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
Integral dependence and integral extensions
We discuss the notion of an integral element of a ring S over a ring R. We dene integral
and module-nite extensions and discuss the relationship between these two notions. We
dene the integral closure of a ring
Categories and functors, the Zariski topology, and the functor Spec
We do not want to dwell too much on set-theoretic issues but they arise naturally here.
We shall allow a class of all sets. Typically, classes are very large and are not allowed to
be ele
Ane algebraic geometry
Closed algebraic sets in ane space
We assume that the reader has basic familiarity with the theory of dimension (i.e.,
Krull dimension) for Noetherian rings, and we also assume the Hilbert basis theorem,
which asserts that a nitely
Noether normalization and Hilberts Nullstellensatz
We prove the Noether normalization theorem over a eld and, more generally, over an
integral domain. We then deduce Hilberts Nullstellensatz.
The following result implies that, after a change of variables,
Math 615, Winter 2010
Problem Set #1: Solutions
1. It is clear that R/m K, so that m is maximal. In Rm , every 1 xj is a unit,
=
and if i < j, the relation xi (1 xj ) = 0 then implies that xi /1 = 0 in Rm . Thus,
Rm Rm /mRm K, and it follows that m is min
Math 615, Winter 2010
Problem Set #3 Solutions
1. Let cfw_S be the direct limit system of R-algebras, where h, : S S if
and h : S S, where S is the direct limit. In the unramiied case, given two dierent
liftings f, g : S T , there is some element s S su
Math 615, Winter 2010
Problem Set #4 Solutions
1. (a) For every subring A of R nitely generated over the prime ring Im (Z R), we
may consider the localization (B, PB ) of B at the contraction of P to B. Clearly, (R, P )
is the directed union of the rings
Math 615, Winter 2010
Problem Set #5 Solutions
1. A separable algebra is the directed union of its nitely generated separable subalgebras,
and a direct limit of reduced rings is reduced. Hence, we may assume that each of the
separable algebras is nitely g
Math 615, Winter 2010
Problem Set #2: Solutions
k
1. We may replace n by a large power of 2, say 2k , and assume that J 2 = 0 for some k.
We use induction on k. The case k = 1 follows from the denition. For the inductive step,
k+1
we have, if J 2
= 0, tha
Dimension theory and systems of parameters
Krulls principal ideal theorem
Our next objective is to study dimension theory in Noetherian rings. There was initially
amazement that the results that follow hold in an arbitrary Noetherian ring.
Theorem (Krulls