Nakayama oriented pullbacks and stably hereditary algebras

Abstract

We introduce a new class of algebras, the Nakayama oriented pullbacks, obtained from pullbacks of surjective morphisms of algebras A C and B C. We prove that such a pullback is tilted when A and B are hereditary. We also show that stably hereditary algebras respecting the clock condition are Nakayama oriented pullbacks, and we use results about these pullbacks to show when a stably hereditary algebra is tilted or iterated tilted. MSC: 16G20; 16G70; 16S50 IntroductionModule categories and pullbacks of rings and algebras have been studied from many points of view (see, for instance, [1,6,11,12,15,18]), but not from the tilting point of view. In this paper, we introduce a particular class of pullbacks of algebras over an algebraically closed field K , which we call Nakayama oriented pullbacks. These are pullbacks of surjective morphisms of algebras A C and B C with C a hereditary Nakayama algebra. We construct a tilting module T over this kind of pullback, and we compute the endomorphism algebra End T of this module when B is hereditary (see 2.4.6). A first consequence of this result is the principal theorem of this paper:Theorem. Let R be a Nakayama oriented pullback of K -algebra surjective morphisms A C and B C. Suppose that A and B are hereditary. Then R is tilted.We next show that a stably hereditary algebra respecting the clock condition, that is, such that the number of clockwise oriented relations on each cycle of its bound quiver equals the number of counterclockwise oriented relations, can be expressed as a Nakayama oriented pullback. As a second consequence of the main theorem above we give a new proof of Theorem 2.6 of [14] which characterizes iterated tilted and tilted stably hereditary algebras.This paper consists of three sections. The first is devoted to preliminaries, the second to Nakayama oriented pullbacks, and the third to stably hereditary algebras. * Tel.

Tree Oriented Pullback

“…Our main result generalizes for this new class of algebras a similar result proven by Lévesque in [7]. Theorem.…”

“…In these notes, we will only use the one of the pushout of quivers. We recall the following description given in [7] (see also [4]). Lemma 1.1.…”

“…Pullbacks of rings and algebras have been studied from many points of view, see for instance [1,6,7,9]. Given two morphisms of algebras f A A → B and f C C → B, we can get a new algebra R defined by a c ∈ A × C f A a = f C c .…”

“…A natural question is to look at properties of A and C which can be transferred to R. In [6], for instance, Igusa, Platzeck, Todorov, and Zacharia have shown a characterization of the bounded quiver of R in terms of the bounded quivers of A, B and C. In [7], Lévesque gave a similar description using the notion of pushout of quivers.…”

Differential Modules over Quadratic Monomial Algebras

“…This allows us to construct, over an arbitrary quadratic monomial algebra, an object in the triangulated hull which cannot belong to the orbit category, under the assumption that Λ contravenes the so-called clock condition. The argument is then completed by results of [2,14] which reveal that if the Λ in our theorem is not piecewise hereditary, then it does violate said condition.Overview, conventions and acknowledgements. This short note starts with a reminder on periodic complexes and the triangulated hull, which is used to prove the first above cited theorem.…”