Relative homological algebra via truncations
This paper gives a homotopy-theoretical interpretation of classic work of \textit{N. Spaltenstein} [Compos. Math. 65, No. 2, 121--154 (1988; Zbl 0636.18006)], constructing injective-style resolutions in categories of unbounded chain complexes. Specifically, the paper provides a Quillen adjoint pair relating a category of unbounded chain complexes (in a general abelian category \(\mathcal{A}\)) with a category of towers of truncated chain complexes built from a given class of ``injective objects'' \(\mathcal{I}\) in \(\mathcal{A}\). With these resolutions in hand, the authors construct a relative derived category \(\mathcal{D(A; I)}\), at least for classes \(\mathcal{I}\) satisfying a certain condition. Many examples are provided. For fixed \(\mathcal{A}\) and \(\mathcal{I}\), the authors define relative monomorphisms and relative quasi-isomorphisms (relative to the class \(\mathcal{I}\)). The latter become isomorphisms in the relative derived category \(\mathcal{D(A; I)}\). The machinery of model approximations is used to provide an explicit recipe for resolutions, at least for well-behaved categories \(\mathcal{A}\), like the category of left modules over a Noetherian ring \(R\) of finite Krull dimension. The authors also exhibit explicit examples demonstrating that their results do not work for an arbitrary ring \(R\). All in all, this paper builds on the existing literature regarding relative homological algebra (and its homotopy theory) in an important way. The explicit resolutions provided by this paper are sure to be of use to many future researchers. The paper is written with a fairly general audience in mind, and includes an excellent exposition of the basics of relative homological algebra, model approximations, resolutions, and towers. Helpful examples are sprinkled throughout. Lastly, the appendices provide elementary exposition to make the paper easier to read for both homotopy theorists and algebraists. This paper should be required reading for graduate students interested in getting into the subject of relative homological algebra, and also contains plenty of results for researchers desiring to carry out concrete computations.
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