A Study of Tate Homology via the Approximation Theory with Applications to the Depth Formula

Acta Mathematica Sinica, English Series - In this paper we are concerned with absolute, relative and Tate Tor modules. In the first part of the paper we generalize a result of Avramov and...     

Syzygies and tensor product of modules

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.     

Hochster’s theta pairing and algebraic equivalence

We define a variant of Hochster’s θ pairing and prove that it is constant in flat families of modules over hypersurfaces with isolated singularities. As a consequence, we show that the θ pairing factors through the Grothendieck group modulo algebraic equivalence. Moreover, our result allows us, in certain situations, to translate the properties of the θ pairing in characteristic zero [established in Moore et al. (Adv Math, 226(2):1692–1714, 2011) and Polishchuk and Vaintrob (Chern characters and Hirzebruch–Riemann-Roch formula for matrix factorizations. Preprint, arXiv:1002, 2010)] to the characteristic p setting. We also give an application of our result to the rigidity of Tor over hypersurfaces.     

Brauer–Thrall for Totally Reflexive Modules over Local Rings of Higher Dimension

Let R be a commutative Noetherian local ring. Assume that R has a pair {x,y} of exact zerodivisors such that dim R/(x,y)?≥?2 and all totally reflexive R/(x)-modules are free. We show that the first and second Brauer–Thrall type theorems hold for the category of totally reflexive R-modules. More precisely, we prove that, for infinitely many integers n, there exists an indecomposable totally reflexive R-module of multiplicity n. Moreover, if the residue field of R is infinite, we prove that there exist infinitely many isomorphism classes of indecomposable totally reflexive R-modules of multiplicity n.