Injective dimensions of local cohomology modules
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Assume that $R$ is a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ is an ideal of $R$, $X$ is an $R$--module, and $t$ is a non-negative integer. In this paper, we present upper bounds for the injective dimension of $X$ in terms of the injective dimensions of its local cohomology modules and an upper bound for the injective dimension of $\H_\mathfrak{a}^t(X)$ in terms of the injective dimensions of the modules $\H_\mathfrak{a}^i(X)$, $i\not= t$, and that of $X$. As a consequence, we observe that $R$ is Gorenstein whenever $\H^{i}_\mathfrak{a}(R)$ is of finite injective dimension for all $i$.
Abstract
Assume that $R$ is a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ is an ideal of $R$, $X$ is an $R$--module, and $t$ is a non-negative integer. In this paper, we present upper bounds for the injective dimension of $X$ in terms of the injective dimensions of its local cohomology modules and an upper bound for the injective dimension of $\H_\mathfrak{a}^t(X)$ in terms of the injective dimensions of the modules $\H_\mathfrak{a}^i(X)$, $i\not= t$, and that of $X$. As a consequence, we observe that $R$ is Gorenstein whenever $\H^{i}_\mathfrak{a}(R)$ is of finite injective dimension for all $i$.
참고문헌 (9)
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