Paulo Brumatti

The module of derivations of a Stanley-Reisner ring

An explicit description is given of the module Der(k[X]/I, k[X]/I) of the derivations of the residue ring k[X]/I, where I is an ideal generated by monomials whose exponents are prime to the characteristic of the field k (this includes the case of square free monomials in any characteristic and the case of arbitrary monomials in characteristic zero). In the case where I is generated by square free monomials, this description is interpreted in terms of the corresponding abstract simplicial complex A. Sharp bounds for the depth of this module are obtained in terms of the depths of the face rings of certain subcomplexes Ai related to the stars of the vertices vi of A. The case of a Cohen-Macaulay simplicial complex A is discussed in some detail: it is shown that Der(k[A], k[A]) is a Cohen-Macaulay module if and only if depthAi > dim A 1 for every vertex vi . A measure of triviality of the complexes Ai is introduced in terms of certain star corners of vi . A curious corollary of the main structural result is an affirmative answer in the present context to the conjecture of Herzog-Vasconcelos on the finite projective dimension of the k[X]/I-module Der(k[X]/I, k[X]/I).

A note on the Nakai conjecture

The conjectures of Zariski-Lipman and of Nakai are still open in general in the class of rings essentially of finite type over a field of characteristic zero. However, they have long been known to be true in dimension one. Here we give counterexamples to both conjectures in the class of one-dimensional pseudo-geometric local domains that contain a field of characteristic zero. Likewise, in connection with a recent result of Traves on the Nakai conjecture, we also show that their hypothesis of finite generation of the integral closure cannot be removed even in the class of local domains containing a field of characteristic zero.

Differential simplicity, Cohen-Macaulayness and formal prime divisors

Abstract We construct counter-examples, in any characteristic of the conjecture that differentially simple local domains should be Cohen-Macaulay. Such counter examples can be made henselian.

Combinatorics of a certain ideal in the Segre coordinate ring

We focus on a “fat” model of an ideal in the class of the canonical ideal of the Segre coordinate ring, looking at its Rees algebra and related arithmetical questions.