Quillen model structures for relative homological algebra

Quillen Model Structures for Relative Homological Algebra

An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen’s homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen’s framework. Speci...

Relative Homological Algebra via Truncations

To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spaltenstein solved this problem for chain complexes of R-modules by truncating further and further to the left, resolving the pieces, and gluing back the partial resolutions. Our aim is to give a homotopy theoretical interpretation of this procedure, which may be extended to a re...

Homological Algebra

Relative Homological Algebra, Waldhausen K-theory, and Quasi-frobenius Conditions

We study the question of the existence of a Waldhausen category on any (relative) abelian category in which the contractible objects are the (relatively) projective objects. The associated K-theory groups are “stable algebraic G-theory,” which in degree zero form a certain stable representation group. We prove both some existence and nonexistence results about such Waldhausen category structure...

Homological Algebra for Schwartz Algebras

Let G be a reductive group over a non-Archimedean local field. For two tempered smooth representations, it makes no difference for the Ext-groups whether we work in the category of tempered smooth representations of G or of all smooth representations of G. Similar results hold for certain discrete groups. We explain the basic ideas from functional analysis and geometric group theory that are ne...