Noetherian rings of low global dimension and syzygetic prime ideals

Let R be a Noetherian ring. We prove that R has global dimension at most two if, and only if, every prime ideal of R is of linear type. Similarly, we show that R has global dimension at most three if, and only if, every prime ideal of R is syzygetic. As a consequence, we derive a characterization of these rings using the André-Quillen homology.This work is partially supported by the Catalan grant 2014 SGR-634.Peer ReviewedPostprint (author's final draft

The relation type of affine algebras and algebraic varieties

We introduce the notion of relation type of an affine algebra and prove that it is well defined by using the Jacobi-Zariski exact sequence of Andre-Quillen homology. In particular, the relation type,is an invariant of an affine algebraic variety. Also as a consequence of the invariance, we show that in order to calculate the relation type of an ideal in a polynomial ring one can reduce the problem to trinomial ideals. When the relation type is at least two, the extreme equidimensional components play no role. This leads to the non-existence of affine algebras of embedding dimension three and relation type two. (C) 2015 Elsevier Inc. All rights reserved.Peer ReviewedPostprint (author's final draft

On the vanishing and non-rigidity of the André-Quillen (co)homology

Let I be an ideal of a commutative ring A, B=A/I. Given n>=2, we characterize the vanishing of the André-Quillen homology modules H_{p}(A,B,W) for all $B$-module $W$ and for all p, 2<=p<=n, in terms of some canonical morphisms. As a corollary, we obtain a new proof of a theorem of André. Finally, we construct an example of an ideal $I$ of a commutative ring $A$ such that H_{2}(A,B,W)=0 and H_{3}(A,B,W)=W for all B-module W

The equations of Rees algebras of equimultiple ideals of deviation one.

We describe the equations of the Rees algebra R(I) of an equimultiple ideal I of deviation one provided that I has a reduction generated by a regular sequence x1, . . . , xs such that the initial forms x∗ 1, . . . , x∗ s−1 are a regular sequence in the associated graded ring. In particular, we prove that there is a single equation of maximum degree in a minimal generating set of the equations of R(I), which recovers some previous known results.Postprint (published version

Ideals of Herzog-Northcott type

This paper takes a new look at ideals generated by 2×2 minors of 2×3 matrices whose entries are powers of three elements not necessarily forming a regular sequence. A special case of this is the ideals determining monomial curves in three-dimensional space, which were studied by Herzog. In the broader context studied here, these ideals are identified as Northcott ideals in the sense of Vasconcelos, and so their liaison properties are displayed. It is shown that they are set-theoretically complete intersections, revisiting the work of Bresinsky and of Valla. Even when the three elements are taken to be variables in a polynomial ring in three variables over a field, this point of view gives a larger class of ideals than just the defining ideals of monomial curves. We then characterize when the ideals in this larger class are prime, we show that they are usually radical and, using the theory of multiplicities, we give upper bounds on the number of their minimal prime ideals, one of these primes being a uniquely determined prime ideal of definition of a monomial curve. Finally, we provide examples of characteristic-dependent minimal prime and primary structures for these ideals.Postprint (published version

Divisors of expected Jacobian type

Divisors whose Jacobian ideal is of linear type have received a lot of attention recently because of its connections with the theory of D-modules. In this work we are interested on divisors of expected Jacobian type, that is, divisors whose gradient ideal is of linear type and the relation type of its Jacobian ideal coincides with the reduction number with respect to the gradient ideal plus one. We provide conditions in order to be able to describe precisely the equations of the Rees algebra of the Jacobian ideal. We also relate the relation type of the Jacobian ideal to some D-module theoretic invariant given by the degree of the Kashiwara operator.Peer ReviewedPostprint (author's final draft

Noncomplete intersection prime ideals in dimension 3

We describe prime ideals of height 2 minimally generated by three elements in a Gorenstein, Nagata local ring of Krull dimension 3 and multiplicity at most 3. This subject is related to a conjecture of Y. Shimoda and to a long-standing problem of J. Sally.Peer ReviewedPostprint (author’s final draft