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

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

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

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

Mean Dimension & Jaworski-type Theorems

According to the celebrated Jaworski Theorem, a finite dimensional aperiodic dynamical system $(X,T)$ embeds in the $1$-dimensional cubical shift $([0,1]^{\mathbb{Z}},shift)$. If $X$ admits periodic points (still assuming $\dim(X)<\infty$) then we show in this paper that periodic dimension $perdim(X,T)<\frac{d}{2}$ implies that $(X,T)$ embeds in the $d$-dimensional cubical shift $(([0,1]^{d})^{\mathbb{Z}},shift)$. This verifies a conjecture by Lindenstrauss and Tsukamoto for finite dimensional systems. Moreover for an infinite dimensional dynamical system, with the same periodic dimension assumption, the set of periodic points can be equivariantly immersed in $(([0,1]^{d})^{\mathbb{Z}},shift)$. Furthermore we introduce a notion of markers for general topological dynamical systems, and use a generalized version of the Bonatti-Crovisier tower theorem, to show that an extension $(X,T)$ of an aperiodic finite-dimensional system whose mean dimension obeys $mdim(X,T)<\frac{d}{16}$ embeds in the $(d+1)$-cubical shift $(([0,1]^{d+1})^{\mathbb{Z}},shift)$.Comment: To appear in Proceedings of the London Mathematical Societ