Cohomological and projective dimensions

In this paper we give an upper bound, in characteristic 0, for the cohomological dimension of a graded ideal in a polynomial ring such that the quotient has depth at least 3. In positive characteristic the same bound holds true by a well-known theorem of Peskine and Szpiro. As a corollary, we give new examples of prime ideals that are not set-theoretically Cohen-Macaulay.Comment: 6 pages. Corrected some typos and added details in Example 3.9. Added Example 2.3 and rearranged the proof of Proposition 3.1. To appear in Compositio Mat

Symbolic Powers and Matroids

We prove that all the symbolic powers of a Stanley-Reisner ideal are Cohen-Macaulay if and only if the associated simplicial complex is a matroid.Comment: 10 pages, some minor change

Koszulness, Krull Dimension and Other Properties of Graph-Related Algebras

The algebra of basic covers of a graph G, denoted by \A(G), was introduced by Juergen Herzog as a suitable quotient of the vertex cover algebra. In this paper we show that if the graph is bipartite then \A(G) is a homogeneous algebra with straightening laws and thus is Koszul. Furthermore, we compute the Krull dimension of \A(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. Finally, we characterize the Cohen-Macaulay property and the Castelnuovo-Mumford regularity of the edge ideal of a certain class of graphs.Comment: 23 page

Partitions of single exterior type

We characterize the irreducible representations of the general linear group GL(V) that have multiplicity 1 in the direct sum of all Schur modules of a given exterior power of V. These have come up in connection with the relations of the lower order minors of a generic matrix. We show that the minimal relations conjectured by Bruns, Conca and Varbaro are exactly those coming from partitions of single exterior type