed subscheme of $S$ is catenary as well.


properties-lemma-scheme-CM-iff-all-Sk

Let $X$ be a locally Noetherian scheme. Then $X$ is Cohen-Macaulay if and only if $X$ has $(S_k)$ for all $k \geq 0$.


properties-lemma-criterion-reduced

Let $X$ be a locally Noetherian scheme. Then $X$ is reduced if and only if $X$ has properties $(S_1)$ and $(R_0)$.


properties-lemma-criterion-normal

Let $X$ be a locally Noetherian scheme. Then $X$ is normal if and only if $X$ has properties $(S_2)$ and $(R_1)$.


properties-lemma-normal-dimension-1-regular

Let $X$ be a locally Noetherian scheme which is normal and has dimension $\leq 1$. Then $X$ is regular.


properties-lemma-normal-dimension-2-Cohen-Macaulay

Let $X$ be a locally Noetherian scheme which is normal and has dimension $\leq 2$. Then $X$ is Cohen-Macaulay.


properties-lemma-nagata-locally-Noetherian

A Nagata scheme is locally Noetherian.


properties-lemma-locally-Japanese

Let $X$ be an integral scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is Japanese. \item For every affine open $U \subset X$ the domain $\mathcal{O}_X(U)$ is Japanese. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is Japanese. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is Japanese. \end{enumerate} Moreover, if $X$ is Japanese then every open subscheme is Japanese.


properties-lemma-locally-nagata

Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is Nagata. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is Nagata. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is Nagata. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is Nagata. \end{enumerate} Moreover, if $X$ is Nagata then every open subscheme is Nagata.


properties-lemma-normal-geometrically-unibranch

A normal scheme is geometrically unibranch.


properties-lemma-number-of-branches-irreducible-components

Let $X$ be a scheme and $x \in X$. Let $X_i$, $i \in I$ be the irreducible components of $X$ passing through $x$. Then the number of (geometric) branches of $X$ at $x$ is the sum over $i \in I$ of the number of (geometric) branches of $X_i$ at $x$.


properties-lemma-number-of-branches-1

Let $X$ be a scheme. Let $x \in X$. \begin{enumerate} \item The number of branches of $X$ at $x$ is $1$ if and only if $X$ is unibranch at $x$. \item The number of geometric branches of $X$ at $x$ is $1$ if and only if $X$ is geometrically unibranch at $x$. \end{enumerate}


properties-lemma-finite-type-module

Let $X = \Spec(R)$ be an affine scheme. The quasi-coherent sheaf of $\mathcal{O}_X$-modules $\widetilde M$ is a finite type $\mathcal{O}_X$-module if and only if $M$ is a finite $R$-module.


properties-lemma-finite-presentation-module

Let $X = \Spec(R)$ be an affine scheme. The quasi-coherent sheaf of $\mathcal{O}_X$-modules $\widetilde M$ is an $\mathcal{O}_X$-module of finite presentation if and only if $M$ is an $R$-module of finite presentation.


properties-lemma-invert-f-sections

\begin{slogan} Sections of quasi-coherent sheaves have only meromorphic singularities at infinity. \end{slogan} Let $X$ be a scheme. Let $f \in \Gamma(X, \mathcal{O}_X)$. Denote $X_f \subset X$ the open where $f$ is invertible, see Schemes, Lemma \ref{schemes-lemma-f-open}. If $X$ is quasi-compact and quasi-separated, the canonical map $$ \Gamma(X, \mathcal{O}_X)_f \longrightarrow \Gamma(X_f, \mathcal{O}_X) $$ is an isomorphism. Moreover, if $\mathcal{F}$ is a quasi-coherent sheaf of $\mathcal{O}_X$-modules the map $$ \Gamma(X, \mathcal{F})_f \longrightarrow \Gamma(X_f, \mathcal{F}) $$ is an isomorphism.


