Commutative and homological algebra study algebraic structures, such as modules over commutative rings, in terms of their generators and relations. They provide some of the most powerful algebraic tools for applications in algebraic and differential geometry, number theory, algebraic topology, and related fields. This course gives a quick introduction to these techniques. It is designed as a starting point for those who intend to study algebraic geometry and related topics.
Prerequisites
Abstract Algebra I (tensor products, multilinear maps); some experience in geometry and topology is desirable, but not essential.
Curriculum:
- Polynomial ideals and algebraic varieties. Noetherian rings. Hilbert’s basis theorem.
- Integer ring extensions. Gauss-Kronecker-Dedekind lemma. Finitely generated algebras over a field.
- Noether normalization theorem. Hilbert’s Nullstellensatz.
- Geometry of ring homomorphisms: category of affine schemes.
- Category of modules. Generators and relations. Exact sequences. Grothendieck group.
- Categories and functors. Representable functors. Natural transformations. Adjoint functors.
- The Hom functor and tensor products. Projective, flat, and injective modules. Frobenius duality.
- Linear, additive, and Abelian categories. Complexes and homology.
- Canonical resolutions, bar construction, classical derived functors; Ext and Tor.
- Composition of derived functors. Spectral sequences.
- Koszul complex. Hilbert’s syzygy theorem. Duality.
- Category of sheaves. Čech cohomology. Leray spectral sequence.
- Sheaves of modules on algebraic varieties. Locally free resolutions. Ext functors.
- Serre duality.
- Category of coherent sheaves on projective space. Serre theorem. Beilinson theorem.
- 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. Macdonald, 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.