Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2311.11788

On Certain Gluing of semigroup rings and indispensable resolution of semigroup rings

Sanjay Kumar Singh·Pranjal Srivastava
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, our aim is twofold: First, by using the technique of gluing semigroups, we give infinitely many families of a projective closure with the Cohen-Macaulay (Gorenstein) property. Also, we give an effective technique for constructing large families of one dimensional Gorenstein local rings associated to monomial curves, which supports Rossi question, saying that every Gorenstein local ring has a non-decreasing Hilbert function. In the second part, we study strong indispensable minimal free resolutions of semigroup rings, focusing on the operation of the join of affine semigroups, which provide class of examples supporting Charalambous and Thoma question on the class of lattice ideal which has a strong indispensable free resolution.

Explore related subjects

math.AC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sanjay Kumar Singh, Pranjal Srivastava. 2023-11-20. On Certain Gluing of semigroup rings and indispensable resolution of semigroup rings. https://arxiv.org/abs/2311.11788

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Unified Common-Root and Interpolation Bounds Based on Leading Monomial Data

For general fields the footprint bound from Gröbner basis theory estimates the number of common affine roots of any set of multivariate polynomials using information on their leading monomials. In this paper we develop an interpolation bound with a similar flavor extending a previously known result for only a single polynomial to any prescribed number of polynomials. Surprisingly, our interpolation theorem and the footprint bound can be shown to be two sides of the same coin, solving similar problems, but for dual spaces. As discussed the footprint bound compares well with the improved Alon-Füredi bound and for finite fields the presented interpolation theorem is sharp. Our work can be viewed as a comment to a question raised by Tao in [Tao, 2014]

math.AC↗
arXiv2026

Poincaré Duality and Quadratic Refinements over Laurent Rings

We develop a Poincaré duality theory for defects of nondegenerate sesquilinear pairings over Laurent polynomial rings. A key ingredient is a novel flat resolution of the character module, constructed from a triangulation of the sphere at infinity associated with a fan. The cup product on this resolution turns Poincaré duality on the sphere into canonical pairings between the resulting defect modules. In middle degrees, we construct distinguished quadratic refinements using equivariant cohomology of the sphere with the antipodal action. The effective replacement of the sphere with a projective space provides a geometric substitute for division by two. Applied to translation-invariant Pauli stabilizer codes, our results establish the nondegeneracy of higher-dimensional braiding pairings. They extend the two-dimensional T-junction formula for topological spin to higher dimensions, while giving it a geometric interpretation.

math.AC↗