properties-lemma-invert-s-sections

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible sheaf on $X$. Let $s \in \Gamma(X, \mathcal{L})$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. \begin{enumerate} \item If $X$ is quasi-compact, then (\ref{equation-module-invert-s}) is injective, and \item if $X$ is quasi-compact and quasi-separated, then (\ref{equation-module-invert-s}) is an isomorphism. \end{enumerate} In particular, the canonical map $$ \Gamma_*(X, \mathcal{L})_{(s)} \longrightarrow \Gamma(X_s, \mathcal{O}_X),\quad a/s^n \longmapsto a \otimes s^{-n} $$ is an isomorphism if $X$ is quasi-compact and quasi-separated.


properties-lemma-quasi-coherent-quasi-affine

Let $A$ be a ring and let $U \subset \Spec(A)$ be a quasi-compact open subscheme. For $\mathcal{F}$ quasi-coherent on $U$ the canonical map $$ \widetilde{H^0(U, \mathcal{F})}|_U \to \mathcal{F} $$ is an isomorphism.


properties-lemma-invert-f-affine

Let $X$ be a scheme. Let $f \in \Gamma(X, \mathcal{O}_X)$. Assume $X$ is quasi-compact and quasi-separated and assume that $X_f$ is affine. Then the canonical morphism $$ j : X \longrightarrow \Spec(\Gamma(X, \mathcal{O}_X)) $$ from Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} induces an isomorphism of $X_f = j^{-1}(D(f))$ onto the standard affine open $D(f) \subset \Spec(\Gamma(X, \mathcal{O}_X))$.


properties-lemma-quasi-affine

Let $X$ be a scheme. Then $X$ is quasi-affine if and only if the canonical morphism $$ X \longrightarrow \Spec(\Gamma(X, \mathcal{O}_X)) $$ from Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} is a quasi-compact open immersion.


properties-lemma-cartesian-diagram-quasi-affine

Let $U \to V$ be an open immersion of quasi-affine schemes. Then $$ \xymatrix{ U \ar[d] \ar[rr]_-j & & \Spec(\Gamma(U, \mathcal{O}_U)) \ar[d] \\ U \ar[r] & V \ar[r]^-{j'} & \Spec(\Gamma(V, \mathcal{O}_V)) } $$ is cartesian.


properties-lemma-quasi-affine-presentation

Let $X$ be a quasi-affine scheme. There exists an integer $n \geq 0$, an affine scheme $T$, and a morphism $T \to X$ such that for every morphism $X' \to X$ with $X'$ affine the fibre product $X' \times_X T$ is isomorphic to $\mathbf{A}^n_{X'}$ over $X'$.


properties-lemma-flat-module

\begin{slogan} Flatness is the same for modules and sheaves. \end{slogan} Let $X = \Spec(R)$ be an affine scheme. Let $\mathcal{F} = \widetilde{M}$ for some $R$-module $M$. The quasi-coherent sheaf $\mathcal{F}$ is a flat $\mathcal{O}_X$-module if and only if $M$ is a flat $R$-module.


properties-lemma-locally-free-module

Let $X = \Spec(R)$ be an affine scheme. Let $\mathcal{F} = \widetilde{M}$ for some $R$-module $M$. The quasi-coherent sheaf $\mathcal{F}$ is a (finite) locally free $\mathcal{O}_X$-module of if and only if $M$ is a (finite) locally free $R$-module.


properties-lemma-finite-locally-free

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The following are equivalent: \begin{enumerate} \item $\mathcal{F}$ is a flat $\mathcal{O}_X$-module of finite presentation, \item $\mathcal{F}$ is $\mathcal{O}_X$-module of finite presentation and for all $x \in X$ the stalk $\mathcal{F}_x$ is a free $\mathcal{O}_{X, x}$-module, \item $\mathcal{F}$ is a locally free, finite type $\mathcal{O}_X$-module, \item $\mathcal{F}$ is a finite locally free $\mathcal{O}_X$-module, and \item $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite type, for every $x \in X$ the stalk $\mathcal{F}_x$ is a free $\mathcal{O}_{X, x}$-module, and the function $$ \rho_\mathcal{F} : X \to \mathbf{Z}, \quad x \longmapsto \dim_{\kappa(x)} \mathcal{F}_x \otimes_{\mathcal{O}_{X, x}} \kappa(x) $$ is locally constant in the Zariski topology on $X$. \end{enumerate}


properties-lemma-locally-projective

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is locally projective, and \item there exists an affine open covering $X = \bigcup U_i$ such that the $\mathcal{O}_X(U_i)$-module $\mathcal{F}(U_i)$ is projective for every $i$. \end{enumerate} In particular, if $X = \Spec(A)$ and $\mathcal{F} = \widetilde{M}$ then $\mathcal{F}$ is locally projective if and only if $M$ is a projective $A$-module.


