The Scheme of Monogenic Generators and Its Twists

Home , Noetherian scheme

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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 monogenic 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) automatically form sheaves in the fpqc, fppf, étale, 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 étale (unramified), this connection relates M to log geometry and Example 5.3 shows the étale fundamental group of M1 yields a well-known action of the Grothendieck-Teichmüller 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 quotients. 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. If every fiber of the map S′ S is → monogenic, does this imply the map S′ S is monogenic? Zariski-locally on S, it does. The classical notion of common index→ divisors addresses the same question. One gets a different notion if only the fibers of S′ S over geometric points are monogenic. We conclude with concrete deformation→ theory for monogenerators, applicable in the case where S is a geometric point. M In Example 1.5, the scheme S′/S coincides with the classical configuration space of n points in C. Section 5 generalizes this phenomenon to any étale map S′ S. Theorem 5.2 establishes the correspondence in general and the section culminates→ in a variety of sample applications in other areas of mathematics. Finding monogenerators for S′ S is equivalent to finding rational points M → on the scheme S′/S. Section 6 applies techniques for finding rational points to M, specifically local-to-global methods. To glue local monogenerators to a global monogenerator, one must control how two monogenerators can differ. We consider 1 symmetries of monogenerators given by actions of Gm or Aff and define twisted monogenerators analogous to Cartier divisors. Twisted monogenerators relate to “affine equivalence classes” of monogenerators the same way Cartier divisors relate to principal Cartier divisors. There are cohomological obstructions to lifting a given twisted monogenerator to a global monogenerator. For any group G, we define G-twisted monogenerators and show there is a moduli space parameterizing them. The Picard group, Brauer group, and Shafarevich-Tate groups can all obstruct gluing G-twisted monogenerators to a global monogenerator. Theorem 6.20 states that a number field K has class number one if and only if all twisted monogenic extensions of ZK are monogenic. Section 7 gives a variety of concrete examples of the scheme of monogenerators M, including: separable and inseparable field extensions, a variety of orders in number fields including Dedekind’s non-monogenic cubic, jet spaces, curves, and completions. These explicit equations put into practice classical theory in addition to the theory we have built. We encourage the reader to consult these examples to complement the earlier sections. Appendix A reviews torsors and stack quotients for the reader’s convenience. This establishes conventions and contains requisite material for Section 6.

1.3. Guide to notions of “Monogeneity”. THESCHEMEOFMONOGENICGENERATORS 5

(Globally) Monogenic

Gm-Twisted Monogenic

Aﬀ1-Twisted Monogenic

§6.2 Monogenic at completionsThm. 4.1 Zariski-Locally MonogenicThm. 4.1 Monogenic at points

Etale-Locally´ Monogenic

Monogenic at geometric points

1.4. Summary of Previous Results. The question of which rings of integers are monogenic was posed to the London Mathematical Society in the 1960’s by Helmut Hasse. Hence the study of monogeneity is sometimes known as Hasse’s problem. For an in-depth look at monogeneity with a focus on algorithms for solving index form equations, see Gaál’s book [Gaá19]. Evertse and Gy˝ory’s book [EG17] provides background with a special focus on the relevant Diophantine equations. For another bibliography of monogeneity, see Narkiewicz’s texts [Nar04, pages 79- 81] and [Nar18, pages 75-77]. The prototypical examples of number rings are monogenic over Z, such as quadratic and cyclotomic rings of integers. Dedekind [Ded78] produced the first example of a non-monogenic number ring (see Example 7.6). Dedekind used the splitting of (2) to show that the field obtained by adjoining a root of x3 x2 2x 8 to Q is not monogenic over Z. Hensel [Hen94] showed that local obstructions− − to− monogeneity come from primes whose splitting cannot be accommodated by the local factorization of polynomials. See [Ser79, Proposition III.12] and also [Ple74] which is discussed briefly below. Global obstructions also preclude monogeneity. For a number field L/Q, the field index is the greatest common divisor gcdα Z [ZL : Z[α]]. A number field L ∈ L can have field index 1 and ZL may still not be monogenic, for example the ring of integers of Q(√3 25 7); see [Nar04, page 65] or Example 7.9. Define the minimal · index to be minα ZL [ZL : Z[α]]. The monogeneity of ZL/Z is equivalent to having minimal index equal∈ to 1. An early result of Hall [Hal37] shows that there exist cubic fields with arbitrarily large minimal indices. In [SYY16], this is generalized to show that every cube-free integer occurs as the minimal index of infinitely many radical cubic fields. The monogeneity of a given extension of Z is encoded by a Diophantine equation called the index form equation. Gy˝ory made the initial breakthrough regarding the resolution of index form equations and related equations in the series of papers [Gy˝o73], [Gy˝o74], [Gy˝o76], [Gy˝o78a], and [Gy˝o78b]. These papers investigate monogeneity and prove effective finiteness results for affine inequivalent monogenerators in a variety of number theoretic contexts. For inequivalent monogenic generators 6 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH one should also consult [EG85], [BEG13], and the survey [Eve11]. Specializing families of polynomials to obtain monogenic extensions is investigated in [Kön18]. In large part due to the group in Debrecen, there is a vast literature involving relative monogeneity: [Gy˝o80], [Gy˝o81], [Gaá01], [GP00], [GS13], [GRS16], [GR19b], and [GR19a]. Pleasants [Ple74] bounds the number of generators needed for a field of degree n by log2(n) , with equality if 2 splits completely. This upper bound is a consequence of a⌈ precise⌉ description of exactly when an extension is locally k-genic. For number rings, Pleasants answers the question of what the minimal positive integer k> 1 is such that ZL is k-genic. Global obstructions only appear in the case of monogeneity: for k> 1, Pleasants shows that local k-geneity is equivalent to global k-geneity. A brief context of the above is detailed in [EG17, Chapter 11], where it is shown that given an order O of a finite étale Q-algebra, one can effectively compute the smallest k such that O is k-genic. In the spirit of the previous work of Gy˝ory, one can also effectively compute the k generators of O over Z. Monogeneity has recently been viewed from the perspective of arithmetic sta- tistics: Bhargava, Shankar, and Wang [BSW16] have shown that the density of monic, irreducible polynomials in Z[x] such that a root is a monogenerator is 6 1 π2 = ζ(2)− 60.79%. That is, about 61% of monic, integer polynomials correspond to monogenerators.≈ They also show the density of monic integer polynomials with square-free discriminants (a sufficient condition for a root to be a monogenerator) is

1 (p 1)2 1 + − 35.82%. − p p2(p + 1) ≈ p Y Thus these polynomials only account for slightly more than half of the polynomials yielding a monogenerator. Using elliptic curves, [ABS20] shows that a positive proportion of cubic number fields are not monogenic despite having no local obstructions (common index divisors). In a pair of papers that investigate a variety of questions ([Sia20a] and [Sia20b]), Siad shows that monogeneity appears to increase the average amount of 2-torsion in the class group of number fields. In particular, monogeneity has a doubling effect on the average amount of 2-torsion in the class group of odd degree number fields. Previously, [BHS20] had established this result in the case of cubic fields.

1.5. Acknowledgements. The third author would like to thank Gebhard Martin, the mathoverﬂow community for [Herb] and [Hera], Tommaso de Fernex, Robert Hines, and Sam Molcho. Tommaso de Fernex looked over a draft and made helpful suggestions about jet spaces. The fourth author would like to thank Henri Johnston and Tommy Hofmann for help with computing a particularly devious relative integral basis in Magma. All four authors would like to thank their graduate advisors Katherine Stange (ﬁrst and fourth authors) and Jonathan Wise (second and third authors). This project grew out of the fourth author trying to explain his thesis to the second author in geometric terms. For numerous computations throughout, we were very thankful to be able to employ Magma [BCP97] and SageMath [Sage]. For a number of examples the [LMFDB] was invaluable. THESCHEMEOFMONOGENICGENERATORS 7

2. The Scheme of Monogenic Generators M We now use Remark 1.3 to construct the scheme of monogenic generators S′/S, our geometric reinterpretation of the classical question of monogeneity. We arrive at M a scheme k,ZL/ZK over Spec ZK whose ZK -points are in bijection with generating M M k-tuples for ZL over ZK . When k = 1, ZL/ZK := 1,ZL/ZK is the scheme of monogenerators. k The map S′ := Spec ZL S := Spec ZK is finite locally free and X = A → ZK → Spec ZK is quasiprojective; these properties suffice for our purposes. This permits analogues of monogeneity such as embeddings into Pk, which are more natural when S′ S is a map of proper varieties. We invite the reader to picture S′ as → 1 the ring of integers Spec ZL in a number field, S = Spec ZK , and X as A . We return to X = A1, Ak in Section 3, and will see there that our approach recovers the well-known index form equations.

Situation 2.1. Let S′ S be a finite locally free morphism of constant degree n 1 with S locally noetherian.→ Consider a quasiprojective morphism f : X S ≥ → and write X′ := X S′. ×S The constant degree assumption is for simplicity; the reader may remove it by working separately on each connected component of S. Leave n = 0 to scholars of the empty set. We allow S to be an algebraic stack in Sections 5 and 6, though the morphism S′ S is always representable. Write (Sch/S) for the big étale → M site, the category of S-schemes with the étale topology. We now define X,S′/S by describing the functor of maps into it and then showing it is representable. Definition 2.2. In Situation 2.1, consider the presheaf on (Sch/S) which sends an S-scheme T to the set of morphisms s fitting into the diagram:

s S′ T X T ×S ×S (2) T   7→      T    with restriction given by pullback. Refer to this presheaf either as the relative hom R presheaf HomS(S′,X′)[Stacks, p. 0D19] or the Weil Restriction X′,S′/S [BLR12, §7.6]. To see the collection of monogenerators within this presheaf, define a subpresheaf M R X,S′/S ⊆ X′,S′/S whose sections are diagrams exactly as above, but with s a closed immersion. We M R k M M R R write k,S′/S , k,S′/S for X = AS and S′/S = 1,S′/S, S′/S = 1,S′/S. When S′ S or X is understood, we may drop “S′/S” or “X” from the notation. If → S′ = Spec B and S = Spec A are affine, we write Mk,B/A or MB/A instead. We offer a few examples from [grg] for those unfamiliar with Weil Restrictions:

Example 2.3. One can always view a complex n-dimensional manifold X′ as a real 2n-dimensional manifold. If S′ = Spec C, S = Spec R, the Weil Restriction R X′,S′/S is simply X′ considered as a variety over R instead of C. For example, R 2 2 Spec C,S′/S = R = AR. 8 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

Take X′ = Gm = Spec C[z1,z2]/(z1z2 1). One replaces the complex variables with pairs of real variables: −

z1 = x1 + iy1; z2 = x2 + iy2.

The equation z1z2 = 1 becomes x x y y = 1; x y + y x =0. 1 2 − 1 2 1 2 1 2 R The Weil Restriction Gm,C/R is Spec R[x , x ,y ,y ]/(x x y y 1, x y + y x ). 1 2 1 2 1 2 − 1 2 − 1 2 1 2 If S′ = Spec L, S = Spec K for a degree-n finite extension L/K and X′ = R Spec L[z1,...,zs]/(f1,...,fr), one obtains X′,L/K the same way. Choose a basis n L = Kei and split the variables zi into n variables over K: z = x e + + x e . L i i1 1 ··· in n Substitute into the fj’s and split them accordingly f (x)= g e + g e j j1 1 ··· jn n to get R X′,L/K = Spec K[x]/(g). Suppose L/K is finite separable of degree d and X an L-scheme. Let L be the sep normal closure of L in a separable closure K with d embeddings σi : L L. e ⊆ This gives d different base changes of X′ to L, denoted Xσ′ i . The Galois group R e G = G(L/K) acts freely on Xσ′ 1 Xσ′ d and X′,L/K is the quotient. ×···× e Remark 1.3 expresses the presheaf M as e k,S′/S k Mk,S /S(T )= (θ1,...,θk) Γ(T S S′, OT S )⊕ OT S = OT [θ1,...,θk] ′ ∈ × ×S ′ | ×S ′ M An S-point of 1 is said to be a monogenerator of S′/S and we say S′/S is monogenic if such a point exists. This recovers the definition of monogeneity of algebras when S is affine. These presheaves are representable: M R Proposition 2.4. The presheaves X X′,S′/S are both representable by quasiprojective S-schemes and the inclusion⊆ is a quasicompact open immersion. R If f : X S is smooth, unramified, or étale, the same is true for X′,S′/S S and M → S. → X → R Proof. The Weil restriction X′,S′/S is representable by a scheme quasiprojective over S [Ji+17, Theorem 1.3, Proposition 2.10]. If X′ S′ is a finite-type affine R → morphism, the same is true for X′,S′/S S by locally applying [Ji+17, Propo- M R→ sition 2.2(1), (2)]. The inclusion X X′,S′/S is open by [Stacks, 05XA] and R⊆ automatically quasicompact because X′,S′/S is locally noetherian. The second statement is immediate. Corollary 2.5. M R The presheaves X , X′,S′/S are sheaves in the Zariski, Nisnevich, étale, fppf, and fpqc topologies on (Sch/S). Definition 2.6. Let τ be a subcanonical Grothendieck topology on schemes, for example the Zariski, Nisnevich, étale, fppf, or fpqc topologies. We say that S′/S is M τ-locally k-genic if the sheaf k,S′/S is locally non-empty in the topology τ. I.e., M there is a τ-cover Ui S i I of S such that k,S′/S(Ui) is non-empty for all i I. By default, we{ use→ the} étale∈ topology. ∈ THESCHEMEOFMONOGENICGENERATORS 9

A k-genic extension S′/S is τ-locally k-genic. If τ1 is a finer topology than τ2, then τ2-locally monogenic implies τ1-locally monogenic. Remark 2.7. We pose a related moduli problem F in § 2.2.1 that parameterizes a choice of finite flat map S′ S together with a monogenerator. It is also repre- → sentable by a scheme. The mere choice of a finite flat map S′ S is representable by an algebraic stack, as shown in [Poo06] and recalled in Section→ 2.2.

Our main example of S′ S comes from rings of integers in number fields → ZL/ZK , but here is another: n Example 2.8. Let S = Spec Z and Sn′ = Spec Z[ε]/ε . The Weil Restriction R is better known as the jet space J [Voj04]. For any ring A, (n 1)- X′,Sn′ /S X,n 1 jets are maps − − Spec A[ε]/εn X. → Jet spaces are usually considered over a field k by base changing from S. The monogeneity space M J parametrizes embedded (n 1)-jets, whose Sn′ /S,X X,n 1 n ⊆ − − map Spec A[ε]/ε X is a closed embedding. If n = 2, JX,1 is the Zariski tangent bundle and M ⊆ is the complement of the zero section. X,S2′ /S The truncation maps JX,n JX,n 1 restrict to maps MX,S /S MX,S /S. → − n′ +1 → n′ The inverse limit limn JX,n sends rings A to maps called arcs Spec AJtK X → according to [Bha16, Remark 4.6, Theorem 4.1]. Under this identification, the limit limn M parametrizes those arcs that are closed immersions into X Spec A. X,Sn′ /S × Compare embedded (n 1)-jets to “regular” ones. An (n 1)-jet f : n − − Spec k[ε]/ε X over a field k is called regular [Dem91, §5] if f ′(0) = 0. I.e., → 6 the truncation of higher order terms f : Spec k[ε]/ε2 X is a closed immersion. → Regular (n 1)-jets are precisely the pullback MX,S /S J JX,n. − 1′ × 1,X Extensions of number rings are generically étale, with a divisor of ramification. The finite flat map S′ S in jet spaces is the opposite, ramified everywhere. n → Remark 2.9. R M If X′,S′/S is affine, then X is quasi-affine. Even for number fields, Mk need not be affine; see Theorem 3.8. A sanity check for our new definition:

Example 2.10. Suppose π : S′ ∼ S is the identity, so n = 1. Then R = → X′,S′/S X′ = X and all sections of the separated X S are closed: M = R . → X X′ Remark 2.11 (Steinitz Classes). We have assumed that π : S′ S is ﬁnite locally → free of rank n, so π OS is a locally free OS-module of rank n. By taking an nth ∗ ′ exterior power, one obtains a locally free OS-module n det π OS := π OS . ∗ ′ ∗ ′ of rank 1 [Har77, Chapter II, Exercise 6.11].^ The Steinitz class of π : S′ S is the → isomorphism class of det π OS in Pic(S). ∗ ′ R The Weil Restriction k,S′/S is precisely the rank kn vector bundle with sheaf of sections π Ok . To see this, recall that the set of T -points of the S-scheme R S′ k,S′/S ∗ k is by deﬁnition the set of T -morphisms S′ T A . By the universal property ×S → T 10 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

k O of AT , such morphisms are in bijection with k-tuples of elements of Γ(T, S′ S T ). R O k × It follows that HomS( , k,S′/S) (π S′ ) as quasicoherent sheaves on S. When n − ≃ ∗ π OS OS ei is trivial, so is RS /S. ∗ ′ ≃ · ′ ExampleL 2.12. The Steinitz class of the jet space of Ak is the trivial vector bundle: k(n+1) Jn,Ak = A [Voj04, Corollary 5.2] 1 R M Remark 2.13. Although A and hence S′/S are ring objects, 1 is neither closed under addition nor multiplication. For multiplication, note that a power of O a generator is typically a non-generator. For addition, note that if θ S′ is a generator, then so is θ, but θ + ( θ)=0 is not for n = 1. ∈ − − 6 O Lemma 2.14. Let θ Γ(S′, S′ ) be any element. There is a canonical monic polynomial m (t) Γ(S,∈ O )[t] of degree n such that m (θ) = 0. θ ∈ S θ Proof. We begin by constructing mθ(t) locally, following [AM69, Proposition 2.4]. n Assume ﬁrst that π : S′ S is such that π OS O⊕ as an OS-module. → ∗ ′ ≃ S Choose a Γ(S, OS)-basis x1,...,xn of Γ(S′, OS ). For each i = 1,...,n, write n { } ′ θx = a x where a Γ(S′, O ). Now we let i j=1 ij j ij ∈ S′ P m (t) = det(δ t a ). θ ij − ij As in the proof of [AM69, Proposition 2.4], mθ(t) has coeﬃcients in Γ(S, OS ), is monic of degree n, and mθ(θ) = 0. Moreover, mθ(t) does not depend on the basis chosen since mθ(t) is computed by a determinant. O Now for general π : S′ S, choose an open cover Ui of S on which π S′ trivializes. On each open→ set, the construction of the{ previous} paragraph yields∗ O a monic polynomial mi(t) S(Ui)[t] of degree n vanishing on θ Ui . Since the construction of the polynomials∈ commutes with restriction and is inde| pendent of choice of basis, we have

mi(t) U U = mj (t) U U . | i∩ j | i∩ j We conclude by the sheaf property.

Remark 2.15. R n Lemma 2.14 deﬁnes a map m : S′/S AS sending an element O → θ Γ(S′, S′ ) to the coeﬃcients (bn 1,...,b0) of the universal canonical monic minimal∈ polynomial −

n n 1 mθ(t)= t + bn 1t − + + b0. − ··· n The preimage of a point of AS is the set of roots of the corresponding polynomial O R n in S′ . Locally in S, S′/S = AS and mθ(t) is given by elementary symmetric polynomials, up to sign. R n n Trivializing S′/S locally, this map becomes the map AS AS parametrizing the n elementary symmetric functions, the coarse quotient space→ for the natural n action of the symmetric group Σn AS. This action need not globalize. For example, take the 3-power map [3] : Gm Gm for S′ S. We have more to say in Section 5. → →

2.1. Functoriality of M . In Situation 2.1, let T S be a map from a locally X → noetherian scheme. Write X := X T , T ′ := S′ T , leading to a pullback T ×S ×S THESCHEMEOFMONOGENICGENERATORS 11 square M XT ,T ′/T T p (3) µ M X,S′/S S. M In this way, X,S′/S is functorial in S. What about X? A closed immersion X X gives 1 ⊆ 2 M M . X1,S′/S → X2,S′/S Warning 2.16. M Beware that X,S′/S is not functorial for all commutative squares

S1′ S2′

S1 S2, R even though the Weil Restriction X′,S′/S is. Similarly, there’s no natural map M 99K M X,S′/S X,S′′/S, M unless S′′ S′ happens to be a closed immersion. Neither is X,S′/S functorial in X except→ for closed immersions.

