arXiv subjects
Publications and source records attributed to Hipolito Treffinger.
In this paper we study the behaviour of modules over finite dimensional algebras whose endomorphism algebra is a division ring. We show that there are finitely many such modules in the module category of an algebra if and only if the length of all such modules is bounded.
In this paper we first show that every non-zero $τ$-rigid $A$-module induces at least one stratifying system in the module category of $A$. Moreover, we show that each of these stratifying systems can be seen as a signed $τ$-exceptional sequence.
In this paper we study the stability functions on abelian categories introduced by Rudakov in \cite{Ru} and their relation with torsion classes and maximal green sequences. Moreover we introduce a new kind of stability function which is compatible with the wall and chamber of the category.
We use $τ$-tilting theory to give a description of the wall and chamber structure of a finite dimensional algebra. We also study $\mathfrak{D}$-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 $\mathfrak{D}$-generic path.
Given a finite dimensional algebra $A$ over an algebraically closed field, we consider the $c$-vectors such as defined by Fu in \cite{Fu2017} and we give a new proof of its sign-coherence. Moreover, we characterise the modules whose dimension vectors are $c$-vectors as bricks respecting a functorially finiteness condition.
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.
Comparing the module categories of an algebra and of the endomorphism algebra of a given support $τ$-tilting module, we give a generalization of the Brenner-Butler's tilting theorem in the framework of $τ$-tilting theory. Afterwards we define $τ$-slices and prove that complete slices of tilted algebras and local slices of cluster tilted algebras are examples of complete $τ$-slices. Then we apply this concept to the study of simply connected tilted algebras. Finally, we study the one-point extensions and the split-by-nilpotent extensions of an algebra with $τ$-slices.