properties-lemma-locally-projective-pullback

Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module. If $\mathcal{G}$ is locally projective on $Y$, then $f^*\mathcal{G}$ is locally projective on $X$.


properties-lemma-extend-trivial

Let $j : U \to X$ be a quasi-compact open immersion of schemes. \begin{enumerate} \item Any quasi-coherent sheaf on $U$ extends to a quasi-coherent sheaf on $X$. \item Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\mathcal{G} \subset \mathcal{F}|_U$ be a quasi-coherent subsheaf. There exists a quasi-coherent subsheaf $\mathcal{H}$ of $\mathcal{F}$ such that $\mathcal{H}|_U = \mathcal{G}$ as subsheaves of $\mathcal{F}|_U$. \item Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $U$. Let $\varphi : \mathcal{G} \to \mathcal{F}|_U$ be a morphism of $\mathcal{O}_U$-modules. There exists a quasi-coherent sheaf $\mathcal{H}$ of $\mathcal{O}_X$-modules and a map $\psi : \mathcal{H} \to \mathcal{F}$ such that $\mathcal{H}|_U = \mathcal{G}$ and that $\psi|_U = \varphi$. \end{enumerate}


properties-lemma-extend

Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be a quasi-compact open. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{G} \subset \mathcal{F}|_U$ be a quasi-coherent $\mathcal{O}_U$-submodule which is of finite type. Then there exists a quasi-coherent submodule $\mathcal{G}' \subset \mathcal{F}$ which is of finite type such that $\mathcal{G}'|_U = \mathcal{G}$.


properties-lemma-quasi-coherent-colimit-finite-type

Let $X$ be a quasi-compact and quasi-separated scheme. Any quasi-coherent sheaf of $\mathcal{O}_X$-modules is the directed colimit of its quasi-coherent $\mathcal{O}_X$-submodules which are of finite type.


properties-lemma-extend-finite-presentation

Let $X$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $U \subset X$ be a quasi-compact open. Let $\mathcal{G}$ be an $\mathcal{O}_U$-module which is of finite presentation. Let $\varphi : \mathcal{G} \to \mathcal{F}|_U$ be a morphism of $\mathcal{O}_U$-modules. Then there exists an $\mathcal{O}_X$-module $\mathcal{G}'$ of finite presentation, and a morphism of $\mathcal{O}_X$-modules $\varphi' : \mathcal{G}' \to \mathcal{F}$ such that $\mathcal{G}'|_U = \mathcal{G}$ and such that $\varphi'|_U = \varphi$.


properties-lemma-lift-finite-presentation

Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be a quasi-compact open. Let $\mathcal{G}$ be an $\mathcal{O}_U$-module. \begin{enumerate} \item If $\mathcal{G}$ is quasi-coherent and of finite type, then there exists a quasi-coherent $\mathcal{O}_X$-module $\mathcal{G}'$ of finite type such that $\mathcal{G}'|_U = \mathcal{G}$. \item If $\mathcal{G}$ is of finite presentation, then there exists an $\mathcal{O}_X$-module $\mathcal{G}'$ of finite presentation such that $\mathcal{G}'|_U = \mathcal{G}$. \end{enumerate}