If g : S′′ S′ and f : S′ S are as in Situation 2.1, consider a commutative diagram → → i

i′ S′′ X S S′ X × p

S′ S.

If i is a closed immersion, so is i′ because S′ S is separated [Stacks, 07RK]. This yields a map →

f ∗ : MX,S /S S S′ MX S ,S /S ′′ × → ×S ′ ′′ ′ Corollary 2.17 (Relative Extensions of k-genic Extensions are k-genic). As above, if S′′ S is k-genic, then S′′ S′ is k-genic. → → Proof. Since S′′ S is k-genic, M has an S-point. Apply f ∗ to the corre- → k,S′′/S sponding S′-point of M S′ to get an S′-point of M . k,S′′/S ×S k,S′′/S

In particular, if M/L/K is a tower of number ﬁelds and ZM /ZK is k-genic, then Z /Z is k-genic. More prosaically, if A B C and C = A[θ ,...,θ ], then M L ⊆ ⊆ 1 k C = B[θ1,...,θk].

Remark 2.18. If S′ = Si′ is a ﬁnite disjoint union of ﬁnite locally free maps Si′ S, the pullback T S S′ is the disjoint union T S Si′. Write Xi′ = X′ S Si′, it follows→ from the universal×F property of coproducts and× the above that the× Weil Restriction decomposes as F R = R . X′,S′/S Xi′,Si′/S i Y 12 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

The monogeneity space is not the product M = M . Rather, we S′/S,X Si′/S,X claim a map 6 Q S′ X i → is a closed immersion if and only if eachG map

S′ X i → is a closed immersion and the closed immersions are disjoint:

S′ S′ = ∅ i ×X j for all i = j. To see the claim, we may check aﬃne locally on X, where it reduces to the statement6 that A B is surjective if and only if A B is surjective → i i → i for each i and Bi A Bj = 0 whenever i = j. This follows quickly in turn from the Chinese remainder⊗ theorem.Q 6 M Remark 2.19. Not only is 1,S′/S functorial in S, but we show its normalization and reduction can be performed on S. If X S is smooth and T S is the normalization of S, one uses [Stacks, → M → 03GV] and properties of X,S′/S in Proposition 2.4 to see that µ in Diagram (3) is also normalization. If a map Y Z is smooth, the square is cartesian → Yred Y

Zred Z. We need only check Z Y is reduced using [Stacks, 034E], since a surjective red ×Z closed immersion of a reduced scheme must be Yred. For smooth X S and M M → T := Sred, the pullback X,S′/S S Sred XS ,T ′/Sred is the reduction of M × ≃ red X,S′/S. 2.2. Relation to the Hilbert Scheme. With this section we pause our develop- ment of M to relate our construction to other well-known objects. Though the rest of the paper does not use this section, it behooves us to situate our work in the existing literature. M At the possible cost of representability of X,S′/S, let X S be any morphism of schemes in this section. Recall the Hilbert scheme of points→ [Stacks, 0B94] Hilbn (T ) := closed embeddings Z X T Z T ﬁnite locally free, deg. n . X/S ⊆ ×S → There is an algebraic moduli stack An of ﬁnite locally free maps of degree n

[Poo06, Definition 3.2], with universal finite flat map Zn An. Any map S′ S in Situation 2.1 is pulled back from Z A . We restrict→ to S-schemes without→ n → n further mention: An = An S. Recall Poonen’s description× of A : A finite locally free map π : Z T is n → equivalent to the data of a finite locally free OT -algebra Q given by π OZ . Suppose Q ∗ Q O n for the sake of exposition that a locally free algebra has a global basis T⊕ . The algebra structure is a multiplication map ≃ 2 Q⊗ Q → that can be written as a matrix using the basis. Conditions of associativity and commutativity are polynomial on the entries of this matrix. We get an affine scheme THESCHEMEOFMONOGENICGENERATORS 13 of finite type Bn parametrizing matrices satisfying the polynomial conditions, or equivalently multiplication laws on globally free finite modules [Poo06, Proposition O n Q O n 1.1]. Two different choices of global basis T⊕ T⊕ differ by an element of ≃ ≃ GLn(OT ). Taking the stack quotient by this action GLn Bn erases the need for a global basis and gives An. There is a map Hilbn A sending a closed embedding Z X to the finite X/S → n ⊆ |T flat map Z T . The fibers of this map are exactly monogeneity spaces: → M X,S′/S S p S′/S

d HilbX/S An.

Conversely, the monogeneity space of the universal finite flat map Zn A is isomorphic to the Hilbert Scheme → M Hilbn X,Zn/An ≃ X/S k over A. The space hn(A ) of [Poo06, §4] is Mk for the universal finite flat map to Bn. M Proposition 2.4 shows S′/S(X) S is smooth for X smooth, and likewise for unramified or étale. This means the→ map Hilbn A is smooth, unramified, or X/S → n étale if X S is. Suppose→X S flat to identify the Chow variety of dimension 0, degree n subvarieties of X→ with SymnX [Ryd08] and take S equidimensional. The Hilbert- Chow morphism Hilbequi n SymnX sending a finite, flat, equidimensional Z S X/S → → to the pushforward of its fundamental class [Z] in Chow A (X) restricts to MX,S /S. ∗ ′ When S′/S is étale, we will see the restriction of the Hilbert-Chow morphism M SymnX is an open embedding by hand in Section 5. X,S′/S → n Question 2.20. Can known cohomology computations of HilbX/S offer obstructions to monogeneity under this relationship? 2.2.1. Finite flat algebras with monogenerators. We mention a related moduli problem and how it fits into the present schema. We encourage the reader to skip to Section 3 on a first reading; this moduli problem will not arise in the sequel. We rely on a classical representability result: Theorem 2.21 ([Fan+05, Theorem 5.23]). If X S is flat and projective and → Y S quasiprojective over a locally noetherian base S, the functor HomS (X, Y ) is representable→ by an S-scheme.

The scheme HomS(X, Y ) is a potentially infinite disjoint union of quasiprojective S-schemes. Fix a flat, projective map C S and quasiprojective X S. Assign to any S-scheme T the groupoid of finite→ flat maps Y C T of→ degree n. One can → ×S think of this as a T -indexed family of finite flat maps Yt C. This problem is represented by → R An,C/S := HomS(C, An).

We study moduli of ﬁnite ﬂat maps Y C S T together with a choice of monogenerator: → × 14 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

Deﬁnition 2.22. The moduli problem F on S-schemes (Sch/S) has T -points given by:

A finite, flat family Y C S T of degree n, • A closed embedding Y → X× C T over C T . • ⊆ ×S ×S ×S These data form a fibered category via pullback. Define a variant F ′ parameterizing Q O n Q the data above together with a global basis T⊕ for the finite, flat algebra corresponding to Y C T . ≃ → ×S The map F ′ F forgetting the basis is a torsor for the smooth group scheme → F HomS(C, GLn). Let XC denote the pullback X S C. The stack is the Weil Re- × n R n striction Hilb ,C/S of the Hilbert Scheme HilbXC /C for XC C along the map XC /C → C S. Both are therefore representable by schemes using the theorem. One must → n use caution: HilbXC /C is an infinite disjoint union of projective schemes indexed by Hilbert polynomials and not itself projective, but this suffices for representability. There are universal finite flat maps

Z Y

e e C Hom (C, B ) C Hom (C, A ), ×S S n ×S S n n with and without a global basis Q O⊕ . The sheaf F may also be obtained by ≃ T the Weil Restriction along C S HomS(C, An) HomS (C, An) of the monogeneity M × → F space X,Y/C Hom (C,A ). The same construction of ′ can be obtained with ×S S n Bn in placee of An. We argue HomS(C, An) is also representable by an algebraic stack. Olsson’s result [Ols06, Theorem 1.1] does not apply here because An is not separated. This means the diagonal ∆An is not proper, and this diagonal is a pseudotorsor for automorphisms of the universal finite flat algebra. The automorphism sheaf Aut(Q) 2 of some finite flat algebras is not proper: take the 2-adic integers Z2, Q = Z2[x]/x , and the map Q Q sending x 2x. This is an automorphism over the generic 1→ 7→ point Q2 = Z2[ 2 ] and the zero map over the special point Z/2Z = Z2/2Z2. The scheme Bn on the other hand is a closed subscheme of an affine space, hence separated. The Weil Restriction HomS(C, Bn) is a scheme by the above theorem and the map Hom (C, B ) Hom (C, A ) S n → S n is again a torsor for the smooth group scheme HomS(C, GLn). Therefore HomS(C, An) is algebraic.

The diagonal ∆An even fails to be quasifinite because some finite flat algebras have infinitely many automorphisms: Example 2.23 (Infinite automorphisms). The dual numbers k[ε]/εn have an action of Gm by ε u ε for a unit u Gm(k). For another7→ example,· let k be∈ an infinite field of characteristic three and consider Q = k[x, y]/(x3,y3 1). Because (y+x)3 = y3, there are automorphisms y y+ux for any u k. − 7→ ∈ The reader may define stable algebras Q as those with unramified automorphism group [Stacks, 0DSN]. There is a universal open, Deligne-Mumford substack A n ⊆ e THESCHEMEOFMONOGENICGENERATORS 15

An of stable algebras [Stacks, 0DSL]. This locus consists of points where the action GLn Bn has unramiﬁed stabilizers [Poo06, §2]. 3. The Local Index Form and Construction of M M R This section describes equations for the monogeneity space 1,S′/S inside = 1 R HomS(S′, A ) by working with the universal homomorphism over . In the classical case ZL/ZK we recover the well-known index form equation. Section 3.2 gives M examples, while 3.3 generalizes the equations to k-geneity k,S′/S. Remark 3.1 (Representable functors). Recall that if a functor F : Cop Set is represented by an object X, then there is an element ξ of F (X) corresponding→ to the identity morphism X id X, called the universal element of F . The proof of the Yoneda lemma shows that→ for all objects Y and elements y F (Y ), there is a morphism f : Y X such that y is obtained by applying F (f ∈) to ξ. y → y ♯ For a map f : X Y of schemes, we write f : OY OX Y for the map of sheaves and its kin. → → | Explicit Equations for the Scheme M. R R 3.1. The scheme = S′/S = 1 1 HomS(S′, A ) is a “moduli space” of maps S′ A . For any T S, every 1 → R → morphism S′ S T AT is pulled back along some T from the universal homomorphism× → → u 1 S′ R AR . ×S

R We want explicit equations for M R. S′/S ⊆ Let t be the coordinate function on A1. The map u corresponds to an element ♯ θ = u (t) Γ(OS R ). Let m(t) be the polynomial of Lemma 2.14 for θ, i.e. m(t) ∈ ′×S is a monic polynomial in Γ(OR )[t] of degree n such that m(θ) = 0. Deﬁnition 3.2. We call the polynomial m(t) the universal minimal polynomial of 1 θ. Let V (m(t)) be the closed subscheme of AR cut out by m(t). The universal map u factors through this closed subscheme:

v 1 S′ R V (m(t)) AR . ×S (4) τ π R

1 Since V (m(t)) AR is a closed immersion, the locus in R over which u restricts to a closed immersion→ agrees with the locus over which v is a closed immersion. R O n 1 O Remark 3.3. The map V (m(t)) is ﬁnite globally free τ V (m(t)) i=0− R i → ∗ ≃ · t . The map v : S′ R V (m(t)) comes from a map ×S → L ♯ v : τ OV (m(t)) π OS R ∗ → ∗ ′×S of ﬁnite locally free OR -modules. Locally, it is an n n matrix. The determinant of this matrix is a unit when it is full rank, i.e. when×v is a closed immersion. The i O R ith column is θ , written out in terms of the local basis of π S′ S . We work this out explicitly to get equations for M R. ∗ × ⊆ 16 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

Remark 2.19 lets us find equations locally. Suppose S′ = Spec B and S = Spec A, n where B = i=1 A ei is a finite free A-algebra of rank n with basis e1,...,en. Let · 1 I = 1,...,n . Write t for the coordinate function of A and write xI as shorthand { L} for n variables xi indexed by i I. R ∈1 n The scheme = HomS (S′, A ) is the affine scheme AS = Spec A[xI ] and Dia- gram (4) becomes

v♯ B[xI ] A[xI ,t]/(m(t)) A[xI ,t]

τ ♯ π♯ A[xI ]

♯ The A[xI ]-homomorphism v sends t θ := x e + + x e . 7→ 1 1 ··· n n Note that A[xI ,t]/(m(t)) has an A[xI ] basis given by the equivalence classes of n 1 1,t,...,t − and B[xI ] has an A[xI ]-basis given by e1,...,en. With respect to this basis, v♯ is represented by the matrix of coeﬃcients

(5) M(e1,...,en) = [aij ]1 i,j n. ≤ ≤ j 1 n where aij A[xI ] are the unique coeﬃcients such that θ − = i=1 aij ei for each j =1,...,n∈. P n Example 3.4. Let S′ := Spec C S := Spec C as in Example 1.5. Let → n e1,e2,...,en be the standard basis vectors of C . The monogenerators of S′ over S are precisely the closed immersions: 1 S′ AS

S n 1 Identify R A . Let t denote the coordinate of A n , and let x1, x2, ..., xn denote S AS ≃ n the coordinates of AS. The analogue of Diagram (4) for this case is: n 1 S′ S AS V (m(t)) AAn × S

n AS 1 We write down mθ(t), as in Lemma 2.14. The coordinate t of A n maps to the AS universal element θ = x e + x e + + x e . Since e = (δ )n Cn, we 1 1 2 2 ··· n n i ij j=1 ∈ have θei = xiei. Computing the minimal polynomial, mθ(t) = det(δij t aij ) = n − i=1(t ai). − i n 1 i Notice that e e = δ e . It follows that θ = − x e . There- Q i j ij i j=0 j j fore M(e1,...,en) is the Vandermonde matrix with ith row given by 2 n 1 P 1 x x x − . i i ··· i Deﬁnition 3.5. With notation as above, let i(e ,...,e ) = det(M(e ,...,e )) 1 n 1 n ∈ A[xI ]. We call this element a local index form for S′ over S. When the basis is clear from context, we may omit the basis elements from the notation. THESCHEMEOFMONOGENICGENERATORS 17

Proposition 3.6. Suppose S′ S is finite free and S is affine. With notation as above, M is the distinguished→ affine subscheme D(i(e ,...,e )) inside R 1 n ≃ Spec A[xI ]. Proof. By Proposition 2.4, M is an open subscheme of R. Therefore it suffices to check that D(i(e1,...,en)) and M have the same points. Let j : y R be the inclusion of a point with residue field k(y). Then →

j factors through M j∗u is a closed immersion, where u is the univ. hom. ⇐⇒ j∗v is a closed immersion, for v as in (4) ⇐⇒ v♯ k(y) is surjective ⇐⇒ ⊗A j♯(M(e ,...,e )) is full rank ⇐⇒ 1 n j♯(i(e ,...,e )) is nonzero. ⇐⇒ 1 n This establishes the claim.

Remark 3.7. If B is a free A-algebra with basis e1,...,en , it follows that B is { } n monogenic over A if and only if there is a solution (x1,...,xn) A to one of the equations ∈ i(e1,...,en)(x1,...,xn)= a as a varies over the units of A. These are the well-known index form equations. In the case that A is a number ring there are only finitely many units, so only finitely many equations need be considered. This perspective gives the set of global monogenerators the flavor of a closed subscheme of R even though M1 is an open subscheme. Corollary 3.8. The map M S classifying monogenerators is affine. 1,S′/S → M Proof. Proposition 3.6 shows that S possesses an affine cover on which 1,S′/S restricts to a single distinguished affine subset of the affine scheme R = Spec A[xI ].

Remark 3.9. Consider a local index form i(e1,...,en) = det(M(e1,...,en)) de- n 1 ﬁned by a basis B − A e as above. Suppose e ,..., e is a second basis ≃ i=0 · i 1 n of B over A and M is the matrix representation of v♯ with respect to the bases n 1 L 1,...,t − and e1,..., en . Then det(M) = u det(eM) fore some unit u of A. Although{ the} determinantf{ of }M may not glue to a global datum S, this shows the ideal that it generatese does.e f N I Deﬁnition 3.10 (Non-monogenerators S′/S). Let S′/S be the locally principal ideal sheaf on R generated locally by local index forms. We call this the index form N R I ideal. Let S′/S be the closed subscheme of cut out by the vanishing of S′/S. We call this the scheme of non-monogenerators, since its support is the complement M R of S′/S inside of .

Remark 3.11. The local index forms i(e1,...,en) are homogeneous with respect to the grading on A[xI ]. To see this, note that since the ith column of M(e1,...,en) i 1 represents θ − , its entries are of degree i 1 in x1,...,xn. The Leibnitz formula for the determinant − det(a )= ( 1)σ a ij − · σ(i)i σ Σ X∈ n Y 18 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

i n i(i 1) shows that (e1,...,en) is homogeneous of degree i=1(i 1) = −2 . The transition functions induced by change of basis respect this grading,− so the index form I P ideal S′/S is a sheaf of homogeneous ideals. 3.2. First examples. To illustrate our machinery and gain some intuition, we compute the set and the scheme of monogenerators for quadratic extensions of number ﬁelds.

Example 3.12 (Quadratic Number Fields). Let K = Q(√d), for any square-free integer d. It is well-known that the ring of integers ZK is monogenic: ZL Z[√d] 1+√d ≃ or Z[ 2 ], depending on d mod 4. We will conﬁrm this using our framework, and determine the scheme M1 of monogenic generators. Let α denote the known generator of O , either √d or 1+√d . Let us take 1, α L 2 { } as the basis e1,...,en. The universal map diagram (4) becomes:

Z[a,b,α] Z[a,b,t]/(m(t)) Z[a,b,t]

Z[a,b] where the map Z[a,b,t]/(m(t)) Z[a,b,α] is given by t a + bα. The universal minimal polynomial m(t) is given→ by t2 Tr(a + bα)t + N(7→a + bα). This diagram encapsulates all choices− of generators as follows. The elements of Z[α] are all of the form a + b α for α ,β Z. Integers α ,β Z are in bijection 0 0 0 0 ∈ 0 0 ∈ with maps φ : Z[a,b] Z. Applying the functor Z φ,Z[a,b] to the diagram above yields a diagram → ⊗ − Z[α] Z[t]/(m(t)) Z[t]

Z where the map Z[t]/(m(t)) Z[α] takes t a + b α. The image is precisely → 7→ 0 0 Z[a0 + b0α], and the index form that we are about to compute detects whether this is all of Z[α]. Returning to the universal situation, the matrix representation of the map Z[a,b,t]/(m(t)) Z[a,b,α] (what we have been calling the matrix of coeﬃcients (5)) is given by → 1 a . 0 b Notice that we did not need to compute m(t) to get this matrix. The determinant, 1 b, is the local index form associated to the basis 1, α . Therefore M Z[a,b,b− ]. { } 1 ≃ Taking Z-points of M1, we learn that a + bα (a,b Z) is a monogenic generator precisely when b is a unit, i.e. b = 1. ∈ ±

Proposition 3.16 generalizes this example to any degree-two S′ S. → 1 n Example 3.13 (Jets in A ). Let S = Spec Z and Sn′ = Spec Z[ε]/ε , as in Exam- ple 2.8. We explicitly describe M1,S /S R = Jn 1,A1 . n′ ⊆ − THESCHEMEOFMONOGENICGENERATORS 19

n 1 n Choose the basis 1,ε,...,ε − for Z[ε]/ε . With respect to this basis, we may write the universal map diagram as

n Z[x1,...,xn,ε]/ε Z[x1,...,xn,t]/(m(t)) Z[x1,...,xn,t]

Z[x1,...,xn]

n 1 where t x1 + x2ε + + xnε − . Change7→ coordinates··· by t t x so that the image of t is 7→ − 1 n 1 t θ = x ε + + x ε − . 7→ 2 ··· n Update m(t) accordingly: m(t) = tn. Our next task is to compute the represen- j n 1 tation of θ in 1,ε,...,ε − -coordinates for j = 0,...,n 1. The multinomial theorem yields { } − n 2 n 1 j j it (t 1)it (x2ε + x3ε + + xnε − ) = xt ε − . ··· i2,...,in i2+i3+ +in=j t=2 X··· Y The coeﬃcient of εp is n j it xt . i2,...,in i2+i3+ +in=j t=2 i +2i + X···(n 1)i =p Y 2 3 ··· − n

