Models for homotopy categories of injectives and Gorenstein injectives. Issue 6 (3rd June 2017)

• Record Type: Journal Article
• Authors: Gillespie, James
• Is Part Of: Communications in algebra. Volume 45:Issue 6(2017)
• Page Start: 2520
• Page End: 2545
• Publication Date: 2017-06-03
• Subjects: Abelian model category -- Grothendieck category -- recollement; 18E15 -- 18G25 -- 18G35 -- 55U35; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2016.1233215
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: A natural generalization of locally noetherian and locally coherent categories leads us to define locally type F P ∞ 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 ( 𝒜 𝒞 ), 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 ( 𝒜 𝒞 ) 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 .