## Syzygies

**Lecture hours/location: MW 14:00-15:15**

**Instructor**: Manoj Kummini.
Email: mkummini AT cmi DOT ac DOT in.
http://www.cmi.ac.in/~mkummini/index.html

**Office**: First floor.

**Office Hours**: MW 11:50-13:00

**Important information**:
- Moodle.
I'll add this information and details on Moodle later;
After you log in, search for `Syzygies' under
`Mathematics /Jan - Apr 2012'.

Course description.
A `syzygy'
is a linear relation among the generators of a
finitely generated module or an ideal. In this course, we will
study a selection of topics from D. Eisenbud's `Geometry of
Syzygies' dealing with the syzygies of the defining ideals of projective
varieties (embedded in some projective space). We will begin with sheaf
cohomology and local cohomology (derived functors, Cech cohomology,
vanishing/non-vanishing theorems, Cohen--Macaulay and Gorenstein rings,
local/Serre duality) and embedding of algebraic varieties into projective
spaces. Following this, we will study free resolutions, and some numerical
invariants associated to them. As specific examples, we will look at points
in projective plane, rational normal curves, curves embedded by high degree
and by the canonical divisor.
A somewhat long explanation: For an algebraic variety (say, a curve) we can
ask whether it can be embedded as a closed subvariety of a projective
space. If such embeddings exist, then the variety can be described in terms
of polynomial equations. In this course, we will look at some measures of
how `nice' the defining ideal of an algebraic variety (embedded in a
projective space) is. My current plan is to talk about the following:

- Regular local rings and polynomial rings.
- Free resolutions, depth, projective dimension and (Castelnuovo-Mumford)
regularity.
- Cohen-Macaulay and Gorenstein rings.
- Cohomology of coherent sheaves, local cohomology
- Line bundles, embeddings into projective spaces.
- Points in P^2
- Line bundles on smooth curves: Canonical bundle, Riemann-Roch theorem.
- Embedding of smooth curves: rational normal curves, curves of high
degree and canonical embedding.
- If time permits: Eisenbud-Goto conjecture and the
Gruson-Lasarzfeld-Peskine theorem.

** Textbook**:
- D. Eisenbud,
*Geoemtry of syzygies*, Springer. This is the main
textbook.
- D. Eisenbud,
*Commutative algebra with a view towards algebraic
geometry*, Springer.
- W. Bruns and J. Herzog,
*Cohen-Macaulay Rings*, Cambridge Univ
Press.
- R. Hartshorne,
*Algebraic geometry*, Springer.
- J.-P. Serre,
*Faisceaux Algebriques Coherents*, *Ann. Math*,
1955.
- R. Hartshorne,
*Local cohomology, a seminar given by
A. Grothendieck*, Springer LNM 41.
- S. Iyengar et. al.,
*Twenty-four hours of local cohomology*,
Amer Math Soc.

Prerequisite: Some knowledge of commutative algebra at the level of
Atiyah-Macdonald and algebraic geometry at the level of Hartshorne
Chapter I (only!) If you are considering taking this course,
talk to me this term (Aug-Dec 2011) itself.
It would help me to know the background
of the students before the course begins.

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Last modified:
Tue Aug 16 15:02:08 IST 2011