# 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

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.

Edit:

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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