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The module of Valabrega-Valla of the Jacobian ideal of a reduced projective variety $V$ is the torsion of the Aluffi algebra. One considers the problem of its vanishing in the case of where $V$ is a reduced set of points in the projective…

We study properties of the resolution of almost Gorenstein artinian algebras $R/I,$ i.e. algebras defined by ideals $I$ such that $I=J+(f),$ with $J$ Gorenstein ideal and $f\in R.$ Such algebras generalize the well known almost complete…

Let F be a locally compact nonarchimedean field with residue characteristic p and G the group of F-rational points of a connected split reductive group over F. For k an arbitrary field, we study the homological properties of the…

Let $R$ be an affine algebra over an algebraically closed field of characteristic $0$ with dim$(R)=n$. Let $P$ be a projective $A=R[T_1,\cdots,T_k]$-module of rank $n$ with determinant $L$. Suppose $I$ is an ideal of $A$ of height $n$ such…

Let G be a perfect graph and let J be its ideal of vertex covers. We show that the Rees algebra of J is normal and that this algebra is Gorenstein if G is unmixed. Then we give a description--in terms of cliques--of the symbolic Rees…

For a regular normal element in an arbitrary ring, we study the category of its module factorizations. The cokernel functor relates module factorizations with Gorenstein projective components to Gorenstein projective modules over the…

Many results are known about test ideals and $F$-singularities for ${\bf Q}$-Gorenstein rings. In this paper we generalize many of these results to the case when the symbolic Rees algebra $O_X \oplus O_X(-K_X) \oplus O_X(-2K_X) \oplus ...$…

Let $X$ be a normal arithmetically Gorenstein scheme in ${\mathbb P}^n$. We give a criterion for all codimension two ACM subschemes of $X$ to be in the same Gorenstein biliaison class on $X$, in terms of the category of ACM sheaves on $X$.…

Let $A$ be a simple abelian variety of dimension $g$ defined over a finite field $\mathbb{F}_q$ with Frobenius endomorphism $\pi$. This paper describes the structure of the group of rational points $A(\mathbb{F}_{q^n})$, for all $n \geq 1$,…

A ladder determinantal module is an arbitrary direct sum of ideals of maximal minors of a generic ladder matrix. In this article, we give necessary and sufficient conditions for the special fiber ring of such modules to be Gorenstein. These…

Let $R$ be a local Gorenstein ring with infinite residue field of arbitrary characteristic. Let $I$ be an $R$--ideal with $g=\height I >0$, analytic spread $\ell$, and let $J$ be a minimal reduction of $I$. We further assume that $I$…

Let $R$ be a polynomial ring over a field and $I \subset R$ be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the saturated special fiber ring of…

Let $H$ be a generic affine Hecke algebra (Iwahori-Matsumoto definition) over a polynomial algebra with a finite number of indeterminates over the ring of integers. We prove the existence of an integral Bernstein-Lusztig basis related to…

A minor is principal means it is defined by the same row and column indices. Let $X$ be a square generic matrix, $K[X]$ the polynomial ring in entries of $X$, over an algebraically closed field, $K$. For fixed $t\leq n$, let $\mathfrak P_t$…

Consider a group acting on a polynomial ring S over a field K by degree-preserving K-algebra automorphisms. Several key properties of the invariant ring can be deduced by studying the nullcone of the action, that is, the vanishing locus of…

In Commutative Algebra structure results on minimal free resolutions of Gorenstein modules are of classical interest. We define Gorenstein modules of finite length over the weighted polynomial ring via symmetric matrices in divided powers.…

For a nonempty topological space X, the ring of all real-valued functions on $X$ with pointwise addition and multiplication is denoted by $F(X)$ and continuous members of $F(X)$ is denoted by $C(X)$. Let $A(X)$ be a subring of $F(X)$ and…

We study the finite generation of the intersection algebra of two principal ideals I and J in a unique factorization domain R. We provide an algorithm that produces a list of generators of this algebra over R. In the special case that R is…

Let $SU_X(n,L)$ be the moduli space of rank n semistable vector bundles with fixed determinant L on a smooth projective genus g curve X. Let $SU_X^s(n,L)$ denote the open subset parametrizing stable bundles. We show that if g>3 and n > 1,…

In this note we construct a series of singular vectors in universal affine vertex operator algebras associated to $D_{\ell}^{(1)}$ of levels $n-\ell+1$, for $n \in \Z_{>0}$. For $n=1$, we study the representation theory of the quotient…