Related papers: On the hit problem for the polynomial algebra and …

We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) equivalence of gradings. We show that for each equivalence class of (co)module algebra structures on a…

In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…

We study rings which have Noetherian cohomology under the action of a ring of cohomology operators. The main result is a criterion for a complex of modules over such a ring to have finite injective dimension. This criterion generalizes, by…

Consider a finite dimensional H-module Lie algebra L over a field of characteristic 0 where H is a Hopf algebra. We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial H-identities of L under some…

We study the transference through finite index extensions of the notion of equational coherence, as well as its effective counterpart. We deduce an explicit algorithm for solving the following algorithmic problem about size two integral…

$ \newcommand{\inparen}[1]{\left( #1 \right)} \newcommand{\pfrac}[2]{\inparen{\frac{1}{2}}} \newcommand{\ilog}[1]{\log^{\circ #1}} \newcommand{\F}{\mathbb{F}} $The Polynomial Identity Lemma (also called the "Schwartz--Zippel lemma") states…

The quantization of the superclassical system used in the proof of the index theorem results in a factor of $\hbar^{2}R/8 $ in the Hamiltonian. The path integral expression of the kernel is analyzed up to and including 2-loop order. The…

We introduce the notion of integrable modules over $\imath$quantum groups (a.k.a. quantum symmetric pair coideal subalgebras). After determining a presentation of such modules, we prove that each integrable module over a quantum group is…

We address a conjecture (referred to as sur in the literature) in the representation theory of a reductive p-adic Lie group G which has important implications for the relationship between mod-p smooth representations and pro-p Iwahori-Hecke…

In order to determine the Hilbert function of the ideal of a fat point subscheme of projective space, we show that it is enough to determine, both for the subscheme itself and the subschemes obtained from it by successively adjoining to it…

We prove that the grades of simple modules indexed by boolean permutations, over the incidence algebra of the symmetric group with respect to the Bruhat order, are given by Lusztig's a-function. Our arguments are combinatorial, and include…

We develop the categorical algebra of the noncommutative base change of a comodule category by means of a Grothendieck category $\mathfrak S$. We describe when the resulting category of comodules is locally finitely generated, locally…

Let H be a coFrobenius Hopf algebra over a field k. Let A be a right H-comodule algebra over k. We recall that the category of right H-comodules admits a certain model structure whose homotopy category is equivalent to the stable category…

For a commutative ring $\mathbf k$ with unit, we describe and study various differential graded $\mathbf k$-modules and $ \mathbf k$-algebras which are models for the cohomology of polyhedral products $(\underline{CX},\underline X)^K$.…

Let $\mathbf{H}$ be the cartesian product of a family of finite abelian groups indexed by a finite set $\Omega$. A given poset (i.e., partially ordered set) $\mathbf{P}=(\Omega,\preccurlyeq_{\mathbf{P}})$ gives rise to a poset metric on…

Let $\lambda$ be a general length function for modules over a Noetherian ring R. We use $\lambda$ to introduce Hilbert series and polynomials for R[X]-modules, measuring the growth rate of~$\lambda$. We show that the leading term $\mu$ of…

Motivated by the study of invariant rings of finite groups on the first Weyl algebras $A_{1}$ (\cite{AHV}) and finding interesting families of new noetherian rings, a class of algebras similar to $U(sl_{2})$ were introduced and studied by…

We initiate a study of Hilbert modules over the polynomial algebra A=C[z_1,...,z_d] that are obtained by completing A with respect to an inner product having certain natural properties. A standard Hilbert module is a finite multiplicity…

We demonstrate a close connection between the classic planar Singer difference sets and certain norm equation systems arising from projective norm graphs. This, on the one hand leads to a novel description of planar Singer difference sets…

The purpose of this paper is to initiate a new attack on Arveson's resistant conjecture, that all graded submodules of the $d$-shift Hilbert module $H^2$ are essentially normal. We introduce the stable division property for modules (and…