Dilip P. Patil, IISc Bangalore

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Kriti Goel, IITGN, Gandhinagar

T 2 : Dimension Theory : (6 Lectures (90 Min.) ; ref : Murthy’s Notes Ch. 2 and Ch. 3)

1.Krull-dimension and Examples, Dimension of an integral extensions (without proofs).
2.Noether’s Normalization Lemma and its consequences.
3.Graded rings and modules, Hilbert functions and series, Hilbert’s Theorem.
4.Hilbert-Samuel functions and polynomials, System of parameters, Dimension Theorem.
5.Dimension of polynomial rings over noetherian rings.
6.Normal rings and their characterizations and properties. Finiteness of integral closure.

T 5 : Complete local rings : (6 Lectures (90 Min.) ;
ref : Murthy’s Notes Ch. 4 and Ch. 9)

1.Derivations and the module of Kähler differentials.
2.Completion of a local ring.
3.Complete local rings, Examples and Properties.
4.Formal smoothness.
5.Formal smoothness and regularity.
6.Cohen’s Structure Theorem for complete local rings.

Jugal Verma, IIT Bombay

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Sudeshna Roy, CMI, Chennai

T 4 : Cohen-Macaulay rings : (6 Lectures (90 Min.) ;
ref : Murthy’s Notes Ch. 5 and Ch. 6)

1.Regular sequences and Koszul complexes.
2.Quasi-regular sequences.
3.Homological characterisation of depth.
4.Depth lemma, properties of grade of an ideal.
5.Depth and dimension, Cohen-Macaulay modules and rings
6.Properties of CM modules, regular local rings are Cohen-Macaulay.

T 6 : Gorenstein rings : (6 Lectures (90 Min.) ;
ref : Matsumura Ch.6, § 18)
1.Homological characterisation of injective dimension of modules.
2.Characterisation of Gorenstein rings.
3.Basic properties of Goretnstein rings.
4.Examples of Gorenstein rings.
5.Structure of injective modules.
6.Matlis duality.

Indranath Sengupta, IIT Gandhinagar

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Joydip Saha, ISI Kolkata

T 1 : Homological Algebra : (6 Lectures (90 Min.) ;
ref : Bosch : § 4.2, § 4.3, § 5.1, § 5.3 and Singh : Ch. 18)

1.Flat and faithfully flat modules.
2.Extension of coefficients, faithfully flat descent and fibres of flat maps.
3.Complexes, homology and cohomology, the Tor modules, flatness using Tor.
4.Injective resolutions and the Ext modules.
5.Injective dimension and Ext, Projective dimension, Global dimension.
6.Projective dimension over local ring.

T 3 : Regular local rings : (6 Lectures (90 Min.) ;
ref : Murthy’s Notes Ch. 7)
1.Regular local rings.
2.Homological characterisation of regular local rings.
3.Regular rings, Regular local ring is a UFD.
4.The Jacobian criterion for geometric regularity.
5.Flat extensions of regular local rings.
6.Serre’s normality criterion, regular local rings are Cohen-Macaulay and normal.