Relative Homological Algebra

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...

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...

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...

Categories and Homological Algebra

The aim of these Notes is to introduce the reader to the language of categories with emphazis on homological algebra. We treat with some details basic homological algebra, that is, categories of complexes in additive and abelian categories and construct with some care the derived functors. We also introduce the reader to the more sophisticated concepts of triangulated and derived categories. Ou...