First-year graduateAlgebraLesson available
Homological algebra
Chain complexes and homology, long exact sequences, projective and injective resolutions, the derived functors Ext and Tor, and their first applications in algebra and topology.
math.algebra.homological-algebra
Builds on
Required prerequisites, all the way down: 33
You will learn to
- Compute homology of chain complexes and derive the long exact sequence of a short exact sequence of complexes via the snake lemma.
- Construct projective and injective resolutions and compute Ext and Tor over ℤ and k[x].
- Explain derived functors as measuring the failure of a functor to be exact.
Definition
- A chain complex is a sequence of modules with maps dₙ satisfying dₙ ∘ dₙ₊₁ = 0; its homology is Hₙ = ker dₙ / im dₙ₊₁.
- A quasi-isomorphism is a chain map that induces isomorphisms on all homology groups.
- Ext and Tor are the derived functors of Hom and of the tensor product.
Theorem
- The snake lemma, and the long exact sequence in homology.
- Ext and Tor do not depend, up to canonical isomorphism, on the resolutions chosen.
- Ext¹(A, B) classifies extensions 0 → B → E → A → 0 up to equivalence.
Common misconceptions
- Thinking every quasi-isomorphism has an inverse chain map; in general it does not.
- Believing a complex is determined up to isomorphism by its homology.
Leads to
Optional discoveries
- Abelian categories Abelian categories axiomatise the setting in which homological algebra works; Grothendieck's Tôhoku paper (1957) developed derived functors in that generality.
- Algebraic topology Singular homology is the homology of the singular chain complex of a space.
- ← Commutative algebra
Learning pathways
Further reading
- Charles A. Weibel, An Introduction to Homological Algebra (1994)
- Henri Cartan and Samuel Eilenberg, Homological Algebra (1956)