We use τ -tilting theory to give a description of the wall and chamber structure of a finite dimensional algebra. We also study D-generic paths in the wall and chamber structure of an algebra A and show that every maximal green sequence in mod A is induced by a D-generic path.
In this article we study chain of torsion classes in an abelian category A and we show several of their properties, generalising the well-known properties of stability conditions. First, we show that every chain of torsion classes induce a Harder-Narasimhan filtration for every non-zero object in A. Secondly, we characterise the slicings introduced by Bridgeland in [4] in terms of chain of torsion classes. Later we show that many properties of stability conditions can be deduced from the results in the first sections of this article. Finally, following ideas of Bridgeland, we finish the paper by showing that all chain of torsion classes form a topological space.
Extending the notion of maximal green sequences to an abelian category, we characterize the stability functions, as defined by Rudakov, that induce a maximal green sequence in an abelian length category. Furthermore, we use τ -tilting theory to give a description of the wall and chamber structure of any finite dimensional algebra. Finally we introduce the notion of green paths in the wall and chamber structure of an algebra and show that green paths serve as geometrical generalization of maximal green sequences in this context.
Given a finite dimensional algebra A over an algebraically closed field, we consider the c-vectors such as defined by Fu in [18] and we give a new proof of its signcoherence. Moreover, we characterise the modules whose dimension vectors are c-vectors as bricks respecting a functorially finiteness condition.
The main theme of this paper is to study τ -tilting subcategories in an abelian category A with enough projective objects. We introduce the notion of τ -cotorsion torsion triples and show a bijection between the collection of τ -cotorsion torsion triples in A and the collection of τ -tilting subcategories of A , generalizing the bijection by Bauer, Botnan, Oppermann and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of A . General definitions and results are exemplified using persistent modules. If A = Mod-R, where R is an unitary associative ring, we characterize all support τ -tilting, resp. all support τ − -tilting, subcategories of Mod-R in term of finendo quasitilting, resp. quasicotilting, modules. As a result, it will be shown that every silting module, respectively every cosilting module, induces a support τ -tilting, respectively support τ − -tilting, subcategory of Mod-R. We also study the theory in Rep(Q, A ), where Q is a finite and acyclic quiver. In particular, we give an algorithm to construct support τ -tilting subcategories in Rep(Q, A ) from certain support τ -tilting subcategories of A and present a systematic way to construct (n + 1)-tilting subcategories in Rep(Q, A ) from n-tilting subcategories in A .
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