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The Scheme of Monogenic Generators and Its Twists

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The Scheme of Monogenic Generators and Its Twists THE SCHEME OF MONOGENIC GENERATORS AND ITS TWISTS SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH Abstract. Given an extension of algebras B/A, when is B generated by a M single element θ ∈ B over A? We show there is a scheme B/A parameterizing the choice of a generator θ ∈ B, a “moduli space” of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples. M A choice of a generator θ is a point of the scheme B/A. This inspires a local-to-global study of monogeneity, piecing together monogenerators over points, completions, open sets, and so on. Local generators may not come from global ones, but they often glue to twisted monogenerators that we de- fine. We show a number ring has class number one if and only if each twisted monogenerator is in fact a global generator θ. The moduli spaces of various M twisted monogenerators are either a Proj or stack quotient of B/A by natural symmetries. The various moduli spaces defined can be used to apply cohomo- logical tools and other geometric methods for finding rational points to the classical problem of monogenic algebra extensions. Contents 1. Introduction 2 1.1. Twists 3 1.2. Summary of the paper 4 1.3. Guide to notions of “Monogeneity” 4 1.4. Summary of Previous Results 5 1.5. Acknowledgements 6 2. The Scheme of Monogenic Generators 7 2.1. Functoriality of MX 10 2.2. Relation to the Hilbert Scheme 12 M arXiv:2108.07185v1 [math.AG] 16 Aug 2021 3. The Local Index Form and Construction of 15 3.1. Explicit Equations for the Scheme M 15 3.2. First examples 18 3.3. Explicit Equations for Polygenerators Mk 20 4. Local monogeneity 23 4.1. Zariski-local monogeneity 23 4.2. Monogeneity over Geometric Points 24 4.3. Monogeneity Over Points 26 5. Etale´ Maps, Configuration Spaces, and Monogeneity 29 5.1. The case of ´etale S′ S 29 5.2. When is a map ´etale?→ 36 Date: August 17, 2021. 2020 Mathematics Subject Classification. Primary 14D20, Secondary 11R04, 13E15. 1 2 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH 6. Twisted Monogeneity 37 6.1. Gm-Twisted Monogenerators and Proj of the Weil Restriction 39 6.2. Aff1-Twisted Monogenerators and Affine Equivalence 41 6.3. Twisting in general 46 7. Examples of the scheme of monogenerators 50 7.1. Field extensions 50 7.2. Orders in number rings 53 7.3. Other examples 57 Appendix A. Basics of Torsors and Stack Quotients 61 References 65 1. Introduction The theorem of the primitive element states that given a finite separable field extension L/K, there is an element θ of L such that L = K(θ). This holds for any extension of number fields L/K. By contrast, the extension of their rings of integers Z /Z may require up to log ([L : K]) elements of Z to generate Z L K ⌈ 2 ⌉ L L as a ZK -algebra [Ple74]. Question 1.1. Which extensions ZL/ZK are generated by a single element over ZK ? More generally, which finite locally free algebras B/A are generated by a single element over A? How many elements does it take to generate B over A otherwise? Definition 1.2. A finite locally free A-algebra B is monogenic1 if there is an element θ B such that B = A[θ]. The element θ is called a monogenic generator or monogenerator∈ of B over A. If there are elements θ ,...,θ B such that B = A[θ ,...,θ ], then B/A is 1 k ∈ 1 k k-genic and (θ1,...,θk) is a generating k-tuple. Monogeneity of an algebra can be restated geometrically: Remark 1.3. Let A B be an inclusion of rings. A choice of element θ B is equivalent to a commutative⊆ triangle ∈ sθ 1 Spec B AA, (1) Spec A where sθ is the map induced by the A-algebra homomorphism A[t] B taking t to θ. → The element θ is a monogenerator if and only if the map sθ is a closed immersion. Likewise, a k-tuple θ~ = (θ ,...,θ ) Bk determines a corresponding map s : 1 k ∈ θ~ Spec B Ak induced by the A-algebra homomorphism A[t ,...,t ] B taking → A 1 k → ti θi. The tuple (θ1,...,θk) generates B over A if and only if sθ~ is a closed immersion.7→ 1 The literature often uses the phrase “L is monogenic over K” to mean ZL/ZK is mono- genic as above. We prefer “ZL is monogenic over ZK ” in order to treat fields and more exotic rings uniformly. If B is monogenic of degree n over A with monogenerator