Section 47.10 (0BJD): Local cohomology for Noetherian rings

47.10 Local cohomology for Noetherian rings

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Set $Z = V(I) \subset \mathop{\mathrm{Spec}}(A)$ . Recall that (47.8.3.1) is the functor

$D(I^\infty \text{-torsion}) \to D_{I^\infty \text{-torsion}}(A)$

In fact, there is a natural transformation of functors

47.10.0.1

$\begin{equation} \label{dualizing-equation-compare-torsion-functors} (0A6N) \circ R\Gamma _ I(-) \longrightarrow R\Gamma _ Z(-) \end{equation}$

Namely, given a complex of $A$ -modules $K^\bullet$ the canonical map $R\Gamma _ I(K^\bullet ) \to K^\bullet$ in $D(A)$ factors (uniquely) through $R\Gamma _ Z(K^\bullet )$ as $R\Gamma _ I(K^\bullet )$ has $I$ -power torsion cohomology modules (see Lemma 47.8.1). In general this map is not an isomorphism (we've seen this in Lemma 47.8.4).

Lemma 47.10.1. Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal.

the adjunction $R\Gamma _ I(K) \to K$ is an isomorphism for $K \in D_{I^\infty \text{-torsion}}(A)$ ,
the functor (47.8.3.1) $D(I^\infty \text{-torsion}) \to D_{I^\infty \text{-torsion}}(A)$ is an equivalence,
the transformation of functors (47.10.0.1) is an isomorphism, in other words $R\Gamma _ I(K) = R\Gamma _ Z(K)$ for $K \in D(A)$ .

Proof. A formal argument, which we omit, shows that it suffices to prove (1).

Let $M$ be an $I$ -power torsion $A$ -module. Choose an embedding $M \to J$ into an injective $A$ -module. Then $J[I^\infty ]$ is an injective $A$ -module, see Lemma 47.3.9, and we obtain an embedding $M \to J[I^\infty ]$ . Thus every $I$ -power torsion module has an injective resolution $M \to J^\bullet$ with $J^ n$ also $I$ -power torsion. It follows that $R\Gamma _ I(M) = M$ (this is not a triviality and this is not true in general if $A$ is not Noetherian). Next, suppose that $K \in D_{I^\infty \text{-torsion}}^+(A)$ . Then the spectral sequence

$R^ q\Gamma _ I(H^ p(K)) \Rightarrow R^{p + q}\Gamma _ I(K)$

(Derived Categories, Lemma 13.21.3) converges and above we have seen that only the terms with $q = 0$ are nonzero. Thus we see that $R\Gamma _ I(K) \to K$ is an isomorphism.

Suppose $K$ is an arbitrary object of $D_{I^\infty \text{-torsion}}(A)$ . We have

$R^ q\Gamma _ I(K) = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Ext}}\nolimits ^ q_ A(A/I^ n, K)$

by Lemma 47.8.2. Choose $f_1, \ldots , f_ r \in A$ generating $I$ . Let $K_ n^\bullet = K(A, f_1^ n, \ldots , f_ r^ n)$ be the Koszul complex with terms in degrees $-r, \ldots , 0$ . Since the pro-objects $\{ A/I^ n\}$ and $\{ K_ n^\bullet \}$ in $D(A)$ are the same by More on Algebra, Lemma 15.94.1, we see that

$R^ q\Gamma _ I(K) = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Ext}}\nolimits ^ q_ A(K_ n^\bullet , K)$

Pick any complex $K^\bullet$ of $A$ -modules representing $K$ . Since $K_ n^\bullet$ is a finite complex of finite free modules we see that

$\mathop{\mathrm{Ext}}\nolimits ^ q_ A(K_ n, K) = H^ q(\text{Tot}((K_ n^\bullet )^\vee \otimes _ A K^\bullet ))$

where $(K_ n^\bullet )^\vee$ is the dual of the complex $K_ n^\bullet$ . See More on Algebra, Lemma 15.73.2. As $(K_ n^\bullet )^\vee$ is a complex of finite free $A$ -modules sitting in degrees $0, \ldots , r$ we see that the terms of the complex $\text{Tot}((K_ n^\bullet )^\vee \otimes _ A K^\bullet )$ are the same as the terms of the complex $\text{Tot}((K_ n^\bullet )^\vee \otimes _ A \tau _{\geq q - r - 2} K^\bullet )$ in degrees $q - 1$ and higher. Hence we see that

$\mathop{\mathrm{Ext}}\nolimits ^ q_ A(K_ n, K) = \text{Ext}^ q_ A(K_ n, \tau _{\geq q - r - 2}K)$

for all $n$ . It follows that

$R^ q\Gamma _ I(K) = R^ q\Gamma _ I(\tau _{\geq q - r - 2}K) = H^ q(\tau _{\geq q - r - 2}K) = H^ q(K)$

Thus we see that the map $R\Gamma _ I(K) \to K$ is an isomorphism. $\square$

Lemma 47.10.2. Let $A$ be a Noetherian ring and let $I = (f_1, \ldots , f_ r)$ be an ideal of $A$ . Set $Z = V(I) \subset \mathop{\mathrm{Spec}}(A)$ . 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 $D(A)$ .

Proof. This follows from Lemma 47.10.1 and the computation of the functor $R\Gamma _ Z$ in Lemma 47.9.1. $\square$

Lemma 47.10.3. 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 $Y = V(IB) \subset \mathop{\mathrm{Spec}}(B)$ .

Proof. Combine Lemmas 47.10.2 and 47.9.3. $\square$

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47.10 Local cohomology for Noetherian rings

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