Homological algebra

Homological algebra is the study of chain complexes and their derived categories. While its origins lie in classical algebra and algebraic topology, it is deeply intertwined with stable homotopy theory: the derived category of an abelian category is a stable \(\infty \)-category, and many of its key structures, like exact sequences and Ext-groups, are instances of general phenomena in stable \(\infty \)-categories.

In this chapter, we develop some of the core aspects of homological algebra from the \(\infty \)-categorical perspective. We begin in Section 6.1 by constructing the derived \(\infty \)-category \(\D (\Aa )\) of an abelian category \(\Aa \) as a localization of the category of chain complexes at quasi-isomorphisms. Using the general framework from Section 2.2, we show that \(\D (\Aa )\) is a stable \(\infty \)-category. We then specialize to \(\D (\Z )\) in Section 6.2, where Eilenberg–MacLane spectra provide the intrinsic construction of ordinary homology and cohomology promised in Chapter 3.

In Section 6.3, we introduce t-structures, an axiomatic framework that captures the interplay between stable homotopy theory and classical algebra. A t-structure comes with classes of connective and coconnective objects, which behave like connective and coconnective spectra. Their intersection turns out to be a 1-category, and in fact an abelian category, called the heart of the t-structure. The derived category \(\D (\Aa )\) carries a canonical t-structure whose heart recovers the original abelian category \(\Aa \).

The remaining sections treat projective and flat objects, Ext- and Tor-groups, and derived functors. In Section 6.4, we define Ext-groups as homotopy groups of mapping spectra and show that they can be computed by projective resolutions. In Section 6.5, we develop the parallel theory of flat objects and Tor-groups. In Section 6.6, we define derived functors, compute them by projective replacement, and construct the derived tensor product and derived internal hom. Finally, Section 6.7 proves the Künneth formula and the universal coefficient theorems, first algebraically for complexes over a principal ideal domain and then for the ordinary homology and cohomology of animae.

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