Some generalizations of preprojective algebras and their properties

Abstract

Abstract. In this note we consider a notion of relative Frobenius pairs of commutative rings S/R. To such a pair, we associate an N-graded R-algebra Π R (S) which has a simple description and coincides with the preprojective algebra of a quiver with a single central node and several outgoing edges in the split case. If the rank of S over R is 4 and R is noetherian, we prove that Π R (S) is itself noetherian and finite over its center and that each Π R (S) d is finitely generated projective. We also prove that Π R (S) is of finite global dimension if R and S are regular.

Construction of non-commutative surfaces with exceptional collections of length 4

“…The proofs of both lemma 37 and lemma 39 are based on local computations: it was shown in [14, §3] that noncommutative P 1 -bundles can be studied locally using the theory of generalized preprojective algebras Π C (D) associated to a relative Frobenius pair D/C of finite rank as in [15]. We recall this theory in section 3.3.…”

“…In this section we introduce generalized preprojective algebras as in [15]. First we introduce some short hand notation.…”

“…We suggest that there exists a construction of a noncommutative P 1 -bundle which does not use a finite morphism f : P 1 → P 1 of degree 4, but rather a finite morphism f : C → P 1 of degree 4 where C is the singular conic in the net of conics associated to the point which is being blown up. Then the formalism from [15] yields an abelian category of infinite global dimension, which should be the same as the category associated to the non-maximal order of infinite global dimension by methods similar to the ones used in this paper. Now the choice of a maximal order (of finite global dimension) containing the pullback gives the construction from the top row in table 5.…”

Frobenius Degenerations of Preprojective Algebras

“…This paper explores a further generalization to decorated quivers -quivers with vertices labelled by k-algebras and arrows labelled by bimodules. An original motivation for this work is [5], which in our language considers the affine D 4 case, while we focus on the Dynkin case.…”

“…The conjecture was proven in the case Q = D4 for commutative Frobenius algebras A i in [5], by computing an explicit basis for the maximally folded and maximally degenerated decoration…”