97f:13020 13D03 13D05

Avramov, Luchezar L. ^{(}1-PURD^{)}

Modules with extremal resolutions. Math. Res. Lett. 3 (1996), no. 3, 319–328. Introduction: “Let R be a commutative Noetherian local ring with maximal ideal 𝔪 and residue field k = R/𝔪. The size of a minimal free resolution of a finite R-module M is given by its Betti numbers β_{n}^{R}(M) = rank_{k}Ext_{R}^{n}(M, k). Dually, that of a minimal injective resolution is measured by the Bass numbers μ_{R}^{n}(M) = rank_{k}Ext_{R}^{n}(k, M).

“These sizes may be estimated asymptotically on a natural, a polynomial, and an exponential scale. The first scale yields the classical homological dimensions. The second produces known notions of complexity, which distinguish between modules of infinite homological dimensions. The third leads to new concepts of homological curvatures, which discriminate among modules with infinite complexities.

“It is well known that k has maximal homological dimensions among all finite R-modules. An elementary computation shows that its complexities and curvatures are maximal as well. By analyzing the representations on Ext_{R}^{∗}(M, k) and Ext_{R}^{∗}(k, M) of the homotopy Lie algebra π^{∗} of the local ring R, we show that modules with extremal homological invariants occur with unexpected frequency.”

Alex Martsinkovsky (1-NORE)

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