properties-lemma-directed-colimit-diagram-finite-presentation

\begin{slogan} Quasi-coherent modules on quasi-compact and quasi-separated schemes are filtered colimits of finitely presented modules. \end{slogan} Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. There exist \begin{enumerate} \item a filtered index category $\mathcal{I}$ (see Categories, Definition \ref{categories-definition-directed}), \item a diagram $\mathcal{I} \to \textit{Mod}(\mathcal{O}_X)$ (see Categories, Section \ref{categories-section-limits}), $i \mapsto \mathcal{F}_i$, \item morphisms of $\mathcal{O}_X$-modules $\varphi_i : \mathcal{F}_i \to \mathcal{F}$ \end{enumerate} such that each $\mathcal{F}_i$ is of finite presentation and such that the morphisms $\varphi_i$ induce an isomorphism $$ \colim_i \mathcal{F}_i = \mathcal{F}. $$


properties-lemma-directed-colimit-finite-presentation

Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. There exist \begin{enumerate} \item a directed set $I$ (see Categories, Definition \ref{categories-definition-directed-set}), \item a system $(\mathcal{F}_i, \varphi_{ii'})$ over $I$ in $\textit{Mod}(\mathcal{O}_X)$ (see Categories, Definition \ref{categories-definition-system-over-poset}) \item morphisms of $\mathcal{O}_X$-modules $\varphi_i : \mathcal{F}_i \to \mathcal{F}$ \end{enumerate} such that each $\mathcal{F}_i$ is of finite presentation and such that the morphisms $\varphi_i$ induce an isomorphism $$ \colim_i \mathcal{F}_i = \mathcal{F}. $$


properties-lemma-finite-directed-colimit-surjective-maps

Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Then we can write $\mathcal{F} = \colim \mathcal{F}_i$ with $\mathcal{F}_i$ of finite presentation and all transition maps $\mathcal{F}_i \to \mathcal{F}_{i'}$ surjective.


properties-lemma-algebra-directed-colimit-finite-presentation

Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_X$-algebra. There exist \begin{enumerate} \item a directed set $I$ (see Categories, Definition \ref{categories-definition-directed-set}), \item a system $(\mathcal{A}_i, \varphi_{ii'})$ over $I$ in the category of $\mathcal{O}_X$-algebras, \item morphisms of $\mathcal{O}_X$-algebras $\varphi_i : \mathcal{A}_i \to \mathcal{A}$ \end{enumerate} such that each $\mathcal{A}_i$ is a quasi-coherent $\mathcal{O}_X$-algebra of finite presentation and such that the morphisms $\varphi_i$ induce an isomorphism $$ \colim_i \mathcal{A}_i = \mathcal{A}. $$


properties-lemma-algebra-directed-colimit-finite-type

Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_X$-algebra. Then $\mathcal{A}$ is the directed colimit of its finite type quasi-coherent $\mathcal{O}_X$-subalgebras.


properties-lemma-finite-algebra-directed-colimit-finite-finitely-presented

Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{A}$ be a finite quasi-coherent $\mathcal{O}_X$-algebra. Then $\mathcal{A} = \colim \mathcal{A}_i$ is a directed colimit of finite and finitely presented quasi-coherent $\mathcal{O}_X$-algebras such that all transition maps $\mathcal{A}_{i'} \to \mathcal{A}_i$ are surjective.


properties-lemma-integral-algebra-directed-colimit-finite

Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{A}$ be an integral quasi-coherent $\mathcal{O}_X$-algebra. Then \begin{enumerate} \item $\mathcal{A}$ is the directed colimit of its finite quasi-coherent $\mathcal{O}_X$-subalgebras, and \item $\mathcal{A}$ is a direct colimit of finite and finitely presented quasi-coherent $\mathcal{O}_X$-algebras. \end{enumerate}


properties-lemma-set-of-iso-classes

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\kappa$ be a cardinal. There exists a set $T$ and a family $(\mathcal{F}_t)_{t \in T}$ of $\kappa$-generated $\mathcal{O}_X$-modules such that every $\kappa$-generated $\mathcal{O}_X$-module is isomorphic to one of the $\mathcal{F}_t$.


properties-lemma-colimit-kappa

Let $X$ be a scheme. There exists a cardinal $\kappa$ such that every quasi-coherent module $\mathcal{F}$ is the directed colimit of its quasi-coherent $\kappa$-generated quasi-coherent subsheaves.