The matrix of coeﬃcients in Figure 3.1 represents the Z[x1,...,xn]-linear map n n 1 i from Z[x1,...,xn,t]/(m(t)) to Z[x1,...,xn,ε]/ε = i=0− Z[x1,...,xn] ε . The coeﬃcient of εp above appears in the (j + 1)st column and (p + 1)st row.· L

10 0 00 0 ···   0 x2 0 00 0  ···   2   0 x3 x2 0 0 0   ···     3  M =  0 x4 2x2x3 x2 0 0  n  ···     0 x 2x x + x2 3x2x x4 0   5 2 4 3 2 3 2   ···     ......   ......       n 1   0 xn x2 −   ············    Figure 3.1. The matrix determined by an (n 1)-jet. −

n(n 1) Since M is lower triangular, it has determinant x 2− . An (n 1)-jet thereby n 2 − belongs to M 1 if and only if the coeﬃcient x is a unit. The x are natu- A ,Sn′ /S 2 i 1 rally coordinates of the jet space, yielding MA1,S /S = Spec Z[x1,...,xn, x− ] n′ 2 ⊆ RS /S = Jn 1,A1 . n′ − 20 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

3.3. Explicit Equations for Polygenerators Mk. The work above readily generalizes to describe Mk. Fix a number k N. We now construct explicit equations for Mk as a subscheme ∈ k of R = Hom (S′, A ) when S′ S is free and S is affine. These hypotheses hold k S → Zariski locally on S, so by Remark 2.19, this gives a construction for Mk locally on S in the general case. n Let S′ = Spec B and S = Spec A, where B = A e is a finite free A-algebra i=1 · i of rank n with basis e1,...,en. Let J = 1,...,k and I = 1,...,n . Write tJ { kL } { } for the J coordinate functions t1,...,tk of A and write xI J as shorthand for I J variables| | x indexed by (i, j) I J. × | × | ij ∈ × The scheme Rk is represented by the affine scheme Spec A[xI J ] and the uni- × versal map for Rk is the commutative triangle

u k S′ S Rk AR × k

Rk where the horizontal arrow u is induced by the ring map A[xI J ,tJ ] B[xI J ] sending × → × t θ := x e . j 7→ j ij i i I X∈ Notice that S′ R R is in Situation 2.1. Apply Lemma 2.14 to ﬁnd monic ×S k → k degree n polynomials mj (tj ) A[xI J ,tJ ] such that mj (θj ) = 0 in B[xI J ]. Write ∈ × × v♯ for the unique map ♯ v : A[xI J ,tJ ]/(mj (tj ): j J) B[xI J ] × ∈ −→ × ♯ factoring u : A[xI J ,tJ ] B[xI J ]. × → × k Now, A[xI J ,tJ ]/(mj(tj ): j J) is a free A[xI J ]-module of rank n with × ∈ × r r basis given by the equivalence classes of the products t 1 t k as the powers r 1 ··· k j vary between 0 and n 1. Since B[xI J ] is also a free A[xI J ]-module with basis − × r ×r e ,...,e , we may choose an ordering of the powers t 1 t k and represent the 1 n 1 ··· k map v♯ by an nk n matrix M. For each subset C 1,...,kn of size n, let M × ⊆{ } C be the submatrix of M whose columns are indexed by C and let det(MC) be the determinant.

Proposition 3.14. Suppose S′ S is finite free and S is affine. Then with → notation as above, Mk is the union of the distinguished affines D(det(MC )) inside Y . Proof. Check on points as in Proposition 3.6.

The scheme of k-generators Mk need not be aﬃne. Even for the Gaussian integers k k Z[i]/Z, we have that Mk = A (A ~0 ). We prove this in Proposition 3.16, after a small lemma. The second× factor\{ begs} to be quotiented by group actions of Gm, Σn, or GLn, which is the topic of Sections 5 and 6. O O Lemma 3.15. Locally on S, the ring S′ has an S-basis in which one basis element is 1. Proof. This is part of the proof of [Poo06, Proposition 3.1]. THESCHEMEOFMONOGENICGENERATORS 21

~ k Proposition 3.16. Suppose S′ S has degree 2 and let 0 AS be the zero section. Then affine locally on S we→ have an isomorphism ∈ M Ak (Ak ~0). k,S′/S ≃ S × S \ Proof. Working affine locally and applying Lemma 3.15, we may take S′ = Spec B and S = Spec A, where B has an A-basis of the form 1,e . Write b ,...,b for { } 1 k the coordinates on the second Ak , so ~0= V (b ,...,b ) Ak . S 1 k ⊆ S In the notation preceding Proposition 3.14, we may take xi,1 = ai, xi,2 = bi, and t a + b e. The matrix of coefficients will have columns given by the 1,e -basis i 7→ i i { } representation of the images of 1,ti, and titj as i, j vary over distinct integers in 1,...,k . Write e2 = c + de where c, d A. Then the column of the matrix {representing} 1 is ∈ 1 , 0 the columns representing the images of the ti are a i , b i and the columns representing the images of titj are a a + b b c i j i j . a b + a b + b b d i j j i i j Among the determinants of the 2 2 minors of this matrix are b1,...,bk, coming from the submatrices × 1 a i . 0 b i The remaining determinants all lie in the ideal (b1,...,bk) since all elements of the second row of the matrix lie in this ideal. We conclude by Proposition 3.14 that M k,S′/S is the union of the open subsets D(bi) of Spec A[a1,b1,...,ak,bk]. This gives us the result.

As an alternative to taking the union of k-determinants, we can use a generalization of the determinant first introducted by Cayley, later rediscovered and generalized by Gel’fand, Kapranov and Zelevinsky: Question 3.17. The map v♯ above is a multilinear map from the tensor product of k free modules A[xI J ,tj ]/mj(tj ) of rank n over A[xI J ] to rank-n free B[xI J ]. × × ♯ × For k = 1, M1 is the complement of the determinant of v . In general, the map v♯ is locally given by a hypermatrix of format (n 1,...,n 1) [Ott13]. This − − n n n-hypercube of elements of A[xI J ] describes a multilinear map the same× way×···× ordinary n n matrices describe a× linear map. What locus does the × hyperdeterminant cut out in Rk? Example 7.14 addresses the case k = 2 for jet spaces. Remark 3.18. We compare our local index form to the classical number theoretic situation. For references see [EG17] and [Gaá19]. If K is a number field and L is an extension of finite degree n, then there are n distinct embeddings of L into an algebraic closure that fix K. Denote them σ1,...,σn. Let Tr denote the trace 22 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH from L to K. The discriminant of L over K is defined to be the ideal Disc(L/K) generated by the set of elements of the form 2 (det[σi(ωj )]1 i,j n) = det[Tr(ωiωj )]1 i,j n, ≤ ≤ ≤ ≤ where we vary over all K-bases for L, ω1,...,ωn , with each ωi ZK . If α is any element of L, then the discriminant of{α over K is} defined to be ∈ 2 j 1 DiscL/K(α)= det σi α − 1 i,j n ≤ ≤ i 1 j 1 = det Tr α − α− 1 i,j n ≤ ≤ = (σ (α) σ(α))2 , i − j 1 i

One deﬁnes the index form of ZL over ZK be to Disc (α) I = [Z : Z [α]] = L/K . ZL/ZK L K Disc(L/K) s

Confer [EG17, Equation 5.2.2]. In the case where ω1,...,ωn is a ZK -basis for ZL, { } I employing some linear algebra [EG17, Equation (1.5.3)], one finds ZL/ZK is, up to an element of ZK∗ , the determinant of the change of basis matrix from ω1,...,ωn n 1 { } to 1,α,...,α − . The{ matrix in Equation} (5) is just such a matrix and its determinant coincides up to a unit with the index form in situations where the index form is typically defined. The generality of our setup affords us some flexibility that is not immediate from the definition of the classical index form equation. Proposition 3.19. N When the scheme of non-monogenerators S′/S is an effective Cartier divisor (equivalently, when none of the local index forms are zero divisors), N R 1 the divisor class of S′/S in = HomS(S′, A ) is the same as the pullback of the Steinitz class of S′/S from S.

1 Proof. Recall that V (m(t)) is the vanishing of m(t) in AR , where m(t) is the generic minimal polynomial for S′/S. Let τ be the natural map τ : V (m(t)) R. Consider the morphism → ♯ v : τ OV (m(t)) π OS R ∗ → ∗ ′×S of sheaves on R. The ﬁrst sheaf is free of rank n since m(t) is a monic polynomial: n 1 there is a basis given by the images of 1,t,...,t − . Therefore, taking nth wedge products in the previous equation, we have a map

OR det(τ OV (m(t))) det (π OS R ) . ≃ ∗ → ∗ ′×S By construction, this map is locally given by a local index form, i(e1,...,en). N Since we have assumed that S′/S is an effective Cartier divisor, the determinant is locally a nonzero-divisor. Therefore, the determinant identifies a non-zero section of π OS R . By definition of NS /S, we may identify det(π OS R ) with O(NS /S). ∗ ′×S ′ ∗ ′×S ′ THESCHEMEOFMONOGENICGENERATORS 23

Writing ψ : R S for the structure map, we also have that → det(π OS R ) ψ∗ det(π OS ). ∗ ′×S ≃ ∗ ′ since taking a determinental line bundle commutes with arbitrary base change and O π commutes with base change for ﬂat maps. The class of the line bundle det(π S′ ) in∗ Pic(S) is by deﬁnition the Steinitz class. ∗

4. Local monogeneity 4.1. Zariski-local monogeneity. This section shows Zariski-local monogeneity can be detected over points and completions, as spelled out in Remark 4.3. Theorem 4.1. The following are equivalent:

(1) π : S′ S is Zariski-locally monogenic. (2) There→ exists a family of maps f : U S such that { i i → } (a) the fi are jointly surjective; 1 (b) for each point p S, there is an index i and point qp fi− (p) so that f induces an isomorphism∈ k(p) k(q ); ∈ i → i (c) S′ U U is monogenic for all i. ×S i → i (3) π : S′ S is monogenic over points, i.e., S′ S Spec k(p) Spec k(p) is monogenic→ for each point p S. × → ∈ Proof. (1) = (2): Choose the Ui to be a Zariski cover on which S′ S is monogenic. ⇒ → (2) = (3): Suppose such a cover fi : Ui S is given. For each i, let σ : Spec⇒k(p) U be the section through{ q →. Monogeneity} is preserved by p → i p pullback on the base, so pulling back S′ S Ui Ui along σp implies (3). (3)= (1): Let p S be a point with residue× → field k(p) and let θ be a monogener- ⇒ ∈ ator for S′ S Spec k(p) Spec k(p). We claim that θ extends to a monogenerator over an open× subset U → S containing p, from which (1) follows. The claim is ⊆ O n O Zariski local, so assume S = Spec A and π S′ i=1 Sei is globally free. The R ∗ ≃ n Weil Restriction S′/S is then isomorphic to affine space AS . R L We first extend θ to a section of S′/S. The monogenerator entails a point θ : Spec k(p) M R An, i.e. n elements x ,..., x k(p). Choose → S′/S ⊆ S′/S ≃ S 1 n ∈ arbitrary lifts xi A(p) of xi. The n elements xi must have a common denominator, ∈ 1 n so we have x ,...,x A f − for some f. Thus our point θ : Spec k(p) A 1 n ∈ → S extends to θ : D(f) R for some distinguished open neighborhood D(f) S S′/S containing p. → ⊆ Finally,e we restrict θ to a section of M1. The monogeneity space M1 is an R R open subscheme of the Weil Restriction S′/S, so θ : D(f) S′/S restricts to a 1 → monogenerator θ U : U e M1 where U = θ− (M1) D(f). By hypothesis, p U, | → e⊂ ∈ so θ U is the desired extension of θ. | e e Remark 4.2. The same proof shows that S′ S is Zariski-locally k-genic if and e → only if its fibers S′ Spec k(p) Spec k(p) are k-genic. ×S → Remark 4.3. Item (2) of Theorem 4.1 implies the following are also equivalent to Zariski-local monogeneity:

(1) S′ S is “monogenic at local rings,” i.e, for each point p of S, S′ S Spec→O Spec O is monogenic. × S,p → S,p 24 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

(2) S′ S is “monogenic at completions,” i.e., for each point p of S, S′ → ×S Spec O Spec O is monogenic, where O denotes the completion of S,p → S,p S,p OS,p with respect to its maximal ideal. (3) S′ bS is locally monogenicb in the Nisnevichb topology as in Deﬁnition 2.6. → Corollary 4.4 ([Ser79, Proposition III.6.12]). Let S′ S be an extension of local → rings inducing a separable extension of residue ﬁelds. Then S′ is monogenic over S. Proof. Use the equivalence of item (1) in Remark 4.3 and (3) in Theorem 4.1.

Deﬁnition 4.5. Let ZL/ZK be an extension of number rings. A non-zero prime 2 I of p ZK is a common index divisor for the extension ZL/ZK if ZL/ZK (θ) p for every⊂ θ Z generating L/K. ∈ ∈ L As mentioned in Section 1, common index divisors are exactly the primes p whose splitting in ZL cannot be mirrored by irreducible polynomials in k(p)[x]; see [Hen94] and [Ple74]. Restating the property of being monogenic at points in terms of the index form, we obtain a generalization of the notion of having no common index divisors:

Proposition 4.6. S′ S is monogenic over points if and only if for each point p of S and local index→ form i around p, there is a tuple (x ,...,x ) k(p)n such 1 n ∈ that i(x1,...,xn) is nonzero in k(p).

Proof. By Proposition 3.6 and functoriality of M1, S′ Spec k(p) Spec k(p) is × 1 → monogenic precisely when Spec k(p)[x1,...,xn, i(x1,...,xn)− ] has a k(p) point. Immediately we recover an explicit corollary validating the generalization:

Corollary 4.7. Suppose S′ S is induced by an extension of number rings → Z /Z . Then S′ S is Zariski-locally monogenic if and only if there are no L K → common index divisors for ZL/ZK . Example 4.8. There are extensions of number rings that are locally monogenic but not monogenic. Narkiewicz [Nar04, Page 65] gives the following family. Let L = Q(√3 m) with m = ab2, ab square-free, 3 ∤ m, and m 1 mod 9. The number 6≡ ± ring ZL is not monogenic over Q despite having no common index divisors. To be even more explicit in Example 7.9 we consider the case ab2 =7 52. · 4.2. Monogeneity over Geometric Points.

Deﬁnition 4.9. Say that S′ over S is monogenic over geometric points if, for each morphism Spec k S where k is an algebraically closed ﬁeld, S′ S Spec k Spec k is monogenic. → × → While it is a weaker condition than monogeneity over points in general, it is equivalent to some conditions that might seem more natural. Theorem 4.10. The following are equivalent:

2Common index divisors are also called essential discriminant divisors and inessential or nonessential discriminant divisors. The shortcomings of the English nomenclature likely come from what Neukirch [Neu99, page 207] calls “the untranslatable German catch phrase [...] ‘außer- wesentliche Diskriminantenteile.’” Our nomenclature is closer to Fricke’s ‘st¨andiger Indexteiler.’ THESCHEMEOFMONOGENICGENERATORS 25

(1) the local index forms for S′/S are nonzero on each fiber of R S; (2) for each point p S with residue field k(p), there is a finite Galois→ extension ∈ L/k(p) such that S′ S Spec L Spec L is monogenic, and this extension may be chosen to be× trivial if k→(p) is an infinite field; (3) S′ S is monogenic over geometric points; → (4) there is a jointly surjective collection of maps U S so that S′ U { i → } ×S i → Ui is monogenic for each i; (5) M S is surjective; 1 → If, in addition, N is a Cartier divisor in R (i.e., the local index forms are non-zero divisors), the above are also equivalent to: (6) N S is flat. → To see some of the subtleties one can compare item 1 above with Lemma 4.6 and item (4) above with item (2) of Theorem 4.1. Proof. The assertions are Zariski-local, so we may choose local coordinates as usual (S = Spec A, S′ = Spec B, coordinates xI ). (1) = (2): Suppose first that p S is a point with k(p) an infinite field. Let ⇒ ∈ p be the corresponding prime of A and write i for the restriction of the local index form modulo p. Recall that since k(p) is infinite, polynomials in k(p)[x1,...,xn] are n determined by their values on (xi) k(p) . Since i is nonzero, i(a1,...,an) must be ∈n nonzero for some (a1,...,an) k(p) . This shows that S′ S Spec k(p) Spec k(p) is monogenic, so we have (2).∈ × → Next suppose k(p) is a finite field. Then, since i is nonzero, there is a finite field extension L (necessarily Galois) of k(p) such that there exists (a ,...,a ) Ln 1 n ∈ with i(a1,...,an) = 0. This shows that S′ S Spec L Spec L is monogenic, so we have (2) again.6 × → (2) = (3): Let k be an algebraically closed field and Spec k S a map. Let p be the image⇒ of Spec k, and let L be the field extension given by (→2). Then pullback along Spec k Spec L implies that S′ Spec k Spec k is monogenic. → ×S → (3) = (4): Take Ui S to be Spec k(p) S p S. (4) =⇒ (1): For{ each→ point} p {S, choose→ an index} ∈ i and a point q U ⇒ ∈ p ∈ i mapping to p. Let i be an index form around p. By pullback, S′ q q is ×S p → p monogenic, so i pulls back to a nonzero function over k(qp). Therefore i is nonzero over k(p) as well. (5) = (4): Take M S as the cover. (2) =⇒ (5): For each→ point p S, the Spec L point of M witnessing mono- ⇒ ∈ 1 geneity of S′ S Spec L Spec L is a preimage of p. (1) (×6): The sequence→ ⇐⇒ 0 IS /S OR ON 0 → ′ → → S′/S → may be written as i 0 A[x ] A[x ] A[x ]/i 0 → I → I → I → where i is a local index form for S′/S. Recall that an A[xI ]-module M is flat if and only if for each prime p of A and ideal q of A[xI ] lying over p, Mq is flat over Ap. Therefore, by the local criterion N for flatness[Stacks, 00MK], S′/S is flat over S if and only if Ap i Tor1 (Ap/pAp, (A[xI ]/ )q)=0 26 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

N for all such ideals p and q. Therefore, S′/S is flat if and only if i(mod p) A[x ]q/pA[x ]q A[x ]q/pA[x ]q I I −→ I I is injective for all p and q as above. All of these maps are injective if and only if the maps of A[xI ]-modules i(mod p) Ap[x ]/pAp[x ] Ap[x ]/pAp[x ] I I −→ I I are all injective as p varies over the prime ideals of A. Since Ap[x ]/pAp[x ] I I ≃ (Ap/pAp)[xI ] is an integral domain for each p, injectivity fails if and only if i reduces to 0 in the fiber over some p. We conclude that (1) holds if and only if (6) holds. Corollary 4.11. If all of the points of S have infinite residue fields, then the following are equivalent:

(1) S′ S is monogenic over geometric points; → (2) S′/S is Zariski-locally monogenic. Remark 4.12. This conclusion fails dramatically if S has finite residue fields. For S′ S coming from an extension of number rings condition (1) always holds (see Corollary→ 4.18 below), yet there are extensions that are not locally monogenic. In this sense, monogeneity is more subtle in the arithmetic context than the geometric one. For an example of an extension that is monogenic over geometric points but is not monogenic over points see Example 7.6. 4.3. Monogeneity Over Points. In light of Theorem 4.1, it is particularly interesting to work out the problem of monogeneity in the case that S is the spectrum of a field k. In this case S′ is the spectrum of an n-dimensional k-algebra B. Such algebras are Artinian rings, and a well-known structure theorem implies that B is a direct product of local Artinian rings Bi. We will exploit this to give a complete characterization of Zariski-local monogeneity. The result in the case that both S′ and S are spectra of fields is well-known. Theorem 4.13 (Theorem of the primitive element). Let ℓ/k be a finite field extension. Then Spec ℓ Spec k is monogenic if and only if there are finitely many → intermediate subfields ℓ/ℓ′/k. In particular, a finite separable extension of fields is monogenic.

