Ascent properties for test modules

We investigate modules for which vanishing of Tor-modules implies finiteness of homological dimensions (e.g., projective dimension and G-dimension). In particular, we answer a question of O. Celikbas and Sather-Wagstaff about ascent properties of such modules over residually algebraic flat local ring homomorphisms. To accomplish this, we consider ascent and descent properties over local ring homomorphisms of finite flat dimension, and for flat extensions of finite dimensional differential graded algebras.

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.

Local homology, finiteness of Tor modules and cofiniteness

Let [Formula: see text] be an ideal of a commutative noetherian ring [Formula: see text] with unity and [Formula: see text] an [Formula: see text]-module supported at [Formula: see text]. Let [Formula: see text] be the supermum of the integers [Formula: see text] for which [Formula: see text]. We show that [Formula: see text] is [Formula: see text]-cofinite if and only if the [Formula: see text]-module [Formula: see text] is finitely generated for every [Formula: see text]. This provides a hands-on and computable finitely-many-steps criterion to examine [Formula: see text]-confiniteness. Our approach relies heavily on the theory of local homology which demonstrates the effectiveness and indispensability of this tool.

A Finiteness Result for Attached Primes of Certain Tor-Modules

In this paper we introduce the notion of width in dimension > s for Artinian modules and give a finiteness result for attached primes of certain Tor-modules.