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 2 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 upn where u is a unit and n 2 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 ´etaleand other important classes of ring extensions, such as smooth extensions and unramified extensiions. (The ´etaleextensions are the extensions that are both smooth and unramified.) There is a beautiful structure theory for 1 2 these classes of extensions. Because ´etaleextensions 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. Etale´ extensions have numerous applications to geometry: they are used to remedy the fact that the implicit function theorem does not hold in the allgebraic context in the same sense that it is does when working with C1 or analytic functions. As an example, we shall later use the theory of ´etaleextensions to establish a relationship that is not obvi- ous between intersection multiplicities defined algebraically and intersection multiplicities defined quite geometrically. The structure theorems we want to prove depend on an algebraic result known as Zariski's Main Theorem, or ZMT. It has many applications in commutative algebra and algebraic geometry. In our formal treatment, we shall first prove Zariski's Main Theorem, and then define and analyze the structure of smooth, ´etale,and unramified homomorphisms. We shall dis- cuss Henselization, Artin approximation, and applications in which one reduces questions about arbitrary Noetherian rings to the case of algebras finitely generated over a field or a discrete valuation ring (DVR). Another tool that we introduce provides a method for reducing many questions about finitely generated algebras over a field of characteristic 0 to corresponding questions for finitely generated algebras over a field of characteristic p > 0: in fact to the case where that field is finite! It may be surprising that one can do this: it turns out to be a very powerful technique. I do want to emphasize that the theory we build here shows that the study of finitely generated algebras over a field or DVR is absolutely central to the study of arbitrary Noetherian rings. Before stating Zariski's Main Theorem, we review some facts from commutative algebra that we assume in the sequel. Following the review, we state the algebraic form of the theorem, review some basic algebraic geometry, and then give a geometric version of ZMT. We explain how to deduce the geometric version from the algebraic version, and then go to work on the proof of the algebraic version, which is rather long and difficult. A prime ideal of R is a a proper ideal such that R=P is an integral domain. The (0) ideal is prime if and only if R is an integral domain. The unit ideal is never prime. The set of prime ideals of R, denoted Spec (R) is a topological space in the Zariski topology, which is characterized by the fact that a set of primes is closed if and only if it has the the form V(I) = fP 2 Spec (R): I ⊆ P g. I may be any subset of R, but V(I) is unchanged by replacing I by the ideal it generates, so that one may assume that I is an ideal. When I is an ideal, V (I) is unchanged by replacing I by Rad (I) = fr 2 R : for some integer n > 0; rn 2 Ig. The closed sets of Spec (R) are in bijective order-reversing correspondence with the radical ideals of R. The closure of the point given by the prime ideal P is V(P ), so that P is a closed point if and only if P is a maximal ideal of R. Note that Spec (R) is not, in general, T1. 3 If f 2 R, Df denotes Spec (R) − V(Rf), the set of prime ideals of R not containing f. The sets Df are a basis for the open sets of the Zariski topology on R. Note that Dfg = Df [ Dg. Let h : R ! S be a ring homomorphism. Then h∗ or Spec (h) denotes the map Spec (S) ! Spec (R) whose value on Q 2 Spec (S) is the inverse image h−1(Q) under h. This inverse image is also called the contraction of Q to R. Note that if R ⊆ S, the contraction of Q to R is simply Q \ R. We assume some familiarity with categories and functors. Spec is a contavariant functor from the category of commutative rings and ring homomorphisms to the category of topological spaces and continuous maps. (Very briefly, functors assign values to objects and morphisms in a category in such a way that identity maps are preserved, and composition is either preserved or reversed. Functors preserving composition are called covariant, while those reversing composition are called contravariant.) A multiplicative system W in a ring R is a subset that contains 1 is closed under multiplication. The localization of R at W , denoted W −1R, is an R-algebra in which every element of W becomes invertible. Every R-module M also has a localization at W , denoted W −1M, which is a W −1R-module. (W −1M may be defined as equivalence classes of pairs (m; w) 2 M × W where (m; w) is equivalent to (m0; w0) if there exists v 2 W such that v(w0m − wm0) = 0. The equivalent class of (m; w) is denote m=w. W −1M and an W −1R-module. Addition and multiplication by scalars are such that (m=w) + (m0=w0) = (w0m + wm0)=(ww0), r(m=w) = (rm)=w, and (r=v)(m=w) = (rm)=((vw). There is an R-linear map M ! W −1M that sends m 7! m=1. This map need not be injective. In fact, the kernel consists of all elements m 2 M such that wm = 0 for some w 2 W . These remarks include the case M = R. Note that W −1R is a ring, and the multiplication satisfies (r=w)(r=w0) = (rr0)=(ww0). M ! W −1M is injective if and only if no element of W is a zerodivisor on M, i.e., multiplication by every w 2 W gives an injective map M ! M. Note also that a homomorphism R ! S can be factored R ! W −1R ! S if and only if the image of W in S consists entirely of units, in which case the factorization is unique. This is referred to as the universal mapping property of localization. The notation MW is used as an alternative to W −1M, but we will not use this notation in these notes. −1 −1 There is a canonical isomorphism W R ⊗R M ! W M such that (r=w) ⊗ m 7! (rm)=w and, under the inverse isomorphism, m=w 7! (1=w) ⊗ m. M 7! W −1M is a covariant exact functor from R-modules to W −1R-modules: if f : M ! N, there is a unique map W −1M ! W −1N such that m=w 7! f(m)=w.