Introduction to Commutative and Homological Algebra

Commutative and homological algebra studies algebraic structures, say, modules over commutative rings, in terms of their generators and relations. It provides the most powerful algebraic tools for applications in algebraic and differential geometry, number theory, algebraic topology, etc. This course gives a quick introduction to these techniques. It is designed as the starting point for those who intend to study algebraic geometry and related topics.

Prerequisites: basic linear and multilinear algebra (tensor products, multilinear maps), some experience in geometry and topology is desirable but not essential.


Curriculum:

  1. Polynomial ideals and algebraic varieties. Noetherian rings. Hilbert base theorem.
  2. Integer ring extensions. Gauss-Kronecker-Dedekind lemma. Finitely generated algebras over a field.
  3. Noether’s normalization theorem. Hilbert’s Nullstellensatz.
  4. Geometry of ring homorphisms: category of affine schemes.
  5. Category of modules. Generators and relations. Exact sequences. Grothendieck group.
  6. Categories and functors. Representable functors. Natural transformations. Adjoint functors.
  7. The Hom-functor and tensor products. Projective, flat and injective modules. Frobenius duality.
  8. Linear, additive and Abelian categories. Complexes and homology.
  9. Canonical resolvents, bar-construction, classical derived functors; Ext and Tor
  10. Composition of derived functors, spectral sequences.
  11. Koszul complex. Hilbert’s syzygies theorem. duality.
  12. Category of sheaves. Cech cohomology. Leray spectral sequence.
  13. Sheaves of modules on algebraic varieties. Locally free resolvents. Ext-functors.
  14. Serre duality.
  15. Category of coherent sheaves on projective space. Serre theorem. Beilinson theorem.
  16. If time allows: Resultants and determinants of multidimensional format, the ideal of a canonical curve and the Green problem, moduli of coherent sheaves on projective spaces.

Textbooks

  • M. F. Atiyah, I. G. McDonald, Introduction to commutative algebra, Addison-Wesley, 1969.
  • S. I. Gelfand, Yu. I. Manin, Methods of homological algebra, I.
  • D.Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Springer, 1995.