Hema Srinivasan

Algebra structures on some canonical resolutions

The study of algebra structures on finite free resolutions of cyclic modules begins with Buchsbaum and Eisenbud [B-El]. The classes of cyclic modules whose minimal resolutions are known to admit an algebra structure include the residue field [Gu], complete intersections (Koszul complex), modules of homological dimension atmost 3 [B-El], Gorenstein of codimension four [K-M], and Herzog algebras [K-M23. Avramov [Al] gave examples to show that there are cyclic modules whose minimal resolutions do not admit an algebra structure. Let R be a noetherian local ring with maximal ideal m. In this paper we construct algebra structures on the minimal resolutions of two classes of cyclic modules R/I, namely, when Z is of the form Jk, where J is an ideal generated by a regular sequence, and when Z is the ideal of maximal minors of a generic n x m matrix, provided R contains the rationals. In [Al] Avramov defined certain obstructions to the existance of an algebra structure on the minimal resolution of a module and then produced modules with non-zero obstructions. The general question is whether the vanishing of these obstructions is also sufficient for the existence of a minimal algebra resolution of a cyclic module. When Z is an ideal generated by a regular sequence in R and M= R/Z” for any positive integer k, then these obstructions are all zero. Therefore, Avramov and Schlessinger asked whether the minimal resolutions of R/Z’ admit an algebra structure. Corollary 3.6 of this paper provides an affirmative answer to this question.

Asymptotic growth of algebras associated to powers of ideals

We study generalized symbolic powers and form ideals of powers and compare their growth with the growth of ordinary powers, and we discuss the question of when the graded rings attached to symbolic powers or to form ideals of powers are finitely generated.

Minimal algebra resolutions for cyclic modules defined by Huneke-Ulrich ideals

In [H-U], Huneke and Ulrich defined a class of non-trivial deviation two Gorenstein ideals, which were the first large class of such ideals to be defined. These ideals were subsequently studied by Kustin [Kl] and he constructed the minimal resolutions for these ideals. In this paper, we construct an algebra structure on the minimal resolutions of the cyclic modules defined by these ideals. Throughout this paper R will denote a commutative noetherian local ring with maximal ideal m and Z, an ideal of R. A free resolution

Periodic occurrence of complete intersection monomial curves

We study the complete intersection property of monomial curves in the family

On Unimodality of Hilbert Functions of Gorenstein Artin Algebras of Embedding Dimension Four

\Gamma_{\aa + \jj} = {(t^{a_0 + j}, t^{a_1+j},..., t^{a_n + j}) ~ | ~ j \geq 0, ~ a_0 < a_1 <...< a_n}

A note on the subadditivity of Syzygies

. We prove that if

A Class of Gorenstein Artin Algebras of Embedding Dimension Four

\Gamma_{\aa+\jj}

A note on Gorenstein monomial curves

is a complete intersection for

Ranks of syzygies of perfect modules

j \gg0

Gorenstein Hilbert coefficients

, then