Matlis reflexive and generalized local cohomology modules

Amir Mafi

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 4, page 1095-1102
  • ISSN: 0011-4642

Abstract

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Let ( R , 𝔪 ) be a complete local ring, 𝔞 an ideal of R and N and L two Matlis reflexive R -modules with Supp ( L ) ⊆ V ( 𝔞 ) . We prove that if M is a finitely generated R -module, then Ext R i ( L , H 𝔞 j ( M , N ) ) is Matlis reflexive for all i and j in the following cases: (a) dim R / 𝔞 = 1 ; (b) cd ( 𝔞 ) = 1 ; where cd is the cohomological dimension of 𝔞 in R ; (c) dim R ≤ 2 . In these cases we also prove that the Bass numbers of H 𝔞 j ( M , N ) are finite.

How to cite

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Mafi, Amir. "Matlis reflexive and generalized local cohomology modules." Czechoslovak Mathematical Journal 59.4 (2009): 1095-1102. <http://eudml.org/doc/37980>.

@article{Mafi2009,
abstract = {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) $\mathop \{\{\rm dim\}\} R\le 2$. In these cases we also prove that the Bass numbers of $H_\{\mathfrak \{a\} \}^j(M,N)$ are finite.},
author = {Mafi, Amir},
journal = {Czechoslovak Mathematical Journal},
keywords = {Bass numbers; generalized local cohomology modules; Matlis reflexive; Bass number; generalized local cohomology module; Matlis reflexive},
language = {eng},
number = {4},
pages = {1095-1102},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Matlis reflexive and generalized local cohomology modules},
url = {http://eudml.org/doc/37980},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Mafi, Amir
TI - Matlis reflexive and generalized local cohomology modules
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 4
SP - 1095
EP - 1102
AB - 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) $\mathop {{\rm dim}} R\le 2$. In these cases we also prove that the Bass numbers of $H_{\mathfrak {a} }^j(M,N)$ are finite.
LA - eng
KW - Bass numbers; generalized local cohomology modules; Matlis reflexive; Bass number; generalized local cohomology module; Matlis reflexive
UR - http://eudml.org/doc/37980
ER -

References

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