**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 |