Santiago Zarzuela

Burch's inequality and the depth of the blow up rings of an ideal

Abstract Let (A, m ) be a local noetherian ring with infinite residue field and I an ideal of A. Consider RA(I) and GA(I), respectively, the Rees algebra and the associated graded ring of I, and denote by l(I) the analytic spread of I. Burchs inequality says that l(I)+ inf { depth A/I n , n≥1}≤ dim (A) , and it is well known that equality holds if GA(I) is Cohen–Macaulay. Thus, in that case one can compute the depth of the associated graded ring of I as depth G A (I)=l(I)+ inf { depth A/I n , n≥1} . We study when such an equality is also valid when GA(I) is not necessarily Cohen–Macaulay, and we obtain positive results for ideals with analytic deviation less or equal than one and reduction number at most two. In those cases we may also give the value of depth R A (I) .

Balanced big Cohen-Macaulay modules and flat extensions of rings

Let A be a (commutative, Noetherian) local ring. The big Cohen-Macaulay conjecture asserts that if a 1 ,…, a n is a system of parameters for A there exists an A -module M such that a 1 ,…, a n is an M -sequence. Then we say that M is a big Cohen-Macaulay module with respect to a 1 ,…, a n . This conjecture implies some important conjectures in commutative algebra and has been established affirmatively by M. Hochster for any ring containing a field as a subring (see [9] for further information).

On Equimultiple Modules

We study the class of equimultiple modules. In particular, we prove several criteria for an equimultiple module to be of the principal class or complete intersection and prove the openness of the equimultiple locus of an ideal module.

ON THE SPECTRUM OF THE GROUP RING

In this paper we extend the study of the prime ideal structure of group rings initiated by Gilmer (1974), Brewer-Costa-Lady (1975), and Anderson-Bouvier-Dobbs-Fontana-Kabbaj (1988). Of particular inter-est is the transfer from A to A[G] of certain properties which are linked to the prime spectrum such as S-domain or Jaffard domain. Their study will follow the same lines as the usual approach to the study of prime ideal structure in polynomial rings.

Bounds for the multiplicity of almost complete intersections

Let (A, m) be a regular local ring and I an almost complete intersection ideal (a.c.i. for short) in the sense that p(I) = ht(I) + 1, where #(I) is the least number of generators of I. A quite natural question is : When is A / I a Cohen-Macaulay ring, in particular, if I is a prime ideal. For almost complete intersection prime ideals ~ in a regular local ring A containing a field, the following positive cases are wellknown, s. [5], (5.1): (i) If ht(~3)= 2, then A/~ is Cohen-Macaulay. (ii) If d imA _ _ 2.