James Gillespie

The flat model structure on ()

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.

The flat model structure on complexes of sheaves

Let Ch(O) be the category of chain complexes of 0-modules on a topological space T (where O is a sheaf of rings on T). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on Ch(O). As a corollary, we have a general framework for doing homological algebra in the category Sh(0) of 0-modules. I.e., we have a natural way to define the functors Ext and Tor in Sh(O).

GORENSTEIN MODEL STRUCTURES AND GENERALIZED DERIVED CATEGORIES

In a paper from 2002, Hovey introduced the Gorenstein projective and Gorenstein injective model structures on R -Mod, the category of R -modules, where R is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce: the Gorenstein flat model structure. The homotopy category with respect to each of these is called the stable module category of R . If such a ring R has finite global dimension, the graded ring R [ x ]/( x 2 ) is Gorenstein and the three associated Gorenstein model structures on R [ x ]/( x 2 )-Mod, the category of graded R [ x ]/( x 2 )-modules, are nothing more than the usual projective, injective and flat model structures on Ch( R ), the category of chain complexes of R -modules. Although these correspondences only recover these model structures on Ch( R ) when R has finite global dimension, we can set R = ℤ and use general techniques from model category theory to lift the projective model structure from Ch(ℤ) to Ch( R ) for an arbitrary ring R . This shows that homological algebra is a special case of Gorenstein homological algebra. Moreover, this method of constructing and lifting model structures carries through when ℤ[ x ]/( x 2 ) is replaced by many other graded Gorenstein rings (or Hopf algebras, which lead to monoidal model structures). This gives us a natural way to generalize both chain complexes over a ring R and the derived category of R and we give some examples of such generalizations.

Absolutely Clean, Level, and Gorenstein AC-Injective Complexes

Absolutely clean and level R-modules were introduced in [2] and used to show how Gorenstein homological algebra can be extended to an arbitrary ring R. This led to the notion of Gorenstein AC-injective and Gorenstein AC-projective R-modules. Here we study these concepts in the category of chain complexes of R-modules. We define, characterize and deduce properties of absolutely clean, level, Gorenstein AC-injective, and Gorenstein AC-projective chain complexes. We show that the category Ch(R) of chain complexes has a cofibrantly generated model structure where every object is cofibrant and the fibrant objects are exactly the Gorenstein AC-injective chain complexes.

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.

Cotorsion Pairs of Modules from Left Adjoints

Let R be a ring with 1 and M a right R-module. In this article, we will see that the functor F = M ⊗ R – gives rise to a complete hereditary cotorsion pair where the left class consists of the F-acyclic objects. This cotorsion pair induces a Quillen model structure on Ch(R) which recovers the derived functors . An F-acyclic resolution is as good as a cofibrant replacement in this model structure. So in short, we formalize the fact that can be computed using F-acyclic resolutions.