## MATH 615, WINTER 2019

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Administrative Matters

Schedule form

This is a tentative and preliminary version of these lecture notes and should

not be used as a reference. Some of the material covered will change from what is here.

Lecture Notes for Math 615, Winter, 2019

Prerequisite material:

Lecture Notes for Math 614, Fall, 2017

Problem sets: 1
2
3
4
5

Solutions: 1
2
3
4
5

References for tight closure:

Tight closure and characteristic *p* methods

Fundamentals of tight closure theory

Reference for multiplicities, Tor, spectral sequences, and Koszul homology:

Lectures on Commutative Algebra II

Supplements:

Categories and functors, the Zariski topology, and the functor Spec

Integral extensions and integral dependence

Hilbert functions and Hilbert polynomials

Noether normalization and Hilbert's Nullstellensatz

Dimension theory and systems of parameters

Affine algebraic geometry

Formal power series rings, inverse limits, and *I*-adic completions of rings

The local nature of an element of a ring or module

Exact sequences with a flat cokernel and a sketch of properties of Tor

Regular rings and finite projective resolutions

The structure theory of complete local rings