Corrigendum to “New algebraic properties of an amalgamated algebra along an ideal”

At some point, after publication, we realized that Proposition 4.1(2) and Theorem 4.4 in [2 D’Anna, M., Finocchiaro, C. A., Fontana, M. (2016). New algebraic properties of an amalgamated algebra along an ideal. Commun. Algebra 44(5):1836–1851.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] hold under the assumption (not explicitly declared) that B = f(A)+J. Furthermore, we provide here the exact value for the embedding dimension of A?^fJ, also when B≠f(A)+J, under the hypothesis that J is finitely generated as an ideal of the ring f(A)+J.     

The Relative Transpose over Cohen-Macaulay Finite Artin Algebras

The relative transpose via Gorenstein projective modules is introduced, and some corresponding results on the Auslander-Reiten sequences and the Auslander-Reiten formula to this relative version are generalized.     

Reducing system of parameters and the Cohen-Macaulay property

Let R be a local ring and let (x ₁, …, x _r) be part of a system of parameters of a finitely generated R-module M, where r < dim_R M. We will show that if (y ₁, …, y _r) is part of a reducing system of parameters of M with (y ₁, …, y _r) M = (x ₁, …, x _r) M then (x ₁, …, x _r) is already reducing. Moreover, there is such a part of a reducing system of parameters of M iff for all primes P ε Supp M ∩ V _R(x ₁, …, x _r) with dim_R R/P = dim_R M − r the localization M _P of M at P is an r-dimensional Cohen-Macaulay module over R _P. Furthermore, we will show that M is a Cohen-Macaulay module iff y _d is a non zero divisor on M/(y ₁, …, y _d−1) M, where (y ₁, …, y _d) is a reducing system of parameters of M (d:= dim_R M).     

KRS and Powers of determinantal ideas

The goal of this paper is to determine Göbner bases of powers of determinantal ideals and to show that the Rees algebras of (products of) determinantal ideals are normal and Cohen–Macaulay if the characteristic of the base field is non-exceptional. Our main combinatorial result is a generalization of Schensted's Theorem on the Knuth–Robinson–Schensted correspondence.     

On the Cohen-Macaulayness of fiber cones

Let be a Noetherian local ring with the maximal ideal and an ideal of Denote by the fiber cone of This paper characterizes the multiplicity and the Cohen-Macaulayness of fiber cones in terms of minimal reductions of ideals.

     

Big Cohen-Macaulay algebras and seeds

In this article, we delve into the properties possessed by algebras, which we have termed seeds, that map to big Cohen-Macaulay algebras. We will show that over a complete local domain of positive characteristic any two big Cohen-Macaulay algebras map to a common big Cohen-Macaulay algebra. We will also strengthen Hochster and Huneke's ``weakly functorial" existence result for big Cohen-Macaulay algebras by showing that the seed property is stable under base change between complete local domains of positive characteristic. We also show that every seed over a positive characteristic ring maps to a balanced big Cohen-Macaulay -algebra that is an absolutely integrally closed, -adically separated, quasilocal domain.

     

Integral Closures of Cohen-Macaulay Monomial Ideals

ABSTRACT

The purpose of this paper is to present a family of Cohen-Macaulay monomial ideals such that their integral closures have embedded components and hence are not Cohen-Macaulay.     

Hilbert schemes, polygraphs and the Macdonald positivity conjecture

We study the isospectral Hilbert scheme , defined as the reduced fiber product of with the Hilbert scheme of points in the plane , over the symmetric power . By a theorem of Fogarty, is smooth. We prove that is normal, Cohen-Macaulay and Gorenstein, and hence flat over . We derive two important consequences.

(1) We prove the strong form of the conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients . This establishes the Macdonald positivity conjecture, namely that .

(2) We show that the Hilbert scheme is isomorphic to the -Hilbert scheme of Nakamura, in such a way that is identified with the universal family over . From this point of view, describes the fiber of a character sheaf at a torus-fixed point of corresponding to .

The proofs rely on a study of certain subspace arrangements , called polygraphs, whose coordinate rings carry geometric information about . The key result is that is a free module over the polynomial ring in one set of coordinates on . This is proven by an intricate inductive argument based on elementary commutative algebra.