We analyze stable homology over associative rings and obtain results over Artin algebras and commutative noetherian rings. Our study develops similarly for these classes; for simplicity we only discuss the latter here. Stable homology is a broad generalization of Tate homology. Vanishing of stable homology detects classes of rings---among them Gorenstein rings, the original domain of Tate homology. Closely related to gorensteinness of rings is Auslander's G-dimension for modules. We show that vanishing of stable homology detects modules of finite G-dimension. This is the first characterization of such modules in terms of vanishing of (co)homology alone. Stable homology, like absolute homology, Tor, is a theory in two variables. It can be computed from a flat resolution of one module together with an injective resolution of the other. This betrays that stable homology is not balanced in the way Tor is balanced. In fact, we prove that a ring is Gorenstein if and only if stable homology is balanced.

We compare two generalizations of Tate homology to the realm of associative rings: stable homology and the J-completion of Tor, also known as complete homology. For finitely generated modules, we show that the two theories agree over Artin algebras and over commutative noetherian rings that are Gorenstein, or local and complete.

AbstractTensor products usually have nonzero torsion. This is a central theme of Auslander's 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modules N with the property that M ⊗R N has nonzero torsion unless M is very special. An important example of such a module N is the Frobenius power peR over a complete intersection domain R of characteristic p > 0.

In this paper we exploit properties of Dao's eta-pairing as well as techniques of Huneke, Jorgensen, and Wiegand to study the vanishing of Tor_i(M,N) of finitely generated modules M, N over complete intersections. We prove vanishing of Tor_i(M, N) for all positive integers i under depth conditions on M, N, and their tensor product. Our arguments improve a result of Dao and establish a new connection between the vanishing of Tor and the depth of tensor products.

We give an application of the New Intersection Theorem and prove the following: let \(R\) be a local complete intersection ring of codimension \(c\) and let \(M\) and \(N\) be nonzero finitely generated \(R\)-modules. Assume \(n\) is a nonnegative integer and that the tensor product \(M\otimes _{R}N\) is an \((n+c)\)th syzygy of some finitely generated \(R\)-module. If \({{\mathrm{Tor}}}^{R}_{>0}(M,N)=0\), then both \(M\) and \(N\) are \(n\)th syzygies of some finitely generated \(R\)-modules.

We study H. Dao’s invariant \({\eta_c^R}\) of pairs of modules defined over a complete intersection ring R of codimension c having an isolated singularity. Our main result is that \({\eta_c^R}\) vanishes for all pairs of modules when R is a graded complete intersection ring of codimension c > 1 having an isolated singularity. A consequence of this result is that all pairs of modules over such a ring are c-Tor-rigid.

For finitely generated modules M and N over a complete intersection R, the vanishing of Tor_i^R(M,N) for all i> 0 gives a tight relationship among depth properties of M, N and their tensor product. Here we concentrate on the converse and show, under mild conditions, that the tensor product of M and N being torsion-free (or satisfying higher Serre conditions) forces vanishing of Tor. Special attention is paid to the case of tensor powers of a single module.

Let R be an isolated hypersurface singularity, and let M and N be finitely generated R-modules. As R is a hypersurface, the torsion modules of M against N are eventually periodic of period two (i.e., Tor_i^R(M,N) is isomorphic to Tor_{i+2}^R(M,N) for i sufficiently large). Since R has only an isolated singularity, these torsion modules are of finite length for i sufficiently large. The theta invariant of the pair (M,N) is defined by Hochster to be length(Tor_{2i}^R(M,N)) - length(Tor_{2i+1}^R(M,N)) for i sufficiently large. H. Dao has conjectured that the theta invariant is zero for all pairs (M,N) when R has even dimension and contains a field. This paper proves this conjecture under the additional assumption that R is graded with its irrelevant maximal ideal giving the isolated singularity. We also give a careful analysis of the theta pairing when the dimension of R is odd, and relate it to a classical pairing on the smooth variety Proj(R).

In 1987 Roberts completed the proof of the New Intersection Theorem (NIT) by settling the mixed characteristic case using local Chern characters, as developed by Fulton and also by Roberts. His proof has been the only one recorded of the NIT in mixed characteristic. This paper gives a new proof of this theorem, one which mostly parallels Roberts' original proof, but avoids the use of local Chern characters. Instead, the proof here uses Adams operations on K-theory with supports as developed by Gillet and Soulé.

Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive R-modules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple hypersurface singularity). The crux of our proof is to argue that if the residue field has a totally reflexive cover, then R is Gorenstein or every totally reflexive R-module is free.

In recent work, Roberts and Srinivas use geometric arguments to establish the existence of modules with unusual intersection properties. These modules are of finite length and finite projective dimension. They have Euler characteristic intersection pairings matching those of a cycle class in the kernel of the hyperplane section on a Chow group of a smooth projective scheme. More generally, a perfect complex, rather than an individual module of finite length and finite projective dimension, can be found with the correct Euler characteristic. This paper addresses how to construct perfect complexes with intersection properties prescribed by cycles in a Chow group.