| It is known that the powers mn of the maximal ideal m of a local Noetherian ring R share certain homological properties for all sufficiently large integers n. When M is a finite R-module, Levin proved that the induced maps TorRi&parl0; mnM,k&parr0;→Tor Ri&parl0;m n-1M,k&parr0; are zero for all large n and all i. In Chapter 1 we show that these maps are zero for all n > pol reg M, where pol reg M denotes the Castelnuovo-Mumford regularity of the associated graded module grmM over the symmetric algebra Symk( m/m2 ). We also give a new application to the theory of Auslander's delta invariants, by showing that diR&parl0;mn M&parr0; = 0 for all i ≥ 0 and all n > pol reg M; this extends and gives an effective version of a theorem of Yoshino. In Chapter 2 we deal with the base change in (co)homology induced by the natural ring homomorphisms R → R/ mn . These maps are known to be Golod, respectively, small, for all large n. We determine bounds on the values of n for which these properties begin to hold. When R is a complete intersection, Avramov and Buchweitz proved that the asymptotic vanishing of ExtnR (-, -) is symmetric in the module variables and raised the question whether this property holds for all Gorenstein rings. Recently, Huneke and Jorgensen gave a positive answer for Gorenstein rings of minimal multiplicity. In Chapter 3 we answer the question positively for all Gorenstein rings of codimension at most 4. |
