Commutative Algebra seminar talk


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1 Commutative Algebra seminar talk Reeve Garrett May 22, Ultrafilters Definition 1.1 Given an infinite set W, a collection U of subsets of W which does not contain the empty set and is closed under finite intersection is called a filter. If U is a filter such that for all D W we have either D U or D c U, then we say U is an ultrafilter. Ultrafilters are precisely the maximal filters. We now fix an infinite set W and an arbitrary ultrafilter U for our discussion. Observation 1.2 Our definition has two easy consequences: (1) If D U and E D is arbitrary, then E U: indeed, if not, then E c U, meaning = D E c U, a contradiction. (2) If A B U, then A U or B U: indeed, if A / U and B / U, then A c U and B c U, so (A B) c = A c B c U, so = (A B) (A B) c U, a contradiction. Example 1.3 Take an ultrafilter U containing a finite set {a 1,..., a n }. Then, by repeatedly applying (2) above, we see that {a i } U for some i. Moreover, by (1), U is precisely defined to be all subsets of W containing a i. Thus, it makes sense to call such an ultrafilter principal. In most commutative algebra settings, we are not interested in principal ultrafilters because we like to operate under the assumption that ultrafilter sets and cofinite sets are big. In fact, an ultrafilter is nonprincipal if and only if it contains all cofinite sets (and thus, necessarily, all ultrafilter sets are infinite). Henceforth, in this section and all remaining sections, we will abide by the convention that any ultrafilter considered is nonprincipal, unless otherwise noted. Remark 1.4 Ultrafilters provide us with a decision procedure on the collection of subsets of W : those in U are large, and those not in U are small. More precisely, defining a nonprincipal ultrafilter U is equivalent to defining a finitely (but not countably) additive measure m on W taking values in {0, 1}, where m(a) = 1 if and only if A U and m(a) = 0 if and only if A / U. In this case, we may say that if a property holds for an ultrafilter set, then it holds almost everywhere. To conclude this section, we state and prove a simple but important lemma. Lemma 1.5 [1, Corollary A.2] Given an infinite set W and an infinite subset V of W, there exists a nonprincipal ultrafilter F on W such that V F. Proof. If V is cofinite, then use any ultrafilter containing the cofinite subsets (which is necessarily nonprincipal), which we apply Zorn s lemma to construct. If not, then clearly we can construct the filter F of all subsets of W containing V and apply Zorn s lemma to the family of all filters containing F. 2 The Ultrafilter Topology on the Spectrum of a Commutative Ring Let R be a commutative ring. Recall from algebraic geometry that for an ideal I of R we set V (I) := {P Spec(R) : I P } and that V (a) is shorthand for V (ar). 1
2 Definition 2.1 Let C Spec(R) and U be an ultrafilter on C. Set P U := {a R : V (a) C U}. We call P U an ultrafilter limit point of C. Theorem 2.2 P U is a prime ideal. Proof. Let a, b R be such that ab P U. Then, V (ab) C U. However, from [5, Lemma II.2.1], we know that V (ab) = V (a) V (b), so we have (V (a) C) (V (b) C) U. Thus, by our observation in Section 1, V (a) C U and V (b) C U, meaning a P U or b P U, as desired. If U is a principal ultrafilter on C, then there is a prime P C such that U is all subsets of C containing P, and thus P U = P C. So, we have another instance in which principal ultrafilters seem to prohibit interesting behavior from occurring; in this case, our socalled limit points don t really close up anything! Passing to nonprincipal ultrafilters, however, it is possible that a set C may not contain all its ultrafilter limit points, which motivates us to form a topology, since we know it won t be trivial to consider. Definition/Theorem 2.3 Let R and C be as above. We say that C is ultrafilter closed if it contains all its ultrafilter limit points. In fact, the ultrafilter closed sets of R form the closed sets of a topology on Spec(R), the ultrafilter topology on Spec(R). Proof. We wish to show the following: (i) Given ultrafilter closed subsets C 1,..., C n of Spec(R), C := C 1 C n is also ultrafilter closed. (ii) Given {C λ : λ Λ} a collection of ultrafilter closed subsets of Spec(R), then D := λ Λ C λ is also ultrafilter closed. To prove (i), let U be an ultrafilter