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Relative $Q$-shaped homological algebra

Anastasios Slaftsos, Jorge Vitória

Abstract

Exact categories are a natural generalisation of abelian categories and provide a fertile ground to develop relative homological algebra. In this paper, starting from a class of relative Gorenstein projective objects in an exact category $(\mathcal{A},\mathscr{E})$, we define exact model structures on $\mathcal{A}$ and cohomology functors that detect trivial objects and weak equivalences. Moreover, we show that varying the exact structure on $\mathcal{A}$ induces Bousfield (co)localisation sequences between the corresponding homotopy categories. We use these techniques to study the category ${}_{Q,A}\operatorname{Mod}$ of ${}_{A}\operatorname{Mod}$-valued representations, for a ring $A$, of a suitable $\Bbbk$-linear small category $Q$, where we apply our results to a range of objectwise exact structures, ranging from the split exact structure to the abelian one. In particular, we recover the $Q$-shaped derived category of Holm and Jorgensen and construct an intermediate $Q$-shaped homotopy category, analogous to the homotopy category of complexes. Finally, we show that the $Q$-shaped derived category is a Verdier quotient of the $Q$-shaped homotopy category, and that this quotient functor is part of recollement - generalising results of Verdier, Krause, and Iyama-Kato-Miyachi for complexes and $N$-complexes, respectively.

Relative $Q$-shaped homological algebra

Abstract

Exact categories are a natural generalisation of abelian categories and provide a fertile ground to develop relative homological algebra. In this paper, starting from a class of relative Gorenstein projective objects in an exact category , we define exact model structures on and cohomology functors that detect trivial objects and weak equivalences. Moreover, we show that varying the exact structure on induces Bousfield (co)localisation sequences between the corresponding homotopy categories. We use these techniques to study the category of -valued representations, for a ring , of a suitable -linear small category , where we apply our results to a range of objectwise exact structures, ranging from the split exact structure to the abelian one. In particular, we recover the -shaped derived category of Holm and Jorgensen and construct an intermediate -shaped homotopy category, analogous to the homotopy category of complexes. Finally, we show that the -shaped derived category is a Verdier quotient of the -shaped homotopy category, and that this quotient functor is part of recollement - generalising results of Verdier, Krause, and Iyama-Kato-Miyachi for complexes and -complexes, respectively.
Paper Structure (21 sections, 41 theorems, 98 equations)

This paper contains 21 sections, 41 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Efficient exact categories
  4. Cotorsion pairs
  5. Exact model structures
  6. Gorenstein projective objects
  7. Exact model structures from Gorenstein projective objects
  8. Objectwise exact structure on Q,A-Mod
  9. Preliminaries on Q-shaped derived categories
  10. Properties of objectwise exact structures on Q,A-Mod
  11. Exact functors
  12. Projective and injective objects
  13. Gorenstein projective and Gorenstein injective objects
  14. Examples
  15. The canonical totally acyclic complex of projectives / injectives
  16. ...and 6 more sections

Key Result

Lemma 2.2

Let $(\mathcal{A},\mathcal{B})$ be a cotorsion pair in a weakly idempotent complete exact category $\mathcal{E}$, with $\mathcal{A}$ being generating and $\mathcal{B}$ cogenerating in $\mathcal{E}$. The following conditions are equivalent:

Theorems & Definitions (99)

  • Remark 2.1
  • Lemma 2.2: Šťovı́ček Stovicek2013
  • Theorem 2.3: Gillespie MR2811572
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7: HZZ
  • proof
  • Lemma 2.8
  • ...and 89 more