Commutative algebra and algebraic
geometry
Fall 2009
Meeting time and place: MW 4:10pm-5:25pm,
room 507. Attendance is strongly encouraged.
Office hours: Mon 11am-noon, Wed 1-2pm in room
613, email: mfedorch@math
TA: Daniel Disegni, email: disegni@math, roundtable: Tue 10am
Texts:
Undergraduate Commutative Algebra by Miles Reid,
Introduction to Commutative Algebra by Michael Atiyah
and Ian Macdonald,
The Red Book of Varieties and Schemes, 2nd Ed., by David
Mumford
References:
Commutative Algebra with A View Toward Algebraic Geometry by David Eisenbud
Algebraic Geometry by Robin Hartshorne
Topics covered:
Commutative algebra: rings and modules;
spectrum of a ring; primary decomposition; flatness and localization; Noetherian rings; integral dependence and normalization;
dimension theory; completions and I-adic topology.
Sheaf theory: presheaves
and sheaves; sheaf of regular functions on a variety.
Varieties: Hilbert's Nullstellensatz
and algebraic sets; affine and projective varieties; morphisms
and rational maps; elimination theory, resultants and discriminants;
classical varieties: Grassmanians, etc.
Attributes of varieties: dimension; degree of a projective variety; Bezout's theorem; tangent spaces, smooth and singular points; resolution of singularities for curves.
Lecture schedule (subject
to change)
Lecture notes (these are updated periodically)
Assignments
Weekly assignments are due in the lecture.
Problem Set 1 due Wed, 9/16
Problem Set 2 due Wed, 9/23
Problem Set 3 due Wed, 9/30
Problem Set 4 due Wed, 10/07
Problem Set 5 due Wed, 10/21
Problem Set 6 due Wed, 10/28
Problem Set 7 due Mon, 11/09
Problem Set 8 due Mon, 11/16
Problem Set 9 due Mod, 11/23
Problem Set 10 due Mon, 12/07 (extended to 12/14)