Mutations of simple-minded collections revisited

Abstract

Simple-minded collections in the bounded derived category ${\mathsf{D}}^{\mathrm{b}}(\mathrm{mod}A)$ are necessary tools in tilting theory of finite dimensional algebras, as well as silting complexes in the perfect derived category ${\mathsf{K}}^{\mathrm{b}}(\mathrm{proj}A)$. In this paper, we give an explicit mutation formula of simple-minded collections at arbitary direct summands, which is compatible with that of silting complexes. Though this mutation formula may be known to experts, we provide a thorough proof for completeness and future reference. Then we use our formula to revisit $\tau$-tilting theory in terms of mutations of 2-term simple-minded collections, including their relationship with maximal/minimal completions of 2-term presilting complexes and $\tau$-tilting reduction.

arXiv comment27 pages