AIS Commutative Algebra (2015) - Speakers and Syllabus

Syllabus:

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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

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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.

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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.

References:
(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.

 Speakers:

  • 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
SUNDAY
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
 SUNDAY
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