Models for homotopy categories of injectives and Gorenstein injectives

ABSTRACT

A natural generalization of locally noetherian and locally coherent categories leads us to define locally type FP_∞ categories. They include not just all categories of modules over a ring, but also the category of sheaves over any concentrated scheme. In this setting we generalize and study the absolutely clean objects recently introduced in [5]. We show that 𝒟(𝒜𝒞), the derived category of absolutely clean objects, is always compactly generated and that it is embedded in K(Inj), the chain homotopy category of injectives, as a full subcategory containing the DG-injectives. Assuming the ground category 𝒢 has a set of generators satisfying a certain vanishing property, we also show that there is a recollement relating 𝒟(𝒜𝒞) to the (also compactly generated) derived category 𝒟(𝒢). Finally, we generalize the Gorenstein AC-injectives of [5], showing that they are the fibrant objects of a cofibrantly generated model structure on 𝒢.

KEYWORDS:

• Abelian model category
• Grothendieck category
• recollement

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 18E15
• 18G25
• 18G35
• 55U35

Acknowledgments

I owe a debt of gratitude to my coauthors of [5], Daniel Bravo and Mark Hovey. The original idea of this paper was simply to generalize some of the results of that paper to a more general setting. To this end, the paper started as notes in the Summer of 2014 while preparing the talk [12] for the ASTA conference in Spineto, Italy. I would like to thank the conference organizers for the invitation to speak. I also thank Jan Stovicek for the paper [35]. The final outcome of this paper was highly influenced by the remarkable results appearing in that paper. Finally, I thank the referee for a careful reading of the manuscript and useful suggestions.