11 research outputs found
Vanishing of (co)homology, freeness criteria, and the Auslander-Reiten conjecture for Cohen-Macaulay Burch rings
We establish new results on (co)homology vanishing and Ext-Tor dualities, and
derive a number of freeness criteria for finite modules over Cohen-Macaulay
local rings. In the main application, we settle the long-standing
Auslander-Reiten conjecture for the class of Cohen-Macaulay Burch rings, among
other results toward this and related problems, e.g., the Tachikawa and
Huneke-Wiegand conjectures. We also derive results on further topics of
interest such as Cohen-Macaulayness of tensor products and Tor-independence,
and inspired by a paper of Huneke and Leuschke we obtain characterizations of
when a local ring is regular, or a complete intersection, or Gorenstein; for
the regular case, we describe progress on some classical differential problems,
e.g., the strong version of the Zariski-Lipman conjecture. Along the way, we
generalize several results from the literature and propose various questions.Comment: 25 pages. Submitted for publicatio
Dao's question on the asymptotic behaviour of fullness
For a local ring (R, \M) of infinite residue field and positive depth, we
address the question raised by H. Dao on how to control the asymptotic
behaviour of the \M-full, full, and weakly \M-full properties of certain
ideals (such notions were first investigated by D. Rees and J. Watanabe), by
means of bounding appropriate numbers which express such behaviour. We
establish upper bounds, and in certain cases even formulas for such invariants.
The main tools used in our results are reduction numbers along with
Ratliff-Rush closure of ideals, and also the Castelnuovo-Mumford regularity of
the Rees algebra of \M.Comment: 11 pages. Submitted for publicatio
Tensor products and solutions to two homological conjectures for Ulrich modules
We address the problem of when the tensor product of two finitely generated
modules over a Cohen-Macaulay local ring is Ulrich in the generalized sense of
Goto et al., and in particular in the original sense from the 80's. As
applications, besides freeness criteria for modules, characterizations of
complete intersections, and an Ulrich-based approach to the long-standing
Berger's conjecture, we show that two celebrated homological conjectures,
namely the Auslander-Reiten and the Huneke-Wiegand problems, are true for the
class of Ulrich modules.Comment: 12 page
A family of reflexive vector bundles of reduction number one
A difficult issue in modern commutative algebra asks for examples of modules (more interestingly, reflexive vector bundles) having prescribed reduction number . The problem is even subtler if in addition we are interested in good properties for the Rees algebra. In this note we consider the case . Precisely, we show that the module of logarithmic vector fields of the Fermat divisor of any degree in projective -space is a reflexive vector bundle of reduction number and Gorenstein Rees ring
Vector fields and a family of linear type modules related to free divisors
AbstractThis paper has three main goals. We start describing a method for computing the polynomial vector fields tangent to a given algebraic variety; this is of interest, for instance, in view of (effective) foliation theory. We then pass to furnishing a family of modules of linear type (that is, the Rees algebra equals the symmetric algebra), formed with vector fields related to suitable pairs of algebraic varieties, one of them being a free divisor in the sense of K. Saito. Finally, we derive freeness criteria for modules retaining a certain tangency feature, so that, in particular, well-known criteria for free divisors are recovered
Generalized local duality, canonical modules, and prescribed bound on projective dimension
We present various approaches to J. Herzog's theory of generalized local
cohomology and explore its main aspects, e.g., (non-)vanishing results as well
as a general local duality theorem which extends, to a much broader class of
rings, previous results by Herzog-Zamani and Suzuki. As an application, we
establish a prescribed upper bound for the projective dimension of a module
satisfying suitable cohomological conditions, and we derive some freeness
criteria and questions of Auslander-Reiten type. Along the way, we prove a new
characterization of Cohen-Macaulay modules which truly relies on generalized
local cohomology, and in addition we introduce and study a generalization of
the notion of canonical module.Comment: Final version, to appear in J. Pure Appl. Algebr