Simply connected right multipeak algebras and the separation property

Abstract

Abstract. Let R = k(Q, I) be a finite-dimensional algebra over a field k determined by a bound quiver (Q, I). We show that if R is a simply connected right multipeak algebra which is chord-free and A-free in the sense defined below then R has the separation property and there exists a preprojective component of the Auslander-Reiten quiver of the category prin(R) of prinjective R-modules. As a consequence we get in 4.6 a criterion for finite representation type of prin(R) in terms of the prinjective Tits quadratic form of R.1. Introduction. Let k be a field. We consider triangular simply connected right multipeak algebras R = kQ/I, where Q is a finite quiver and I is an admissible ideal in the path algebra kQ. Triangularity means that the ordinary quiver Q of R has no oriented cycles. Following [13] we say that R is a right multipeak algebra if the right socle soc(R R ) of R is R-projective. The main objective of the paper is a criterion for R to have the separation property [2]. We prove in Section 4 that R has the separation property when R is chord-free (see 2.5) and A-free as a right multipeak algebra. Our main result, Theorem 4.5, is analogous to [21, Theorem 4.1] and [1, 1.2]; cf. [6].Recall from [1, 1.2] that if R is schurian, triangular, simply connected and does not contain any full subcategory (see 2.2) isomorphic to k A m , m ≥ 1, then R has the separation property. Our result is a version of this statement: algebras considered are right multipeak algebras and the requirement that R is A-free as a right multipeak algebra is a weaker version of A-freeness considered in [1], [3]. The condition that R is chord-free plays a similar role as the assumption that R is schurian. Note that the arguments used in [1] do not work in our situation: our assumptions on R do not imply that R is schurian. Moreover, R (viewed as a k-category) admits full subcategories isomorphic to k A m for some m ≥ 1, although R is A-free as a right multipeak algebra (see the Example in 2.5). Hence the arguments used in [3, 2.3] and [5, 2.9] do not apply here.