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Derived projective covers and Koszul duality of simple-minded and silting collections

Lukas Bonfert

Abstract

We introduce derived projective covers and explain how they are related to the notion of enough derived projectives. This provides an if-and-only-if criterion for when derived projective covers form a silting collection. We prove moreover a Koszul duality result for silting and simple-minded collections.

Derived projective covers and Koszul duality of simple-minded and silting collections

Abstract

We introduce derived projective covers and explain how they are related to the notion of enough derived projectives. This provides an if-and-only-if criterion for when derived projective covers form a silting collection. We prove moreover a Koszul duality result for silting and simple-minded collections.
Paper Structure (17 sections, 30 theorems, 22 equations)

This paper contains 17 sections, 30 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. t-structures
  4. Weight structures
  5. Simple-minded collections
  6. Silting collections
  7. Adjacency and orthogonality
  8. Silting collections and derived projectives
  9. Derived projective objects
  10. Derived projective covers
  11. Silting collections as derived projective covers
  12. The WT correspondence revisited
  13. Koszul duality between simple-minded and silting collections
  14. dg Koszul duality
  15. Koszul duality between simple-minded and silting collections
  16. ...and 2 more sections

Key Result

Theorem A

Let $t$ be a non-degenerate t-structure on $\mathscr{D}$ with finite-length heart. Let $\mathcal{L}$ be a full set of isomorphism representatives of the simple objects in $\heartsuit_t$ and $\mathcal{P}$ a full set of isomorphism representatives of the indecomposable derived projectives. Then the fo

Theorems & Definitions (85)

  • Theorem A: \ref{['siltingderprojconditions']}
  • Theorem B: \ref{['koszuldualitygeneral']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • ...and 75 more