On Syzygy Modules over Laurent Polynomial Rings

In this paper, we present a dynamical method for computing the syzygy module of multivariate Laurent polynomials with coefficients in a Dedekind ring (with zero divisors) by reducing the computation over Laurent polynomial rings to calculations over a polynomial ring via an appropriate isomorphism.

Some criteria for regular and Gorenstein local rings via syzygy modules

Let [Formula: see text] be a Cohen–Macaulay local ring. We prove that the [Formula: see text]th syzygy module of a maximal Cohen–Macaulay [Formula: see text]-module cannot have a semidualizing direct summand for every [Formula: see text]. In particular, it follows that [Formula: see text] is Gorenstein if and only if some syzygy of a canonical module of [Formula: see text] has a nonzero free direct summand. We also give a number of necessary and sufficient conditions for a Cohen–Macaulay local ring of minimal multiplicity to be regular or Gorenstein. These criteria are based on vanishing of certain Exts or Tors involving syzygy modules of the residue field.

VANISHING OF (CO)HOMOLOGY OVER DEFORMATIONS OF COHEN-MACAULAY LOCAL RINGS OF MINIMAL MULTIPLICITY

Let R be a d-dimensional Cohen–Macaulay (CM) local ring of minimal multiplicity. Set S := R/(f), where f := f1,. . .,fc is an R-regular sequence. Suppose M and N are maximal CM S-modules. It is shown that if ExtSi(M, N) = 0 for some (d + c + 1) consecutive values of i ⩾ 2, then ExtSi(M, N) = 0 for all i ⩾ 1. Moreover, if this holds true, then either projdimR(M) or injdimR(N) is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.