FIELD PATCHING, FACTORIZATION, AND LOCAL-GLOBAL PRINCIPLES

DANIEL KRASHEN

1. Introduction

The goal of this paper is twofold - first to give an introduction to the method of field patching as first presented in [HH], and later used in [HHKa], paying special attention to the relationship between factorization and local-global principles and second, to extend the basic factorization result in [HHKa] to the case of retract rational groups, thereby answering a question posed to the author by J. L. Colliot-Thélène.

Throughout, we fix a complete discrete valuation ring T with field of fractions K and residue field k. Let t ∈ T be a uniformizer. Let X/K be a smooth projective curve and F its function field.

Broadly speaking, the method of field patching is a procedure for constructing new fields F_{ξ} which will be in certain ways simpler than F, and to reduce problems concerning F to problems about the various F_{ξ}. Overall, there are two ways in which this is done. Let us suppose that we are interested in studying a particular type of arithmetic object, such as a quadratic form, a central simple algebra, etc. Constructive Strategy (Patching) : This consists in showing that under suitable hypotheses, algebraic objects defined over the fields F_{ξ} which are “compatible,” exactly correspond to objects defined over F (see Theorem 3.2.3). One may then use this idea to construct new examples and counterexamples of such objects by building them “locally.” Deconstructive Strategy (Local-global principle) : We say that a particular type of algebraic object satisfies a local global principle if whenever an object defined over F becomes “trivial” when scalars are extended to each F_{ξ}, it must in fact have been trivial to begin with (see Section 2.2).

In this paper, we will not focus on these applications, which are discussed for example in [HH, HHKa, HHKb, CTPS09]. Instead, we focus on elucidating and extending the underlying methods used.

The author is grateful to David Harbater and Julia Hartmann for numerous helpful comments and corrections in the preparation of this manuscript.

2. Patches and local-global principles

2.1. Fields associated to patches. The fields F_{ξ} are not canonically defined - they depend on a number of choices, beginning with the choice of a model for X over T.

Definition 2.1.1 Models A model for the scheme X/K is defined to be a connected normal projective ℙ_{T}^{1} -scheme X̂ such that

(1) the structure morphism f : X̂ → ℙ_{T}^{1} is finite,

(2) considered as a T -scheme, the generic fiber X̂_{K} is isomorphic to X,

(3) The reduced closed fiber X̂_{k}^{red} is a normal crossings divisor in X̂,

(4) f^{−1}(∞) contains all the singular points of the reduced closed fiber X̂_{k}^{red}.

Given a model X̂ (we will generally suppress the morphism f : X̂ → ℙ_{T}^{1} from the notation), we let S(X̂) denote the set of closed points in f^{−1}(∞) and 𝒰(X̂) denote the set of connected (or equivalently, irreducible) components of X̂_{k}^{red} \ S(X̂). These sets play a critical role in what follows.

Warning 2.1.2 In other sources such as [HH, HHKa, HHKb], X̂ is not given the structure of a ℙ_{T}^{1} -scheme, but rather the structure of a T scheme together with a distinguished set of closed points S. In this context, one is allowed more general sets S. The reader must keep in mind that a model X̂ comes with the extra structure of a morphism to ℙ^{1} throughout!

It is perhaps a bit odd to include the finite morphism to ℙ_{T}^{1} as part of the definition of a model — by comparison, in [HH], it is only assumed that one should start with a projective T -curve with a set S of closed points such that there exists a finite T -morphism to a curve with smooth reduced closed fiber and such that the set S is the inverse image of a set of closed points under this morphism. We include the morphism to ℙ_{T}^{1} as part of our definition simply as a matter of convenience of exposition. The following lemma shows that it is not much of an extra assumption, however:

Lemma 2.1.3 [HH], Proposition 6.6 Suppose X̂ is a projective T -curve and S ⊂ X̂ a finite set of closed points. Then there exists a finite morphism f : X̂ → ℙ^{1} such that S ⊂ f^{−1}(∞).

