Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras

TL;DR

This paper establishes bijective correspondences between silting objects, simple-minded collections, t-structures, and co-t-structures for finite-dimensional algebras, facilitating the computation of stability conditions.

Contribution

It introduces new bijections between key algebraic structures and demonstrates their compatibility with mutations, advancing understanding of derived categories.

Findings

Bijective correspondences between four algebraic structures

Compatibility of these correspondences with mutations

Application to computing Bridgeland stability conditions

Abstract

Bijective correspondences are established between (1) silting objects, (2) simple-minded collections, (3) bounded $t$-structures with length heart and (4) bounded co-$t$-structures. These correspondences are shown to commute with mutations. The results are valid for finite-dimensional algebras. A concrete example is given to illustrate how these correspondences help to compute the space of Bridgeland's stability conditions.