Gorenstein algebras and Hochschild cohomology

For homomorphism K-->S of commutative rings, where K is Gorenstein and S is essentially of finite type and flat as a K-module, the property that all non-trivial fiber rings of K-->S are Gorenstein is characterized in terms of properties of the cohomology modules Ext_n^{S\otimes_KS}S{S\otimes_KS}.Comment: This is the published version, except for updates to references and bibliography. Sections 3, 4 and 8 have been removed from the preceding version, arXiv:0704.3761v2. Substantial generalizations of results in those sections are proved in our paper with Joseph Lipman and Suresh Nayak, arXiv:0904.400

Cohomology over complete intersections via exterior algebras

A general method for establishing results over a commutative complete intersection local ring by passing to differential graded modules over a graded exterior algebra is described. It is used to deduce, in a uniform way, results on the growth of resolutions of complexes over such local rings.Comment: 18 pages; to appear in "Triangulated categories (Leeds, 2006)", LMS lecture notes series

Constructing modules with prescribed cohomological support

A cohomological support, Supp_A(M), is defined for finitely generated modules M over an left noetherian ring R, with respect to a ring A of central cohomology operations on the derived category of R-modules. It is proved that if the A-module Ext^R(M,M) is noetherian and Ext_i^R(M,R)=0 for i>>0, then every closed subset of Supp_A(M) is the support of some finitely generated R-module. This theorem specializes to known realizability results for varieties of modules over group algebras, over local complete intersections, and over finite dimensional algebras over a field. The theorem is also used to produce large families of finitely generated modules of finite projective dimension over commutative local noetherian rings.Comment: To appear in the Illinois Journal of Mathematics, the issue honoring Phillip Griffith. Revised version has 18 pages. A word (the first one) has been added to the title and the material has been reorganized into seven sections, in place of the original six. There are, however, no changes of any substanc

Homology over local homomorphisms

The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism f: R --> S. Various techniques are developed to study the new invariants and to establish their basic properties. In several cases they are computed in closed form. Applications go in several directions. One is to identify new classes of finite R-modules whose classical Betti numbers or Bass numbers have extremal growth. Another is to transfer ring theoretical properties between R and S in situations where S may have infinite flat dimension over R. A third is to obtain criteria for a ring equipped with a `contracting' endomorphism -- such as the Frobenius endomorphism -- to be regular or complete intersection; these results represent broad generalizations of Kunz's characterization of regularity in prime characteristic.Comment: To appear in the American Journal of Mathematics; new version has minor changes in the presentation; table of content removed; 52 page