Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2017; 54(4): 1331-1336

Online first article April 6, 2017      Printed July 31, 2017

https://doi.org/10.4134/BKMS.b160571

Copyright © The Korean Mathematical Society.

Injective dimensions of local cohomology modules

Alireza Vahidi

Payame Noor University (PNU)

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 $i$.

Keywords: Gorenstein rings, injective dimensions, local cohomology modu\-les

MSC numbers: 13D05, 13D45, 13H10

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