Homological Conjectures in representation theory
of finite-dimensional algebras (reference [Ha])
(Report, Sherbrooke 1990)
Note that this is a rather old report, thus some assertions are out-dated.
We will point out a few instances in which one distinguishes different behaviors in global dimension (gldim) 1 vs. gldim 2, or in gldim 2 vs. gldim 3:
Abstract: By definition, the projective spectrum Proj(R) of a positively Z-graded algebra R is obtained by Serre construction from the category of finitely presented R-modules as its quotient category by the Serre subcategory of finite length modules. (R may be noncommutative. In order to obtain a reasonable theory, we assume R to be coherent and moreover all simple R-modules finitely presented.) In the sense of noncommutative algebraic geometry, this quotient category is considered as the category of coherent sheaves on a noncommutative (projective) scheme. We focus on the case where R is either the free algebra F=k< X_1,...,X_n > in n≥ 3 variables or, alternatively, the preprojective algebra S of a wild, finite, connected quiver Q. (Let A=kQ be the path algebra of Q over a field k then, homologically, we may think of S as the tensor algebra of the (A,A)-bimodule Ext1(DA,A).) Both F and S are naturally graded, F has global dimension one while S has global dimension two. It is known for some time (Lenzing, '86) that the projective spectrum for S is an abelian category with nice properties: it is hereditary, has Serre duality in the (one-dimensional) form, has almost-split sequences, and has a tilting object T with endomorphism ring A. More recently, for Q a wild Kronecker quiver the category Proj(S) has attracted the interest of several noncommutative algebraic geometers: J.J. Zhang, D. Piantovskii, Minamoto, Mori.
In a recent workshop at Banff, Paul Smith made a comparison of the two examples, pointing out that the projective spectrum of F has very exotic properties making it much more complicated than the projective spectrum of S. The surprising conclusion was that - in NCAG - global dimension one may be more difficult than global dimension two. In this talk we are going to analyse this question, offering a definitive conclusion.
Abstract: My talk is based on work by and with Claire Amiot.
Claire Amiot has introduced cluster categories for algebras of global dimension 2 (so-called Amiot-cluster categories). This construction will be recalled in Idun Reiten's talk.
We call two algebras of global dimension at most two "cluster equivalent" if their Amiot-cluster categories are equivalent. It follows from the definition that derived equivalent algebras are cluster equivalent. In my talk I will show examples that the converse is not true. Thus cluster equivalence provides a more coarse way of organizing algebras of global dimension two than derived equivalence.
I will then present a result saying that, under certain technical conditions (which seem to always be satisfied for the small examples we checked), cluster equivalence is equivalent to graded derived equivalence, that is the two algebras in question admit gradings such that the categories of graded modules are derived equivalent.
Abstract: We discuss some of the occurrances of algebras of global dimension 2 in the theory of 2-Calabi-Yau categories. This includes passing from tilted to cluster tilted algebras by Assem-Bruestle-Schiffler. Amiot's construction of generalized cluster categories and Keller's construction of quivers with potential.
Algebras of global dimension 2 are constructed from 2-Calabi-Yau categories associated with elements in Coxeter groups (with Amiot-Todorov) and for some stable categories of Cohen-Macaulay modules (with Amiot-Iyama). This is the first step for relating these categories to the generalized cluster categories of Amiot.