θ, the elements {1,θ,θ2,...,θn−1} are elsewhere referred to as a “power A-integral basis.” THESCHEMEOFMONOGENICGENERATORS 3 We ask if there is a scheme that represents such commutative triangles as in moduli theory [Fan+05, Chapter 1]. For extensions of number rings or any other algebras fitting into Situation 2.1 below, there is a representing scheme MB/A over Spec A called the scheme of monogenerators. There is an analogous scheme of generating k-tuples or scheme of k-generators denoted by Mk,B/A. When the extension B/A is implied, we simply write M or Mk as appropriate. Theorem 1.4. Let B/A be an extension of rings such that Spec B Spec A is → in Situation 2.1. There is an affine A-scheme MB/A and finite type quasiaffine A-schemes M , for k 1, with natural bijections k,B/A ≥ Hom (Spec A, M ) θ B θ is a monogenerator for B over A (Sch/A) B/A ≃{ ∈ | } and Hom (Spec A, M ) θ~ Bk θ~ is a generating k-tuple for B over A . (Sch/A) k,B/A ≃ ∈ | n o We compute explicit affine charts for Mk,B/A and deduce affineness for MB/A. The problem of finding generating k-tuples of ZL/ZK is thereby identified with M M that of finding ZK -points of the schemes k,ZL/ZK . The functors Hom( , k) au- tomatically form sheaves in the fpqc, fppf, ´etale, and Zariski topologies,− permitting monogeneity to be studied locally. These spaces are already interesting for trivial covers of a point: Example 1.5. Let A = C and B = Cn. The complex points of the monogeneity space are naturally homeomorphic to the configuration space of n distinct points in C: M (C) Conf (C) := (x ,...,x ) Cn x = x for i = j . 1,B/A ≃ n { 1 n ∈ | i 6 j 6 } Monogeneity generalizes configuration spaces by conceiving of B/A as “families 1 of points” to be configured in A . We spell out the relation to both Confn and the Hilbert scheme. When B/A is ´etale (unramified), this connection relates M to log geometry and Example 5.3 shows the ´etale fundamental group of M1 yields a well-known action of the Grothendieck-Teichm¨uller group GT on the profinite completion of the braid group. Similarly, Example 5.6 gives an isomorphism of a quotient of M with the moduli space of n-pointed genus zero curvesd M0,n. 1.1. Twists. The space of monogenerators admits several natural group actions, 1 such as affine transformations Aff or scaling by units Gm. Two monogenerators θ ,θ Z are “affine equivalent” if there are a Z∗ ,b Z such that 1 2 ∈ L ∈ K ∈ K aθ1 + b = θ2. Section 6 studies the twists of MB/A by this action using the stack quotient. The points of one of the twisted spaces are then commutative triangles like Diagram 1 (1), but with AA replaced by a line bundle L. The line bundle L Spec A has a class α Pic(A). → We say∈ B/A is twisted monogenic of class α if such a triangle exists. Twisted monogeneity of an extension of algebras can also be characterized without the language of schemes, see Definition 6.1. Likewise, we construct a space of twisted monogenerators that is itself a twisted form of the space of ordinary monogenerators MB/A. For a precise study, see Section 6. To see a practical example, consult Example 6.22. Twisted monogenic extensions are a strong invariant of a number field: 4 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH Theorem 1.6 (=Theorem 6.20). A number field K has class number one if and only if all twisted monogenic extensions of ZK are in fact monogenic. Warning 1.7. The reader with a background in classical monogeneity or number theory is advised to turn directly to Sections 3 and 7 for concrete worked examples. Appendix A is similarly directed at those unfamiliar with torsors and stack quo- tients. It contains a minimum for the purposes of Sections 5 and 6. More thorough expository accounts of stacks include [Fan+05]. M 1.2. Summary of the paper. Section 2 defines our main object of study S′/S for a finite flat map S′ S. This scheme parametrizes choices of monogenerators →O O for the algebra extension S′ / S. We offer basic properties, functoriality, examples, and a relation with the classical Hilbert scheme and the work of Poonen [Poo06]. M In Section 3, we obtain explicit equations for S′/S using index forms. The O O O index form tells whether a given section θ S′ is a monogenerator for S′ / S or not. We generalize to k-generators. ∈ Section 4 concerns local monogeneity.
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