We prove that the generic quantized coordinate ringOq(G) is Auslander-regular, Cohen-Macaulay, and catenary for every connected semisimple Lie group G. This answers questions raised by Brown, Lenagan, and the first author. We also prove that under certain hypotheses concerning the existence of normal elements, a noetherian Hopf algebra is Auslander-Gorenstein and Cohen-Macaulay. This provides a...

Let C ⊂ N be an affine semigroup, and R = K[C] its semigroup ring. This paper is a collection of various results on “C-graded” R-modules M = ⊕ c∈C Mc, especially, monomial ideals of R. For example, we show the following: If R is normal and I ⊂ R is a radicalmonomial ideal (i.e., R/I is a generalization of Stanley-Reisner rings), then the sequentially Cohen-Macaulay property of R/I is a topologi...

A recent framework for generalizing the Erdős-KoRado Theorem, due to Holroyd, Spencer, and Talbot, defines the Erdős-Ko-Rado property for a graph in terms of the graph’s independent sets. Since the family of all independent sets of a graph forms a simplicial complex, it is natural to further generalize the Erdős-Ko-Rado property to an arbitrary simplicial complex. An advantage of working in sim...

The algebra of basic covers of a graph G, denoted by Ā(G), was introduced by Herzog as a suitable quotient of the vertex cover algebra. In this paper we compute the Krull dimension of Ā(G) in terms of the combinatorics of G. As a consequence, we get new upper bounds on the arithmetical rank of monomial ideals of pure codimension 2. Furthermore, we show that if the graph is bipartite, then Ā(G) ...

We describe some recent work concerning Gorenstein liaison of codimension two subschemes of a projective variety. Applications make use of the algebraic theory of maximal Cohen–Macaulay modules, which we review in an Appendix.

Pinched Veronese rings are formed by removing an algebra generator from a subring of polynomial ring. We study the homological properties such rings, including Cohen-Macaulay, Gorenstein, and complete intersection properties. Greco Martino classified Cohen-Macaulayness pinched maximum entry exponent vector monomial; we re-prove their results with semigroup methods correct omission small class e...

This paper considers the following conjecture: If R is an unmixed, equidimensional local ring that is a homomorphic image of a Cohen-Macaulay local ring, then for any ideal J generated by a system of parameters, the Chern coefficient e1(J) < 0 is equivalent to R being non Cohen-Macaulay. The conjecture is established if R is a homomorphic image of a Gorenstein ring, and for all universally cate...

Two formulas for the multiplicity of the fiber cone F (I) = ⊕∞n=0I /mI of an m-primary ideal of a d-dimensional Cohen-Macaulay local ring (R,m) are derived in terms of the mixed multiplicity ed−1(m|I), the multiplicity e(I) and superficial elements. As a consequence, the Cohen-Macaulay property of F (I) when I has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized ...

In this paper, we introduce the notion of “extension” of a toric variety and study its fundamental properties. This gives rise to infinitely many toric varieties with a special property, such as being set theoretic complete intersection or arithmetically Cohen-Macaulay (Gorenstein) and having a Cohen-Macaulay tangent cone or a local ring with non-decreasing Hilbert function, from just one singl...

In this paper we study Cohen-Macaulay local rings of dimension d, multiplicity e and second Hilbert coefficient e2 in the case e2=e1−e+1. Let h=μ(m)−d. If e2≠0 then our can prove that type(A)≥e−h−1. type(A)=e−h−1 show associated graded ring G(A) is Cohen-Macaulay. next when type(A)=e−h determine all possible series A. depthG(A) completely determines Series