In this paper we present results concerning the following question: If M is a finitely-generated module with finite local cohomologies over a Noetherian local ring (A,m), does there exist an integer l such that every parameter ideal for M contained in m has the same index of reducibility? We show that the answer is yes if dimM = 1 or if dimM = 2 and depthM > 0. This research is closely related ...

In this note we exhibit a condition under which a sequence of elements a19 , an in a commutative noetherian local ring A form an Asequence, and derive a number of corollaries. Recall that if M is an A-module, then a19 , an is an M-sequence if 1) ai+1 is a nonzerodivisor on M/(a19 9a1)M for i = 0, 9n — 1,. and 2) M Φ (al9 . . . , α n ) M . If a19 , an is an A-sequence and / is the ideal generate...

A notion of stratification is introduced for any compactly generated triangulated category T endowed with an action of a graded commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences which follow when T is stratified by R. Among them are a classification of the localizing subcategories of T in terms of subsets of the set of prime ideals in...

We use the notion of quasilength to introduce a nonnegative real number hx(M) that is intended heuristically as a “measure” of the local cohomology module H I (M). It is defined as a limit or lim inf of normalized quasilengths: see §2. When x is understood from context we shall also refer to this real number as the h-content of M . The positivity of hx(R) gives a necessary condition for there t...

In this paper we prove that ifM is a finitely-generated module of dimension d with finite local cohomologies over a Noetherian local ring (A,m), and if Him (M) = 0 except possibly for i ∈ {0, r, d} with some 0 ≤ r ≤ d, then there exists an integer l such that every parameter ideal for M contained in m has the same index of reducibility. This theorem generalizes earlier work of the second author...

We organize this set of notes around a few theorems of Weierstrass. Write OCn for the sheaf of holomorphic functions on C. In the first three sections, we deduce the following “algebraic” results as consequences of the Weierstrass theorems: (i) for each open set Ω ⊂ C, the ring OCn(Ω) is not Noetherian, (ii) the local ring OCn,0 is factorial, (iii) the local ring OCn,0 is Noetherian. The two la...

In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results est...

Let (R,m) be a d-dimensional Noetherian local ring and E a finitely generated R-submodule of a free module R. In this work we introduce a Buchsbaum-Rim multiplicity sequence ck(E), k = 0, . . . , d + p − 1 for E that generalize the Buchsbaum-Rim multiplicity defined when E has finite colength in R as well as the Achilles-Manaresi multiplicity sequence that applies when E ⊆ R is an ideal. Our ma...

Finitely generated modules with finite F -representation type (or FFRT for short) over Noetherian (local) rings of prime characteristic p are studied. If a ring R has FFRT or, more generally, if a faithful R-module has FFRT, then tight closure commutes with localizations over R. We also define F -contributors and use them to give an effective way to characterize tight closure. Then we show lime...

Let A be a commutative Noetherian ring of dimension n (n ≥ 3). Let I be a local complete intersection ideal in A[T ] of height n. Suppose I/I is free A[T ]/I-module of rank n and (A[T ]/I) is torsion inK0(A[T ]). It is proved in this paper that I is a set theoretic complete intersection ideal in A[T ] if one of the following conditions holds: (1) n ≥ 5, odd; (2) n is even, and A contains the fi...