A note on the injective dimension of local cohomology modules

Hellus, M.

doi:10.1090/S0002-9939-08-09198-3

A note on the injective dimension of local cohomology modules
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Proc. Amer. Math. Soc. 136 (2008), 2313-2321 Request permission

Abstract:

For a Noetherian ring $R$ we call an $R$-module $M$ cofinite if there exists an ideal $I$ of $R$ such that $M$ is $I$-cofinite; we show that every cofinite module $M$ satisfies $\dim _R(M)\leq \operatorname {injdim}_R(M)$. As an application we study the question which local cohomology modules $H^i_I(R)$ satisfy $\operatorname {injdim}_R(H^i_I(R))=\dim _R(H^i_I(R))$. There are two situations where the answer is positive. On the other hand, we present two counterexamples, the failure in these two examples coming from different reasons.

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