MATH 615 LECTURE NOTES, WINTER, 2010
by Mel Hochster
ZARISKIS MAIN THEOREM, STRUCTURE OF SMOOTH, UNRAMIFIED, AND
ETALE HOMOMORPHISMS, HENSELIAN RINGS AND HENSELIZATION,
ARTIN APPROXIMATION, AND REDUCTION TO CHARACTERISTIC p
Lecture of January 6, 2010
Thro
Lecture 1
1
Lecture 1
1 Three theorems of McCoy
R is always a commutative ring with unity 1R . U (R) is the group of units of R.
C (R) is the set of regular elements of R, i.e. the set of non-zero-divisors. Z (R) is
the set of zero-divisors, the complemen
The CRing Project
A collaborative, open source textbook on commutative algebra.
http:/people.fas.harvard.edu/~amathew/cr.html
The following people have contributed to this work.
Shishir Agrawal
Eva Belmont
Zev Chonoles
Rankeya Datta
Anton Geraschenko
Sher
COMMUTATIVE ALGEBRA
PETE L. CLARK
Contents
Introduction
0.1. What is Commutative Algebra?
0.2. Why study Commutative Algebra?
0.3. Some themes of Commutative Algebra
1. Commutative rings
1.1. Fixing terminology
1.2. Adjoining elements
1.3. Ideals and quot
MATH 8020 CHAPTER 1: COMMUTATIVE RINGS
PETE L. CLARK
Contents
1. Commutative rings
1.1. Fixing terminology
1.2. Adjoining elements
1.3. Ideals and quotient rings
1.4. The monoid of ideals of R
1.5. Pushing and pulling ideals
1.6. Maximal and prime ideals
MATH 8020 CHAPTER 3: MODULES
PETE L. CLARK
Contents
3. The category of modules over a ring
3.1. Basic denitions
3.2. Finitely presented modules
3.3. Torsion and torsionfree modules
3.4. Tensor products of modules
3.5. Projective modules
3.6. Injective mod