The Stacks project

47.9 Local cohomology

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Set $Z = V(I) \subset \mathop{\mathrm{Spec}}(A)$. We will construct a functor

47.9.0.1
\begin{equation} \label{dualizing-equation-local-cohomology} R\Gamma _ Z : D(A) \longrightarrow D_{I^\infty \text{-torsion}}(A). \end{equation}

which is right adjoint to the inclusion functor. For notation see Section 47.8. The cohomology modules of $R\Gamma _ Z(K)$ are the local cohomology groups of $K$ with respect to $Z$. By Lemma 47.8.4 this functor will in general not be equal to $R\Gamma _ I( - )$ even viewed as functors into $D(A)$. In Section 47.10 we will show that if $A$ is Noetherian, then the two agree.

We will continue the discussion of local cohomology in the chapter on local cohomology, see Local Cohomology, Section 51.1. For example, there we will show that $R\Gamma _ Z$ computes cohomology with support in $Z$ for the associated complex of quasi-coherent sheaves on $\mathop{\mathrm{Spec}}(A)$. See Local Cohomology, Lemma 51.2.1.

Lemma 47.9.1. Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. There exists a right adjoint $R\Gamma _ Z$ (47.9.0.1) 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 $K \in D(A)$.

Proof. Say $I = (f_1, \ldots , f_ r)$ is an ideal. Let $K^\bullet $ be a complex of $A$-modules. There is a canonical map of complexes

\[ (A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r}) \longrightarrow A. \]

from the extended Čech complex to $A$. Tensoring with $K^\bullet $, taking associated total complex, we get a map

\[ \text{Tot}\left( K^\bullet \otimes _ A (A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r})\right) \longrightarrow K^\bullet \]

in $D(A)$. We claim the cohomology modules of the complex on the left are $I$-power torsion, i.e., the LHS is an object of $D_{I^\infty \text{-torsion}}(A)$. Namely, we have

\[ (A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r}) = \mathop{\mathrm{colim}}\nolimits K(A, f_1^ n, \ldots , f_ r^ n) \]

by More on Algebra, Lemma 15.29.6. Moreover, multiplication by $f_ i^ n$ on the complex $K(A, f_1^ n, \ldots , f_ r^ n)$ is homotopic to zero by More on Algebra, Lemma 15.28.6. Since

\[ H^ q\left( LHS \right) = \mathop{\mathrm{colim}}\nolimits H^ q(\text{Tot}(K^\bullet \otimes _ A K(A, f_1^ n, \ldots , f_ r^ n))) \]

we obtain our claim. On the other hand, if $K^\bullet $ is an object of $D_{I^\infty \text{-torsion}}(A)$, then the complexes $K^\bullet \otimes _ A A_{f_{i_0} \ldots f_{i_ p}}$ have vanishing cohomology. Hence in this case the map $LHS \to K^\bullet $ is an isomorphism in $D(A)$. The construction

\[ R\Gamma _ Z(K^\bullet ) = \text{Tot}\left( K^\bullet \otimes _ A (A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r})\right) \]

is functorial in $K^\bullet $ and defines an exact functor $D(A) \to D_{I^\infty \text{-torsion}}(A)$ between triangulated categories. It follows formally from the existence of the natural transformation $R\Gamma _ Z \to \text{id}$ given above and the fact that this evaluates to an isomorphism on $K^\bullet $ in the subcategory, that $R\Gamma _ Z$ is the desired right adjoint. $\square$

Lemma 47.9.2. 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 $D_{J^\infty \text{-torsion}}(B) \to D_{I^\infty \text{-torsion}}(A)$.

