Report on the finiteness of silting objects.(English)Zbl 1505.16013
Silting theory, a generalisation of tilting theory, is currently a highly active area of representation theory. Silting objects provide information on the relationship between derived categories of different finite-dimensional algebras, and when the silting theory of an algebra is sufficiently well-behaved (precisely, the algebra is silting-discrete), the stability manifold of its derived category is known to be contractible [D. Pauksztello et al., Forum Math. 30, No. 5, 1255–1263 (2018;Zbl 1407.18013)]. Silting-discreteness implies, in particular, the existence of only finitely many \(2\)-term silting objects.
In this paper, the authors demonstrate how abstract results from other sources can be used to control the silting theory of certain classes of algebras. Most of the results concern the derived category, with the authors showing that for particular families of algebras this category has finitely many \(2\)-term silting objects (up to additive equivalence). They also show that singularity categories of Iwanaga-Gorenstein algebras have no silting objects at all, and give classes of algebras for which finiteness of the number of \(2\)-term silting objects is equivalent to being representation-finite.
The main technique is the use of reduction theorems for silting or \(\tau\)-tilting objects, as in [O. Iyama andD. Yang, Trans. Am. Math. Soc. 370, No. 11, 7861–7898 (2018;Zbl 1443.18006)] and [F. Eisele et al., Math. Z. 290, No. 3–4, 1377–1413 (2018;Zbl 1433.16011)].
Some of the families of algebras considered in this paper have complete classifications, and the authors compute the precise number of \(2\)-term silting objects in their derived categories.
In this paper, the authors demonstrate how abstract results from other sources can be used to control the silting theory of certain classes of algebras. Most of the results concern the derived category, with the authors showing that for particular families of algebras this category has finitely many \(2\)-term silting objects (up to additive equivalence). They also show that singularity categories of Iwanaga-Gorenstein algebras have no silting objects at all, and give classes of algebras for which finiteness of the number of \(2\)-term silting objects is equivalent to being representation-finite.
The main technique is the use of reduction theorems for silting or \(\tau\)-tilting objects, as in [O. Iyama andD. Yang, Trans. Am. Math. Soc. 370, No. 11, 7861–7898 (2018;Zbl 1443.18006)] and [F. Eisele et al., Math. Z. 290, No. 3–4, 1377–1413 (2018;Zbl 1433.16011)].
Some of the families of algebras considered in this paper have complete classifications, and the authors compute the precise number of \(2\)-term silting objects in their derived categories.
Reviewer: Matthew Pressland (Glasgow)
MSC:
| 16G20 | Representations of quivers and partially ordered sets |
| 16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
Cite
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