We next consider the monogeneity of S′ S when S′ is a nilpotent thickening of Spec ℓ, leaving S = Spec k fixed. A key ingredient→ is a study of the square zero extensions of Spec ℓ. We remark that the proof below does not consider whether monogenerators extend to a deformation of S′ S along a nilpotent thickening of the base S → → S. In fact, if S S is a nilpotent closed immersion with S′ = S S′, any → ×S monogenerator θ : S M extends to S locally in the étale topology.e This → S′/S resultse from the smoothnesse of M S. e S′/S → e e e Theorem 4.14. Let S′ S be induced by k B where k is a field and B is a → e → local Artinian k-algebra with residue field ℓ and maximal ideal m. Then S′ S is monogenic if and only if → (1) Spec ℓ Spec k is monogenic → THESCHEMEOFMONOGENICGENERATORS 27

2 (2) dimℓ m/m 1. ≤2 (3) If dimℓ m/m = 1 and ℓ/k is inseparable, then

0 m/m2 B/m2 ℓ 0 → → → → is a non-split extension.

2 Proof. If the tangent space (m/m )∨ has dimension greater than 1, then no map 1 S′ A can be injective on tangent vectors as is required for a closed immersion. → S Conversely, if the tangent space of S′ has dimension 0, we have B = ℓ, and the result is true by hypothesis. 1 Now suppose the tangent space of S′ has dimension 1. A morphism S′ AS is a closed immersion if and only if it is universally closed, universally injectiv→e, and unramified[Stacks, Tag 04XV]. 1 Choose a closed immersion Spec ℓ Ak. Equivalently, write ℓ = k[t]/(f(t)) where f(t) is the monic minimal polynomial→ of some element θ ℓ. Since Spec ℓ ∈ →1 S′ is a universal homeomorphism[Stacks, Tag 054M], any extension of Spec ℓ Ak 1 → to S′ Ak inherits the properties of being universally injective and universally → 1 closed from Spec ℓ Ak. → 1 Whether such an extension S′ Ak is ramified can be checked on the level → 1 of tangent vectors. It follows that a morphism S′ Ak is a closed immersion if and only if its restriction to the vanishing of m2 →is. On the other hand, any 2 1 1 map V (m ) A extends to S′ A (choose a lift of the image of t arbitrarily). → k → k Therefore, it suffices to consider the case that S′ is a square zero extension of Spec ℓ. By hypothesis, we have a presentation of ℓ as k[t]/(f(t)). We conclude with some elementary deformation theory, see for example [Ser06, §1.1]. We have a square zero extension of ℓ 0 (f(t))/(f(t)2) k[t]/(f(t))2 ℓ 0. → → → → By assumption, B is also a square zero extension of ℓ:

0 m B ℓ 0. → → → → By [Ser06, Proposition 1.1.7], there is a morphism of k-algebras φ : k[t]/(f(t))2 B inducing the identity on ℓ. Since (f(t))/(f(t)2) ℓ asa k[t]/(f(t)2) module, φ either→ restricts to an isomorphism f(t)/(f(t))2 m ≃or else the zero map. In the former case, the composite k[t] k[t]/(f(t))2 →B is a surjection, and we are done. In the → → latter case, B is the pushout of the extension k[t]/(f(t))2 along (f(t))/(f(t)2) 0 m, so B is the split extension ℓ[ε]/ε2. → If ℓ/k is separable, then the extension k[t]/(f(t))2 is itself split [Ser06, Proposi- tion B.1, Theorem 1.1.10], i.e. there an isomorphism k[t]/(f(t))2 ℓ[ε]/ε2. Com- posing with k[t] k[t]/(f(t)2) gives the required monogenerator. ≃ → 2 If ℓ/k is inseparable and B ℓ[ε]/ε , we will show that S′ S is not monogenic. Any generator for ℓ[ε]/ε2 over≃ k must also be a generator→ for ℓ[ε]/ε2 over the maximal separable subextension k′ of ℓ/k, so we may assume that ℓ/k is purely inseparable. Moreover, any generator θ for ℓ[ε]/ε2 over k must reduce modulo ε to a generator θ of ℓ/k. Since ℓ/k is purely inseparable, the minimal polynomial f(t) of θ satisﬁes f ′(t) = 0. Note θ = θ + cε for some c ℓ. Since θ is assumed to be a monogenerator, there is a polynomial g(t) k[t] such∈ that ε = g(θ). Reducing, ∈ 28 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH g(θ)= g(θ)=0, so g(t)= q(t)f(t) for some q(t) k[t]. Then ∈ g(θ)= g(θ)+ g′(θ)cε

=0+ q′(θ)f(θ)cε + q(θ)f ′(θ)cε =0, a contradiction. We conclude that in this case S′ S is not monogenic. → Remark 4.15. In the case that k is perfect, the ﬁrst and third conditions hold automatically. If S′ is regular of dimension 1 the second condition is trivial.

Theorem 4.16. Suppose S′ S is induced by k A where k is a field and B is → → an Artinian k-algebra. Write B = i Bi where the Bi are local artinian k-algebras with respective residue fields ki. Then S′ S is monogenic if and only if Q → (1) Spec B S is monogenic for each i; i → (2) for each finite extension k′ of k, S′ has fewer points with residue field isomorphic to k′ than k has elements that generate k′ over k. Proof. Remark 2.18 tells us that a map from a disjoint union is a closed immersion when it restricts to a closed immersion on each connected component and the components have non-overlapping images. The proof of Theorem 4.14 shows that 1 a closed immersion Spec Bi A can be chosen with image any of the points of 1 → AS with residue field ki. The condition on numbers of points is exactly what we need for the images of the Spec Bis not to overlap without running out of points. (Since topologically, the components are single points.) Remark 4.17. Condition (2) is trivial in the case that the residue fields of S are infinite, highlighting the relative simplicity of monogeneity in the geometric context. If S′ S is instead induced by an extension of number rings, then Remark 4.15 implies condition→ (1) is trivial. In particular, an extension of Z has common index divisors if and only if there is “too much prime splitting” in the sense of condition (2). This recovers the theorem of [Hen94] (also see [Ple74, Cor. to Thm. 3]) that p is a common index divisor if and only if there are more primes in ZL above p of residue class degree f than there are irreducible polynomials of degree f in k(p)[x] for some positive integer f.

Corollary 4.18. If S′ S is induced by an extension of number rings Z /Z , → L K then S′ S is monogenic over geometric points. → Proof. Let p be a point of S. Let A = Z Z k(p) be the ring for the fiber of L ⊗ K S′ over Spec k(p). Note that k(p) is either of characteristic 0 or finite, so k(p) is perfect. Decompose A into a direct product of local Artinian k(p) algebras Ai. Since k(p) is perfect, conditions (1) and (3) of Theorem 4.14 hold for Spec A i → Spec k(p). Condition (2) holds as well since S′ is regular of dimension 1. Therefore Spec A Spec k(p) is monogenic for each i. i → Now consider the base change of S′ to the algebraic closure k(p) of k(p). Write B for the ring of functions Z Z k(p) of this base change, and write B as a product of L ⊗ K local Artinian algebras Bj . For each i we have that Spec Ai k(p) k(p) Spec k(p) is monogenic. Each Spec B is a closed subscheme of exactly⊗ one of the→ Spec A j i ⊗k(p) k(p)s, so by composition, Spec B Spec k(p) is monogenic for each j. This gives j → THESCHEMEOFMONOGENICGENERATORS 29 us condition (1) of Theorem 4.16 for S′ Spec k(p) Spec k(p). Since k(p) is ×S → infinite, condition (2) holds triviallly. We conclude that S′ S Spec k(p) Spec k(p) is monogenic, as required. × →

5. Etale´ Maps, Configuration Spaces, and Monogeneity

This section concerns maps π : S′ S that are étale, or unramified. The monogeneity space becomes a “configuration→ space” classifying arrangements of n points on a given topological space. Example 1.5 showed monogeneity is a configuration space for the trivial cover of C: n M Cn C(C) = Conf (C)= (x ,...,x ) C x = x for i = j . 1, / n { 1 n ∈ | i 6 j 6 } Configuration spaces arise for finite étale covers S′ S more generally. There is a universal finite étale cover whose space of monogenerators→ is a configuration space. M Philosophically, S′/S regards S′ S as a twisted family of points to be con- 1 → M figured in A or any other target. In other words, we may interpret S′/S as an arithmetic refinement of the configuration space. Example 5.3 shows the corresponding action of the absolute Galois group Gal(Q/Q) is known in anabelian geometry. This indicates the power of the topological invariants of M. We end Subsection 5.1 with a handful of exotic applications in other areas. All extensions S′ S sit somewhere between the étale case and the jet spaces of Example 3.13, between→ being totally unramified and totally ramified. The trace pairing gives equations which cut out the locus of ramification, as described in Section 5.2. Specifically, [Poo06, §6] says that our description in the étale case holds precisely away from the vanishing of the discriminant. The discriminant plays a similar role in the classical case when investigating the monogeneity of a polynomial. We end with some remarks on using stacks to promote a ramified cover of curves to an étale cover of stacky curves as in [Cos06].

5.1. The case of étale S′ S. Write n = 1, 2,...,n and n = S → h i { } h iS 1 i n → S for the trivial degree n finite étale cover. Consider the category (Sch/≤ ≤ ) of F schemes over a final scheme equipped with the étale topology. For example,∗ take ∗ = Spec Z or Spec C. Write Σn for the symmetric group on n letters and BΣn for ∗the stack on (Sch/ ) of étale Σ -torsors. ∗ n Etale-locally´ in S, all finite étale morphisms S′ S are isomorphic to n S [Stacks, 04HN]. The sheaf of isomorphisms → h i

P = Isom (S′, n ) S h iS is a left Σn-torsor, classified by some map S BΣd. → Conversely, any such torsor Σn P begets a finite étale n-sheeted cover via

Σn n P := ( n P )/(Σn, ∆) S BΣ BΣn 1, h i ∧ h i× ≃ × n − where we are quotienting by the diagonal action Σn n P . The quotient is a “contracted product” discussed in Appendix A. h i× We therefore have an equivalence between ﬁnite ´etale n-sheeted covers and Σn- torsors [Cos06, Lemma 2.2.1], [HW21, Lemma 3.2]. I.e., the map BΣn 1 BΣn is − → the universal n-sheeted cover. The map BΣn 1 BΣn comes from the inclusion − → Σn 1 Σn obtained by omitting any one of the n letters and does not depend on which− ⊆ letter is omitted up to isomorphism as in Remark A.10. 30 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

Pullback squares

T ′ S′ p

T S induce identifications R R T and M M T. T ′/T ≃ S′/S ×S 1,T ′/T ≃ 1,S′/S ×S Reduce thereby to the universal n-sheeted finite étale cover S′ = BΣn 1, S = BΣn. 1 1 − Each has an affine line ABΣn = [A /Σn] obtained via quotienting by the trivial action. We claim the Weil Restriction is the vector bundle over BΣn coming from the n usual permutation representation Σn A : R n BΣn 1/BΣn = [A /Σn]. − Diagram (2) becomes:

n 1 [A /Σn] S′ AS   =   .          S′    S BΣn S     If we pull back the diagram along BΣ , we get    ∗ → n  n A1 h i   .     ∗   Because n is the trivial cover, we have n sections of A1 over the base , representedh i→∗ by An. An equivalent description is as n ordered points in A1. ∗ The group Σn acts on n-tuples of sections by acting on the source: Σ Hom( n , A1) An. n h i ≃ This translates to the usual action on the coordinates of An. n Identifying the Weil Restriction with [A /Σn] in general requires working over an arbitrary base, arguing on the total space of the torsor by pulling back along BΣn, and checking things are equivariant. ∗ →The case of a point S = Spec K is more classical: Example 5.1 (S = Spec K). Let L/K be a finite separable extension, i.e., an n- sheeted finite étale cover Spec L Spec K which is classified by a map Spec K → → BΣn. The corresponding torsor is the sheaf of isomorphisms Isom (L,Kn) = Isom ( n , Spec L) K Spec K h iK of K-algebras. After base changing to a separable closure Ksep of K (or L if sep sep n sep one wishes), we get L K K = (K ) as algebras and Spec L K K = sep ⊗ ⊗ n sep Spec K as schemes. Then L/K is étale locally an n-sheeted cover, so h iK → we see the Weil Restriction RL/K is the pullback of the universal Weil Restriction R n n BΣn 1/BΣn = [A /Σn]. Such a pullback is simply AK, just like Example 2.3. − THESCHEMEOFMONOGENICGENERATORS 31

Returning to the general discussion, the closed substack of non-monogenerators N R n 1,BΣn 1/BΣn BΣn 1/BΣn = [A /Σn] − ⊆ − n is the quotient by Σn of the “fat diagonal” ∆ A given by the union of all loci V (x x ) An where two coordinates are equal.⊆ This is checked after pulling i − j ⊆ back along BΣ , where the universal diagramb is ∗ → n n s A1. h i

∗ Identifying the diagram with n-tuples of sections of A1 over the base, some of the n sections coincide if and only if some coordinates are equal xi = xj . Remark 2.18 affirms our intuition that such a map s is a closed immersion if and only if the images of two distinct points in n are not the same point. In other words, M h i BΣn 1/BΣn is the space of ordered configurations of n points − |∗ Conf A1 = (x ,...,x ) x = x for i = j . n { 1 n | i 6 j 6 } The stack quotient is likewise the nth unordered configuration space of points in A1: UConf A1 = (x ,...,x ) x = x for i = j /Σ . n { 1 n | i 6 j 6 } n 1 The quotient by Σn can be seen as forgetting the order on n points in A . Moreover, we have a pullback square

1 1 Confn(A ) UConfn(A ) p

BΣ . ∗ n

n The action of Σn A restricts to an action on ∆, and

N n 1,BΣn 1/BΣn = ∆/Σn [bA /Σn]. − ⊆ h i n The fat diagonal ∆ is exactly the locus ofbA where Σn has stabilizers. The coarse n n moduli space of [A /Σn] is precisely A by the fundamental theorem of symmetric functions, with theb composite An [An/Σ ] An; x = (x ,...,x ) (s (x),s (x),...,s (x)) → n → 1 n 7→ 1 2 n given by the elementary symmetric polynomials si(x1,...,xn)[Art11, §16.1-2]. The composite sends a list of n roots to the coeﬃcients of the monic polynomial of degree n vanishing at those roots, up to sign:

n n 1 n 2 (t x ) (t x )= t s (x)t − + s (x)t − s (x). − 1 ··· − n − 1 2 −···± n The assignment is plainly equivariant in Σn relabeling the xi. n n The map to the coarse moduli space [A /Σn] A is an isomorphism precisely M N → n over 1,BΣn 1/BΣn . The image of 1,BΣn 1/BΣn in A is the closed subscheme cut − − 32 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH out by the discriminant of the above polynomial n Disc (t x ) = D(s (x),s (x),...,s (x)) − i 1 2 n i=1 ! Y = (x x )2, i − j i

,Extending along an embedding Q ֒ C does not affect the fundamental group and the étale fundamental group over→C is the profinite completion of the ordinary topological fundamental group: M M top M an π1( XQ )= π1( XC )= π1 ( X (C) )∧. 1 The topological fundamental group of MX (C) = Confn(A ) is famously [Knu18, §1.2] the braid group Bn on n strings generated by σ1,...,σn with relations

σiσi+1σi = σi+1σiσi+1, σ σ = σ σ , if i j > 1. i j j i | − | Write Bn for its proﬁnite completion. The fundamental group π1(Spec Q)= GQ is the absolute Galois group of Q. Chooseb a section σ of XQ Spec Q. The section induces a splitting of the sequence (6) and a continuous→ conjugation action

1 GQ Bn; g.b := σ(g)bσ(g)− .

As discussed in [Fur12], the action of GQ extends to an action of the Grothendieck- b Teichmüller group GT . Conjecturally, GQ = GT . Question 5.4. Fulton and MacPherson defined a natural compactification of con- d d figuration spaces by blowing up the locus in An where the points come together in a specific way [FM94]. Their compactification has a natural Σn action and descends R to a blowup of BΣn 1/BΣn . One could make the same definitions and apply their computations of rational− cohomology to monogeneity. Can their computations tell us whether specific extensions are monogenic? Is there a version when S′ S is not étale? Can it capture the operadic nature of the usual configuration→ spaces over R in an algebraic setting? R The stacks 1,BΣn 1/BΣn we study arise naturally in log geometry as “Artin fans” [Abr+14]. − Example 5.5 (Thanks to Sam Molcho). Consider the stack quotient n n [A /Gm]/Σn of R by the image of the dense torus, an Artin fan [Abr+14]. The quotient is the 3 Artin fan of the stack A /Σ3 , but we can describe a log scheme Y for n = 3 with the same Artin fan. 3 3 1 1 1 Let Z = G A = Spec k[t± ,u± , v± ,x,y,z] over a field k not of characteristic m× 2 and containing a primitive 3rd root of unity ω k. Write G =Σ3 and define an action G Z by ∈ (12).(t,u,v,x,y,z) := (u, t, v,y,x,z) − (123).(t,u,v,x,y,z) := (ω t,ω2 u,v,z,x,y). · · Check the action of (12) has order 2, (123) has order 3, and they satisfy the relation (12).(123).(12).(123).(t,u,v,x,y,z) = (t,u,v,x,y,z) 3 to induce a free action of Σ3. One can replace Gm with any free action of Σ3 instead. 3 3 Endow Z with the natural log structure on A and none on Gm, i.e., the divisorial log structure from x,y,z. Define Y := Z/G, with descended log structure. To 34 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

3 compute its Artin fan, endow A with the natural Σ3 action and note the projection Z A3 is G-equivariant. The Artin fan of Y is that of A3/G , our R above. → n For larger n, use a similar presentation of Σn and the natural action on A , k entwined with a free action on some Gm. The Artin cones for n = 3 show up naturally as the top-dimensional cone of the moduli space of tropical genus-one trop 3-pointed curves M 1,3 corresponding to the triangle with marked points on each vertex.

Example 5.6 (Moduli spaces of curves in genus 0). Let M0,n be the moduli stack of smooth curves of genus 0 (i.e. P1) with n marked points. The evident [Wes11] isomorphism with a quotient of conﬁguration space gives:

1 MP1 Cn C/PGL Conf (P )/PGL M . , / 2 ≃ n 2 ≃ 0,n One can always put the first point at and get equivalent descriptions: ∞ 1 1 M MA1 Cn C/Aff Conf (C)/Aff , 0,n ≃ , / ≃ n 1 1 where Aff is the group of affine transformations Gm ⋉A . The stack quotient classifies local affine equivalence classes of monogenerators, as detailed in Section 6. One can likewise obtain the other moduli spaces of curves by an ad hoc construction. Consider U M the universal connected, proper, genus-g nodal curve, its relative smooth locus→ Usm U, and the monogeneity space ⊆

M sm U , n M/M h i of the trivial cover n over the moduli space M. The monogeneity stack is naturally h i isomorphic to the space of nodal, n-marked curves Mg,n. One can also obtain the open substack of stable curves as the universal Deligne-Mumford locus M g,n ⊆ Mg,n. Example 5.7 (Torsors for finite groups). Let k be a separably closed field and G a finite group, yielding a group scheme Spec k over k. A G-torsor S′ S is, g G → in particular, a finite étale map of degree n∈= #G admitting the above description. F M Notice that the action of G on S′ induces an action of G on S′/S. The map S′ S is classified by a map S BG, the stack of G-torsors A.4. The left → → regular representation G G gives an inclusion G Σn upon ordering the set G. The induced representable map BG BΣ is essentially⊆ independent of the → n ordering as in Proposition A.9. The classifying map S BΣn is the composite S BG BΣ with the left regular representation.→ The universal family is → → n BG, pulled back from BΣn 1 BΣn. Using BG instead of BΣn refines ∗the → transition functions and is closer− → to the original approach of E. Galois, since “groups” were then only defined as subsets of Σn. A similar description locally holds for other finite étale group schemes. For merely finite flat group schemes G such as αp,µp in characteristic p, the group action on the monogeneity space of G-torsors S′ S still holds but the local → decomposition S′ = n S and Σn action do not. Example 5.8 (ExplicitF Elliptic Curve Monogeneity). Example 5.7 applies to iso- genies of elliptic curves over a field K. Let ϕ : E E be a separable isogeny 1 → 2 of degree N, with S′ = E1, S = E2. The map ϕ is a torsor for ker(ϕ). We want THESCHEMEOFMONOGENICGENERATORS 35 monogenerators θ 1 E1 AE2 ,

E2 1 1 but all global maps E1 AE2 are constant on the fibers of AE2 E2. We will have to look for local monogenerators→ instead. → A good source of such monogenerators are the rational functions on E1. Recall that ϕ induces an Galois extension K(E ) K(E ) with Galois group ker ϕ. An 2 ⊆ 1 element s K(E1) = R(Spec K(E2)) is a monogenerator over K(E2) if and only if it lies in∈ no proper subextension, i.e., if it is not invariant under any non-trivial O subgroup or element of ker ϕ. Such an s extends to a regular function s E1 (U) 1 ∈ on some open subset U of E1. We can assume U = ϕ− V is pulled back from some V E . Since M is open in R and s M , we may shrink V to ⊆ 2 1,E1/E2 |Spec K(E2) ∈ 1 ensure s M1(V ). Apply∈ our results to a concrete situation. Consider the elliptic curves over Q defined by the affine equations E : y2 = x3 9x +9 1 − E : y2 = x3 189x 999. 2 − − The rational map x3 6x2 + 45x 72 x3y 9x2y 9xy +9y ϕ(x, y)= − − , − − x2 6x +9 x3 9x2 + 27x 27 − − − gives an isogeny ϕ : E1 E2 with deg(ϕ) = 3 and ker(ϕ)= E1 , (3, 3), (3, 3) . The isogeny gives a torsor→ in the sense of Example 5.7 via the{∞ group action− given} by adding points of the kernel. Writing x3y y3 27x2 27xy +9y2 + 162x + 27y 216 A(x, y) := − − − − , (x 3)3 − the kernel acts via (y 3)2 (x + 3)(x 3)2 (3, 3)+ (x, y)= − − − , A(x, y) E1 (x 3)2 − (y 3)2 (x + 3)(x 3)2 (3, 3)+ (x, y)= − − − , A(x, y) . − E1 (x 3)2 − − We claim that the rational function y on E defines a monogenerator for E E 1 1 → 2 over an open subset V E2. To see that y works on some open subset, all we have to note is that y is not⊆ fixed by the action of ker ϕ on K(E ), since y = A(x, y). 1 6 Better, we can compute V explicitly. Which points of E2 need to be thrown out? The function y fails to define a closed immersion where (1) y has a pole, (2) there is a pair of points (x ,y ), (x ,y ) E with the same y-value y = y 0 0 1 1 ∈ 1 0 1 and the same ϕ-value ϕ(x0,y0)= ϕ(x1,y1).