For the remainder of the section, we will suppose that we are given such a model X̂, and we let F = F (X̂) be its function field. Given any nonempty subset of points Z ⊂ X̂, we define

R_{Z} = {f ∈ F | ∀P ∈ Z, f ∈ 𝒪_{X̂,P} }

We will define fields associated to two particular types of subsets Z:

Definition 2.1.4 Fields associated to closed points Let P ∈ X̂ be a closed point. We define R_{P} = R_{{P }}, R̂_{P} its completion with respect to its maximal ideal, and F_{P} the field of fractions of R̂_{P}.

Definition 2.1.5 Fields associated to open subsets of X̂_{k}^{red} Let U ⊂ X̂_{k}^{red} be a nonempty irreducible Zariski open affine subset of the reduced closed fiber which is disjoint from the singular locus of X̂_{k}^{red}. We let R̂_{U} be the completion of R_{U} with respect to the t-adic valuation, and F_{U} the field of fractions of R̂_{U}.

Note that there are natural maps F ⊂ F_{U}, F_{P} for any such P and U, as well as inclusions F_{U} → F_{V} and F_{U} → F_{P}, whenever V ⊂ U or P ∈ U respectively.

2.2. Some local-global principles. We may now give some examples of local-global principles. For these, we assume that X/K is a smooth projective curve where K a complete discretely valued field with valuation ring T, and that we are given a model X̂ → ℙ^{1}. We let F be the function field of X.

Theorem 2.2.1 Local-global principle for the Brauer group (see [HH], Theorem 4.10) The natural homomorphism

Br(F) → ( ∏_{P ∈S(X̂)} Br(F_{P})) × ( ∏_{U∈𝒰(X̂)} Br(F_{U}))

is injective.

We give a proof of this result on page 5. In fact, we will see later, using patching, that this may be extended to a three term exact sequence by adding a term on the right (see Theorem 3.3.1).

Theorem 2.2.2 Local-global principle for isotropy (see [HHKa], Theorem 4.2) Suppose q is a regular quadratic form of dimension at least 3, and char(F) ≠ 2. If q_{F_{P}} and q_{F_{U}} are isotropic for every P ∈ S(X̂) and U ∈ 𝒰(X̂) then q is also isotropic.

The proof of this is given on page 5. See also Theorem 3.3.2 for a related result.

Both of these principles in fact, may be regarded as special cases of either of the following results, the main new results of this paper:

Definition 2.2.3 Suppose H is a variety over F and G is an algebraic group which acts on H. We say that G acts transitively on H if for every field extension L/F, the group G(L) acts transitively on the set H(L).

The following result generalizes [HHKa], Theorem 3.7 by weakening the hypothesis of rationality to allow for retract rational groups as well:

Theorem 2.2.4 Local-global principle for varieties with transitive actions Suppose G is a connected retract rational algebraic group defined over F, and H is a variety on which G acts transitively. Then H(F) ≠ ∅ if and only if H(F_{P}), H(F_{U}) ≠ ∅ for all P ∈ S(X̂) and U ∈ 𝒰(X̂).

This theorem follows quickly from Theorem 5.1.1, and its proof may be found just after the statement of this theorem on page 14. The proof of this in the case of retract rationality will occupy a good portion of this paper. Along the way, we will explore the connections between this local global principles and the notion of “factorization” for the group G. The following corollary is particularly useful.

Corollary 2.2.5 Suppose G is a retract rational reductive group over F and H a homogeneous variety for G. Then H(F) ≠ ∅ if and only if H(F_{P}), H(F_{U}) ≠ ∅ for all P ∈ S(X̂) and U ∈ 𝒰(X̂).

From these theorems (or even the versions assuming only rationality of G from [HHKa]), we may prove the above local global results concerning the Brauer group and quadratic forms.

proof of Theorem 2.2.1. Let α ∈ Br(F) and suppose α_{F_{P}} = 0 = α_{F_{U}} for every P ∈ S(X̂), U ∈ 𝒰(X̂). We need to show that α = 0.

