Week 1: (12 Lectures)
- L1 (CDC): Basic module theory; free modules; homogeneous Nakayama's lemma
- L2 (AVJ): Integral extensions; normal domains; polynomial rings and UFDs are normal
- L3 (CDC): Noetherian rings and modules; Hilbert basis; recognizing Noetherian rings (e.g., Atiyah-MacDonald Exercise 7.5)
- L4-5 (AVJ): Going up and going down theorems
- L6-7 (CDC): Nullstellensatz and dimension theory
- L8-9 (AVJ): Graded rings, Hilbert series and examples; degree and dimension from Hilbert series
- L10-11 (CDC): Homogeneous systems of parameters; homogeneous Noether normalization; regular sequences
- L12 (AVJ): Identifying normal domains, say in the hypersurface case, using the Jacobian criterion
Week 2: (12 Lectures)
Regular rings (polynomial rings); Hilbert syzygy theorem; Depth, Projective dimension and Auslander-Buchsbaum formula in the graded situation. Benson Proposition 5.4.2 (Recognizing when a finitely generated algebra, over a field of characteristic zero, is a polynomial ring). Cohen-Macaulay rings (in the graded setting); characterizing CM rings as free over a homogeneous Noether normalization. Proving that one homogeneous sop is a regular sequence iff each is; polynomial rings, complete intersections are CM; Gorenstein rings: R is Gorenstein in the graded setting iff Hom_A(R,A) is isomorphic to a shift of R for some/any homogeneous Noether normalization A; the equivalence of some/any can be proved using Hilbert series in the domain case: Bruns-Herzog Corollary 4.4.6(c).
L14,16,18,20,22,24: (KNR): Group actions: when R is a normal domain, so is R^G; how this helps compute various examples e.g., invariants of symmetric and alternating groups. For finite groups and algebraically closed fields, MaxSpec R^G is in bijection with orbits; example where this fails for infinite groups. Finite generation of R^G in characteristic zero AND in positive characteristic.
Week 3: (10 Lectures) (AS & SI).
- L25. Noether bound in characteristic zero; failure in positive characteristic
- L26. Noether bound in positive characteristic (Benson's proof)
- L27. Molien's formula and examples
- L28. Semi-invariants, and Molien for semi-invariants
- L29. Laurent coefficients of Hilbert series: degree and number of pseudoreflections
- L30. Hochster-Eagon theorem and Bertin's example
- L31. Characterize when G is a subgroup of SL in terms of Hilbert series of R^G
- L32. Shephard-Todd Theorem
- L33. Shephard-Todd Theorem ctd.
- L34. Watanabe's theorem
Pre-requisites: Knowledge of basic commutative algebra: localization, Hom and tensor, Noetherian and Artinian rings, associated primes and primary decomposition, integral closure, going up and going down theorems, dimension theory.
(a) D. J. Benson, Polynomial Invariants of Finite Groups Cambridge University Press.
(b) W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge University Press.
(c) D. Eisenbud, Commutative Algebra, with a view towards algebraic geometry, Springer Verlag.
(d) H. Matsumura, Commutative Ring Theory, Cambridge University Press.
(e) J.-P. Serre, Local Algebra, Springer Verlag.
(f) R. P. Stanley, Invariants of nite groups and their applications to combinatorics, Bull. Amer. Math. Soc (N.S.), 1979.
- Clare D'Cruz: (CDC)
- A. V. Jayanthan: (AVJ)
- Manoj Kummini: (MK)
- K. N. Raghavan: (KNR)
- Srikanth Iyengar: (SI)
- Anurag Singh: (AS)