 |
 Indian Institute of Technology Bombay |
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Indian Institute of Technology Bombay |
| Dr. A. V. Jayanthan, TIFR, Mumbai |
| 10.00 a.m. on Aug 23, 30, Sept 6, 13, 20, 27 |
| Macaulay's Theorem on Hilbert functions |
| Macaulay's theorem on Hilbert functions characterizes which sequences can arise as Hilbert functions of graded algebras over fields. We will present Green's proof of this theorem. We will also disucss how Cohen-Macaulayness and reducedness of graded algebras are reflected in their Hilbert function. |
| Prof. Jugal Verma, IIT Bombay |
| 11.30 a. m. on Sept 6, 13, 20, 27, Oct 4, 11 |
| Cohen-Macaulay Modules |
| After discussing fundamental properties of CM-modules, we will describe various ways for testing CM property. We will provide examples of classes of CM rings, such as, rings of invariants, face rings of shellable simplicial complexes and local rings of smooth points of algebraic varieties. |
| Prof. Tony Puthenpurakal, IIT Bombay |
| 11.30 a.m. on Aug 23, 30, Oct 18, 25, Nov 1, 8 |
| Projective and injective resolutions |
| We will discuss basics of projective and injective modules and resolutions. We shall also prove Hilbert-Burch theorem for ideals of projective dimension one and construct injective resolution of the residue field of gorenstein ring after discussing basic properties of Bass numbers. |
| Prof. Balwant Singh, EE Dept, IIT Bombay |
| 10 a.m. on Oct 4, 11, 18, 25, Nov 1, 8 |
| Formal smoothness and Cohen structure theorems |
| The main results will be Cohen's structure theorems for complete local rings (both the equicharacteristic and the inequicharacteristic cases) and normalization lemma for complete local rings. The techniques that go into the proof include formal smoothness and separable extensions, which will be done as preliminaries. If time permits, the relationship of formal smoothness with regularity will also be discussed. |