MATH 615 LECTURE NOTES, WINTER, 2010
by Mel Hochster
ZARISKI’S 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
Throughout these lectures, unless otherwise indicated, all rings are commutative, asso
ciative rings with multiplicative identity and ring homomorphisms are
unital
, i.e., they are
assumed to preserve the identity. If
R
is a ring, a given
R
module
M
is also assumed to
be
unital
, i.e., 1
·
m
=
m
for all
m
∈
M
. We shall use
N
,
Z
,
Q
, and
R
and
C
to denote
the nonnegative integers, the integers, the rational numbers, the real numbers, and the
complex numbers, respectively.
Our focus is very strongly on Noetherian rings, i.e., rings in which every ideal is finitely
generated. Our objective will be to prove results, many of them very deep, that imply that
many questions about arbitrary Noetherian rings can be reduced to the case of finitely
generated algebras over a field (if the original ring contains a field) or over a discrete
valuation ring (DVR), by which we shall always mean a Noetherian discrete valuation
domain.
Such a domain
V
is characterized by having just one maximal ideal, which is
principal, say
pV
, and is such that every nonzero element can be written uniquely in the
form
up
n
where
u
is a unit and
n
∈
N
. The formal power series ring
K
[[
x
]] in one variable
over a field
K
is an example in which
p
=
x
.
Another is the ring of
p
adic integers for
some prime
p >
0, in which case the prime used does, in fact, generate the maximal ideal.
One can make this sort of reduction in steps as follows. First reduce to the problem to
the local case. Then complete, so that one only needs to consider the problem for complete
local rings.
We shall study Henselian rings and the process of Henselization. We shall give numer
ous characterizations of Henselian rings. In good cases, the Henselization consists of the
elements of the completion algebraic over the original ring. The next step is to “approxi
mate” the complete ring in the sense of writing it as a direct limit of Henselian rings that
are Henselizations of local rings of finitely generated algebras over a field or DVR. But
this is done in a “good” way, where many additional conditions are satisfied. The result
needed is referred to as
Artin approximation
.
We are not yet done. Henselizations are constructed as direct limits of localized ´
etale
extensions, and so we are led to study ´
etale and other important classes of ring extensions,
such as smooth extensions and unramified extensiions.
(The ´
etale extensions are the
extensions that are both smooth and unramified.) There is a beautiful structure theory for
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these classes of extensions. Because ´
etale extensions are finitely generated algebras, one
can take the fourth step, which is to replace the Henselian ring by a ring that is finitely
generated over a field or DVR. Carrying out these ideas in detail will take up a large
portion of these notes.
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 Summer '11
 Clark
 Algebra, Approximation, The Land, Algebraic geometry, prime ideal, Zariski

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