×

Modules of finite virtual projective dimension. (English) Zbl 0677.13004

In order to study the asymptotic of the ranks of the free modules in (an infinite) minimal free resolution, the author defines two new notions for a finitely generated module M over a local ring \((R,{\mathfrak m},k):\) the complexity \(cx_ RM\) is d if d-1 is the least degree of a polynomial in n bounding the sequence of Betti numbers \(\{b_ n(M)=\dim_ k Ext^ n_ R(M,k)\}\), and the virtual projective dimension \(vpd_ RM\) is the least value of \(pd_ QM\) for local deformations Q of R having the same residue field (with technical adjustments if R is not complete or the residue field is finite). The central formula is \[ \text{vpd}_ R M = \text{depth}(R) -\text{depth}(M) + cx_ R M,\tag{\(*\)} \] and the principal result is that (\(*\)) holds if M is a non-zero module with \(vpd_ RM<\infty\). In particular (\(*\)) holds for all non-trivial finitely generated modules over a complete intersection. The formula generalizes the classical Auslander-Buchsbaum formula for modules of finite projective dimension (in this case \(vpd_ RM=pd_ RM\) and \(cx_ RM=0).\)
Various “relative” theorems follow: for instance, the author extends the validity of the Eisenbud conjecture that modules with bounded Betti numbers have eventually periodic resolutions by placing the hypothesis \(vpd_ RM<\infty\) on M and eliminating any hypothesis on R (Eisenbud had established the conjecture for all modules over a complete intersection R). The author also gives detailed results on the growth of \(\{b_ n(M)\}\), which imply the much weaker fact that \(\lim_{n\to \infty}b_ n=\infty\) if the sequence is not bounded.
The main body of the paper concerns the construction and interpretation of two algebraic varieties: V(Q,x,M) is defined for a local ring Q that specializes by the regular sequence x to R, where \(pd_ QM<\infty\); and \(V^*_ R(M)\) is defined in terms of the action of a certain subalgebra of the homotopy Lie algebra of R on \(Ext_ R(M,k)\). The first one parameterizes intermediate specializations \(Q\to Q'\to R\) with finite \(pd_{Q'}M\), and has dimension equal to \(cx_ RM\); the second one maps finite-to-one onto the first one and encodes information on all the deformations Q of interest. The author uses structural results for these varieties to prove that all admissible pairs of values of \(cx_ RM\) and \(vpd_ RM\) do in fact occur for R a complete intersection.
The final section concerns applications to representations of k[G] for G a finite abelian p-group and k an algebraically closed field of characteristic \(p>0.\) For known results there are given new proofs and sharpened by relating the existing notions of complexity and cohomological varieties for group rings to the new ones of this paper.
Reviewer: M.Miller

MSC:

13D05 Homological dimension and commutative rings
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13D10 Deformations and infinitesimal methods in commutative ring theory
17B55 Homological methods in Lie (super)algebras
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] [AE] Alperin, J., Evans, L.: Representations, resolutions, and Quillen’s dimension theorem. J. Pure Appl. Algebra22, 1-9 (1981) · Zbl 0469.20008
[2] [AB] Auslander, M., Buchsbaum, D.: Homological dimension in local rings. Trans. Am. Math. Soc.85, 390-405 (1957) · Zbl 0078.02802
[3] [Av1] Avramov, L.L.: Local algebra and rational homotopy. Astérisque113-114, 15-43 (1984)
[4] [Av2] Avramov, L.L.: Homological asymptotics of modules over local rings, in: Proceedings of the Micro-Program in Commutative Algebra, MSRI, Berkeley, 1987 (to appear)
[5] [AGP] Avramov, L.L., Gasharov, V.N., Peeva, I.V.: A periodic module of infinite virtual projective dimension. J. Pure Appl. Algebra (to appear) · Zbl 0694.13007
[6] [AS] Avrunin, G.S., Scott, L.L.: Quillen Stratification for modules. Invent. Math.66, 277-286 (1982) · Zbl 0489.20042
[7] [Ba] Bass, H.: On the ubiquity of Gorenstein rings. Math. Z.82, 8-28 (1963) · Zbl 0112.26604
[8] [Ca1] Carlson, J.F.: The dimensions of periodic modules over modular group algebras, Ill. J. Math.23, 295-306 (1979) · Zbl 0388.16008
[9] [Ca2] Carlson, J.F.: The varieties and the cohomology ring of a module. J Algebra85, 104-143 (1983) · Zbl 0526.20040
[10] [Ca3] Carlson, J.F.: The variety of an indecomposable module is connected. Invent. Math.77, 291-299 (1984) · Zbl 0543.20032
[11] [Ei] Eisenbud, D.: Homological algebra on a complete intersection, with an application to group representations. Trans. Am. Math. Soc.260, 35-64 (1980) · Zbl 0444.13006
[12] [Fo] Foxby, H.-B.: Isomorphisms between complexes with applications to the homological theory of modules. Math. Scand.40, 5-19 (1977) · Zbl 0356.13004
[13] [Gu] Gulliksen, T.H.: A change of rings theorem, with applications to Poincaré series and intersection multiplicity. Math. Scand.34, 167-183 (1974) · Zbl 0292.13009
[14] [Ja] Jacobsson, C.: On local flat homomorphisms and the Yoneda Ext-algebra of the fibre. Astérisque113-114, 227-233 (1984)
[15] [Kr] Kroll, O.: Complexity and elementary abelianp-groups. J. Algebra88, 155-172 (1984) · Zbl 0536.20033
[16] [Na] Nagata, M.: Local rings. New York: Interscience 1962 · Zbl 0123.03402
[17] [Qu] Quillen, D.: The spectrum of an equivariant cohomology ring, I and II. Ann. Math.94, 549-572, 573-602 (1971) · Zbl 0247.57013
[18] [Se] Serre, J.-P.: Algèbre locale. Multiplicités (Lecture Notes in Maths. Vol. 11). Berlin-Heidelberg-New York: Springer 1975
[19] [Sh] Shamash, J.: The Poincaré series of a local ring. J. Algebra12, 453-470 (1969) · Zbl 0189.04004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.