Geodesic
Matlis reflexive and generalized local cohomology modules
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 4, pp. 1095-1102 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $(R,\mathfrak m )$ be a complete local ring, $\mathfrak a $ an ideal of $R$ and $N$ and $L$ two Matlis reflexive $R$-modules with $\mathop{{\rm Supp}} (L)\subseteq V(\mathfrak a )$. We prove that if $M$ is a finitely generated $R$-module, then $\mathop{{\rm Ext}}\nolimits_R^i(L,H_{\mathfrak a }^j(M,N))$ is Matlis reflexive for all $i$ and $j$ in the following cases: (a) $\mathop{{\rm dim}} R/{\mathfrak a }=1$; (b) $\mathop{{\rm cd}} (\mathfrak a )=1$; where $\mathop{{\rm cd}} $ is the cohomological dimension of $\mathfrak a $ in $R$; (c) $\mathop{{\rm dim}} R\leq 2$. In these cases we also prove that the Bass numbers of $H_{\mathfrak a }^j(M,N)$ are finite.
Let $(R,\mathfrak m )$ be a complete local ring, $\mathfrak a $ an ideal of $R$ and $N$ and $L$ two Matlis reflexive $R$-modules with $\mathop{{\rm Supp}} (L)\subseteq V(\mathfrak a )$. We prove that if $M$ is a finitely generated $R$-module, then $\mathop{{\rm Ext}}\nolimits_R^i(L,H_{\mathfrak a }^j(M,N))$ is Matlis reflexive for all $i$ and $j$ in the following cases: (a) $\mathop{{\rm dim}} R/{\mathfrak a }=1$; (b) $\mathop{{\rm cd}} (\mathfrak a )=1$; where $\mathop{{\rm cd}} $ is the cohomological dimension of $\mathfrak a $ in $R$; (c) $\mathop{{\rm dim}} R\leq 2$. In these cases we also prove that the Bass numbers of $H_{\mathfrak a }^j(M,N)$ are finite.
Classification : 13D07, 13D45, 13E99
Keywords: Bass numbers; generalized local cohomology modules; Matlis reflexive 
@article{CMJ_2009_59_4_a17,
     author = {Mafi, Amir},
     title = {Matlis reflexive and generalized local cohomology modules},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1095--1102},
     year = {2009},
     volume = {59},
     number = {4},
     mrnumber = {2563580},
     zbl = {1224.13016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2009_59_4_a17/}
}
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Mafi, Amir. Matlis reflexive and generalized local cohomology modules. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 4, pp. 1095-1102. http://geodesic.mathdoc.fr/item/CMJ_2009_59_4_a17/