Let (R,m) be a local ring, I a proper ideal of R and M a finitely generated R-module of dimension d. We discuss the local homology modules of H I (M). When M is Cohen-Macaulay, it is proved that H d m(M) is co-CohenMacaulay of N.dimension d and H x d (H m(M)) = M̂ where x = (x1, . . . , xd) is a system of parameters for M . 2000 Mathematics Subject Classification: 13C14, 13D45

Given a local Noetherian ring (R, m) of dimension d > 0 and infinite residue field, we study the invariants (dimension and multiplicity) of the Sally module SJ (I) of any m-primary ideal I with respect to a minimal reduction J. As a by-product we obtain an estimate for the Hilbert coefficients of m that generalizes a bound established by J. Elias and G. Valla in a local Cohen-Macaulay setting. ...

Let (R,m) be a Cohen–Macaulay local ring and let I be an ideal. There are at least five algebras built on I whose multiplicity data affect the reduction number r(I) of the ideal. We introduce techniques from the Rees algebra theory of modules to produce estimates for r(I), for classes of ideals of dimension one and two. Previous cases of such estimates were derived for ideals of dimension zero.

We extend a reciprocity theorem of Stanley about enumeration of integer points in polyhedral cones when one exchanges strict and weak inequalities. The proof highlights the roles played by Cohen–Macaulayness and canonical modules. The extension raises the issue of whether a Cohen–Macaulay complex of dimension d embedded piecewise-linearly in R is necessarily a d-ball. This is observed to be tru...