Let A be a central simple F algebra in the class α and let H be the Severi-Brauer variety for A. Note that this is a homogeneous variety for the group GL(A) which is rational, connected and reductive. Recall that for a field extension L/F, H(L) is nonempty exactly when A⊗_{F} L is a split algebra — that is to say, α_{L} = 0. But since α_{F_{P}} , α_{F_{U}} = 0, we have H(F_{P}), H(F_{U}) ≠ ∅ for every U, P. Consequently, by Corollary 2.2.5, it follows that H(F) ≠ ∅ and so α = 0 as desired. □

proof of Theorem 2.2.2. Let q be a quadratic form over F satisfying the hypotheses of the Theorem. We wish to show that q is isotropic. Let H be the quadratic hypersurface of projective space defined by the equation q = 0. Recall that this is a homogeneous variety for the group SO(q) which under the hypotheses is a rational, connected, reductive group (see [KMRT98], page 209, excercise 9). As above, we immediately see that since H(F_{P}), H(F_{U}) are nonempty for each P ∈ S(X̂) and U ∈ 𝒰(X̂), we have by Corollary 2.2.5, H(F) ≠ ∅ as desired. □

3. Patching

The fundamental idea of patching is that defining an algebraic object over the field F is equivalent to defining objects over each of the fields F_{P} for P ∈ S(X̂) and F_{U} for U ∈ 𝒰(X̂), together with the data of how these objects agree on overlaps. This will be stated in this section in terms of an equivalence of categories. We will simply cite the results of [HH] section 6 and 7 for the most part, but we focus more on the equivalence of tensor categories, and explore how to produce other examples of algebraic patching.

Suppose we are given a model X̂ for a curve X/K. Given a point P ∈ S(X̂), the height 1 primes of R_{P} which contain t correspond to the components of X̂_{k}^{red} incident to P. Each such component is the closure of a uniquely determined element U ∈ 𝒰(X̂).

Definition 3.0.6 Branches, and their fields Given such a height 1 prime 𝒫 of R_{P}, corresponding to an element U ∈ 𝒰(X̂), a branch along U at P is an irreducible component of the scheme R̂_{P} /𝒫R̂_{P}. Alternately, these are in correspondence with the height one primes of R̂_{P} containing 𝒫R̂_{P}. Given such a height 1 prime ℘, we let R̂_{℘} be the t-adic completion of the localization of R̂_{P} at ℘, and F_{℘} its field of fractions. We let ℬ(X̂) denote the set of all branches at all points in S(X̂).

The fields F_{P} and F_{U} come equipt with natural inclusions into F_{℘} which we now describe. We note that the natural inclusion R̂_{P} → R̂_{℘} induces an inclusion of fields F_{P} → F_{℘}. Further, we note that R̂_{℘} is a 1 dimensional regular local ring, and hence a DVR, whose valuation is determined by considering order of vanishing along the branch corresponding to ℘. In particular, considering the inclusion F ⊂ F_{P} ⊂ F_{℘}, we find that all the elements of R_{U}, having no poles along U, can also not have poles along any branch lying along U, and in particular, we see we have an inclusion R_{U} ⊂ R̂_{℘}. Since the t-adic topology on R̂_{℘} is the same as the ℘-adic topology, we further find that R̂_{℘} is t-adically complete, and we therefore have an induced inclusion F_{U} → F_{℘}.

3.1. Patching finite dimensional vector spaces.

Definition 3.1.1 Patching problems A patching problem is a collection V_{ξ} for ξ ∈ S(X̂) ∪ 𝒰(X̂), where V_{ξ} is a finite dimensional F_{ξ} vector space together with a collection of isomorphisms ϕ_{℘} : V_{P} ⊗_{F_{P}} F_{℘} → V_{U} ⊗_{F_{U}} F_{℘} of F_{℘} vector spaces for every branch ℘ at P on U. We denote this problem by (V, ϕ). We define a morphism of patching problems f : (V, ϕ) → (W, ψ) to be a collection of homomorphisms f_{ξ} : V_{ξ} → W_{ξ} such that whenever ℘ is a branch at P lying on U, the following diagram commutes:

