arXiv subjects
Publications and source records attributed to Grzegorz Bobiński.
We study various symmetry conditions related to the Auslander-Reiten formula for finite-dimensional algebras. In particular, we define a new class of algebras which we call $E$-symmetric. A theorem by Derksen-Weyman-Zelevinksy implies that all Jacobian algebras are $E$-symmetric. We prove the converse for the class of monomial algebras.
In the representation theory of finite-dimensional algebras, the study of projective presentations of maximal rank is closely related to the study of generically $τ$-regular irreducible components of varieties of modules over such algebras. We show that a module is $τ$-regular if and only if its minimal projective presentation is of maximal rank. This is a refinement of a theorem by Plamondon. We prove that generic extensions of generically $τ$-regular components by simple projective modules are again generically $τ$-regular. This leads to the classification of all generically $τ$-regular components for triangular algebras. We also show that an algebra is hereditary if and only if all irreducible components of its varieties of modules are generically $τ$-regular. Finally, we discuss when the set of generically $τ$-regular components coincides with the set of generically $τ^-$-regular components.
We study projective presentations of finite-dimensional modules over finite-dimensional algebras. We discuss if projective presentations of maximal rank behave additively. More precisely, we ask if the direct sum of copies of a projective presentation of maximal rank is again of maximal rank. The modules which have a projective presentation of maximal rank are exactly the $τ$-regular modules. This class of modules can be seen as a generalization of modules of projective dimension at most one, and of $τ$-rigid modules. The $τ$-regular modules form open subsets of module varieties. Their closures are therefore unions of irreducible components, which are called generically $τ$-regular. We discuss when a $τ$-regular module or a generically $τ$-regular component can be reduced to a module or component of projective dimension at most one, and we show that this is closely related to the question on the additivity of maximal rank presentations.
Assume that $K$ is an algebraically closed field and denote by $KG(R)$ the Krull-Gabriel dimension of $R$, where $R$ is a locally bounded $K$-category (or a bound quiver $K$-algebra). Assume that $C$ is a tilted $K$-algebra and $\widehat{C},\check{C},\widetilde{C}$ are the associated repetitive category, cluster repetitive category and cluster-tilted algebra, respectively. Our first result states that $KG(\widetilde{C})=KG(\check{C})\leq KG(\widehat{C})$. Since the Krull-Gabriel dimensions of tame locally support-finite repetitive categories are known, we further conclude that $KG(\widetilde{C})=KG(\check{C})=KG(\widehat{C})\in\{0,2,\infty\}$. Finally, in the Appendix Grzegorz Bobiński presents a different way of determining the Krull-Gabriel dimension of the cluster-tilted algebras, by applying results of Geigle.
We call a finite-dimensional K-algebra A geometrically irreducible if for all d all connected components of the affine scheme of d-dimensional A-modules are irreducible. We prove that a geometrically irreducible algebra with exactly two simple modules has to be of a very special form, which we describe. Based on this result we prove that every minimal geometrically irreducible algebra without shortcuts in its Gabriel quiver has at most two simple modules.
We call a finite-dimensional K-algebra A geometrically irreducible if for all d, all connected components of the affine scheme of d-dimensional A-modules are irreducible. We show that the geometrically irreducible algebras without loops (this includes all algebras of finite global dimension) are the hereditary algebras. We also show that truncated polynomial rings are the only geometrically irreducible local algebras. Finally, we formulate a conjectural classification of all geometrically irreducible algebras.
We show that the orbit closures for directing modules over tame algebras are normal and Cohen-Macaulay. The proof is based on deformations to normal toric varieties.