on C. We wish to show P U C. By our observation in Section 1, we have that for some i, C i U. Without loss of generality, suppose C 1 U. Then, the collection U 1 := {C 1 B : B U} is an ultrafilter on C 1, as one easily sees, and the ultrafilter limit point it defines for C 1, denoted P U1, is precisely P U. Indeed, suppose d P U1, which by definition means V (d) C 1 U 1. Then, since C 1 U and therefore every set in U 1 also lies in U, V (d) C 1 U. However, since supersets of ultrafilter sets are ultrafilter sets, we see V (d) C U. This means d P U, as desired. The reverse containment is even easier: if d P U, then V (d) C U, so by definition of U 1 we have V (d) C 1 = (V (d) C) C 1 U 1, meaning d P U1. Thus, we have P U = P U1 C 1 C, and since U was arbitrary, we have that C is ultrafilter closed, as desired, proving (i). To prove (ii), let U be an ultrafilter on D, and let P U be the ultrafilter prime associated to U. For each λ Λ, define an ultrafilter U λ := {B C λ : B D U} on C λ. Then, by a routine adaption of our argument showing that P U1 = P U above, we see that P U = P Uλ for each λ, meaning (since each C λ is closed) P U C λ for each λ, so P U D, as desired. Thus, (ii) is proven. Definition 2.4 The patch topology is the smallest topology on Spec(R) for which the sets V (I) and D(a) := Spec(R) \ V (a) are closed for I any ideal of R and any a R. Definition 2.5 The constructible topology is the topology for which the collection {C Spec(R) : C = ϕ 1 (Spec(S)) for some ring homomorphism ϕ : R S where S is a commutative ring} comprises all closed sets. Theorem 2.6 The patch topology, constructible topology, and ultrafilter topology on Spec(R) all coincide. The equivalence of the patch and ultrafilter topologies is proven by Loper and Fontana in [4], and the equivalence of the constructible and ultrafilter topologies is proven in [2]. Remark 2.7 The patch (and therefore constructible and ultrafilter) topology makes Spec(R) a compact, Hausdorff, and totally disconnected space. It is also apparent that this topology is finer than the Zariski topology. 2
3 3 The ultrafilter topology on the ZariskiRiemann space of valuation rings In this section, let K be a field, let A be a ring contained in K, and let Zar(K A) denote the set of valuation domains containing A with quotient field K. We may define a topology on Zar(K A) by declaring its basic open sets to be of the form U(x 1,..., x n ) := {V Zar(K A) : x 1,..., x n V }, where x 1,..., x n K. With this topology, Zar(K A) is called the ZariskiRiemann space of A with respect to K. Also, suppose that Zar(K A) = {V i : i Ω} and that U is an ultrafilter on Ω. Notation 3.1 For d K, define B(d) := {i Ω : d V i }. Define V U := {d K : B(d) U}. Theorem 3.2 V U is a valuation ring, possibly K. Proof. Let x K. If x V U, we re done. If not, then B(x) = {i Ω : x V i } / U. Since U is an ultrafilter and each V i is a valuation ring with quotient field K, we then have Ω \ B(x) = {i Ω : x / V i } = {i Ω : x 1 V i } U, which means x 1 V U, as desired, thus proving V U is a valuation ring. Theorem 3.3 Let C U and consider U C := {B C : B U}. Then, if we index C by its elements, U C is an ultrafilter on C. Moreover, V UC = V U. Like in the previous section, we call V U an ultrafilter limit point, and if C Zar(K) contains all its ultrafilter limit points, then we say C is ultrafilter closed. Also, the ultrafilter closed sets form a topology which we call the ultrafilter topology. In the same spirit as the previous section, it is indeed true that the ultrafilter topology coincides with a yettobedefined constructible topology, which we will define now. Definition 3.4 Let X be a topological space, and set K = {U : U X open and quasicompact}. Let K be the smallest subcollection of the power set of X containing K and closed with respect to finite unions, finite intersections, and complementation. The constructible topology on X is the topology on X with K its basis for open sets. We denote X with the constructible topology by X const. Recall that a spectral space is a topological space that is homeomorphic to the prime spectrum of a ring equipped with the Zariski topology. Suppose X is a spectral space. Then, K is a basis for the topology on X and is closed under finite intersections. Moreover, the constructible topology on X is the coarsest topology for which K is a collection of clopen