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615 - MATH 615 LECTURE NOTES WINTER 2010 by Mel Hochster...

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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 1
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2 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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