The points of E2 which it is necessary to remove are therefore

(1) the point at inﬁnity of E2 and 36 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

(2) the images of the points (x, y) E1 satisfying one of the following pairs of equations: ∈ y = A(x, y), y2 = x3 9x +9, − or y = A(x, y), y2 = x3 9x +9. − − Using a computer algebra system, we solve the first pair of equations: the solutions to y = A(x, y) are y = 3, y = 12 3x, y =3x 6, with x = 3, and plugging these possibilities into the equation for−E : y2 = x3 − 9x + 9, we6 obtain the points: 1 − (0, 3), ( 3, 3), (3 6i, 3+18i), (3+6i, 3 18i). − − − The solutions to the second pair of equations are less tidy; they are approximately (2.43, 1.22), (10.48, 32.66), ( 2.36 1.41i, 5.62+0.72i), − − − − ( 2.36+1.41i, 5.62 0.72i), (0.40 1.78i, 3.35+3.11i), (0.40+1.78i, 3.35 3.11i). − − − − − We have a complete finite set of points (defined over a finite extension of Q) to remove from E1 in order to obtain a closed immersion. To obtain V , we take the complement of the images of these points in E2.

5.2. When is a map étale? We recall from [Poo06, §6] that a map S′ S is étale precisely when the discriminant of the algebra does not vanish. Below→we employ the algebraic moduli stack An of finite locally free algebras and the affine scheme of finite type Bn parametrizing such algebras together with a choice of global basis Q O e , both defined in § 2.2 . ≃ S · i Suppose π : S′ S comes from a finite flat algebra Q with a global basis ϕ : QL n O e→, corresponding to a map S B . There is a trace pairing ≃ i=1 S · i → n Tr : Q OS which we can use to define the discriminant: →L Disc(Q, ϕ) := det [Tr(e e )] Γ(O ) i j ∈ S Changing ϕ changes the function Disc by a unit. The function Disc does not descend to A , but the vanishing locus V (Disc) B does. Writing Bet, Aet n ⊆ n n n for the open complements of the vanishing locus V (Disc), a map π : S′ S is et → étale if and only if S An factors through the open substack An An [Poo06, Proposition 6.1]. → ⊆ Remark 5.9. Most finite flat algebras are not étale, nor are they degenerations of étale algebras. B. Poonen shows the moduli of étale algebras inside of all finite et flat algebras An An cannot be dense by computing dimensions [Poo06, Remark ⊆ et 6.11]. The closure An is nevertheless an irreducible component.

What if S′ S is not étale? Readers familiar with [Cos06] know one can → sometimes endow S and S′ with stack structure S and S′ at the ramification to make S′ S étale in the ramified case. Then all S′ S are Σ -torsors, and not → → n just unramified covers S′ S. We sketch these idease overeC. The ideas in Section → 5.1 applye ine this level of generality. e e Consider (7) y2 = x(x 1)(x λ), − − 1 for some λ C. If C := P and C′ is the projective closure of the above affine ∈ equation, the projection (x, y) x extends to a finite locally free map π : C′ C. This is in Situation 2.1 so our7→ definitions make sense for it. However π is ramified→ THESCHEMEOFMONOGENICGENERATORS 37 at four points, preventing us from interpreting its monogeneity space using the perspective of this section. Nevertheless, we may observe that the function y gives a section of M over C . The section naturally extends to a section of 1,C′/C \ ∞ R 1 P ,C′/C over all of C. 1 Let X := P . If we work over C and endow C′ and C with stack structure to obtain a finite étale cover of stacky curves C′ C as in [Cos06], the stacky finite → étale cover together with the map C′ X is parameterized by a representable map → C [SymnX] to the stack quotient e e → en n e [Sym X] := [X /Σn].

We can similarly allow C′ and C to be nodal families of curves over some base S. Maps from nodal curves C over S entail an S-point of the moduli stack M([SymnX]) of prestable maps to the symmetric product. As in Proposition 2.4, there is an open substack for which the mape from the coarse space C′ to X is a closed immersion. The stack M([SymnX]) splits into components indexed by the ramification profiles of the cover of coarse spaces C′ C. There are some subtleties in characteristic→ p – one cannot treat all ramification as a µn torsor because some ramification is a Z/pZ-torsor in characteristic p. The formalism of tuning stacks [ESZ21] is a substitute in arbitrary characteristic.

6. Twisted Monogeneity The Hasse local-to-global principle is the idea that “local” solutions to a polynomial equation over all the p-adic fields Qp and the real field R can piece together to a single “global” solution over Q. We ask the same for monogeneity: given local monogenerators, say over completions or local in the Zariski or étale topologies, do they piece together to a single global monogenerator? The Hasse principle fails for elliptic curves. Let E be an elliptic curve over a number field K and consider all its places ν. The Shafarevich-Tate group X(E/K) of an elliptic curve sits in an exact sequence

0 X(E/K) H1 (K, E) H1 (K , E). → → et → et ν ν Y Elements of X are genus-one curves with rational points over each completion Kν that do not have a point over K. Similarly, we want sequences of cohomology groups to control when local monogenerators do or do not come from a global monogenerator. For such a sequence, one needs to know how a pair of local monogenerators can diﬀer. One would like a group G or sheaf of groups transitively acting on the set of local monogenerators so that cohomology groups can record the struggle to patch local monogenerators together into a global monogenerator. Suppose B/A is an algebra extension inducing S′ S and θ1,θ2 B are both monogenerators. Then → ∈ θ B = A[θ ], θ B = A[θ ], 1 ∈ 2 2 ∈ 1 so each monogenerator is a polynomial in the other: θ = p (θ ) and θ = p (θ ), with p (x),p (x) A[x]. 1 1 2 2 2 1 1 2 ∈ 38 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

We can think of the pi(x) as transition functions or endomorphisms of the aﬃne 1 line A . Even though p1(p2(θ1)) = θ1, it is doubtful that p1 p2 = idA1 or even 1 ◦ that pi(x) are automorphisms of A . One remedy would be to take G to be the group completion of the noncommuta- =0 1 1 tive monoid End6 (A ) of nonzero endomorphisms of A under composition. Each polynomial pi(x) has image in the group completion. Unfortunately, this group is terribly complicated. It is generally undecidable whether the group completion of a noncommutative monoid is nonzero. The degree homomorphism shows G surjects onto Z, but we are not able to say anything meaningful about the kernel. Moreover, =0 1 the map from End6 (A ) to the group completion is non-injective: for example, the images of x and x must coincide, since the composite of x2 with both polynomials is x2. − Instead of working with the most general group as above, we require our tran- 1 1 sition functions pi(x) to lie in a group G A acting on A . We focus on the two speciﬁc group sheaves with group laws

G (A)= A∗, u u′ := uu′ m · 1 Aff (A)= A∗ ⋉ A, (u, v) (u′, v′) := (uu′, uv′ + v). · Affine transformations Aff1 are essentially polynomials ux + v of degree one under composition. These act on monogenerators:

1 G A : a A∗,θ M(A), a.θ := a θ, m ∈ ∈ · 1 1 Aﬀ A : a A∗,b A, θ M(A), (a,b).θ := aθ + b. ∈ ∈ ∈ Deﬁnition 6.1 (Twisted Monogenerators). A (Gm)-twisted monogenerator for B/A is:

(1) a Zariski open cover Spec A = i D(fi) for elements fi A, 1 ∈ 1 1 (2) a system of “local” monogenerators θ B f − for B[f − ] over A[f − ], S i i i i and ∈ 1 1 (3) units a A f − ,f − ∗ ij ∈ i j such that for all i, j, we have a .θ = θ , • ij j i for all i, j, k, the “cocycle condition” holds: • aij .ajk = aik.

Two such systems (a ), (θ ) , (a′ ), (θ′) are equivalent if they differ by further { ij i } { ij i } refining the cover Spec A = D(f ) or global units u A∗: u a = a′ , u θ = θ′. i ∈ · ij ij · i i Likewise B/A is Aff1-twisted monogenic if there is a cover with θ ’s as above, S i 1 1 but with units (3) replaced by pairs aij ,bij A , such that each aij is a unit ∈ fi fj and aij θj + bij = θi. h i The elements θ may or may not come from a single global monogenerator θ A. i ∈ Nevertheless, the transition functions (aij ) or (aij ,bij ) define an affine bundle L on Spec A with global section θ induced by the θi’s. We say S′/S is “twisted monogenic” to mean there exists a Gm-twisted monogenerator and similarly say “Aff1-twisted” monogenic. Both are clearly Zariski-locally monogenic. Compare with Cartier divisors: THESCHEMEOFMONOGENICGENERATORS 39

twisted monogenerator Cartier divisor global monogenerator rational function 1 Gm / Aff action differing by units. Recall the notion of “multiply monogenic orders” or “affine equivalence” in the literature. Two monogenerators θ ,θ Γ(Spec A, M ) are said to be “affine 1 2 ∈ B/A equivalent” if there are u A∗, v A such that uθ1 +v = θ2. In other words, affine ∈ ∈ 1 equivalence classes are elements of the quotient Γ(Spec A, MB/A)/Aff (A). Under certain hypotheses in Remark 6.13, Aff1-twisted monogeneity is parameterized by 1 the sheaf quotient MB/A/Aff . There is almost an “exact sequence” Aff1(A) Γ(M ) Γ(M /Aff1) H1(Aff1) → B/A → B/A → that dictates whether a twisted monogenerator comes from an affine equivalence class of global monogenerators. Under certain hypotheses, we show: 1 Proposition 6.15: B/A is Gm-twisted monogenic if and only if it is Aff -twisted monogenic, Theorem 6.20: The class number of a number ring ZK is one if and only if all twisted monogenic extensions of number rings ZL/ZK are in fact monogenic, Remark 6.13: There is a local-to-global sequence relating affine equivalence classes of monogenerators with global monogenerators as above, 1 Theorem 6.2: There are moduli spaces of Gm and Aff -twisted monogenerators, Theorem 6.17: There are finitely many twisted monogenerators up to equivalence.

We warm up with a classical approach to Gm-quotients, namely taking Proj . Then we study Aﬀ1-twisted monogenerators before ﬁnally introducing G-twisted monogenerators for arbitrary groups G. There is a moduli space for each notion of twisted monogeneity. We use these moduli spaces now and defer the proof until Theorem 6.27:

1 1 Theorem 6.2 (=Theorem 6.27). Let Gm, Aff act on A on the left in the natural 1 way, inducing a left action on M. The stack quotients [M/Gm] and M/Aff represent G - and Aff1-twisted monogenerators up to equivalence, respectively. m

6.1. Gm-Twisted Monogenerators and Proj of the Weil Restriction. Writ- ing S′ = Spec B and S = Spec A, a twisted monogenerator amounts to a Zariski 1 cover S = Spec A = Ui, a system of closed embeddings θi : S′ A over Ui, Ui ⊆ Ui and elements aij Gm(Ui) such that ∈ S 1 a .θ = θ : S′ A . ij j i Uij → Uij Equivalently, a twisted monogenerator is a line bundle L on S defined by the above cocycle aij and a global embedding θ : S′ L over S. Twisted monogenerators are identified if they differ by global units u ⊆ G (S) or refinements of the cover U , ∈ m i i.e., if the corresponding line bundles L,L′ are isomorphic in a way that identifies the closed embeddings θ,θ′. For number fields L/K with θ Z and a Z , one has aθ Z . If a ∈ L ∈ K ∈ L ∈ ZK∗ , then θ is a monogenerator if and only if aθ is. The multiplication action Gm(ZK ) R(ZK ) corresponds to the global Gm action on the vector bundle R over S. An action of Gm corresponds to a Z-grading on the sheaf of algebras [Stacks, O n O R n 0EKJ]. Locally in S, π S′ S ei and AS. The Gm action ∗ ≃ · ≃ L 40 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH is the diagonal action and corresponds to the total degree of polynomials in O n = O [x ,...,x ]. AS S 1 n The associated projective bundle to the vector bundle R is given by the relative Proj [Stacks, 01NS] PR := Proj OR , S I N with the total-degree grading. The ideal S′/S cutting out the complement 1,S′/S of M R is graded by Remark 3.11, defining a closed subscheme PN PR. 1,S′/S ⊆ ⊆ Definition 6.3. Define the scheme of projective monogenerators

PMS /S PRS /S := Proj OR ′ ⊆ ′ S S′/S N to be the open complement of the closed subscheme P S′/S cut out by the graded I homogeneous ideal S′/S . The reader may deﬁne projective polygenerators in the same fashion.

Lemma 6.4. OR R The vanishing of the irrelevant ideal V ( ,+) of S′/S is contained inside of the non-monogeneity locus N for S′ = S. S′/S 6 Proof. Locally, the lemma states that θ = 0 is not a monogenerator.

Remark 6.5. We relate the Proj construction to stack quotients by Gm according to [Ols16, Example 10.2.8]. The ring OR is generated by elements of degree one. S′/S R n O O Locally, S /S AS and R S[x1,...,xn] is generated by the degree one ′ ≃ S′/S ≃ elements x . Write Spec OR for the relative spectrum [Stacks, 01LQ]. The map i S

/Gm Spec OR V (OR )− Proj OR S \ ,+ S is therefore a stack quotient or Gm-torsor. We have a pullback square

MS /S Spec OR V (OR,+) ′ p S \

PM Proj OR S′/S S M M of Gm-torsors and a stack quotient P S′/S = S′/S /Gm .

Theorem 6.2 states that [M/Gm] represents twisted monogenerators, and now we know the quotient stack is wondrously a scheme: M M Corollary 6.6. The scheme P S′/S = [ /Gm] represents the Gm-twisted mono- M generators of Deﬁnition 6.1. That is, P S′/S is a moduli space for twisted monogenerators. The action Gm M is free.

Warning 6.7. Given a monogenerator θ Z and a pair a Z∗ , β Z , write ∈ L ∈ K ∈ L n 1 β = b0 + b1θ + + bn 1θ − . ··· − One may try to deﬁne a second action

? 2 2 n 1 n 1 a.β := b0 + b1aθ + b2a θ + + bn 1a − θ − ··· − THESCHEMEOFMONOGENICGENERATORS 41 encoding the degree with respect to θ, but this action does not define a grading as it is almost never multiplicative. For example, take ZL = Z[√2] with monogenerator √2 over ZK = Z. Then a.2=2 = a.√2 a.√2. 6 · Jet spaces are the only example where a.x is multiplicative and induces a second action G R = JA1 . The two actions of λ G on a jet m S′/S ,m ∈ m f(ε)= a + a ε + + a εm 0 1 ··· m on A1 are λ.f(ε)= λ f(ε) and λ.f(ε)= f(λε). The Proj of J with respect to · X,m this second Gm action is known as a “Demailly-Semple jet” or a “Green-Griffiths jet” in the literature [Voj04, Definition 6.1]. For certain S′ S, there may be → a distinguished one-parameter subgroup, i.e., the image of Gm AutS(S′), that R → results in a second action Gm S′/S and allows an analogous construction. 6.2. Aff1-Twisted Monogenerators and Affine Equivalence. We enlarge our study to Aff1-twisted monogenerators and the related study of affine equivalence classes of ordinary monogenerators. We delay twisting by general sheaves of groups 1 other than Gm and Aff until the next section. For an S-scheme X, the automorphism sheaf Aut(X) is the subsheaf of automorphisms in HomS(X,X). 1 Remark 6.8. The automorphism sheaf AutS(A ) has a subgroup of affine trans- 1 1 formations Aff under composition. These are identified in turn with A ⋊Gm via (a,b) (x bx + a). 7→ 7→ k The automorphism sheaf can be much larger for other AS. For example, (x, y) (x + y3,y) 7→ is an automorphism of A2. The automorphism sheaf Aut(A1) is not the same as Aff1, though they have the same points over reduced rings. See [Dup13] for some discussion over nonreduced rings. Definition 6.9. The group sheaf Affk Aut(Ak) of affine transformations is the set of functions ⊆ ~x M~x +~b 7→ where M GL and ~b Ak, under composition. Note Affk Ak ⋊ GL . ∈ k ∈ ≃ k k k There is a sub-group scheme A ⋊Gm Aff cut out by the condition that M is a unit multiple of the identity matrix. The⊆ sub-group scheme sits in a short exact sequence 1 Ak ⋊G Affk PGL 1. → m → → k → Recall that two monogenerators θ1,θ2 of an A-algebra B are said to be equivalent if θ1 = uθ2 + v, where u A∗ and v A. Likewise, say that two embeddings θ1,θ2 : S′ L of ∈ 1 ∈ 1 → S′ into an Aff bundle L over S are equivalent if there is in f Aff (S) such that ∈ θ1 = f.θ2. The set of monogenerators up to equivalence is then Γ(S, M)/Γ(S, Aff1). 1 If S′ ∼ S is an isomorphism and n = 1, the action of Aff is trivial. Otherwise, the Aff1→-action is often free: 42 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH Lemma 6.10. The action Aut(S′) M has trivial stabilizers. If S is normal and 1 S′ S is not an isomorphism, the action Aff M has trivial stabilizers as well. → Proof. A stabilizer of the action AutS(S′) MX entails a diagram X X

S′ ∼ S′.

