Journal of Commutative Algebra

Cohomology of finite modules over short Gorenstein rings

Melissa C. Menning and Liana M. Şega

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Abstract

Let $R$ be a Gorenstein local ring with maximal ideal $\mathfrak{m} $ satisfying $\mathfrak{m} ^3=0\ne \mathfrak{m} ^2$. Set $\mathfrak{k} =R/\mathfrak{m} $ and $e=rank _{\mathfrak{k} }(\mathfrak{m} /\mathfrak{m} ^2)$. If $e>2$ and $M$, $N$ are finitely generated $R$-modules, we show that the formal power series \[ \sum _{i=0}^\infty rank _{\mathfrak{k} }\left (Ext ^i_R(M,N)\otimes _R\mathfrak{k} \right )t^i \] and \[ \sum _{i=0}^\infty rank _{\mathfrak{k} }\left (Tor _i^R(M,N)\otimes _R \mathfrak{k} \right )t^i \] are rational, with denominator $1-et+t^2$.

Article information

Source
J. Commut. Algebra, Volume 10, Number 1 (2018), 63-81.

Dates
First available in Project Euclid: 18 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.jca/1526608935

Digital Object Identifier
doi:10.1216/JCA-2018-10-1-63

Mathematical Reviews number (MathSciNet)
MR3804847

Zentralblatt MATH identifier
06875414

Subjects
Primary: 13D07: Homological functors on modules (Tor, Ext, etc.)
Secondary: 13D02: Syzygies, resolutions, complexes 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]

Keywords
Gorensenstein ring Koszul module Poincaré series

Citation

Menning, Melissa C.; Şega, Liana M. Cohomology of finite modules over short Gorenstein rings. J. Commut. Algebra 10 (2018), no. 1, 63--81. doi:10.1216/JCA-2018-10-1-63. https://projecteuclid.org/euclid.jca/1526608935


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