sets, and X const is a compact, Hausdorff space. Theorem 3.5 [2, Corollary 3.6] Zar(K A) is a spectral space; it is homeomorphic to Spec(Kr(K A)), where Kr(K A) := {V (T ) : (V, M V ) Zar(K A)}, T is an indeterminate, and V (T ) denotes the trivial extension of V Zar(K A) to K(T ), i.e. V (T ) := V [T ] MV [T ]. For context, these ideas were developed for and principally applied to the problem of expressing integrally closed domains as intersections of valuation overrings, which sheds some insight on their structure. We now state one positive result in this direction. Proposition 3.6 [2, Proposition 4.1] Let {V i : i Ω} and {W j : j Λ} be two collections in Zar(K A). Suppose both collections have the same closure in the constructible topology. Then, i Ω V i = j Λ W j, i.e. the two intersections represent the same integrally closed domain. Moreover, these considerations have some applications to the theory of Kronecker function rings and semistar operations. First, recall that a vacant domain is a domain with a unique Kronecker function ring. Among other more technical results in [2], we have the following. Corollary 3.7 [2, Corollary 4.11] Let A be an integrally closed domain with quotient field K. If each representation of A (i.e. collection of valuation overrings of A that when intersected yield A) is dense in Zar(K A) with the constructible topology, then A is a vacant domain. Of course, star operations and Kronecker function rings are intricately related, so one might expect that some star operation can be related to a given representation. Indeed, in [2, Theorem 4.13], it is proven that some subsets of Zar(K A) determine finitetype e.a.b. semistar operations. 3
4 4 Ultraproducts and their Applications Definition 4.1 Let W be an infinite set, F be an ultrafilter on W, and {O w } w W be a collection of sets indexed by W. Define an equivalence relation on w W O w as follows: (a w ) w W (b w ) w W if and only if {w : a w = b w } F. In general, depending on the reference, the equivalence classes constructed here are denoted by either [(a w ) w W ] or ulim w a w, and the set of these equivalence classes is called the ultraproduct of the O w and is denoted by either ulim w O w, F O w, or O. In general, if the O w have algebraic structure that their Cartesian product naturally inherits componentwise, then ulim w O w naturally inherits the same algebraic structure. Moreover, if each O w is totally ordered, then ulim w O w is too (see [1, Lemma A3]), and the ultraproduct of fields is a field ([1, Lemma A3]). Example 4.2 In valuation theory, one is generally interested in families (K (w), v (w), Γ (w) ) (w W ) and the resulting trio which we ll denote by (K, v, Γ ), where K and Γ are ultraproducts and v : K Γ is defined by v ([a (w) ]) = [v (w) (a (w) )]. Perhaps unsurprisingly in light of the observations about the ultraproducts in question that we just made, v is indeed a valuation. We call this trio the ultraproduct of the given valued fields. Moreover, the residue field for the valuation ring corresponding to v is isomorphic to the ultraproduct of the residue fields of the v (w). We now collect a few basic facts about ultraproducts of valued fields that we ll use later. The proofs are available in Appendix A of [1]. Theorem 4.3 [1, Theorem A.4] Let all notation be as defined in the above example. Then, (i) for a fixed prime p, char K = p if and only if {s char K (s) = p} F; similarly for the residue fields; (ii) K is C 2 (d) if and only if {s K (s) is C 2 (d)} F; (iii) if all K (s) are Henselian, then so is K ; (iv) for a fixed prime p, if all K (s) are padically closed, then so is K. A few terms in the above theorem are heretofore undefined, so we define them now. Definition 4.4 A field K is padically closed if K is a Henselian valued field such that (i) its residue field is F p, (ii) its value group is discrete (i.e. a direct sum of copies of Z) with v(p) as minimal positive element (which forces char K = 0), and (iii) to every m 2 and every a K there exists b K and ν such that 0 ν m 1 and v(a) = v(p ν ) + mv(b). Definition 4.5 Let K be a field and i and d be positive integers. K is said to have the C i (d)property if every homogeneous polynomial f with coefficients in K in more than d i variables and of total degree d has a nontrivial root in K. For example, all finite fields are C 1 (d) for every d, and if a field k is