Proof. This follows from uniqueness of adjoint functors as both $R\Gamma _ Z((-)_ A)$ and $R\Gamma _ Y(-)_ A$ are right adjoint to the functor $D_{I^\infty \text{-torsion}}(A) \to D(B)$, $K \mapsto K \otimes _ A^\mathbf {L} B$. Alternatively, one can use the description of $R\Gamma _ Z$ and $R\Gamma _ Y$ in terms of alternating Čech complexes (Lemma 47.9.1). Namely, if $I = (f_1, \ldots , f_ r)$ then $J$ is generated by the images $g_1, \ldots , g_ r \in B$ of $f_1, \ldots , f_ r$. Then the statement of the lemma follows from the existence of a canonical isomorphism

\begin{align*} & M_ A \otimes _ A (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}) \\ & = M \otimes _ B (B \to \prod \nolimits _{i_0} B_{g_{i_0}} \to \prod \nolimits _{i_0 < i_1} B_{g_{i_0}g_{i_1}} \to \ldots \to B_{g_1\ldots g_ r}) \end{align*}

for any $B$-module $M$. $\square$

Lemma 47.9.3. 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 $K \in D(A)$.

Proof. Write $I = (f_1, \ldots , f_ r)$. Then $J$ is generated by the images $g_1, \ldots , g_ r \in B$ of $f_1, \ldots , f_ r$. Then we have

\[ (A \to \prod A_{f_{i_0}} \to \ldots \to A_{f_1\ldots f_ r}) \otimes _ A B = (B \to \prod B_{g_{i_0}} \to \ldots \to B_{g_1\ldots g_ r}) \]

as complexes of $B$-modules. Represent $K$ by a K-flat complex $K^\bullet $ of $A$-modules. Since the total complexes associated to

\[ K^\bullet \otimes _ A (A \to \prod A_{f_{i_0}} \to \ldots \to A_{f_1\ldots f_ r}) \otimes _ A B \]

and

\[ K^\bullet \otimes _ A B \otimes _ B (B \to \prod B_{g_{i_0}} \to \ldots \to B_{g_1\ldots g_ r}) \]

represent the left and right hand side of the displayed formula of the lemma (see Lemma 47.9.1) we conclude. $\square$

Lemma 47.9.4. 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 $R\Gamma _ Z(K^\bullet ) = 0$.

Proof. Namely, in this case the cohomology modules of $R\Gamma _ Z(K^\bullet )$ are both $f$-power torsion and $f$ acts by automorphisms. Hence the cohomology modules are zero and hence the object is zero. $\square$

Lemma 47.9.5. 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 $K \otimes _ A^\mathbf {L} L$.

Proof. By Lemma 47.9.1 we know that $R\Gamma _ Z$ is given by $C \otimes ^\mathbf {L} -$ for some $C \in D(A)$. Hence, for $K, L \in D(A)$ general we have

\[ R\Gamma _ Z(K \otimes _ A^\mathbf {L} L) = K \otimes ^\mathbf {L} L \otimes _ A^\mathbf {L} C = K \otimes _ A^\mathbf {L} R\Gamma _ Z(L) \]

The other equalities follow formally from this one. This also implies the last statement of the lemma. $\square$

Lemma 47.9.6. 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 13.22.2.

Proof. There is a bit of abuse of notation in the lemma as strictly speaking we cannot compose $R\Gamma _ Y$ and $R\Gamma _ Z$. The meaning of the statement is simply that we are composing $R\Gamma _ Z$ with the inclusion $D_{I^\infty \text{-torsion}}(A) \to D(A)$ and then with $R\Gamma _ Y$. Then the equality $R\Gamma _ Y \circ R\Gamma _ Z = R\Gamma _{Y \cap Z}$ follows from the fact that

\[ D_{I^\infty \text{-torsion}}(A) \to D(A) \xrightarrow {R\Gamma _ Y} D_{(I + J)^\infty \text{-torsion}}(A) \]

is right adjoint to the inclusion $D_{(I + J)^\infty \text{-torsion}}(A) \to D_{I^\infty \text{-torsion}}(A)$. Alternatively one can prove the formula using Lemma 47.9.1 and the fact that the tensor product of extended Čech complexes on $f_1, \ldots , f_ r$ and $g_1, \ldots , g_ m$ is the extended Čech complex on $f_1, \ldots , f_ n. g_1, \ldots , g_ m$. The final assertion follows from this and the cited lemma. $\square$

The following lemma is the analogue of More on Algebra, Lemma 15.91.24 for complexes with torsion cohomologies.