\xymatrix{V_{P} ⊗_{F_{P}} F_{℘} \ar[r]^{f_{P} ⊗F_{℘}} \ar[d]_{ϕ_{℘}} & W_{P} ⊗_{F_{P}} F_{℘} \ar[d]^{ψ_{℘}} \\ V_{U} ⊗_{F_{U}} F_{℘} \ar[r]_{f_{U} ⊗F_{℘}} & V_{P} ⊗_{F_{P}} F_{℘}}

We see then that patching problems naturally form a category, which we denote by 𝒫𝒫(X̂, S). In fact, this category has a ⊗-structure as well defined by (V, ϕ)⊗(W, ψ) = (V ⊗W, ϕ⊗ψ) where (V ⊗W)_{ξ} = V_{ξ}⊗_{F_{ξ}} W_{ξ} and

(ϕ ⊗ ψ)_{℘} : (V ⊗ W)_{P} → (V ⊗ W)_{U}

is given by ϕ_{℘} ⊗_{F_{℘}} ψ_{℘} via the above identification. One may also verify that this monoidal structure is symmetric and closed (see [ML98], VII.7).

Definition 3.1.2 If V is a vector space over F, we let (Ṽ, 𝕀) denote the patching problem defined by Ṽ_{P} = V_{F_{P}} and Ṽ_{U} = V_{F_{U}} and where 𝕀_{℘} is induced by the natural identifications

(V ⊗_{F} F_{P}) ⊗_{F_{P}} F_{℘} = V_{℘} = (V ⊗_{F} F_{U}) ⊗_{F_{U}} F_{℘}

Theorem 3.1.3 [[HH], Theorem 6.4] Consider the functor

Ω : 𝒱ect_{f.d.}(F) → 𝒫𝒫(X̂, S)

from the category of finite dimensional F -vector spaces to the category of patching problems defined by sending a finite dimensional vector space V to the patching problem (Ṽ, 𝕀). Then Ω is an equivalence of categories.

3.2. Patching algebraic objects.

Definition 3.2.1 A type of algebraic object (generally abbreviated to simply a “type”) is a symmetric closed monoidal category 𝒯 . If 𝒯 is a type and L a field, then an algebraic object of type 𝒯 over L is a strict symmetric closed monoidal functor (see [ML98], §VII.1, §VII.7 and [Hov99] §4.1 for definitions) from the category 𝒯 to the category of finite dimensional vector spaces over L (with its natural symmetric closed monoidal structure). Morphisms between algebraic objects of type 𝒯 are defined simply to be natural transformations between functors. We let 𝒯 (L) denote the category of such objects.

Note that 𝒯 in fact defines a (pseudo-)functor from the category of fields to the 2-category of categories (see [Gra74] for definitions).

Despite the formality of this definition, one may observe that one may interpret an algebraic object of a given type 𝒯 to be given by a vector space, or a collection of vector spaces, together with extra structure encoded by perhaps a collection of morphisms between various tensor powers of the vector spaces satisfying certain axioms, and where morphisms between these objects are given by collections of linear maps satisfying certain compatibilies with the extra structures given. For example, we might consider:

• Lie algebras,

• Alternative (or Jordan) algebras,

• Operads,

• Central simple algebras

• Quadratic forms, where morphisms are isometries,

• Quadratic forms, where morphisms are similarities,

• Separable commutative or noncommutative algebras,

• G-Galois extensions of rings in the sense of [DI71]

• and so on...

In these cases, the category 𝒯 in question is simply given as the symmetric closed monoidal category generated by some set of objects (corresponding to the underlying vector spaces of the structure) and some morphisms (defining the structure of the algebra or form), such that certain diagrams commute which define the structure in question. For example, a central simple algebra is a vector space A together with a bilinear product A ⊗ A → A, and F -algebra structure F → A such that the canonical “sandwich map” of algebras

A ⊗ A^{op} → Hom(A, A)

is an isomorphism (see [DI71], Chapter 2, Theorem 3.4(iii)). In this case, the category 𝒯 is generated by a single element a, a morphism a ⊗ a → a and 𝟙 → a (where 𝟙 is the unit for the monoidal structure), and such that the natural map a ⊗ a → Hom(a, a) (where the Hom is defined by the closed structure) has an inverse.