The fact that S′ X is a monomorphism forces S′ S′ to be the identity. Normality of S⊆ means S is a finite disjoint union≃ of integral schemes [Stacks, 033N]; we assume S is integral without loss of generality. 1 Computing stabilizers of Aff M is local, so we may assume S′ S is induced by a non-identity finite map A B of rings where A is an integrally→ closed domain with field of fractions K. A stabilizer→

a + bθ = θ; a A, b A∗ ∈ ∈ implies (1 b)θ = a. If b = 1, then a = 0 and the stabilizing aﬃne transformation − a is trivial. Otherwise, 1 b K∗ and θ = K. Elements θ B are all − ∈ 1 b ∈ ∈ integral over A. Since A is integrally closed,−θ A. Hence B = A[θ] = A, a contradiction. ∈

Remark 6.11. Suppose given transition functions (aij ,bij ) and local monogener- 1 ators (θi) as in an Aff -twisted monogenerator that may not satisfy the cocycle condition a priori. For normal S with n> 1 as in the lemma, the cocycle condition holds automatically, since Aff1 acts without stabilizers. Corollary 6.12. M 1 If S is normal, the stack quotient 1,S′/S/Aff is represented by the ordinary sheaf quotient M/Aff1. Proof. If G X is a free action, the stack quotient [X/G] coincides with the sheaf quotient X/G. Remark 6.13. If S is normal, Corollary 6.12 tells us that an Aff1-twisted monogenerator is the same as a global section Γ(S, M/Aff1). Equivalence classes of monogenerators are given by the presheaf quotient Γ(S, M)/Γ(S, Aff1). Affine equivalence classes of monogenerators thereby relate to twisted monogenerators in something like an exact sequence: Γ(S, M)/Γ(S, Aff1) Γ(S, M/Aff1) H1(S, Aff1). → → As in sheaf cohomology, the second map takes θ to its torsor of lifts δ(θ) in M: δ(θ)(U)= f M(U): f + Aff1(U)= θ . { ∈ |U } A section of the sheaf quotient θ Γ(S, M/Aff1) ∈ lifts to an affine equivalence class in the presheaf quotient θ Γ(S, M)/Γ(S, Aff1) ∈ if and only if the induced Aff1-torsor is trivial. The exact sequence is analogous to Cartier divisors. If X is an integral scheme with rational function field K(X), the long exact sequence associated to

1 O∗ K(X)∗ K(X)∗/O∗ 1 → X → → X → THESCHEMEOFMONOGENICGENERATORS 43 is analogous to the above.

Remark 6.14. One can do the same with Gm, or any other group. Compare twisted monogenerators PM = [M/Gm] with ordinary monogenerators M up to Gm-equivalence to obtain a sequence Γ(S, M)/Γ(S, G ) Γ(S, M/G ) H1(S, G ). m → m → m We preferred the case of affine equivalence already prevalent in the literature, and the reader can make the necessary modifications in general. Freeness of the action is necessary to identify the stack quotient with the ordinary sheaf quotient. 1 Sometimes, being Gm-twisted monogenic is the same as being Aff -twisted monogenic: Proposition 6.15. If S = Spec A is affine, all Aff1-torsors on S are induced by Gm-torsors: H1(Spec A, G ) H1(Spec A, Aff1). m ≃ The corresponding twisted forms of A1 are the same, so we can furthermore identify 1 Gm-twisted monogenerators with Aff -twisted monogenerators. Proof. The inclusion A1 Aff1; a (a, 1) → 7→ and the projection Aff1 G fit into a short exact sequence → m 0 A1 Aff1 G 1. → → → m → The sheaf Aff1 is not commutative. Cohomology sets Hi(S, Aff1) are nevertheless defined for i =0, 1, 2. Serre Vanishing [Stacks, 01XB] says Hi(Spec A, A1)=0 for 1 i = 0 and Γ(Aff ) Γ(Gm) is surjective, yielding an identification in all nonzero degrees:6 → i 1 i H (Spec A, Aff ) H (Spec A, Gm), i =1, 2.

≃ 1 1 The action Gm A is the restriction of that of Aff , factoring G Aff1 Aut(A1). m ⊆ → The corresponding twisted forms of A1 are the same. The literature abounds with finiteness results on equivalence classes of monogenerators, for example: Theorem 6.16 ([EG17, Theorem 5.4.4]). Let A be an integrally closed integral domain of characteristic zero and finitely generated over Z. Let K be the quotient field of A, Ω a finite étale K-algebra with Ω = K, and B the integral closure of A in Ω. Then there are finitely many equivalence6 classes of monogenic generators of B over A. Assume in addition that Pic(A) is finite, as when A is of the form ZL[x1,...,xn, 1/p]. We have an analogous finiteness result for equivalence classes of Aff1-twisted monogenerators:

Corollary 6.17. Let A, K, Ω, B be as in Theorem 6.16, with S′ S induced from A B. Assume Pic(A) is finite. Then there are finitely many→ equivalence → 1 classes of Aff -twisted monogenerators for S′ S. → 44 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

Proof. We essentially use the sequence Γ(S, M)/Γ(S, Aff1) Γ(S, M/Aff1) H1(S, Aff1) → → of Remark 6.13. If this were a short exact sequence of groups, the outer terms being finite would force the middle term to be; our proof is similar in spirit. Since S is quasicompact and there are finitely many elements of the Picard group 1 1 1 Pic(S) := H (S, Gm)= H (S, Aff ), we can find an affine open cover S = i Ui by finitely many open sets of S that simultaneously trivializes all line bundles on S. S The above sequence of presheaves restricts to the Ui’s in a commutative diagram

Γ(S, M)/Γ(S, Aff1) Γ(S, M/Aff1) H1(S, Aff1)

1 1 1 1 Γ(Ui, M)/Γ(Ui, Aff ) Γ(Ui, M/Aff ) H (Ui, Aff ).

The restrictionQ H1(S, Aff1) HQ1(U , Aff1) is zero byQ construction of the → i U ’s. The restriction ρ : Γ(S, M/Aff1) Γ(U , M/Aff1) is injective by the i Q → i sheaf condition. A diagram chase reveals that the restriction ρ(θ) of any section Q θ Γ(S, M/Aff1) is in the image of Γ(U , M)/Γ(U , Aff1). Theorem 6.16 asserts ∈ i i that each set Γ(U , M)/Γ(U , Aff1) is finite. i i Q We conclude with several consequences of twisted monogeneity and our Theorem 6.20 that shows twisted monogenerators detect class number-one number rings.

Theorem 6.18. Suppose S′ S is twisted monogenic, with an embedding into a line bundle E. Let E be the sheaf→ of sections of E. Then n(n 1) top − det(π OS ) := π OS E− 2 ∗ ′ ∧ ∗ ′ ≃ in Pic(S). In particular, if an extension of number rings ZL/ZK is twisted monogenic, then its Steinitz class is a triangular number power in the class group. d Proof. Write Sym∗E := Sym E for the symmetric algebra. Recall that E d ≃ V(E∨) := Spec Sym∗(E∨). We have a surjection of OS-modules S L Sym∗E∨ ։ π OS , ∗ ′ which we claim factors through the projection of OS-modules Sym∗E∨ n 1 iE → i=0− Sym ∨. Such a factorization is a local question and local factorizations automatically glue because there is at most one. Locally, we may assume S′ S is L → induced by a ring homorphism A B and E∨ is trivialized. We have a factorization of A-modules →

Sym∗E∨ A[t] B ≃

n 1 i n 1 i − Sym E∨ − A t i=0 ≃ i=0 · due to the existence of a monic polynomialL mθ(t) ofL degree n for the image θ of t in n 1 i O (Lemma 2.14). The A-modules − A t and B are abstractly isomorphic, S′ i=0 · L THESCHEMEOFMONOGENICGENERATORS 45 and any surjective endomorphism of a ﬁnitely generated module is an isomorphism [MRB89, Theorem 2.4]. We conclude that globally n 1 − i π OS Sym E∨. ∗ ′ ≃ i=0 M i i Since E is invertible, Sym E∨ = (E∨) . Taking the determinant, n 1 − n 1 n(n 1) i − i − det(π OS ) = det (E∨) = E− Pi=0 = E− 2 . ∗ ′ i=0 ! M

Lemma 6.19. Degree-two extensions are all Aff1-twisted monogenic. If S is affine, they are also Gm-twisted monogenic. O O O Proof. Localize and use Lemma 3.15 to write π S′ S Sθ for some θ O ∗ ≃ ⊕ ∈ Γ( S′ ). Given two such generators θ1,θ2, we may write θ = a + bθ , θ = c + dθ , a,b,c,d O . 1 2 2 1 ∈ S 1 1 Hence bd = 1 are units, and the transition functions come from Aff = A ⋊Gm. By choosing such generators on a cover of S, one obtains a twisted monogenerator. Proposition 6.15 further refines our affine bundle to a line bundle. 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.

Proof. If the class number of K is one, then all line bundles on Spec ZK are trivial and the equivalence is clear. Mann [Man58] has shown that K has quadratic extensions without an integral basis if and only if the class number of K is not one: adjoin the square root of α, where (α)= b2c with b non-principal and c square-free. By Lemma 6.19, such an extension is necessarily Gm-twisted monogenic. As the monogeneity of quadratic extensions is equivalent to the existence of an integral basis, the result follows. Remark 6.21. Theorem 6.20 implies that the ring of integers of a number field is twisted monogenic over Z if and only if it is monogenic over Z. Example 7.9 thus provides an example of a number field which is not twisted monogenic. Example 6.22 (Properly Twisted Monogenic, Not Quadratic). Let K = 3 Q(√5 23) and let p3, p5, and p23 be the unique primes of K above 3, 5, and 23, · 3 2 3 respectively. One can compute p3 = (ρ3 := 1970(√5 23) +9580(√5 23)+46587). 3 · · Consider ZL/ZK , where L = K(√23ρ3). On D(p23), the local index form with re- 3 3 2 3 3 spect to the local basis 1, √23ρ3, (√23ρ3) is b 23ρ3c . On D(p5), we have the 3 2{ 3 } − 3 2 3 2 2 local index form B 5 ρ3C with respect to the local basis 1, 5 ρ3, ( 5 ρ3) . 3 − { } √ 2 2 We transition via 23 5 /23, which is not a global unit, so thep extensionp ZL/ZK is twisted monogenic. · To see what is going on more explicitly, we investigate how the transitions affect the local index forms. We have 52 54 52 54 b3 23ρ c3 = B3 23ρ C3 = B3 ρ C3 = a unit in O . − 3 23 − 232 · 3 23 − 23 3 D(p23) 46 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

If B and C could be chosen to be ZK -integral so that local index form represented 3 2 a unit of ZK , then 5 ρ3 would be a global monogenerator. However, p5-adic 3 2 valuations tell us 5pρ3 is not a monogenerator. One can also apply Dedekind’s index criterion to x3 52ρ . Similarly, we have p − 3 23 232 23 232 B3 52ρ C3 = b3 52ρ c3 = b3 ρ c3 = a unit in O . − 3 52 − 54 · 3 52 − 52 3 D(p5)

If b and c could be chosen to be ZK -integral so that local index form represented a 3 unit of ZK , then √23ρ3 would be a global monogenerator. As above, the p23-adic valuations tell us this cannot be the case. Again, we could also use polynomial- speciﬁc methods. We have shown that ZL/ZK is twisted monogenic, but it remains to show that the 3 twisting is non-trivial. We need to show the ideal p5 = (5, √5 23) is not principal. 3 · On D(p23) it can be generated by √5 23 and on D(p5) it can be generated by · 3 5. We transition between these two generators via √52 232/23, exactly as above. Thus our twisted monogenerators correspond to a non-trivial· ideal class. A computer algebra system can compute a K-integral basis for ZL: 5 2 3 2 1, 2√3 23 5+ √3 23 5 3 23ρ + 3+ √3 23 5 3 23ρ , − · 25 · 3 − 23 · 3 5243 p 2 p 120589 + 5243√3 23 5+ √3 23 5 3 23ρ · 23 · 3 57125 p 2 2 + 22850 + √3 23 5 + 1828 √3 23 5 3 23ρ , 23 · · 3 p with index form:

3 2 3 3 2 2 IZ Z =13796817(√5 23) b 1367479703949(√5 23) b c L/ K · − · + 45179341009193328(√3 5 23)2bc2 + 67103709√3 5 23b3 · · 497537273719431009077(√3 5 23)2c3 6650125342740√3 5 23b2c − · − · + 219702478196413227√3 5 23bc2 2419492830176044167763√3 5 23c3 · − · + 326269891b3 32339923090800b2c + 1068411032584717260bc2 − 11765841517121285321908c3. − Because ZL/ZK is twisted monogenic, there are no common index divisors. Thus I we will always find solutions to ZL/ZK when we reduce modulo a prime of ZK . We do not expect ZL to be monogenic over ZK ; however, showing that there are no values of b,c Z such that IZ Z (b,c) Z∗ appears to be rather difficult. ∈ K L/ K ∈ K Remark 6.23. One can perform the same construction of Example 6.22 with radical cubic number rings other than Q(√3 5 23). Specifically, take any radical 3 3 · 3 3 cubic where (3) = p3 = (α) , ℓ, and q are distinct primes with (ℓ) = l , (q) = q , and neither l nor q principal. The ideas behind this construction can be taken further by making appropriate modifications.

6.3. Twisting in general. Definition 6.1 readily generalizes. Replace Gm by any sheaf of groups G acting on A1 or any left action G X and work in the étale topology. A G-twisted monogenerator for S′ S (into X) is an étale cover U S, → i → THESCHEMEOFMONOGENICGENERATORS 47 closed embeddings θi : S′ XU , and elements gij G(Uij ) such that Ui ⊆ i ∈ gij .θj = θi : S′ XU . Uij → ij Equivalently, the θ ’s glue to a global closed embedding S′ X into a twisted form i ⊆ X of X the same way the Gm-twisted monogenerators give embeddings into a line bundle. By “twisted form,” we mean a scheme X S whichb becomes isomorphic tob X S locally in the étale topology. → → The twisted form X arises from transition functionsb in G, meaning there is a G-torsor P such that X is the contracted product of Definition A.4: b X = X G P := X P/(G, ∆) b ∧ × We have already seen the variant G = Aff1, X = A1. Other interesting cases 1 b n include G = PGL2 P , GLn A , an ellliptic curve E acting on itself E E, etc. Example 6.24. Usually, contracted products are defined for a left action G P and a right action G X by quotienting by the antidiagonal action X G P := X P/( ∆, G) ∧ × − defined by 1 G X P ; g.(x, p) := (x.g− , g.p). × We instead take two left actions and quotient by the diagonal action of G. Remark 1 A.5 equates the two. The literature often turns left actions Gm A into right actions anyway, as in [Bre06, Remark 1.7]. Throughout this section, fix notation as in Situation 2.1 and work in the category (Sch/S) of schemes over S equipped with the étale topology. Group sheaves G beget stacks BG = BGS classifying G-torsors on S-schemes with universal G-torsor S BG; these are reviewed in Appendix A. → Twisted forms X of X are equivalent to torsors for Aut(X), as in [Poo17, Theo- rem 4.5.2]. Given a twisted form X S, we obtain the torsor Isom(X,X) of local isomorphisms. Givenb a Aut(X)-torsor→ P , we define a twisted form via contracted product (Definition A.4): b b X := X Aut(X) P. P ∧ The stack BAut(X) is thereby a moduli space for twisted forms of X with universal family X Aut(X) S = [X/Aut(bX)]. An action G Aut(X) lets one turn a G- ∧BAut(X) → torsor P into a twisted form X := X G P P ∧ classified by the map BG BAut(X). → The automorphism sheaf Aut(Xb) acts on the scheme MX via postcomposition with the embeddings S′ X, yielding a map of sheaves → γ : Aut(X) Aut(M )3. → X Similarly, the automorphism sheaf Aut(S′) acts on MX on the right via precompo- sition: op ξ : Aut(S′) Aut(M ). → X 3 3The map γ need not be injective. Consider S = Spec k a geometric point, S′ = F S the 2 trivial 3-sheeted cover, and X = F S only 2-sheeted. There are no closed immersions S′ ⊆ X, so MX = ∅ has global automorphisms Aut(∅) ≃ {id}, but Aut(X) ≃ Z/2Z is nontrivial. Similar examples abound for non-monogenic S′ → S. 48 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

The induced map H1(S, Aut(X)) H1(S, Aut(M)) sends a twisted form X of M M → X to the twisted form X of X given by looking at closed embeddings into X. M M We package these twistedb forms X into a universal version: Twist . b b b Deﬁnition 6.25 (TwistM). Let TwistM BAut(X) be the BAut(X)-stack whose T -points are given by →

M s Twist S′ S T X    ×    := s is a closed immersion .        b  T BAut(X) T X  b         The ﬁbers of TwistM  BAut(X) are representable because they are twisted M M → forms X of X itself: b M X S b p X b TwistM BAut(X).

The universal torsor over BAut(X) is S, but the universal twisted form is obtained by the contracted product of Deﬁnition A.4 with X over BAut(X):

X Aut(X) S [X/Aut(X)]. ∧BAut(X) ≃ One can exhibit TwistM as an open substack of the Weil Restriction of [X/Aut(X)] BAut(X) as in Proposition 2.4. There is a universal closed embedding over→ TwistM into the universal twisted form of X as in the deﬁnition of M in Section 3.1:

u TwistM S′ [X/Aut(X)]. ×S

TwistM The universal case is concise to describe but unwieldy because Aut(X) need not be ﬁnite, smooth, or well-behaved in any sense. We simplify by specifying our twisted form X T to get a scheme TwistM = T TwistM or by → X ×X,BAut(X) specifying the structure group G. b b Fixing the structureb group G requires X = X G P for the speciﬁed sheaf of groups G and some G-torsor P . These G-twisted forms∧ are parameterized by the pullback b TwistMG TwistM p

BG BAut(X).

The ﬁbers of TwistMG BG over maps T BG are again twisted forms of M . → → X If X = Ak and X is a G-twisted form, we may refer to the existence of sections of M Twist X by saying S′/S is G-twisted k-genic, etc. b b THESCHEMEOFMONOGENICGENERATORS 49

Remark 6.26. Trivializing X is not the same as trivializing the torsor P that induces X unless the group G is Aut(X) itself. For example, take the trivial action G X. b b Theorem 6.27. The stack of twisted monogenerators TwistM is isomorphic to [MX /Aut(X)] over BAut(X). More generally, for any sheaf of groups G X, we have an isomorphism TwistMG [M /G] over BG. ≃ X Proof. Address the second, more general assertion and let T be an S-scheme. Write G XT := X S T , T ′ := T S S′, etc. A T -point of TwistM is a G-torsor P T and a solid× diagram × →

P ′ XT T P p × /∆,G

G T ′ X P, T ∧T

T G with T ′ XT T P a closed immersion. Form P ′ by pullback: P ′ is a left G- torsor with→ an equivariant∧ map to X P with the diagonal action. The map T ×T P ′ X P entails a pair of equivariant maps P ′ P and P ′ X . The → T ×T → → T map P ′ P over T forces P ′ P T T ′. These data form an equivariant map P M →over T , or T [M /G≃]. Reverse× the process to finish the proof. → X → X Example 6.28. Consider S′ = Spec Z[i], S = Spec Z. The space of k-generators is k k Mk = A (A 0) according to Proposition 3.16. Take the quotient by the groups of affine transformations:× \ k k 1 k k k 1 [Mk/A ⋊Gm]= P − , [Mk/Aff ] = [A /GLk] = [P − /PGLk]. These quotients represent twisted monogenerators according to Theorem 6.27. The corresponding PGLk-torsors were classically identified with Azumaya algebras k 1 and Severi-Brauer varieties (see the exposition in [Kol16]), or twisted forms of P − . 2 These yield classes in the Brauer group H (Gm) via the connecting homomorphism from 1 G GL PGL 1. → m → k → k → The same holds locally for any degree-two extension S′ S with S integral using Proposition 3.16. → If G is an abelian variety over a number field S = Spec K, let P S be a G- → torsor inducing a twisted form X of X. Given a twisted monogenerator θ : S′ X, one can try to promote θ to a bona fide monogenerator by trivializing P and⊆ thus X. b b Suppose one is given trivializations of P over the completions Kν at each place. Whether these glue to a global trivialization of P over K and thus a monogenerator S′ X is governed by the Shafarevich-Tate group X(G/K). Given⊆ a G-twisted monogenerator with local trivializations, the Shafarevich-Tate group obstructs lifts of θ to a global monogenerator the same way classes of line bundles in Pic obstruct Gm-twisted monogenerators from being global monogenerators. Theorem 6.20 showed a converse – nontrivial elements of Pic imply twisted monogenerators that are not global monogenerators. 50 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

Question 6.29. Is the same true for X? Does every element of the Shafarevich- Tate group arise this way?

The Shafarevich-Tate group approach is useless for G = Gm or GLn because of Hilbert’s Theorem 90 [Stacks, 03P8]: 1 1 Het(Spec K, G)= HZar(Spec K, G)=0. The same goes for any “special” group with étale and Zariski cohomology identified. The strategy may work better for PGLn or elliptic curves E. Remark 6.30. This section defined G-twisted monogenerators using covers in the étale topology, whereas Definition 6.1 used the Zariski topology. For G = Gm or Aff1, either topology gives the same notion of twisted monogenerators. Observe that Gm has the same Zariski and étale cohomology by Hilbert’s Theorem 90. The 1 1 1 same is true for A by [Stacks, 03P2] and so also Aff = Gm ⋉A . 7. Examples of the scheme of monogenerators We conclude with several examples to further illustrate the nature and variety of the scheme of monogenerators. We will consider situations in which the classical index form of Remark 3.18 is well-studied, such as field extensions and extensions of number rings, as well as more exotic situations, such as covers of curves and jet spaces. We suggest starting with Subsection 3.2, which contains what the authors consider the most approachable examples. We will make frequent reference to computation of the index form using the techniques of Section 3.1.

