Czechoslovak Mathematical Journal, first online, pp. 1-18


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

Thiago H. Freitas, Victor H. Jorge Pérez

Received August 19, 2017.   Published online December 13, 2018.

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 $i\neq t$, 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.
Keywords:  local cohomology; Matlis duality; endomorphism ring
Classification MSC:  13D45; 13C14
DOI:  10.21136/CMJ.2018.0386-17

PDF available at:  Institute of Mathematics CAS

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Affiliations:   Thiago H. Freitas, Universidade Tecnológica Federal do Paraná, Campus Guarapuava CEP 85053-525, Guarapuava, Brazil, e-mail: freitas.thf@gmail.com, Victor Hugo Jorge Pérez, Universidade de São Paulo, ICMC, Caixa Postal 668, 13560-970, São Carlos, Brazil, e-mail: vhjperez@icmc.usp.br


 
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