Homotopy category of projective complexes and complexes of Gorenstein projective modules

Abstract

Abstract. Let R be a ring with identity and C(R) denote the category of complexes of R-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp. Gorenstein injective) modules. We show that the homotopy category of projective complexes over R, denoted K(Prj C(R)), is always well generated and is compactly generated provided K(Prj R) is so. Based on this result, it will be proved that the class of Gorenstein projective complexes is precovering, whenever R is a commutative noetherian ring of finite Krull dimension. Furthermore, it turns out that over such rings the inclusion functor ι : K(GPrj R) ֒→ K(R) has a right adjoint ιρ, where K(GPrj R) is the homotopy category of Gorenstein projective R modules. Similar, or rather dual, results for the injective (resp. Gorenstein injective) complexes will be provided. If R has a dualising complex, a triangle-equivalence between homotopy categories of projective and of injective complexes will be provided. As an application, we obtain an equivalence between the triangulated categories K(GPrj R) and K(GInj R), that restricts to an equivalence between K(Prj R) and K(Inj R), whenever R is commutative, noetherian and admits a dualising complex.

On rings with finite Gorenstein weak global dimension

“…Therefore, as R is left virtually Gorenstein, the subcategory R GP K is closed under direct limits. On the other hand, we know from [18, Corollary 3.4 (1)] that the pair p R GP, R GP K q forms a cotorsion pair (over any ring). Thus, [46,Theorem 6.1] gives that the subcategory R GP K is definable.…”

“…Assume that p R F m , R GIq is an injective cotorsion pair in R-Mod. To see (1), first let R I be any injective left R-module. Using the completeness of the cotorsion pair p R F m , R GIq, one gets a short exact sequence of R-modules…”

“…injective) left R-modules. The study of compactness of KpR-Projq and KpR-Injq and triangle equivalence between them was initiated in 2005 by Krause [40] and Jørgensen [39], and has been extensively investigated during the last years by Iyengar, Krause, Neeman, Chen, Št'ovíček, Asadollahi, Hafezi, Salarian, Positselski and Gillespie [38,42,43,13,48,1,33,45]. Now the relation between such homotopy categories and left Gorenstein rings is made explicit in the following result due to Chen Theorem 1.1.…”

Bounded Derived Categories of Infinite Quivers: Grothendieck Duality, Reflection Functor

“…Observation 3.1 Since K(Prj-R) is compactly generated, there is a right adjoint functor q : K(Flat-R) → K(Prj-R) for the inclusion functor K(Prj-R) → K(Flat-R). Now, for any complex X ∈ K(Flat-R) an argument similar to [AHS,Lemma 5.2.1] shows that…”

Constructing cogenerators in triangulated categories and Brown representability

“…As another consequence we show that the homotopy category of projective representations of a quiver satisfies Brown representability for covariant functors (Theorem 2.6). In particular homotopy category of projective complexes of modules considered in [1] has to satisfy this property (Example 2.7). Finally we note that there is no known example of triangulated categories which are not compactly generated but satisfy the hypothesis of Krause's criterion.…”