Gorenstein complexes and recollements from cotorsion pairs

By James Gillespie. 2012

103 citations recorded by the source corpus.

We describe a general correspondence between injective (resp. projective) recollements of triangulated categories and injective (resp. projective) cotorsion pairs. This provides a model category description of these recollement situations. Our applications focus on displaying several recollements that glue together various full subcategories of K(R), the homotopy category of chain complexes of modules over a general ring R. When R is (left) Noetherian ring, these recollements involve complexes built from the Gorenstein injective modules. When R is a (left) coherent ring for which all flat modules have finite projective dimension we obtain the duals. These results extend to a general ring R by replacing the Gorenstein modules with the Gorenstein AC-modules introduced recently in the work of Bravo-Gillespie-Hovey. We also see that in any abelian category with enough injectives, the Gorenstein injective objects enjoy a maximality property in that they contain every other class making up the right half of an injective cotorsion pair.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c3587b3bec7471cbc53d7e3ecfb6ac3ff82babf9

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Citations in the corpus

The stable module category of a general ring (2014, background): “Inspired by Becker’s work, the second author has given a general study of when these recollements happen in [Gil12].”
Hereditary abelian model categories (2015, background): “And in [5, 31], there appeared an elegant connection between abelian model categories and recollements of triangulated categories.”
On Ding injective, Ding projective, and Ding flat modules and complexes (2015, background): “See [Gil12] for more on injective cotorsion pairs.”
Models for singularity categories (2012, background): “Further extensive treatments of recollements from model structures on abelian and even exact categories can be found in [7,8,4].”
How to construct a Hovey triple from two cotorsion pairs (2014, background): “gories we have particularly constructed the proofs so that they hold in a general weakly idempotent complete exact category. The author again wishes to thank Hanno Becker and the referee of his paper [Gil13]. The construction of Hovey triples that we give here is a direct generalization of a constru”