On perpendicular categories of stones over quiver algebras

Abstract

Let H = kzt be the path algebra of a quiver A, where k denotes an arbitrary field. J will always be assumed to be finite without oriented cycles. We denote by mod H the category of finite dimensional left modules over H. An indecomposable H-module X is called a stone provided Ext}~(X, X) = 0. Note that for a stone X the endomorphism ring End X = k (see for example part 1). For a stone X we denote by X l the full subcategory of mod H of the modules Y with Homn(X, Y) = 0 = Ext,(X, Y). Dually, • consists of those H-modules Y satisfying Homn(Y, X) = 0 = Ext,(Y, X). The categories X • and Xare called the right respectively left perpendicular category of X. They are equivalent to mod Ar respectively mod At, where A r and At are again finite dimensional, hereditary k-algebras, and both have one simple module less than H (compare [GL] and [S1]). We denote by Ar respectively zt t the quivers with kAr = Ar and kJz = A,. Since their introduction to Geigle, Lenzing and Schofield, perpendicular categories have become a powerful tool in the representation theory of finite dimensional algebras. They inherit essential properties from mod H, furnish a reduction procedure and open the possibility for proofs by induction. In general, it is difficult problem to determine the (right or left) perpendicular category of a given stone X. We briefly recall what is known. It is quite easy to see (Proposition 2.1) that the quiver Ar differs from a full subquiver of zl by an admissible change of orientation if and only if X is preprojective or preinjective over H. Moreover, if H is tame, then there are only finitely many regular stones X over H, and the structure of its perpendicular categories is contained in [Rill. Hence we may restrict to the context where H is wild and X is a regular stone. Baer and Straul3 proved [B2], [St] that in this situation the perpendicular categories are wild again, but combining results of Xi IX] and Kerner [K3] it follows that certain wild module categories, namely of those algebras classified in [X], do not arise as perpendicular categories of regular stones over wild algebras. On the other hand, investigations of growth numbers in [K3] prove, that for a fixed hereditary, connected, wild algebra with at least three simple modules, there are infinitely many pairwise non isomorphic algebras Ai such that mod Ai is equivalent to X • for a regular stone X over H.