C i (d) for every d, then its field of formal Laurent series k((x)) is C i+1 (d) for all d. Moreover, each padic field Q p is C 2 (2) and C 2 (3), as proved by D.J. Lewis in This led Emil Artin to form the conjecture that Q p is C 2 (d) for all positive integers d. It turns out that this conjecture is false; however, in 1965, Ax and Kochen showed that this conjecture is almost true, in a sense made precise by introducing an ultrafilter on an appropriate set. Before we sketch the proof, we need a theorem. Theorem 4.6 [1, Theorem 6.1.1] Suppose (K 1, v 1 ) and (K 2, v 2 ) are Henselian fields with the same residue field k of characteristic zero and same value group Γ and that both K 1 and K 2 are ℵ 1 saturated (this is a modeltheoretic concept with a very technical definition that we will not need to refer to in this presentation, so we will not define it in this presentation). Examples of such fields include F p ((X)) and Q p for any prime numbers p and ultraproducts of them, as well as subfields of ℵ 1 saturated fields. Let F be a subfield of K 1, and let σ : F K 2 be a valuepreserving embedding, i.e. v 2 (σ(x)) = v 1 (x) for all x F and σ(x) = x for all x O v1 F, where O v1 denotes the valuation ring in K corresponding to v 1. Assume that (i) F is countable and Henselian with respect to v 1 F and that (ii) Γ/v 1 (F ) is torsionfree. Then, for a 1,..., a m K 1 there exists a valuepreserving extension σ : F K 2 of σ such that F (a 1,..., a m ) F and F again satisfies (i) and (ii). 4
5 Partial Solution to Artin s Conjecture. Fix d N, denote by P the set of all prime natural numbers, and suppose the set A d := {p P : Q p is not C 2 (d)} is infinite. We hope to derive a contradiction from this. Since A d is infinite, by the lemma we concluded Section 1 with, there exists a nonprincipal ultrafilter F on the power set of P such that A d F. Moreover, since Q p and F p ((X)) are Henselian fields with respect to their usual valuations, the ultraproducts K = F Q p and L = F F p((x)) are both Henselian fields with respect to their canonical valuations v and w, respectively. Moreover, they have the same residue field k = F F p, which has characteristic zero, and they have the same value group Γ = F Z, which has infinite rank. Since A d F and Q p is not C 2 (d) for every p A d, we get from [1, Theorem A.4] that K is not C 2 (d). However, since P F and F p ((X)) is C 2 (d) for every p P, we have that L is C 2 (d) by the same theorem. Since the residue fields of K and L have characteristic 0, the identity isomorphism of their prime subfields δ : Q Q is clearly valuepreserving, since v Q and w Q are trivial. Moreover, with these valuations, Q is a Henselian subfield of both K and L. Let h 1 = N i=1 c ix αi K [X 1,..., X d2 +1] be homogeneous of degree d and suppose that h 1 has no nontrivial zero in K. By [1, Theorem 6.1.1], the isomorphism δ can be extended to a subfield F 1 Q(c 1,..., c N ), say to δ 1 : F 1 F 2 L such that δ 1 is still valuepreserving. Since L is C 2 (d), the image of h 1 under δ 1 has a nontrivial zero in L, say the tuple (x 1,..., x d2 +1). Once again, by [1, Theorem 6.1.1], the isomorphism δ 1 1 can be extended to an embedding δ 2 : F 2 (x 1,..., x d2 +1) K. But this means that the tuple (δ 2 (x 1 ),..., δ 2 (x d2 +1)) is a nontrivial zero of h 1 in K, which contradicts the finding that K is not C 2 (d), which followed from the assumption that A d was infinite. References [1] A.J. Engler and A. Prestel, Valued Fields. Springer Monographs in Mathematics. [2] C.A. Finocchiaro, M. Fontana, K.A. Loper, The constructible topology on spaces of valuation domains. Transactions of the AMS, Volume 365, Number 12, December 2013, p [3] C.A. Finocchiaro, M. Fontana, K.A. Loper, Ultrafilter and constructible topologies on spaces of valuation domains. Communications in Algebra, Volume 41, Issue 5, 2013, p [4] M. Fontana and K.A. Loper, The Patch Topology and Ultrafilter Topology on the Prime Spectrum of a Commutative Ring. [5] R. Hartshorne, Algebraic Geometry. Springer Graduate Texts in Mathematics. [6] K.A. Loper and F. Tartarone, A classification of the integrally closed rings of polynomials containing Z[X]. Journal of Commutative Algebra, Volume 1, Number 1, Spring [7] H. Schoutens, The Use of Ultraproducts in Commutative Algebra. Springer Lecture Notes in Mathematics,
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