Dec 18 – 22, 2023
Kavli IPMU, Kashiwa, Japan
Asia/Tokyo timezone

TF equivalence, silting theory and canonical decompositions

Dec 22, 2023, 11:40 AM
50m
Lecture Hall (Kavli IPMU, Kashiwa, Japan)

Lecture Hall

Kavli IPMU, Kashiwa, Japan

Kashiwa, Japan

Speaker

Sota Asai (Tokyo)

Description

This talk is based on joint work with Osamu Iyama. The representation theory of a finite dimensional algebra $A$ deals with the category $\mathsf{mod} A$ of finitely generated $A$-modules. One of the main topics is torsion pairs in $\mathsf{mod} A$. Functorially finite torsion pairs have been well-studied, but they are too few among all torsion pairs. Thus, we are now studying a wider class called semistable torsion pairs introduced by Baumann-Kamnitzer-Tingley associated to elements of the real Grothendieck group $K_0(\mathsf{proj}A)_\mathbb{R}$ of the category of finitely generated projective $A$-modules, which is identified with the Euclidean space whose canonical basis is given by the isoclasses of indecomposable projective $A$-modules. By using semistable torsion pairs, I (Asai) introduced an equivalence relation called TF equivalence on $K_0(\mathsf{proj} A)_\mathbb{R}$. A typical example of TF equivalence classes is the silting cone $C^\circ(U)$ generated by the g-vectors of indecomposable direct summands of each 2-term presilting complex $U$. On the other hand, there can be other TF equivalence classes. To study them, we have found that canonical decompositions introduced by
Derksen-Fei is useful. I would like to explain some important properties of these notions.

Presentation materials

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