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Gluing simple-minded collections in triangulated categories

Yongliang Sun, Yaohua Zhang

Abstract

We provide a technique to glue simple-minded collections along a recollement of Hom-finite Krull-Schmidt triangulated categories over a field. This gluing technique for simple-minded collections is shown to be compatible with those for gluing bounded $t$-structures, silting objects, and co-$t$-structures in the literature. Furthermore, it also enjoys the properties of preserving partial order and commuting with the operation of mutation.

Gluing simple-minded collections in triangulated categories

Abstract

We provide a technique to glue simple-minded collections along a recollement of Hom-finite Krull-Schmidt triangulated categories over a field. This gluing technique for simple-minded collections is shown to be compatible with those for gluing bounded -structures, silting objects, and co--structures in the literature. Furthermore, it also enjoys the properties of preserving partial order and commuting with the operation of mutation.
Paper Structure (12 sections, 22 theorems, 100 equations)

This paper contains 12 sections, 22 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Recollements
  5. $t$-structures
  6. Simple-minded collections
  7. Co-$t$-structures
  8. Gluing simple-minded collections
  9. Partial order and mutations
  10. Gluing partial order
  11. Partial order of mutations
  12. Gluing mutations

Key Result

Theorem 1.1

(CPP2022, J2023, Theorem thm:main, Proposition prop:comp $t$-structure) With the notation as above. The set is a simple-minded collection of $\mathcal{T}$, where $W_j$ lies in the triangle (see Section sec:gluing for the detailed construction) where $\tau^{\leq 0}$ is the truncation at 0 of the associated $t$-structure $(\mathop{\mathrm{\mathsf{Filt}}}\nolimits S_\mathcal{X}[\geq 0], \mathop{\ma

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2: Theorem \ref{['thm:mutation smc']}
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Example 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 34 more