A filtration on the ring of Laurent polynomials and representations of the general linear Lie algebra

We first present a filtration on the ring Ln of Laurent polynomials such that the direct sum decomposition of its associated graded ring grLn agrees with the direct sum decomposition of grLn, as a module over the complex general linear Lie algebra gl(n), into its simple submodules. Next, generalizing the simple modules occurring in the associated graded ring grLn, we give some explicit constructions of weight multiplicity-free irreducible representations of gl(n).

Buchsbaumness of the associated graded rings of filtration

Let [Formula: see text] be a Noetherian local ring of dimension [Formula: see text] and [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text]. In this paper, we discuss a sufficient condition, for the Buchsbaumness of the local ring [Formula: see text] to be passed onto the associated graded ring of filtration. Let [Formula: see text] denote an [Formula: see text]-good filtration. We prove that if [Formula: see text] is Buchsbaum and the [Formula: see text] -invariant, [Formula: see text] and [Formula: see text], coincide then the associated graded ring [Formula: see text] is Buchsbaum. As an application of our result, we indicate an alternative proof of a conjecture, of Corso on certain boundary conditions for Hilbert coefficients.

Homology, mixed multiplies and Hilbert coefficients of the fiber cone

In this paper, we compare the depth of the fiber cone and the associated graded ring. To achieve this, we construct a bi-graded complex corresponding to a bi-graded, Noetherian, Hilbert filtration. The vanishing of the homology modules of this complex helps us to compare the depth of the fiber cone of the filtration and the depth of the corresponding associated graded ring. We also give a formula for the fiber coefficients in terms of the lengths of certain homology modules. We give an upper bound for the first fiber coefficient and show that when this bound is attained, the fiber cone has good depth.

Equimultiple coefficient ideals

Let $(R,\mathfrak {m})$ be a quasi-unmixed local ring and $I$ an equimultiple ideal of $R$ of analytic spread $s$. In this paper, we introduce the equimultiple coefficient ideals. Fix $k\in \{1,\dots ,s\}$. The largest ideal $L$ containing $I$ such that $e_{i}(I_{\mathfrak{p} })=e_{i}(L_{\mathfrak{p} })$ for each $i \in \{1,\dots ,k\}$ and each minimal prime $\mathfrak{p} $ of $I$ is called the $k$-th equimultiple coefficient ideal denoted by $I_{k}$. It is a generalization of the coefficient ideals introduced by Shah for the case of $\mathfrak {m}$-primary ideals. We also see applications of these ideals. For instance, we show that the associated graded ring $G_{I}(R)$ satisfies the $S_{1}$ condition if and only if $I^{n}=(I^{n})_{1}$ for all $n$.

THE STRUCTURE OF THE SALLY MODULE OF INTEGRALLY CLOSED IDEALS

The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra and in algebraic geometry. In this paper we give a complete algebraic structure of the Sally module of integrally closed ideals $I$ in a Cohen–Macaulay local ring $A$ satisfying the equality $\text{e}_{1}(I)=\text{e}_{0}(I)-\ell _{A}(A/I)+\ell _{A}(I^{2}/QI)+1,$ where $Q$ is a minimal reduction of $I$, and $\text{e}_{0}(I)$ and $\text{e}_{1}(I)$ denote the first two Hilbert coefficients of $I,$ respectively, the multiplicity and the Chern number of $I.$ This almost extremal value of $\text{e}_{1}(I)$ with respect to classical inequalities holds a complete description of the homological and the numerical invariants of the associated graded ring. Examples are given.

On the Grothendieck ring of varieties

Let K0(Vark) denote the Grothendieck ring of k-varieties over an algebraically closed field k. Larsen and Lunts asked if two k-varieties having the same class in K0(Vark) are piecewise isomorphic. Gromov asked if a birational self-map of a k-variety can be extended to a piecewise automorphism. We show that these two questions are equivalent over any algebraically closed field. If these two questions admit a positive answer, then we prove that its underlying abelian group is a free abelian group and that the associated graded ring of the Grothendieck ring is the monoid ring $\mathbb{Z}$[$\mathfrak{B}$] where $\mathfrak{B}$ denotes the multiplicative monoid of birational equivalence classes of irreducible k-varieties.

The structure of sally modules and Buchsbaumness of associated graded rings

Let A be a Noetherian local ring with the maximal ideal m, and let I be an m-primary ideal in A. This paper examines the equality on Hilbert coefficients of I first presented by Elias and Valla, but without assuming that A is a Cohen–Macaulay local ring. That equality is related to the Buchsbaumness of the associated graded ring of I.