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Homological algebra front page

Quick description

Homological algebra is the branch of algebra that studies chain complexes, various kinds of resolutions of modules (free, projective, injective, flat) and similar sorts of objects (e.g. sheaves), derived functors (such as Ext, Tor, sheaf cohomology, and group cohomology), derived and triangulated categories (which provide convenient categorical settings for doing various sorts of homological algebra), and various related topics. It can be studied in its own right, but also plays a crucial role in many other branches of mathematics, including algebraic topology (where it provides the basic algebraic framework for describing homology and cohomology of spaces), algebraic and complex analytic geometry (where sheaf cohomology methods play a key role), algebraic number theory (where the group cohomology of Galois groups, often referred to simply as Galois cohomology, is a fundamental tool), and many other fields as well.


Homological algebra techniques are typically taught at the graduate level, and most examples will require a familiarity with graduate level algebra, as well as more specialized knowledge from other graduate level topics, depending on the nature of the particular example.

Homological algebra pages

How to compute derived functors

How to compute group cohomology

How to compute the (co)homology of a space

How to compute the cohomology of a sheaf

How to apply cohomology vanishing results

How to use exact sequences

How to use spectral sequences