7.1. Field extensions. When S′ = Spec L S = Spec K is induced by a field extension L/K, we know that the monogenic→ generators of L over K are precisely the elements of L that do not belong to any proper subfield of L. Therefore, on the level of K-points of M1, we can expect to see that the index form vanishes on precisely the proper subfields of L. However, it has further structure that is better seen after extension to a larger field.

Example 7.1 (A Z/2 Z/2 ﬁeld extension). Let S = Spec Q and S′ = × Spec Q(√2, √3). The isomorphism of groups Q(√2, √3) Q Q√2 Q√3 Q√6 R ≃ ⊕ ⊕ ⊕ identiﬁes the Weil Restriction Q(√2,√3)/Q and its universal maps with Spec of

a+b√2+c√3+d√6 t[ Q[a,b,c,d][√2, √3] ← Q[a,b,c,d][t].

Q[a,b,c,d] R Hence Q(√2,√3)/Q = Q[a,b,c,d] and the universal morphism R 1 u : S′ S Q √ √ Q AR × ( 2, 3)/ → Q(√2,√3)/Q is induced by t a + b√2+ c√3+ d√6. 7→ We expand the images of the powers 1,t,t2,t3 to ﬁnd the matrix of coeﬃcients 1 a a2 +2b2 +3c2 +6d2 a3 +6ab2 +9ac2 + 36bcd + 18ad2 0 b 2ab +6cd 3a2b +2b3 +9bc2 + 18acd + 18bd2 . 0 c 2ac +4bd 3a2c +6b2c +3c3 + 12abd + 18cd2 0 d 2bc +2ad 6abc +3a2d +6b2d +9c2d +6d3      THESCHEMEOFMONOGENICGENERATORS 51

We compute the local index form associated to our chosen basis by taking the determinant: i(a,b,c,d)= 8b4c2 + 12b2c4 + 16b4d2 36c4d2 48b2d4 + 72c2d4 − − − = 4(2b2 3c2)(b2 3d2)(c2 2d2). − − − − Note that this determinant has degree 6. Dropping subscripts, the factorization N R 4 implies that the closed subscheme of non-generators inside AQ has three components of degree 2. ≃ Consider the Q-points of M = R N. These are in bijection with the 1,S′/S − elements a + b√2+ c√3+ d√6 Q(√2, √3) where a,b,c,d are in Q and the index form does not vanish. Equivalently,∈ 2b2 3c2 =0, b2 3d2 =0, and c2 2d2 =0. − 6 − 6 − 6 Let θ = a + b√2+ c√3+ d√6 for some a,b,c,d Q and consider what it would mean to fail one of these conditions. If 2b2 3c∈2 = 0 for b,c Q, it must be that b = c = 0. Then θ Q(√6), a proper− subfield of Q(√2, √∈3). Similarly, if ∈ b2 3d2 = 0 then θ Q(√3), and if c2 2d2 = 0 then θ Q(√2). It follows that − M ∈ − ∈ √ √ the Q-points of 1,S′/S are in bijection with the elements of Q( 2, 3) that do not lie in a proper subfield, as we expect from field theory. See Example 7.8 for an analysis of the monogeneity of some orders contained in the field considered above.

4 Example 7.2 (A Z/4Z-extension). Let S = Spec Q(i) and S′ = Spec Q(i, √2). 3 We have a global Q(i)-basis 1, √4 2, √2, √4 2 for Q(i, √4 2) over Q(i). We may use this basis to write R Spec{ Q(i)[a,b,c,d]} where the universal map from A1 is 4 ≃ 4 t a + b√2+ c√2+ d(√2)3. The matrix of coefficients is 7→ 1 a a2 +2c2 +4bd a3 +6b2c +6ac2 + 12abd + 12cd2 0 b 2ab +4cd 3a2b +6bc2 +6b2d + 12acd +4d3 . 0 c b2 +2ac +2d2 3ab2 +3a2c +2c3 + 12bcd +6ad2 0 d 2bc +2ad b3 +6abc +3a2d +6c2d +6bd2      The determinant yields the local index form with respect to this basis: (b2 2d2)(b4 +8c4 16bc2d +4b2d2 +4d4). − − We note that the first factor vanishes for a,b,c,d Q(i) when a + b√4 2+ c√2+ 3 ∈ d√4 2 Q(i, √2). At first glance the second factor is more mysterious, but after adjoining∈ enough elements, the entire index form factors into distinct linear terms: (b √2d)(b + √2d)(ib (1 + i)√4 2c + √2d)( ib (1 i)√4 2c + √2d) − − − − − ( ib + (1 i)√4 2c + √2d)(ib +(1+ i)√4 2c + √2d). · − − This behavior of factorization into distinct linear factors occurs in general:

Proposition 7.3. Let S′ S be induced by a ﬁnite separable extension of ﬁelds → L/K. Let e1,...,en be a K-basis for L, and let x1,...,xn be the corresponding coordinates for R. Then the local index form i(e1,...,en) factors completely into distinct linear factors in x1,...,xn over the normal closure L of L/K.

e 52 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

Compare this with Example 5.1. Monogenerators correspond to configurations 1 of n points in A and the distinct linear factors of i(e1,...,en) correspond to when pairs of points collide. This is the image of the “fat diagonal” ∆. Some interesting and useful specifics in the case of number fields are investigated in more depth in chapter 7 of [Gaá19]. b

Proof. We may consider i(e1,...,en) as an element of L[x1,...,xn] by pulling back R R to L/K S Spec L L L/L. Our strategy is to compute a second generator of × ≃ ⊗K I e e e the pullback of S′/S with respect to a more convenient basis. By the Chinesee remainder theorem, L L n L. Let e ,..., e be ⊗K ≃ i=1 1 n the standard basis of (L)n, let x ,..., x be the corresponding coordinates on 1 n Q R R e e L L/L Ln/L, and let θ = x1e1 + + xnen. Computinge a matrixe M ⊗K ≃ ··· e e e e e for the map L[x1,..., xn,t]/m(t) e L[x1,...,e xn] sending t θ, we see that it is a e → 7→ Vandermonde matrix with factors x1e,...,e xn, sincee e e e e e k ek k e (x1e1 + + xnen) = x e1 + x en, ··· e e 1 ··· n when computed in the product ring (L)n. Therefore i(e ,..., e ) = det(M) = e e e e e e | e 1e n | | | (x x ) . Applying the L-linear change of basis from x to x , we see | i

3 Example 7.4 (A purely inseparable extension). For F3(α)[β]/(β α) over F3(α), write a,b,c for the universal coefficients of the basis 1,β,β2. In other− words, θ = a + bβ + cβ2. One computes that the index form is then b3 c3α. − To find the monogenic generators of this extension, we look for a,b,c F3(α) so that b3 c3α = 0. Clearly, at least one of b,c must be nonzero. Choose∈b,c arbitrarily so− that one6 is nonzero. Is this enough to ensure we have a monogenerator? 3 3 1/3 Suppose first that b = 0. Then b c α = 0 implies c F3(α ) F3(α), a 6 − 3 3 ∈ 1/3\ contradiction. Symmetrically, if c = 0 and b c α = 0, then b F3(α ) F3(α), a contradiction again. We conclude6 that the set− of monogenerator∈ s is \ M (F (α)) = a + bβ + cβ2 a,b,c F (α) and (b,c) = (0, 0) . 1,S′/S 3 { | ∈ 3 6 } = F (α) F , 3 \ 3 as one expects from field theory. 3 3 The polynomial b c α is irreducible in F3(α)[a,b,c], so the scheme of non- generators N is an irreducible− subscheme of R A3. However, N is not geo- ≃3 metrically reduced: after base extension to F3(α, √α), the index form factors as (b cβ)3. − The factorization noted above is not an isolated phenomenon:

m Proposition 7.5. Let S′ S be induced by a degree n := p completely insep- m m arable extension of ﬁelds K→(α1/p )/K. Then over K(α1/p ), the local index form factors into a repeated linear factor of multiplicity pm. THESCHEMEOFMONOGENICGENERATORS 53

Proof. Consider the local index form i(e1,...,en) as an element of 1/pm K(α )[x1,...,xn] by pulling back to

R m R m m m K(α1/p )/K S S′ K(α1/p ) K(α1/p )/K(α1/p ). × ≃ ⊗K Once again, to arrive at the result, we will compute a second generator of the pull I back of S′/S with respect to another basis. By the Chinese remainder theorem, 1/pm 1/pm pm 1/pm K(α ) K K(α ) K[t]/(t α) K K(α ) ⊗ ≃ m − ⊗ m m K(α1/p )[t]/((t α1/p )p ) ≃ m m− K(α1/p )[ε]/εp , ≃

m where ε = t α1/p . − pm 1 1/pm pm Let b1 = 1,b2 = ε,...,bpm = ε − be a basis for K(α )[ε]/ε over m 1/p R m K(α ), and let y1,...,yn be corresponding coordinates on K(α1/p )/K S S′. We are now in the situation of Example 3.13. Following the calculation there,× we do n 1 a second change of coordinates to the basis c =1,c = ε y ,...,c = (ε y ) − 1 2 − 1 n − 1 R m and let z1,...,zn be the corresponding coordinates on K(α1/p )/K S S′. Tak- 1/pm × ing the determinant of the matrix M of the map K(α )[z1,...,zn,t]/m(t) 1/pm pm → K(α )[ε][z1,...,zn]/ε sending t z1c1 + + zncn with respect to the bases n 1 7→ ··· 1,...,t − and c ,...,c , we obtain { } { 1 n} pm(pm 1) i 2 − (c1,...,cn)= z2 . Applying the change of basis from z ,...,z to x ,...,x , we see that { 1 n} { 1 n} i(e1,...,en) is a power of a linear term. 7.2. Orders in number rings. Example 7.6 (Dedekind’s Non-Monogenic Cubic Field). Let η denote a root of the polynomial X3 X2 2X 8 and consider the field extension L := Q(η) over K := Q. When Dedekind− − constructed− this example [Ded78] it was the first example of a non-monogenic extension of number rings. Indeed two generators are necessary 2 η+η2 η+η2 2 to generate ZL/ZK : take η and 2 , for example. In fact, 1, 2 , η is a Z- { } 2 η+η 2 basis for ZK . The matrix of coefficients with respect to the basis 1, 2 , η is { }

1 a a2 +6b2 + 16bc +8c2 0 b 2ab +7b2 + 24bc + 20c2 . 0 c 2b2 +2ac 8bc 7c2  − − − Taking its determinant, the index form associated to this basis is 2b3 15b2c 31bc2 20c3. − − − − Were the extension monogenic, we would be able to find a,b,c Z so that the index form above is equal to 1. ∈ ± In fact, ZQ(η)/Z is not even locally monogenic. By Lemma 4.6, we may check by reducing at primes. Over the prime 2 the index form reduces to b2c + bc2, 54 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH and iterating through the four possible values of (b,c) (Z/2Z)2 shows that the index form always to reduces to 0. That is, 2 is a common∈ index divisor. Dedekind showed that ZQ(η)/Z is non-monogenic, not by using an index form, but by deriving a contradiction from the ZL-factorization of the ideal 2, which splits into three primes. In our terms, Spec ZQ Spec Z has three points over Spec F , (η) → 2 all with residue field F2. Therefore, condition (2) of Theorem 4.16 for monogeneity at the prime (2) fails, so S′ S is not monogenic. → The example above is minimal in the sense that the smallest a common index divisor can be is 2, and to fail to be monogenic over the integral prime (2) requires at least 3 points over Spec F2 by Theorem 4.16, hence an extension of degree 3. The next smallest that a common index divisor can be is 3. By Theorem 4.16, this requires at least 4 points over the Spec F3, so we will need an extension of at least degree 4. Our next example highlights a number field with common index divisor 3. Example 7.7 (Non-monogenic with common index divisor 3). Let η denote a root of the polynomial X4 X3+2X2+4X+3 and consider the field extension L := Q(η) over K := Q. The integral− prime 3 splits completely in this quartic field, making 3 a common index divisor. Let us check directly that the index form for ZL/ZK admits no solutions modulo 3. 2 3 3 One can compute a (ZK = Z)-module basis for ZL: 1, η , η , (2η + η)/3 . The matrix of coefficients is: { }

1 a a2 4bc 3c2 +2bd 6cd +3d2 A 0 b 2ab 4b2 28−bc −12c2 4bd −42cd +3d2 B  0 c 3b−2 +2ac− 8bc− 2c2 − 8bd − 12cd 5d2 C  − − − − − −  0 d 18bc +9c2 +2ad 6bd + 26cd 9d2 D   − −  where   A = a3 +6b3 12abc + 51b2c 9ac2 + 39bc2 +3c3 +6abd + 15b2d 18acd − − − + 150bcd + 63c2d +9ad2 +3bd2 + 108cd2 9d3; − B =3a2b 12ab2 + 58b3 84abc + 276b2c 36ac2 + 165bc2 +3c3 12abd − − − − + 213b2d 126acd + 834bcd + 279c2d +9ad2 + 237bd2 + 621cd2 + 72d3; − C = 9ab2 + 24b3 +3a2c 24abc + 66b2c 6ac2 + 24bc2 5c3 24abd − − − − − + 102b2d 36acd + 210bcd + 45c2d 15ad2 + 141bd2 + 165cd2 + 63d3; − − D = 27b3 + 54abc 189b2c + 27ac2 135bc2 9c3 +3a2d 18abd − − − − − 81b2d + 78acd 558bcd 222c2d 27ad2 54bd2 405cd2 + 10d3. − − − − − − The index form equation corresponding to monogeneity is obtained by setting the determinant of this matrix equal to a unit of Z, so 1. Since 3 is a known common index divisor, we consider this equation modulo 3:± b4cd + b3c2d b2c3d b4d2 bc3d2 b3d3 b2cd3 bc2d3 + b2d4 + bcd4 + bd5 = 1. − − − − − − − ± Running through the possible values of b,c,d Z/3Z, we see that there are no ∈ solutions to this equation, and thus no monognerators of ZL over Z. Example 7.8 (A non-monogenic order and monogenic maximal order). Consider the extension Z[√2, √3] of Z. Note that Z[√2, √3] is not the maximal order of THESCHEMEOFMONOGENICGENERATORS 55

Q(√2, √3). As we will see below, the maximal order is Z[ √3 + 2]. The isomor- √ √ √ √ √ phism of groups Z[ 2, 3] Z Z 2 Z 3 Z 6 identiﬁesp the Weil Restriction R ≃ ⊕ ⊕ ⊕ Z[√2,√3]/Z and its universal maps with Spec of

a+b√2+c√3+d√6 t[ Z[a,b,c,d][√2, √3] ← Z[a,b,c,d][t].

Z[a,b,c,d] Now, 1 1 7→ t a + b√2+ c√3+ d√6 7→ t2 (a + b√2+ c√3+ d√6)2 7→ t3 (a + b√2+ c√3+ d√6)3 7→ is given by 1 a a2 +2b2 +3c2 +6d2 a3 +6ab2 +9ac2 + 36bcd + 18ad2 0 b 2ab +6cd 3a2b +2b3 +9bc2 + 18acd + 18bd2 . 0 c 2ac +4bd 3a2c +6b2c +3c3 + 12abd + 18cd2 0 d 2bc +2ad 6abc +3a2d +6b2d +9c2d +6d3    Taking the determinant, the index form with respect to our chosen basis is 8b4c2 + 12b2c4 + 16b4d2 36c4d2 48b2d4 + 72c2d4 − − − = 4(2b2 3c2)(b2 3d2)(c2 2d2). − − − − The Z-points of M are in bijection with the tuples (a,b,c,d) Z4 1,Z[√2,√3]/Z ∈ such that the determinant is a unit. Since the determinant is divisible by 2, this never happens. We conclude that Z[√2, √3] is not monogenic over Z. In light of Proposition 4.10, S′ S fails to be monogenic over even geometric points, since → the index form reduces to 0 in the fiber over 2. In terms of Theorem 4.14, S′ S fails to be monogenic since the fiber over 2 consists of a single point with a→ two dimensional tangent space. The non-monogeneity of the order Z[√2, √3] is in marked contrast to the maximal order of Q(√2, √3), which is monogenic. A computation shows that a power integral basis for the maximal order is given by 1, α, α2, α3 , where α is a root of 4 2 { } t 4t +1. One could take α = √3 + 2. Here the Weil Restriction RZ[α]/Z and its− universal maps are identified with Spec of p a+bα+cα2+dα3 t[ Z[a,b,c,d][α] ← Z[a,b,c,d][t].

Z[a,b,c,d] The element-wise computation 1 1 7→ t a + bα + cα2 + dα3 7→ t2 (a + bα + cα2 + dα3)2 7→ t3 (a + bα + cα2 + dα3)3 7→ 56 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

yields the matrix of coeﬃcients 1 a a2 c2 2bd 4d2 A 0 b − 2ab− 2cd− B , 0 c b2 +2ac +4c−2 +8bd + 15d2 C 0 d 2bc +2ad +8cd D   where   A = a3 3b2c 3ac2 4c3 6abd 24bcd 12ad2 45cd2, − − − − − − − B =3a2b 3bc2 3b2d 6acd 12c2d 12bd2 15d3, − − − − − − C =3ab2 +3a2c + 12b2c + 12ac2 + 15c3 + 24abd + 90bcd + 45ad2 + 168cd2, D = b3 +6abc + 12bc2 +3a2d + 12b2d + 24acd + 45c2d + 45bd2 + 56d3. The determinant of this matrix yields the index form (b2 2c2 +6bd +9d2)(b2 6c2 + 10bd + 25d2)(b2 +4bd + d2). − − One can compute that the index of Z[√2, √3] inside of Z[ √3+2] is 2. There- fore the index forms are equivalent away from the prime 2. p Example 7.9 (A locally monogenic, but not twisted-monogenic extension). Here we take a closer look at one member of the family in Example 4.8. Let K = Q, 3 2 3 2 3 2 L = K(√5 7). The ring of integers ZL = Z[√5 7, √5 7 ] is not monogenic · 3 3 · · over Z. Let α = √52 7, β = √5 72. It turns out that 1,α,β is a Z-basis for · · { } ZL, so the universal map may be identiﬁed with a+bα+cβ t[ ZL[a,b,c] ← Z[a,b,c][t].

Z[a,b,c] Expanding 1 1 7→ t a + bα + cβ, 7→ t2 (a + bα + cβ)2 7→ we ﬁnd that the matrix of coeﬃcients is 1 a a2 + 70bc 0 b 2ab +7c2 . 0 c 2ac +5b2 Computing the determinant, we get the index form 5b3 7c3. Thus, for a given choice of a,b,c Z, the primes which divide the value 5b3− 7c3 are precisely the primes at which∈a + bα + cβ will fail to generate the extension.− The values obtained by this index form are 0, 5 modulo 7. Since 5 is a unit in Z/7Z, we do have local monogenerators over{ ±D(7).} Similarly, we have local monogenerators over D(5). Together, D(5) and D(7) form an open cover, and we see that this extension is Zariski-locally monogenic. However, as we can see by reducing modulo 7, the index form cannot be equal to 1. Therefore there are no ± global monogenerators and ZL/Z is not monogenic. This is not a Gm-twisted monogenic extension by Theorem 6.20, because h(Z)= 1 and this extension is not globally monogenic. See Example 6.22 for a properly THESCHEMEOFMONOGENICGENERATORS 57

Gm-twisted monogenic extension and an interesting comparison to the example presented here. 7.3. Other examples. Example 7.10. We investigate the analog of the integers in Example 7.4. We keep 3 the same notation. The base ring is F3[α] and the extension ring is F3[α][x]/(x 3 − α)= F3[β], where β = α.

a+bβ+cβ2 x[ F3[a,b,c][β] ← F3[α][a,b,c][x].

