Chain complexes, derived functors, Ext, Tor, and the fundamental lemmas of homological algebra.
Core idea
This section covers the foundations of homological algebra, providing tools for measuring the failure of exactness and computing invariants of algebraic structures.
Derived functors formalize the idea that a non-exact functor can be “corrected” by replacing objects with resolutions and then taking homology/cohomology.
Throughout, let A,B be abelian categories and F:A→B an additive functor.
Derived functors (see derived functors) turn short exact sequences into long exact sequences in homology/cohomology. Conceptually, this is a systematic abstraction of the snake lemma and its boundary map; see also connecting homomorphisms.
In an abelian category, consider a commutative diagram with exact rows
If f1 is an epimorphism, f2 and f4 are isomorphisms, and f5 is a monomorphism, then f3 is an isomorphism. In particular, if f1,f2,f4,f5 are isomorphisms, then so is f3.
a short exact sequence of R-modules. Given projective resolutions P∙A→A and P∙C→C, the horseshoe lemma provides a projective resolution P∙B→B and a short exact sequence of chain complexes
be a short exact sequence of chain complexes in an abelian category (in particular, this holds for R-modules). Then there is a natural long exact sequence in homology:
⋯→Hn(A)→Hn(B)→Hn(C)δHn−1(A)→Hn−1(B)→⋯
where δ is the connecting homomorphism.
The map δ is natural with respect to morphisms of short exact sequences of complexes.
Let R be a commutative ring, and let M and N be R-modules. The functors HomR(M,−) and HomR(−,N) on R-modules are left exact, with the latter reversing arrows. The functor −⊗RN is right exact. These functors are exact precisely when M is projective, N is injective, and N is flat, respectively.
This long exact sequence is natural in morphisms of short exact sequences of complexes. The maps δn are constructed by a standard diagram chase and can be viewed as arising from the snake lemma; see also connecting homomorphisms.