3,271 research outputs found
Picard groups of punctured spectra of dimension three local hypersurfaces are torsion-free
Let (R,m) be a local ring and U_R=Spec(R) -{m} be the punctured spectrum of
R. Gabber conjectured that if R is a complete intersection of dimension 3, then
the abelian group Pic(U_R) is torsion-free. In this note we prove Gabber's
statement for the hypersurface case. We also point out certain connections
between Gabber's Conjecture, Van den Bergh's notion of non-commutative crepant
resolutions and some well-studied questions in homological algebra over local
rings.Comment: Some statements/typos fixed thanks to corrections from the referees,
main results remain the sam
The radius of a subcategory of modules
We introduce a new invariant for subcategories X of finitely generated
modules over a local ring R which we call the radius of X. We show that if R is
a complete intersection and X is resolving, then finiteness of the radius
forces X to contain only maximal Cohen-Macaulay modules. We also show that the
category of maximal Cohen-Macaulay modules has finite radius when R is a
Cohen-Macaulay complete local ring with perfect coefficient field. We link the
radius to many well-studied notions such as the dimension of the stable
category of maximal Cohen-Macaulay modules, finite/countable Cohen-Macaulay
representation type and the uniform Auslander condition.Comment: Final version, to appear in Algebra and Number Theor
Krull-Schmidt categories and projective covers
Krull-Schmidt categories are additive categories such that each object
decomposes into a finite direct sum of indecomposable objects having local
endomorphism rings. We provide a self-contained introduction which is based on
the concept of a projective cover
Auslander-Reiten conjecture and Auslander-Reiten duality
Motivated by a result of Araya, we extend the Auslander-Reiten duality
theorem to Cohen-Macaulay local rings. We also study the Auslander-Reiten
conjecture, which is rooted in Nakayama's work on finite dimensional algebras.
One of our results detects a certain condition that forces the conjecture to
hold over local rings of positive depth.Comment: 16 pages, to appear in Journal of Algebr
Cluster structures for 2-Calabi-Yau categories and unipotent groups
We investigate cluster tilting objects (and subcategories) in triangulated
2-Calabi-Yau categories and related categories. In particular we construct a
new class of such categories related to preprojective algebras of non Dynkin
quivers associated with elements in the Coxeter group. This class of
2-Calabi-Yau categories contains the cluster categories and the stable
categories of preprojective algebras of Dynkin graphs as special cases. For
these 2-Calabi-Yau categories we construct cluster tilting objects associated
with each reduced expression. The associated quiver is described in terms of
the reduced expression. Motivated by the theory of cluster algebras, we
formulate the notions of (weak) cluster structure and substructure, and give
several illustrations of these concepts. We give applications to cluster
algebras and subcluster algebras related to unipotent groups, both in the
Dynkin and non Dynkin case.Comment: 49 pages. For the third version the presentation is revised,
especially Chapter III replaces the old Chapter III and I
The negative side of cohomology for Calabi-Yau categories
We study integer-graded cohomology rings defined over Calabi-Yau categories.
We show that the cohomology in negative degree is a trivial extension of the
cohomology ring in non-negative degree, provided the latter admits a regular
sequence of central elements of length two. In particular, the product of
elements of negative degrees are zero. As corollaries we apply this to
Tate-Hochschild cohomology rings of symmetric algebras, and to Tate cohomology
rings over group algebras. We also prove similar results for Tate cohomology
rings over commutative local Gorenstein rings.Comment: 14 page
Dynamic Considerations for Control of Closed Life Support Systems
Reliability of closed life support systems depend on their ability to continue supplying the crew's needs during perturbations and equipment failures. The dynamic considerations interact with the basic static design through the sizing of storages, the specification of excess capacities in processors, and the choice of system initial state. A very simple system flow model was used to examine the possibilities for system failures even when there is sufficient storage to buffer the immediate effects of the perturbation. Two control schemes are shown which have different dynamic consequences in response to component failures
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