On band modules and $τ$-tilting finiteness

Schroll, Sibylle; Treffinger, Hipolito; Valdivieso, Yadira

doi:10.48550/arxiv.1911.09021

Cited by 1 publication

(1 citation statement)

References 17 publications

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“…First, as an application of Theorem 5.8, we have the following result, where the equivalence of (a) and (b) has already been proved in [STV,Theorem 5.1]. More explicitly, our results give a proof of (b) ⇒ (a) different from theirs.…”

Section: Key Argumentsmentioning

confidence: 62%

Non-rigid regions of real Grothendieck groups of gentle and special biserial algebras

Asai¹

2022

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In the representation theory of finite-dimensional algebras A over a field, the classification of 2-term (pre)silting complexes is an important problem. One of the useful tool is the g-vector cones associated to the 2-term presilting complexes in the real Grothendieck group K0(proj A) R := K0(proj A) ⊗ Z R. The aim of this paper is to study the complement NR of the union Cone of all g-vector cones, which we call the non-rigid region. By the work of Iyama and us, NR is determined by 2-term presilting complexes and a certain closed subset R0 ⊂ K0(proj A) R , which is called the purely non-rigid region. In this paper, we give an explicit description of R0 for complete special biserial algebras in terms of a finite set of maximal nonzero paths in the Gabriel quiver of A. We also prove that NR has some kind of fractal property and that NR is contained in a union of countably many hyperplanes of codimension one. Thus, any complete special biserial algebra is g-tame, that is, Cone is dense in K0(proj A) R . Contents1. Introduction 1 2. Basic properties of complete special biserial algebras 5 3. Preliminary on silting theory and canonical decompositions 12 4. τ -Tilting reduction 21 5. Descriptions of the non-rigid regions 28 6. Applications 42 Funding 48 Acknowledgment 48 References 49

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Section: Key Argumentsmentioning

confidence: 62%

Non-rigid regions of real Grothendieck groups of gentle and special biserial algebras

Asai¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

On band modules and $τ$-tilting finiteness

Cited by 1 publication

References 17 publications

Non-rigid regions of real Grothendieck groups of gentle and special biserial algebras

Non-rigid regions of real Grothendieck groups of gentle and special biserial algebras

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