properties-lemma-quasi-coherent-finite-type-ideals

Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be an open subscheme. The following are equivalent: \begin{enumerate} \item $U$ is retrocompact in $X$, \item $U$ is quasi-compact, \item $U$ is a finite union of affine opens, and \item there exists a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ such that $X \setminus U = V(\mathcal{I})$ (set theoretically). \end{enumerate}


properties-lemma-sections-annihilated-by-ideal

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


properties-lemma-sections-supported-on-closed-subset

Let $X$ be a scheme. Let $Z \subset X$ be a closed subset. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Consider the sheaf of $\mathcal{O}_X$-modules $\mathcal{F}'$ which associates to every open $U \subset X$ $$ \mathcal{F}'(U) = \{s \in \mathcal{F}(U) \mid \text{the support of }s\text{ is contained in }Z \cap U\} $$ If $X \setminus Z$ is a retrocompact open of $X$, then \begin{enumerate} \item for an affine open $U \subset X$ there exist a finitely generated ideal $I \subset \mathcal{O}_X(U)$ such that $Z \cap U = V(I)$, \item for $U$ and $I$ as in (1) we have $\mathcal{F}'(U) = \{x \in \mathcal{F}(U) \mid I^nx = 0 \text{ for some } n\}$, \item $\mathcal{F}'$ is a quasi-coherent sheaf of $\mathcal{O}_X$-modules. \end{enumerate}


properties-lemma-push-sections-supported-on-closed-subset

Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes. Let $Z \subset Y$ be a closed subset such that $Y \setminus Z$ is retrocompact in $Y$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf of sections supported in $f^{-1}Z$. Then $f_*\mathcal{F}' \subset f_*\mathcal{F}$ is the subsheaf of sections supported in $Z$.


properties-lemma-sections-over-quasi-compact-open-in-affine

Let $A$ be a ring. Let $I \subset A$ be a finitely generated ideal. Let $M$ be an $A$-module. Then there is a canonical map $$ \colim_n \Hom_A(I^n, M) \longrightarrow \Gamma(\Spec(A) \setminus V(I), \widetilde{M}). $$ This map is always injective. If for all $x \in M$ we have $Ix = 0 \Rightarrow x = 0$ then this map is an isomorphism. In general, set $M_n = \{x \in M \mid I^nx = 0\}$, then there is an isomorphism $$ \colim_n \Hom_A(I^n, M/M_n) \longrightarrow \Gamma(\Spec(A) \setminus V(I), \widetilde{M}). $$


properties-lemma-sections-over-quasi-compact-open

Let $X$ be a quasi-compact scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals of finite type. Let $Z \subset X$ be the closed subscheme defined by $\mathcal{I}$ and set $U = X \setminus Z$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The canonical map $$ \colim_n \Hom_{\mathcal{O}_X}(\mathcal{I}^n, \mathcal{F}) \longrightarrow \Gamma(U, \mathcal{F}) $$ is injective. Assume further that $X$ is quasi-separated. Let $\mathcal{F}_n \subset \mathcal{F}$ be subsheaf of sections annihilated by $\mathcal{I}^n$. The canonical map $$ \colim_n \Hom_{\mathcal{O}_X}(\mathcal{I}^n, \mathcal{F}/\mathcal{F}_n) \longrightarrow \Gamma(U, \mathcal{F}) $$ is an isomorphism.


properties-lemma-ample-power-ample

\begin{reference} \cite[II Proposition 4.5.6(i)]{EGA} \end{reference} Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $n \geq 1$. Then $\mathcal{L}$ is ample if and only if $\mathcal{L}^{\otimes n}$ is ample.


properties-lemma-ample-on-closed

Let $X$ be a scheme. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module. For any closed subscheme $Z \subset X$ the restriction of $\mathcal{L}$ to $Z$ is ample.


properties-lemma-affine-cap-s-open

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$. For any affine $U \subset X$ the intersection $U \cap X_s$ is affine.


properties-lemma-ample-tensor-globally-generated

\begin{reference} \cite[II Proposition 4.5.6(ii)]{EGA} \end{reference} Let $X$ be a scheme. Let $\mathcal{L}$ and $\mathcal{M}$ be invertible $\mathcal{O}_X$-modules. If \begin{enumerate} \item $\mathcal{L}$ is ample, and \item the open sets $X_t$ where $t \in \Gamma(X, \mathcal{M}^{\otimes m})$ for $m > 0$ cover $X$, \end{enumerate} then $\mathcal{L} \otimes \mathcal{M}$ is ample.


