Kaplansky classes and derived categories

Abstract

Abstract. We put a monoidal model category structure on the category of chain complexes of quasi-coherent sheaves over a quasi-compact and semiseparated scheme X. The approach generalizes and simplifies the method used by the author in [Gil04] and [Gil06] to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of sheaves over a ringed space. Indeed, much of the paper is dedicated to showing that in any Grothendieck category G, any nice enough class of objects F induces a model structure on the category Ch(G) of chain complexes. The main technical requirement on F is the existence of a regular cardinal κ such that every object F ∈ F satisfies the following property: Each κ-generated subobject of F is contained in another κ-generated subobject S for which S, F/S ∈ F . Such a class F is called a Kaplansky class. Kaplansky classes first appeared in [ELR02] in the context of modules over a ring R. We study in detail the connection between Kaplansky classes and model categories. We also find simple conditions to put on F which will guarantee that our model structure is monoidal. We will see that in several categories the class of flat objects form such Kaplansky classes, and hence induce monoidal model structures on the associated chain complex categories. We will also see that in any Grothendieck category G, the class of all objects is a Kaplansky class which induces the usual (non-monoidal) injective model structure on Ch(G).

Cotorsion pairs and degreewise homological model structures

“…Small cotorsion pairs first appeared in [Hov02]. We also refer the reader to [Gil07] for more detail on small cotorsion pairs. We give the definition of a Kaplansky class in Section 5 but again the reader will probably want to refer to [Gil07] for more details.…”

“…The cofibrant complexes are the dg-flat complexes, the fibrant complexes are the dgcotorsion complexes, and the trivial objects are the exact complexes. In [Gil07] we see that the flat model structure generalizes to the category of complexes of quasicoherent sheaves on a quasi-compact, semi-separated scheme X.…”

“…The author calls them homological model structures since they all are on chain complex categories and since we always require the weak equivalences to be homology isomorphisms. We see that a nice cotorsion pair (A, B) may give rise to two other "degreewise" model structures on Ch(C) besides the type used to construct the flat model structures in [Gil04] and [Gil07]. One of these model structures is such that the cofibrant objects form the class of complexes A for which each A n ∈ A.…”

“…However, we need to assume that A is a Kaplansky class and closed under direct limits. We use the definition of Kaplansky class from [Gil07] which is a categorical version of the definition given by Edgar Enochs in [ELR02]. Section 5 is a little more technical in that the reader will want to refer to [Gil07] for some definitions and theorems that will be used.…”

“…We use the definition of Kaplansky class from [Gil07] which is a categorical version of the definition given by Edgar Enochs in [ELR02]. Section 5 is a little more technical in that the reader will want to refer to [Gil07] for some definitions and theorems that will be used. Throughout Sections 4 and 5 we give examples of cotorsion pairs which induce such model structures.…”