The structure of the inverse system of Gorenstein k-algebras

Macaulay's Inverse System gives an effective method to construct Artinian Gorenstein k-algebras. To date a general structure for Gorenstein k-algebras of any dimension (and codimension) is not understood. In this paper we extend Macaulay's correspondence characterizing the submodules of the divided power ring in one-to-one correspondence with Gorenstein d-dimensional k-algebras. We discuss effective methods for constructing Gorenstein graded rings. Several examples illustrating our results are given.Comment: 19 pages, to appear in Advances in Mathematic

Regularity and linearity defect of modules over local rings

Given a finitely generated module $M$ over a commutative local ring (or a standard graded $k$-algebra) (R,\m,k) we detect its complexity in terms of numerical invariants coming from suitable \m-stable filtrations $\mathbb{M}$ on $M$. We study the Castelnuovo-Mumford regularity of $gr_{\mathbb{M}}(M)$ and the linearity defect of $M,$ denoted \ld_R(M), through a deep investigation based on the theory of standard bases. If $M$ is a graded $R$-module, then \reg_R(gr_{\mathbb{M}}(M)) <\infty implies \reg_R(M)<\infty and the converse holds provided $M$ is of homogenous type. An analogous result can be proved in the local case in terms of the linearity defect. Motivated by a positive answer in the graded case, we present for local rings a partial answer to a question raised by Herzog and Iyengar of whether \ld_R(k)<\infty implies $R$ is Koszul.Comment: 15 pages, to appear in Journal of Commutative Algebr

Poincar\'e series of modules over compressed Gorenstein local rings

Given positive integers e and s we consider Gorenstein Artinian local rings R of embedding dimension e whose maximal ideal $\mathfrak{m}$ satisfies $\mathfrak{m}^s\ne 0=\mathfrak{m}^{s+1}$. We say that R is a compressed Gorenstein local ring when it has maximal length among such rings. It is known that generic Gorenstein Artinian algebras are compressed. If $s\ne 3$, we prove that the Poincare series of all finitely generated modules over a compressed Gorenstein local ring are rational, sharing a common denominator. A formula for the denominator is given. When s is even this formula depends only on the integers e and s. Note that for $s=3$ examples of compressed Gorenstein local rings with transcendental Poincare series exist, due to B{\o}gvad.Comment: revised version, to appear in Adv. Mat

Castelnuovo-Mumford regularity and extended degree

The main result of this paper shows that the Castelnuovo-Mumford regularity of the tangent cone of a local ring is effectively bounded by the dimension and any extended degree. From this it follows that there are only a finite number of Hilbert-Samuel functions of local rings with given dimension and extended degree.Comment: 15 page