28 research outputs found
Generalized Gorenstein Arf rings
In this paper we study generalized Gorenstein Arf rings; a class of
one-dimensional Cohen-Macaulay local Arf rings that is strictly contained in
the class of Gorenstein rings. We obtain new characterizations and examples of
Arf rings, and give applications of our argument to numerical semigroup rings
and certain idealizations. In particular, we generalize a beautiful result of
Barucci and Fr\"oberg concerning Arf numerical semigroup rings.Comment: 15 page
On a generalization of Ulrich modules and its applications
We study a modified version of the classical Ulrich modules, which we call
-Ulrich. Unlike the traditional setting, -Ulrich modules always exist. We
prove that these modules retain many of the essential properties and
applications observed in the literature. Additionally, we reveal their
significance as obstructions to Cohen-Macaulay properties of tensor products.
Leveraging this insight, we show the utility of these modules in testing the
finiteness of homological dimensions across various scenarios.Comment: 22 page
Semidualizing Modules over Numerical Semigroup Rings
A semidualizing module is a generalization of Grothendieck's dualizing
module. For a local Cohen-Macaulay ring , the ring itself and its canonical
module are always realized as (trivial) semidualizing modules. Reasonably, one
might ponder the question; when do nontrivial examples exist? In this paper, we
study this question in the realm of numerical semigroup rings and completely
classify which of these rings with multiplicity at most 9 possess a nontrivial
semidualizing module. Using this classification, we construct numerical
semigroup rings in any multiplicity at least 9 possesses a nontrivial
semidualizing module.Comment: 22 pages, comments welcom
SPINOR STRUCTURES ON FREE RESOLUTIONS OF CODIMENSION FOUR GORENSTEIN IDEALS
We analyze the structure of spinor coordinates on resolutions of Gorenstein ideals of codimension four. As an application we produce a family of such ideals with seven generators which are not specializations of the Kustin-Miller model
Associated graded rings and connected sums
summary:In 2012, Ananthnarayan, Avramov and Moore gave a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. In this article, we investigate conditions on the associated graded ring of a Gorenstein Artin local ring , which force it to be a connected sum over its residue field. In particular, we recover some results regarding short, and stretched, Gorenstein Artin rings. Finally, using these decompositions, we obtain results about the rationality of the Poincaré series of
Weakly Arf rings
In this paper, we introduce and develop the theory of weakly Arf rings, which
is a generalization of Arf rings, initially defined by J. Lipman in 1971. We
provide characterizations of weakly Arf rings and study the relation between
these rings, the Arf rings, and the strict closedness of rings. Furthermore, we
give various examples of weakly Arf rings that come from idealizations, fiber
products, determinantal rings, and invariant subrings.Comment: 57 page