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On the endomorphism ring and Cohen-Macaulayness of local cohomology defined by a pair of ideals

Thiago H. Freitas; Victor H. Jorge Pérez

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 2, page 453-470
  • ISSN: 0011-4642

Abstract

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Let 𝔞 , I , J be ideals of a Noetherian local ring ( R , 𝔪 , k ) . Let M and N be finitely generated R -modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of H I , J t ( M ) and D ( H I , J t ( M ) ) , where t is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and D ( - ) : = Hom R ( - , E R ( k ) ) is the Matlis dual functor. We show that if R is a d -dimensional complete Cohen-Macaulay ring and H I , J i ( R ) = 0 for all i ≠ t , the natural homomorphism R → Hom R ( H I , J t ( K R ) , H I , J t ( K R ) ) is an isomorphism, where K R denotes the canonical module of R . Also, we discuss the depth and Cohen-Macaulayness of the Matlis dual of the top local cohomology modules with respect to a pair of ideals.

How to cite

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Freitas, Thiago H., and Jorge Pérez, Victor H.. "On the endomorphism ring and Cohen-Macaulayness of local cohomology defined by a pair of ideals." Czechoslovak Mathematical Journal 69.2 (2019): 453-470. <http://eudml.org/doc/294425>.

@article{Freitas2019,
abstract = {Let $\mathfrak \{a\}$, $I$, $J$ be ideals of a Noetherian local ring $(R,\mathfrak \{m\},k)$. Let $M$ and $N$ be finitely generated $R$-modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of $H^t_\{I,J\}(M)$ and $D(H^t_\{I,J\}(M))$, where $t$ is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and $D(-):= \{\rm Hom\}_R(-,E_R(k))$ is the Matlis dual functor. We show that if $R$ is a $d$-dimensional complete Cohen-Macaulay ring and $H^i_\{I,J\}(R)=0$ for all $i\ne t$, the natural homomorphism $R\rightarrow \{\rm Hom\}_R(H^t_\{I,J\}(K_R), H^t_\{I,J\}(K_R))$ is an isomorphism, where $K_R$ denotes the canonical module of $R$. Also, we discuss the depth and Cohen-Macaulayness of the Matlis dual of the top local cohomology modules with respect to a pair of ideals.},
author = {Freitas, Thiago H., Jorge Pérez, Victor H.},
journal = {Czechoslovak Mathematical Journal},
keywords = {local cohomology; Matlis duality; endomorphism ring},
language = {eng},
number = {2},
pages = {453-470},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the endomorphism ring and Cohen-Macaulayness of local cohomology defined by a pair of ideals},
url = {http://eudml.org/doc/294425},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Freitas, Thiago H.
AU - Jorge Pérez, Victor H.
TI - On the endomorphism ring and Cohen-Macaulayness of local cohomology defined by a pair of ideals
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 2
SP - 453
EP - 470
AB - Let $\mathfrak {a}$, $I$, $J$ be ideals of a Noetherian local ring $(R,\mathfrak {m},k)$. Let $M$ and $N$ be finitely generated $R$-modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of $H^t_{I,J}(M)$ and $D(H^t_{I,J}(M))$, where $t$ is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and $D(-):= {\rm Hom}_R(-,E_R(k))$ is the Matlis dual functor. We show that if $R$ is a $d$-dimensional complete Cohen-Macaulay ring and $H^i_{I,J}(R)=0$ for all $i\ne t$, the natural homomorphism $R\rightarrow {\rm Hom}_R(H^t_{I,J}(K_R), H^t_{I,J}(K_R))$ is an isomorphism, where $K_R$ denotes the canonical module of $R$. Also, we discuss the depth and Cohen-Macaulayness of the Matlis dual of the top local cohomology modules with respect to a pair of ideals.
LA - eng
KW - local cohomology; Matlis duality; endomorphism ring
UR - http://eudml.org/doc/294425
ER -

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