F3[α][a,b,c]

1 1 7→ x a + bβ + cβ2 7→ x2 (a + bβ + cβ2)2 7→ is given by 1 a a2 +2bcα 0 b c2α +2ab . 0 c b2 +2ac  The determinant is b3 c3α, which is not geometrically reduced: it factors as (b cβ)3. To ﬁnd the− monogenerators of this extension, we set this expression − equal to the units of F3[α]. Since (F3[α])∗ = 1, the only solutions are b = 1, c = 0. Thus ± ±

M F F (F [α]) = a β : a F [α] . 1, 3[β]/ 3[α] 3 { ± ∈ 3 } We can see that, much like number rings, monogeneity imposes a stronger restriction here than it does for the extension of fraction ﬁelds.

Our next examples concern the case that S′ S is a finite map of algebraic curves, which are essentially never monogenic. On→ the other hand, we find explicit examples of G -twisted monogenic S′ S. Theorem 6.18 constrains the possible m → line bundles that we may use to show Gm-twisted monogeneity. We make this precise in the lemma below. Lemma 7.11. Let π : C D be a finite map of smooth projective curves of degree n and let g denote the genus.→ (1) π is only monogenic if it is the identity map; (2) If π is Gm-twisted monogenic, then 1 g(C) n(1 g(D)) is divisible by 1 − − − 2 n(n 1) in Z. Moreover, if π factors through a closed embedding into a line bundle− E with sheaf of sections E, then 1 g(C) n(1 g(D)) deg(E)= − − − . − 1 n(n 1) 2 − Proof. To see (1), note that a map f : C A1 is determined by a global section of → D OC. Since C is a proper variety, the global sections of OC are constant functions. 1 It follows that a map f : C AD is constant on fibers of π. Therefore f cannot be an immersion unless π has→ degree 1, i.e., is the identity. 58 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

Suppose π : C D is Gm-twisted monogenic with an embedding into a line bundle E. By [Stacks→ , 0AYQ] and Riemann-Roch,

deg(det(π OC ))=1 g(C) n(1 g(D)). ∗ − − − By Theorem 6.18, n(n 1) − det(π OS ) E− 2 ∗ ′ ≃ where E is the sheaf of sections of E. Taking degrees of both sides, n(n 1) 1 g(C) n(1 g(D)) = deg(E) − . − − − − · 2 This shows (2). Example 7.12 (Maps P1 P1). Let k be an algebraically closed field and let 1 1 → π : Pk Pk be a finite map of degree n. If n = 1, then π is trivially monogenic. When n→= 2, Lemma 7.11 tells us that π cannot be monogenic, while Lemma 6.19 tells us that π is Aff1-twisted monogenic. Lemma 7.11 tells us that for degrees n> 2 the map π is neither monogenic nor Gm-twisted monogenic, although Theorems 4.1 and 4.16 tell us that π is Zariski-locally monogenic. Working with Z instead of an algebraically closed field, consider the map π : 1 1 2 2 S′ = P S = P given by [a : b] [a : b ]. We will show by direct computation Z → Z 7→ that this map is Gm-twisted monogenic. Write U = Spec Z[x] and V = Spec Z[y] for the standard affine charts of the target P1. The map π is then given on charts by Z[x] Z[a] → x a2 7→ and Z[y] Z[b] → y b2. 7→ Let us compute M1,S /S. Over U, π OP1 has Z[x]-basis 1,a . Let c1,c2 be the ′ ∗ { } coordinates of R = A2, with universal map |U Z[c ,c ,t] Z[c ,c ,a], t c + c a. 1 2 → 1 2 7→ 1 2 The index form associated to this basis is

i(c1,c2)= c2.

Similarly, π OP1 has Z[y]-basis 1,b , R V analogous coordinates d1, d2, and the index form associated∗ to this basis{ is} |

i(d1, d2)= d2. An element of Z[x] (resp. Z[y]) is a unit if and only if it is 1, so ± M (U)= c a c Z[x] 1 { 1 ± | 1 ∈ } M (V )= d b d Z[y] . 1 { 1 ± | 1 ∈ } We can see directly that π is not monogenic: the condition that a monogenerator c a on U glue with a monogenerator d b on V is that 1 ± 1 ± (c1 a) U V = (d1 b) U V . ± | ∩ ± | ∩ 1 But this is impossible to satisfy since a U V = b U− V . | ∩ | ∩ THESCHEMEOFMONOGENICGENERATORS 59

Lemma 7.11 tells us that if S′ S is twisted monogenic, the line bundle into → which S′ embeds must have degree 1. Let us therefore attempt to embed S′ into the line bundle with sheaf of sections OP1 (1). The sheaf OP1 (1) restricts to the trivial line bundle on both U and V , and a section f O glues to a section g O if ∈ U ∈ V y f U V = g U V . · | ∩ | ∩ Embedding S′ into this line bundle is therefore equivalent to finding a monogenerator c a on U, and a monogenerator d b on V such that 1 ± 1 ± y (c1 a) U V = (d1 b) U V . ± | ∩ ± | ∩ 2 2 Bearing in mind that y = b = a− on U V , we find a solution by taking positive signs, c = 0, and d = 0. Therefore π : P∩1 P1 is twisted monogenic. 1 1 Z → Z Lemma 7.11 tells us that we must pass to higher genus to find a Gm-twisted monogenic cover of P1 of degree greater than 2. Here is an example where the source is an elliptic curve. Example 7.13 (Twisted monogenic cover of degree 3). Let E be the Fermat 3 3 3 2 elliptic curve V (x + y z ) PZ. Consider the projection from [0 : 0 : 1], i.e., the map π : E P1 defined− by⊂ [x : y : z] [x : y]. Write U = Spec Z[x] and V = Spec Z[y] for→ the standard affine charts of7→ P1. The map is given on charts by Z[x] Z[x, z]/(x3 +1 z3) and Z[y] Z[y,yz]/(1 + y3 (yz)3). The gluing on → − 1 1 → 1 − overlaps is given by x y− on P and by x y− ,z z on E. 7→ 7→ 7→ 2 We now compute ME/P1 . Note that over U, OE has the Z[x]-basis 1,z,z . The index form associated to 1,z,z2 is i(c ,c ,c )= c3 c3(x3 + 1). 1 2 3 2 − 3 2 2 Over V , OE has the Z[x]-basis 1,yz,y z . The index form associated to this basis is i(d , d , d )= d3 d3(y3 + 1). 1 2 3 2 − 3 An element of Z[x] or Z[y] is a unit if and only if it is 1. This implies that ± M (U)= c z c Z[x] 1 { 1 ± | 1 ∈ } M (V )= d yz d Z[y] . 1 { 1 ± | 1 ∈ } We see that there are no global sections of M1, since coefficients of z cannot match on overlaps. However, if we twist so that we are considering embeddings of E into OP1 (1), then the condition for a monogenerator c1 z on U to glue with a monogenerator d yz on V is that ± 1 ± y (c1 z) U V = (d1 yz) U V . ± | ∩ ± | ∩ This is satisfiable, for example by taking the positive sign for both generators and c = d = 0. Therefore E P1 is twisted monogenic with class 1 Pic(P1 ). 1 1 → ∈ Z Example 7.14 (Jet spaces of A2). Consider an m-jet of A2 = Spec k[t,u] determined as in Example 3.13 by t = a + a ε + a ε2 + a εm 0 1 2 ··· m u = b + b ε + b ε2 + b εm. 0 1 2 ··· m 60 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

m+1 Linear changes of coordinates ensure a0 = b0 = 0 and that our jets satisfy t = um+1 = 0 in k[ε]/εm+1. To ﬁnd the matrix for the induced k-linear map from

k[t,u]/(tm+1,um+1)= k teuf · to the jets k[ε]/εm+1 = k εi, we need the coefficientM of each εp in the expression: · teuf = (a ε + a ε2 + L+ a εm)e(b ε + b ε2 + + b εm)f 1 2 ··· m 1 2 ··· m m r e it =  ε at  · i1,...,im 1 r m i1+i2+ +im=m t=0 ≤ ≤ ···  X i1+2i2+Xmim=r Y   ···  m s f jt  ε bt  · · j1,...,jm 1 s m j +j + +j =m t=0 1 2 ··· m  ≤X≤ j1+2j2+Xmjm=s Y   ···    m p  e f it jt  = ε at bt i1,...,im j1,...,jm 1 p m  i1+i2+ +im=m t=0  ≤X≤  j1+j2+X···+jm=m Y  (i +j )+2(i +j )+··· +m(i +j )=p   1 1 2 2 ··· m m    If e + f >p, the coefficient of εp in teuf is again zero. If e + f >m, all the coefficients are zero. The corresponding m2 m matrix is “lower triangular” in this sense. × Take m = 1 to reduce to A. Cayley’s original situation of a 2 2 2 hypermatrix; 2 2 × R× compute his second hyperdeterminant Det to be a1b1. In this case, 2 is the tangent 2 space of A , the index forms cut out the locus where both a1 and b1 are zero, and the hyperdeterminant cuts out the locus where either a1 or b1 are zero. Computability is a serious constraint for even simple cases. Taking A3 and m =1 yields a 2 2 2 2 hypermatrix. The formula for such a hyperdeterminant is degree 24 and× × has 2,894,276× terms [Ott13, Remark 5.7]. Example 7.15 (Limits and Colimits). Let B be an A-algebra which is complete m with respect to I B. If each Bm := B/I is finite locally free over A and ⊆ M X Spec A is quasiprojective, there are affine restriction maps X,Bm+1/A M → → X,Bm/A. One can define

MX,B/A := lim MX,B /A, m m which is a scheme [Stacks, 01YX]. By [Bha16, Remark 4.6, Theorem 4.1], this limit parametrizes closed embeddings s : Spec B X over Spec A as in Deﬁnition 2.2. The arc space examples kJtK/k, kJx, yK/k were→ mentioned in Example 2.8. We cannot make a similar statement for colimits of algebras. Suppose Bi is a diagram of A-algebras indexed by N. Then for each i < j there is a natural{ } map R R Bi/A Bj /A. Notice that if the image of some θ Bi is a monogenerator of Bj , → R ∈R M then Bi Bj is surjective. It follows that Bi/A Bj /A only takes Bi/A into M → → Bj /A if Spec Bj Spec Bi is a closed immersion over each open set U Spec A M → M ⊆ over which Bi/A is non-empty. Assuming B0/A is locally non-empty, the only diagrams B for which the colimit colim M can even be formed are those for { i} i Bi/A THESCHEMEOFMONOGENICGENERATORS 61 which each Bi Bj is surjective. Since B0 is Noetherian, all such diagrams are eventually constant→ and uninteresting.

Appendix A. Basics of Torsors and Stack Quotients With this appendix our aim is to introduce the minimum prerequisites for torsors and stack quotients. The concepts covered here will be employed throughout the paper, especially in Sections 5 and 6. The reader interested in a more thorough treatment should consult [Fan+05]. Let C be the category (Sch/ ) of schemes over a fixed final scheme such as = Spec Z or Spec C, equipped∗ with the étale topology. One can also∗ take the ∗Zariski, fppf, or fpqc topologies, or any other site for C. Let G be a sheaf of groups 1 on C, which one may take to be Gm, Aff , GLn for concreteness. Definition A.1. A (left) G-pseudotorsor P over an object X C is a sheaf on C/X such that ∈ G P ∼ P P ; (g,p) (gp,p) × → ×X 7→ is an isomorphism. It is called a G-torsor if, for some cover Ui X , P (Ui) is nonempty for each i. { → }

Slogan: for any two sections p,p′ of P , there is a unique section g of G such that gp = p′. If one locally chooses sections p P (U ), the multiplication map i ∈ i G(U ) ∼ P (U ); g g.p i → i 7→ i is a bijection. Locally, but not necessarily globally, P is isomorphic to G – a torsor is called trivial if P is actually globally isomorphic to G. Example A.2. Let L X be a line bundle and P := L X be L without the → 1 \ zero section. Locally in X, L AX and P Gm,X . But two such trivializations 1 ≃ ≃ L AX differ by units in Gm,X and the same holds for P . Regardless, P is a ≃ 1 Gm-torsor, and so is the sheaf of isomorphisms Isom(L, A ). Remark A.3. Define right G-torsors the same way but with a right G-action. Right G-torsors P become left G-torsors P L via 1 g.p := p.g− in the usual way. For abelian groups, these two actions and left and right G-torsors coincide. We exclusively use left torsors, even with contracted products: Definition A.4. Let G be a sheaf of groups on X. Let BG be the X-stack whose T points are BG(T ) := left G -torsors on T , { |T } 1 where maps are given by G-equivariant isomorphisms. Remark that H (T, G T ) BG(T )/isomorphism. | ≃ If G X is a left action and G P a left torsor, form the contracted product X G P := X P/(∆, G) ∧ × as the quotient of X P by the diagonal action: × g.(x, p) := (g.x, g.p). 62 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH

The same can be done for a fiber product over a base Z: X G P := X P/(∆, G). ∧Z ×Z For example, let ϕ : G H be a morphism of sheaves of groups and P a G- torsor. Obtain an action G→ H and form the contracted products H G P to get a map ∧ BG BH; P P G H. → 7→ ∧ which is well-defined. If G is abelian, the multiplication map m : G G G is a group homomorphism and we have a group structure on BG via contracted× → product along m. If G X and H Y are two actions and f : X Y is “equivariant along” ϕ : G H, i.e. f(g.x) = ϕ(g).f(x), then there is a→ unique H-equivariant map H G X→99K Y such that X H G X 99K Y is f. ∧ → ∧ Remark A.5. Contracted products are usually defined for a left action G P and a right action G X by taking the quotient X G P := X P/( ∆, G) ∧ × − by the antidiagonal action ∆: − 1 G X P ; g.(x, p) := (x.g− , g.p). × Turning the right action G X into a left action XL as in Remark A.3, this is the same as quotienting by the diagonal action X G P = XL P/(∆, G)= X P/( ∆, G). ∧ × × − Definition A.6. Let G X be an action on a scheme or a sheaf X. Let T BG parametrize a G -torsor P T and define the quotient stack by → |T → [X/G] := f : P X f is G-equivariant . { → | } T BG P In other words, a map T [X/G] from a scheme to the quotient stack → parametrizes a G T -torsor P T and an equivariant morphism P X. If is the final sheaf with| trivial G→action, [ /G] = BG. If f : X Y is→ equivariant∗ along ϕ : G H as in A.4, there is a map∗ → → [X/G] [Y/H] → similarly induced by contracted product. Example A.7. If ϕ : G M is a group homomorphism, the nonempty preimages 1 → ϕ− (m) for m M become torsors (“cosets”) for ker ϕ. This applies to sheaves of groups as well,∈ giving possibly nontrivial torsors. From a short exact sequence of sheaves of groups ϕ 0 K G M 0 → → → → and a section s Γ(M), define the K-torsor P by ∈ s P (U) := t G(U) s = ϕ(t) . s { ∈ | } The connecting map Γ(M) H1(K) sends s Γ(M) to the isomorphism class of P H1(K). This leads to the→ usual long exact∈ sequence s ∈ 1 Γ(K) Γ(G) Γ(M) H1(K) H1(G) H1(M) → → → → → → → · · · THESCHEMEOFMONOGENICGENERATORS 63

Remark A.8 ([Stacks, 003O]). Suppose given functors f : F H , g : G H → → between stacks over C. The T -points of the fiber product F H G are: × a pair of points x F (T ), y G (T ) and • an isomorphism β∈: f(x) g(∈y) between their images in H . • ≃ Proposition A.9. Let G, H be finite groups, and suppose = Spec k where k is separably closed. Then the category Hom(BG,BH) is equivalent∗ to the category whose objects are homomorphisms φ : G H of groups, and where a morphism → 1 f : (φ : G H) (ψ : G H) is an element h H such that ψ(g)= h− φ(g)h → → → f ∈ f f for all g G. ∈ Proof. By definition, a morphism BG BH is equivalent to an H-torsor P BG. The map BG is an étale cover because→ G is finite. Since BH is a stack,→ an H-torsor P∗ →on BG is uniquely determined by its associated descent datum on the cover BG. Conversely, every such H-torsor arises from such a descent datum. Consider∗ → a descent datum for BH with respect to the cover BG. In the notation of [Fan+05, Section 4.1.2], we have ∗ → (1) an H-torsor P ; ∗ →∗ (2) an isomorphism of H-torsors α : pr1∗P pr2∗P on BG ; ∗ → ∗ ∗× ∗ such that the cocycle condition

pr∗ α pr∗ α = pr∗ α 23 ◦ 12 13 holds on BG BG . We argue∗× that∗× the ﬁber∗ products of the cover BG are: ∗ → pr G G23 G × pr13 pr 12 pr3 pr3 G pr2 ∗

pr2 pr1 G . ∗

pr1 BG ∗ An H-torsor P over a point can only be the trivial torsor H since we have assumed our∗ base→∗ separably closed: has no non-trivial étale covers,→∗ so for P to be étale-locally a trivial torsor, P must∗ itself be a trivial torsor. ∗ ∗ We next consider the pullbacks of H to BG as in Remark A.8. In our case, (T ) is the one-element set and its→∗ lonely∗× element∗ maps to the trivial torsor ∗ G T T in BG(T ). Therefore ( BG )(T ) may be identified with the set of automorphisms× → β of G T T . Such∗× automorphisms∗ are in bijection with sections of G T T : one only× has→ to choose an element of G by which to translate in each× fiber.→ Thus the two pullbacks pr1∗P and pr2∗P may both be identified with the trivial H-torsor H G G. To complete∗ a descent∗ datum, we ask what isomorphisms α : H G × H→ G satisfy the cocycle condition. Suppose that α is such a morphism.× Then→ since× α is an isomorphism of torsors, its action on points takes the form α : (h,g) (h φ(g),g), 7→ · 64 SARAH ARPIN, SEBASTIAN BOZLEE, LEO HERR, HANSON SMITH where φ : G H is the function taking g G to the H-coordinate of α(1,g). It remains→ to check the implications of the∈ cocycle condition. Arguing similarly to , we may identify the T -points of with pairs of isomor- ∗×BG ∗ ∗×BG ∗×BG ∗ phisms (β,γ) of the trivial torsor, whose three projections to BG are given by pr ((β,γ)) = β, pr ((β,γ)) = γ, and pr ((β,γ)) = γ β.∗× Since∗ such isomor- 12 23 13 ◦ phisms can be identified with elements of G, we may identify BG BG with G G. The projections then become ∗× ∗× ∗ × pr : G G G 12 × → (g ,g ) g 12 23 7→ 12 pr : G G G 23 × → (g ,g ) g 12 23 7→ 23 pr : G G G 12 × → (g ,g ) g g 12 23 7→ 12 23

Therefore, the pullbacks of α to may be identiﬁed with ∗×BG ∗×BG ∗ pr∗ α : H G G G 12 × × 7→ (h,g ,g ) (h φ(g ),g ,g ) 1 2 7→ · 1 1 2 pr∗ α : H G G G 23 × × 7→ (h,g ,g ) (h φ(g ),g ,g ) 1 2 7→ · 2 1 2 pr∗ α : H G G G 13 × × 7→ (h,g ,g ) (h φ(g g ),g ,g ). 1 2 7→ · 1 2 1 2 We conclude that an isomorphism α satisﬁes the cocycle condition if and only if φ(g )φ(g )= φ(g g ), for all g ,g G. 1 2 1 2 1 2 ∈ That is, if and only if φ is a group homorphism. This establishes our claim on the elements of Hom(BG,BH). Now suppose that P BG and P BG are H-torsors with associated φ → ψ → group homomorphisms φ : G H, ψ : G H. A morphism f : Pφ Pψ is equivalent to a morphism of the→ associated descent→ data, i.e., a morphism→ between the restrictions of P and P to that commutes with the gluing morphisms. Now, φ ψ ∗ since Pφ and Pφ are both trivial torsors over a point, the morphisms Pφ Pψ are in bijection|∗ with|∗ elements of H: one just has to choose what element|∗ of→H to|∗ translate by. Write hf for the element associated to f : Pφ Pψ. Compatibility with gluing morphisms translates to → (hφ(g)h ,g) = (h h ψ(g),g) for all h H,g G. f · f ∈ ∈ Equivalently, 1 ψ(g)= h− φ(g)h for all g G. f f ∈

Remark A.10. When H = Σn and G is a ﬁnite group, the proposition describes the category Hom(BG,BΣn) as: Ob: Objects are group homomorphisms G Σn, or actions G n . Mor: Morphisms are reorderings of the set →n on which G acts. h i h i REFERENCES 65

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