Lemma 47.9.7. 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 $D_{I^\infty \text{-torsion}}(A) = D_{(IB)^\infty \text{-torsion}}(B)$.

Proof. More precisely the functors are $K \mapsto K \otimes _ A^\mathbf {L} B$ for $K$ in $D_{I^\infty \text{-torsion}}(A)$ and $M \mapsto M_ A$ for $M$ in $D_{(IB)^\infty \text{-torsion}}(B)$. The reason this works is that $H^ i(K \otimes _ A^\mathbf {L} B) = H^ i(K) \otimes _ A B = H^ i(K)$. The first equality holds as $A \to B$ is flat and the second by More on Algebra, Lemma 15.89.2. $\square$

The following lemma was shown for $\mathop{\mathrm{Hom}}\nolimits $ and $\mathop{\mathrm{Ext}}\nolimits ^1$ of modules in More on Algebra, Lemmas 15.89.3 and 15.89.8.

Lemma 47.9.8. Let $A \to B$ be a flat ring map and let $I \subset A$ be a finitely generated ideal such that $A/I \to B/IB$ is an isomorphism. For $K \in D_{I^\infty \text{-torsion}}(A)$ and $L \in D(A)$ the map

\[ R\mathop{\mathrm{Hom}}\nolimits _ A(K, L) \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ B(K \otimes _ A B, L \otimes _ A B) \]

is a quasi-isomorphism. In particular, if $M$, $N$ are $A$-modules and $M$ is $I$-power torsion, then the canonical map

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ A(M, N) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_ B(M \otimes _ A B, N \otimes _ A B) \]

is an isomorphism for all $i$.

Proof. Let $Z = V(I) \subset \mathop{\mathrm{Spec}}(A)$ and $Y = V(IB) \subset \mathop{\mathrm{Spec}}(B)$. Since the cohomology modules of $K$ are $I$ power torsion, the canonical map $R\Gamma _ Z(L) \to L$ induces an isomorphism

\[ R\mathop{\mathrm{Hom}}\nolimits _ A(K, R\Gamma _ Z(L)) \to R\mathop{\mathrm{Hom}}\nolimits _ A(K, L) \]

in $D(A)$. Similarly, the cohomology modules of $K \otimes _ A B$ are $IB$ power torsion and we have an isomorphism

\[ R\mathop{\mathrm{Hom}}\nolimits _ B(K \otimes _ A B, R\Gamma _ Y(L \otimes _ A B)) \to R\mathop{\mathrm{Hom}}\nolimits _ B(K \otimes _ A B, L \otimes _ A B) \]

in $D(B)$. By Lemma 47.9.3 we have $R\Gamma _ Z(L) \otimes _ A B = R\Gamma _ Y(L \otimes _ A B)$. Hence it suffices to show that the map

\[ R\mathop{\mathrm{Hom}}\nolimits _ A(K, R\Gamma _ Z(L)) \to R\mathop{\mathrm{Hom}}\nolimits _ B(K \otimes _ A B, R\Gamma _ Z(L) \otimes _ A B) \]

is a quasi-isomorphism. This follows from Lemma 47.9.7. $\square$

Comments (2)

Comment #1324 by Tom on

Hey Johan,

Three lines below displayed equation (44.8.4.1). "quasi-comherent" should be quasi-coherent. (If I'd known how hard it was going to be to make this comment, I would have ignored it.) Also, now I notice that the last line of the section contains "quasi-isommorphism." You might want to tighten up your "m" key.

Comment #1346 by on

Hey Tom! Which Tom? Sorry you found it hard to leave a comment. I fixed the typos here.

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