Stable categories of higher preprojective algebras

We introduce (n+1)$(n+1)$-preprojective algebras of algebras of global dimension n$n$. We show that if an algebra is n$n$-representation-finite then its (n+1)$(n+1)$-preprojective algebra is self-injective. In this situation, we show that the stable module category of the (n+1)$(n+1)$-preprojective algebra is (n+1)$(n+1)$-Calabi–Yau, and, more precisely, it is the (n+1)$(n+1)$-Amiot cluster category of the stable n$n$-Auslander algebra of the original algebra. In particular this stable category contains an (n+1)$(n+1)$-cluster tilting object. We show that even if the (n+1)$(n+1)$-preprojective algebra is not self-injective, under certain assumptions (which are always satisfied for n∈{1,2}$n\in \{1,2\}$) the results above still hold for the stable category of Cohen–Macaulay modules.     

Exceptional collections of line bundles on the Beauville surface

We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface S$S$. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on S$S$. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections are spires of two helices.     

Finitely generated relatively universal varieties of Heyting algebras

Given distinct varieties and of the same type, we say that is relatively -universal if there exists an embedding :K from a universal categoryK such that for every pairA, B ofK-objects, a homomorphismf:A B has the formf=_g for someK-morphismg:A B if and only if Im(f) . Finitely generated relatively -universal varieties of Heyting algebras are described for the variety of Boolean algebras, the variety generated by a three element chain, and for the variety generated by the four element Boolean algebra with an added greatest element.Dedicated to the memory of Alan DayPresented by J. Sichler.The support of the NSERC is gratefully acknowledged.     

Recollements of Singularity Categories and Monomorphism Categories

We generalize results on existence of recollement situations of singularity categories of lower triangular Gorenstein algebras and stable monomorphism categories of Cohen–Macaulay modules.     

New Braided Crossed Categories and Drinfel'd Quantum Double for Weak Hopf Group Coalgebras

A simple and nice structure theorem for orthogroups was given by Petrich in 1987. In this paper, we consider a generalized orthogroup, that is, a quasi-completely regular semigroup with a band of idempotents in which its set of regular elements, namely, RegS, forms an ideal of S. A method of construction of such semigroups is provided and as a result, the Petrich structure theorem of orthogroups becomes an immediate corollary of our theorem on generalized orthogroups. An example of such generalized orthogroup is also constructed. This example provides some useful information for the construction of various kinds of quasi-completely regular semigroups.     

Translation cokernels of rational modules

A fundamental approach to the homological algebra of rational modules in positive characteristic p should begin with the translation properties of injective modules. This point of view leads to the study of functors related to coherent translation and their effect on the injective indecomposable modules indexed by p-regular dominant weights. The main result shows that the calculation can often be reduced to a corresponding calculation on the principal indecomposable modules of the Frobenius kernel. ^{1 1}1991 Mathematics Subject Classification: 20G05      

Coquasitriangular Hopf Group Algebras and Drinfel'd Co-Doubles

A ring R is called “semicommutative” if any right annihilator over R is an ideal of R. We show that special subrings of upper triangular matrix rings over a reduced ring are maximal semicommutative. Consequently, new families of semicommutative rings are presented.     

Braid Action on Derived Category Nakayama Algebras

We construct an action of a braid group associated to a complete graph on the derived category of a certain symmetric Nakayama algebra which is also a Brauer star algebra with no exceptional vertex. We connect this action with the affine braid group action on Brauer star algebras defined by Schaps and Zakay–Illouz. We show that for Brauer star algebras with no exceptional vertex, the action is faithful.     

Chase’s lemma and its context

Chase’s lemma provides a powerful tool for translating properties of (co)products in abelian categories into chain conditions. This note discusses the context in which the lemma is used, making explicit what is often neglected in the literature because of its technical nature.