Does local cohomology commutes with direct sums?

Let $A$ be a commutative noetherian ring, $Isubseteq A$ an ideal, $M_alpha$ be $A$-modules, $forallalphain J$. It is easily seen that the $I$-torsion commutes with direct sums:
$$Gamma_I(bigoplus_{alphain J}M_alpha)=bigoplus_{alphain J}Gamma_I(M_alpha).$$
This is because, those elements in the direct sum annihilated by a power of $I$ also have each of its components annihilated by the same power of $I$, and conversely we can annihilate the direct sum of these components by a large enough power of $I$, too.

Since the local cohomology $H_I^n$ is defined as the right derived functors of $Gamma_I$, I am wondering whether we can similarly show
$$H_I^n(bigoplus_{alphain J}M_alpha)cong bigoplus_{alphain J}H_I^n(M_alpha).$$

I have seen some proofs of a more general result about local cohomology commuting with direct limits, but I am looking for a straight-forward proof here.

Thank you very much for your help!

Mathematics Asked by Ivon on November 21, 2021

2 Answers

2 Answers

Convince yourself first that $H_I^n(M) = varinjlim_k operatorname{Ext}_R^n(R / I^k, M);$ then, use the fact that Ext commutes with finite direct sums in the second component, i.e., $$operatorname{Ext}_R^n(R / I^k, oplus_{i = 1}^m M_i) cong oplus_{i = 1}^m operatorname{Ext}_R^n(R / I^k, M_i).$$

For the first fact, use the definition of the local cohomology modules as the right-derived functors of $Gamma_I(M).$ Convince yourself that $Gamma_I(M) cong varinjlim_k operatorname{Hom}_R(R / I^k, M);$ then, use the facts that (1.) direct limits commute with cohomology and (2.) Ext is the right-derived functor of Hom.

Unfortunately, I am not aware of a more straightforward proof than this.

Answered by Dylan C. Beck on November 21, 2021

Yes because homology commutes with direct sums. Alternatively you could use the formulation $$H_{mathfrak{a}}^{n}(-)simeq varinjlim_{t}text{Ext}_{R}^{n}(R/mathfrak{a}^{t},-)$$ combined with the fact that $R/mathfrak{a}^{t}$ is finitely generated to show that local cohomology commutes with all direct limits; in particular it will commute with direct sums.


Since $R/mathfrak{a}^{t}$ is finitely generated, there are isomorphisms $$text{Ext}_{R}^{n}(R/mathfrak{a}^{t},varinjlim_{J}N_{j})simeq varinjlim_{J}text{Ext}_{R}^{n}(R/mathfrak{a}^{t},N_{j})$$ for any directed system ${N_{j}}_{J}$ of modules and $ngeq 0$. Consequently one has isomorphisms $$begin{align*} H_{mathfrak{a}}^{n}(varinjlim_{J}N_{j})&simeq varinjlim_{t}text{Ext}_{R}^{n}(R/mathfrak{a}^{t},varinjlim_{J}N_{j}) \ &simeq varinjlim_{t} varinjlim_{J}text{Ext}_{R}^{n}(R/mathfrak{a}^{t},N_{j}) \ &simeq varinjlim_{J} varinjlim_{t} text{Ext}_{R}^{n}(R/mathfrak{a}^{t},N_{j}) \ &simeq varinjlim_{J} H_{mathfrak{a}}^{n}(N_{j}) end{align*}$$ for every directed system and $ngeq 0$.

Answered by Zeek on November 21, 2021

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