Cohomology of finite modules over short Gorenstein rings. (English) Zbl 1436.13031

Menning, Melissa C.; Şega, Liana M.

J. Commut. Algebra 10, No. 1, 63-81 (2018).

Let \((R,\mathfrak{m},k)\) be a Gorenstein local ring with \(\mathfrak{m}^3=0\neq\mathfrak{m}^2\) and \(e=\mathrm{rank}_k(\mathfrak{m}/\mathfrak{m}^2)\). In the present paper, as mentioned in the abstract, the authors shows that if \(e>2\) and \(M, N\) are finitely generated \(R\)-modules, then the power series \((1-et+t^2)\sum_{i=0}^{\infty}\mathrm{rank}_k \left(\mathrm{Ext}_R^i(M,N)\otimes_Rk\right)t^i\) and \((1-et+t^2)\sum_{i=0}^{\infty}\mathrm{rank}_k \left(\mathrm{Tor}_i^R(M,N)\otimes_Rk\right)t^i\) belong to \(\mathbb{Z}[t]\).

Reviewer: Mohammad-Reza Doustimehr (Tabriz)

MSC:

13D07	Homological functors on modules of commutative rings (Tor, Ext, etc.)
13D02	Syzygies, resolutions, complexes and commutative rings
13H10	Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)

Keywords:

Gorenstein ring; Koszul module; Poincaré series

References:

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