AIS Commutative Algebra (2015) - Speakers and Syllabus


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)

L13,15,17,19,21,23 (MK):
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)

Tentative time-table:

Date 9:30-11:10 11:30-1:00 2:30-3:30 4:00-500
14-12-2015 CDC AVJ T1 T2
15-12-2015 CDC AVJ T3 T4
16-12-2015 AVJ CDC T5 T6
17-12-2015 CDC AVJ T7 T8
18-12-2015 AVJ CDC T9 T10
19-12-2015 CDC AVJ T11 T12
21-12-2015 MK KNR T13 T14
22-12-2015 MK KNR T15 T16
23-12-2015 MK KNR T17 T18
24-12-2015 MK KNR T19 T20
25-12-2015 MK KNR T21 T22
26-12-2015 MK KNR T23 T24
28-12-2015 AS SI T25 T26
29-12-2015 AS SI T27 T28
30-12-2015 AS SI T29 T30
31-12-2015 AS SI T31 T32
01-01-2016 AS SI T33 T34