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Prerequisite material:   Lecture Notes for Math 614, Fall, 2020

Foundations of tight closure theory, 2007 version

Partial version of new lecture notes, to be modified on a continuing basis as lectures are given.

Foundations of tight closure theory, 2022 version

Local cohomology

Tight closure of parameter ideals and splitting in module-finite extensions

Tight closure and strongly F-regular rings

Some basic exposition about tight closure:

Tight closure and characteristic p methods

Problem sets:      1             2             3             4             5            

Solutions:            1             2             3             4             5

Some recorded lectures

02/02    02/04    02/07    02/09    02/11    02/14    02/16    02/18    02/21    02/23   

02/25    03/07    03/09    03/11    03/14    03/16    03/18    03/21    03/23    03/25   

03/28    03/30    04/01    04/04    04/06    04/08    04/11    04/13    04/15    04/18

Supplements. Most of these review 614 material. The last four are probably new to many.

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

Integral extensions and integral dependence

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

Regular rings and finite projective resolutions

The structure theory of complete local 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