Abstract. Given a cotorsion pair (A, B) in an abelian category C with enough A objects and enough B objects, we define two cotorsion pairs in the category Ch(C) of unbounded chain complexes. We see that these two cotorsion pairs are related in a nice way when (A, B) is hereditary. We then show that both of these induced cotorsion pairs are complete when (A, B) is the "flat" cotorsion pair of R-modules. This proves the flat cover conjecture for (possibly unbounded) chain complexes and also gives us a new "flat" model category structure on Ch(R). In the last section we use the theory of model categories to show that we can define Ext n R (M, N ) using a flat resolution of M and a cotorsion coresolution of N .
a b s t r a c tWe define model structures on exact categories, which we call exact model structures. We look at the relationship between these model structures and cotorsion pairs on the exact category. In particular, when the underlying category is weakly idempotent complete, we get Hovey's one-to-one correspondence between model structures and complete cotorsion pairs. We classify the right and the left homotopy relation in terms of the cotorsion pairs and look at examples of exact model structures. In particular, we see that given any hereditary abelian model category, the full subcategories of cofibrant, fibrant and cofibrant-fibrant subobjects each has natural exact model structures equivalent to the original model structure. These model structures each has interesting characteristics. For example, the cofibrant-fibrant subobjects form a Frobenius category, whose stable category is the same as the homotopy category of its model structure.
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).
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