Let A be a two-dimensional excellent normal Gorenstein local domain. In this paper, we characterize elliptic ideals for its normal tangent cone to be Gorenstein. Moreover, we classify all those ideals in a Gorenstein elliptic singularity in the characteristic zero case.
We prove a formula for the Chow groups of Quot schemes that resolve degeneracy loci of a map between vector bundles, under expected dimension conditions. This formula simultaneously generalizes the formulas for projective bundles, Grassmannian bundles, blowups, Cayley's trick, projectivizations, flops from Springer-type resolutions, and Grassmannian-type flips and flops. We also apply the formula to study the Chow groups of (i) blowups of determinantal ideals; (ii) moduli spaces of linear series on curves; and (iii) (nested) Hilbert schemes of points on surfaces.
We study the vanishing of (co)homology along ring homomorphisms for modules that admit certain filtrations and generalize a theorem of Celikbas-Takahashi. Our work produces new classes of rigid and test modules, particularly over local rings of prime characteristic. Additionally, it provides applications in the study of torsion in tensor products of modules, including a conjecture of Huneke-Wiegand.
For a pair (algebra, module) with equidimensional and isolated singularity we establish the existence of a versal henselian deformation. Obstruction theory in terms of an André-Quillen cohomology for pairs is a central ingredient in the Artin theory used. In particular we give a long exact sequence relating the algebra cohomology and the module cohomology with the cohomology of the pair and define a Kodaira-Spencer class for pairs.
We generalize the notion of an NS-algebra, which was previously only considered for associative, Lie and Leibniz algebras, to arbitrary categories of binary algebras with one operation. We do this by defining these algebras using a bimodule property, as we did in our earlier work for defining the notions of a dendriform and tridendriform algebra for such categories of algebras. We show that several types of operators lead to NS-algebras: Nijenhuis operators, twisted Rota-Baxter operators and relative Rota-Baxter operators of arbitrary weight. We thus provide a general framework in which several known results and constructions for associative, Lie and Leibniz-NS-algebras are unified, along with some new examples and constructions that we also present.
We study a Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the group ring of finitely generated abelian groups. The Batalin-Vilkovisky algebra structure for finite abelian groups comes from the fact that the group ring of finite groups is a symmetric algebra, and the Batalin-Vilkovisky algebra structure for free abelian groups of finite rank comes from the fact that its group ring is a Calabi-Yau algebra.
Let be a pair of a smooth variety X over an algebraically closed field k of characteristic and its Frobenius morphism F. Given a Frobenius -lifting of the pair for , Nori and Srinivas [9] determined the obstruction to Frobenius -lifting of in terms of Čech cohomology. The extension representing has been only known for , which uses the Cartier operator. In this paper, we interpret in terms of Kato's version of de Rham-Witt Cartier operator [8] and determine the extension representing for .
In this paper we adapt the notion of the nucleus defined by Benson, Carlson, and Robinson to compact Lie groups in non-modular characteristic. We show that it describes the singularities of the projective scheme of the cohomology of its classifying space. A notion of support for singularity categories of ring spectra (in the sense of Greenlees and Stevenson) is established, and is shown to be precisely the nucleus in this case, consistent with a conjecture of Benson and Greenlees for finite groups.
The set of minimal primes of a ring is a very important set as far the spectrum of a ring is concerned as every prime contains a minimal prime. So, knowing the minimal primes is the first (important and difficult) step in describing the spectrum. In the algebraic geometry, the minimal primes of the algebra of regular functions on an algebraic variety determine/correspond to the irreducible components of the variety. The aim of the paper is to obtain several descriptions of the set of minimal prime ideals of localizations of rings under several natural assumptions. In particular, the following cases are considered: a localization of a semiprime ring with finite set of minimal primes; a localization of a prime rich ring where the localization respects the ideal structure of primes and primeness of certain minimal primes; a localization of a ring at a left denominator set generated by normal elements, and others. As an application, for a semiprime ring with finitely many minimal primes, a description of the minimal primes of its largest left/right quotient ring is obtained.
For a semiprime ring R with finitely many minimal primes , criteria are given for the map being a well-defined map or surjective where is the centre of R.
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