properties-lemma-affine-s-opens

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume the open sets $X_s$, where $s \in \Gamma(X, \mathcal{L}^{\otimes n})$ and $n \geq 1$, form a basis for the topology on $X$. Then among those opens, the open sets $X_s$ which are affine form a basis for the topology on $X$.


properties-lemma-affine-s-opens-cover-quasi-separated

Let $X$ be a scheme and $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume for every point $x$ of $X$ there exists $n \geq 1$ and $s \in \Gamma(X, \mathcal{L}^{\otimes n})$ such that $x \in X_s$ and $X_s$ is affine. Then $X$ is separated.


properties-lemma-ample-separated

Let $X$ be a scheme. If there exists an ample invertible sheaf on $X$ then $X$ is separated.


properties-lemma-map-into-proj

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Set $S = \Gamma_*(X, \mathcal{L})$ as a graded ring. If every point of $X$ is contained in one of the open subschemes $X_s$, for some $s \in S_{+}$ homogeneous, then there is a canonical morphism of schemes $$ f : X \longrightarrow Y = \text{Proj}(S), $$ to the homogeneous spectrum of $S$ (see Constructions, Section \ref{constructions-section-proj}). This morphism has the following properties \begin{enumerate} \item $f^{-1}(D_{+}(s)) = X_s$ for any $s \in S_{+}$ homogeneous, \item there are $\mathcal{O}_X$-module maps $f^*\mathcal{O}_Y(n) \to \mathcal{L}^{\otimes n}$ compatible with multiplication maps, see Constructions, Equation (\ref{constructions-equation-multiply}), \item the composition $S_n \to \Gamma(Y, \mathcal{O}_Y(n)) \to \Gamma(X, \mathcal{L}^{\otimes n})$ is the identity map, and \item for every $x \in X$ there is an integer $d \geq 1$ and an open neighbourhood $U \subset X$ of $x$ such that $f^*\mathcal{O}_Y(dn)|_U \to \mathcal{L}^{\otimes dn}|_U$ is an isomorphism for all $n \in \mathbf{Z}$. \end{enumerate}


properties-lemma-map-into-proj-quasi-compact

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Set $S = \Gamma_*(X, \mathcal{L})$. Assume (a) every point of $X$ is contained in one of the open subschemes $X_s$, for some $s \in S_{+}$ homogeneous, and (b) $X$ is quasi-compact. Then the canonical morphism of schemes $f : X \longrightarrow \text{Proj}(S)$ of Lemma \ref{lemma-map-into-proj} above is quasi-compact with dense image.


properties-lemma-ample-immersion-into-proj

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Set $S = \Gamma_*(X, \mathcal{L})$. Assume $\mathcal{L}$ is ample. Then the canonical morphism of schemes $f : X \longrightarrow \text{Proj}(S)$ of Lemma \ref{lemma-map-into-proj} is an open immersion with dense image.


properties-lemma-open-in-proj-ample

Let $X$ be a scheme. Let $S$ be a graded ring. Assume $X$ is quasi-compact, and assume there exists an open immersion $$ j : X \longrightarrow Y = \text{Proj}(S). $$ Then $j^*\mathcal{O}_Y(d)$ is an invertible ample sheaf for some $d > 0$.


properties-lemma-ample-on-locally-closed

Let $X$ be a scheme. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module. Let $i : X' \to X$ be a morphism of schemes. Assume at least one of the following conditions holds \begin{enumerate} \item $i$ is a quasi-compact immersion, \item $X'$ is quasi-compact and $i$ is an immersion, \item $i$ is quasi-compact and induces a homeomorphism between $X'$ and $i(X')$, \item $X'$ is quasi-compact and $i$ induces a homeomorphism between $X'$ and $i(X')$. \end{enumerate} Then $i^*\mathcal{L}$ is ample on $X'$.