## MATH 615, FALL 2014

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Administrative Matters

This is a preliminary, partial version of these lecture notes (through April 11) and should not

be used as a reference. I expect to change several of the topics in these notes and discuss

others instead. Treatments of supplementary background material will be posted as needed.

Lecture Notes for Math 615, Winter, 2014

Problem sets: 1
2
3
4
5

Solutions:
1
2
3
4
5

Supplements:

Integral Closure in a Finite Separable Algebraic Extension

Faithful Flatness

The Functor Tor

Regular Rings, Finite Projective Resolutions, and Grothendieck Groups

The Structure Theory of Complete Local Rings

Module-finiite Extensions of Complete Local Rings

Ideals of minors and Fitting invariants

Hilbert Functions

Cohen-Macaulay rings

Test elements using the Lipman-Sathaye theorem

Expository tight closure references:

* Tight closure and characteristic p methods *

The paper, with an Appendix by Graham J. Leuschke, appeared in Trends in Commutative Algebra,

MSRI Pubications 51, Cambridge University Press, Cambridge, England, 2004, pp. 181-210.

*The notion of tight closure in equal characteristic zero*

Appeared in Proc. of the CBMS Conference on Tight Closure and Its Applications, Fargo, North Dakota,

July, 1995), Appendix to the notes on the main lectures by Craig Huneke, C.B.M.S. Regional Conference

Series, A.M.S., Providence, R.I., 1996.

* Tight closure in equal characteristic, big Cohen-Macaulay algebras, and solid closure *

Appeared in Commutative Algebra: Syzygies, Multiplicities and Birational Algebra, Contemp. Math. 159,

Amer. Math. Soc., Providence, R. I., 1994, 173-196.