To see quadratic forms and isometries in this way, one may simply let the category 𝒯 be generated by a single element v a morphism v ⊗ v → 𝟙, assumed to commute with the morphism switching the order of the v’s. In the case of similarities instead of isometries, one may add a new object ℓ, and replace v ⊗ v → 𝟙 with a morphism v ⊗ v → ℓ. To force ℓ to correspond to a 1-dimensional vector space, one may then add to this category an inverse to the natural morphism 𝟙 → ℓ ⊗ ℓ^{∗} ≅ Hom(ℓ, ℓ).

Definition 3.2.2 Patching problems Let 𝒯 be a type of algebraic object. A patching problem of objects of type 𝒯 is a collection A_{ξ} for ξ ∈ S(X̂) ∪ 𝒰(X̂), where A_{ξ} is an object of type 𝒯 over F_{ξ}, together with a collection of isomorphisms ϕ_{℘} : A_{P} ⊗_{F_{P}} F_{℘} → A_{U} ⊗_{F_{U}} F_{℘} in 𝒯 (F_{℘}). We denote this problem by (A, ϕ).

Just as with vector spaces, we may define morphisms of patching problems of objects of type 𝒯 , and again find that these form a tensor category, which we denote 𝒫𝒫_{𝒯} (X̂). Again as before, if A is an algebraic object of type 𝒯 over F, we may form a natural patching problem (Ã, 𝕀), and obtain a functor from 𝒯 (F) to 𝒫𝒫_{𝒯} (X̂).

Theorem 3.2.3 Consider the functor

Ω_{𝒯} : 𝒯 (F) → 𝒫𝒫_{𝒯} (X̂)

defined by sending an algebraic object A to the patching problem (A, 𝕀). Then Ω_{𝒯} is an equivalence of categories.

Proof. Since we have an equivalence of categories 𝒱ect_{f.d.}(F) ≅ 𝒫𝒫(X̂) by Theorem 3.1.3, it is immediate that this equivalence also induces an equivalence of functor categories

𝒯 (F) = F un(𝒯 , 𝒱ect_{f.d.}(F) ≅ F un(𝒯 , 𝒫𝒫(X̂)) ≅ 𝒫𝒫_{𝒯} (X̂).

One may now check that this gives the desired equivalence. □

Remark 3.2.4 It would be interesting to know if one could extend this to equivalences of other kinds of objects. In particular, infinite dimensional vector spaces, finitely generated commutative algebras, or perhaps even to (some suitably restricted) categories of schemes. None of these fall under the definition of an algebraic object given above, and it is therefore not at all clear if the conclusions of Theorem 3.2.3 will still hold.

3.3. Central simple algebras and quadratic forms. For the following results, we suppose we are given X̂ a normal, connected, projective, finite ℙ_{T}^{1} -scheme. The machinery of patching gives the exactness of various exact sequences relating to field invariants derived from algebraic objects.

Theorem 3.3.1 (see [HH], Theorem 7.2) We have an exact sequence:

0 → Br(F) → ( ∏_{P ∈S(X̂)} Br(F_{P})) × ( ∏_{U∈𝒰(X̂)} Br(F_{U})) → ∏_{℘∈ℬ(X̂)} Br(F_{℘})

Proof. Exactness on the left was noted in Theorem 2.2.1. To see exactness in the middle, suppose we have classes α_{P} , α_{U} such that (α_{U})_{F_{℘}} ≅ (α_{P})_{F_{℘}} whenever ℘ is a branch at P on U. Since there are only a finite number of points and components, we may choose an integer n such that each of the Brauer classes α_{U}, α_{P} may be represented by central simple algebras A_{U}, A_{P} of degree n. Now, by hypothesis, for each branch ℘ as above, we may find an isomorphism of central simple algebras ϕ_{℘} : (A_{P})_{F_{℘}} → (A_{U})_{F_{℘}}. But this gives the data of a patching problem for central simple algebras, and therefore we may find a central simple F -algebra A such that A_{F_{P}} ≅ A_{P} and A_{F_{U}} ≅ A_{U} as desired. □

Theorem 3.3.2 We have an exact sequence:

W(F) → ( ∏_{P ∈S(X̂)} W(F_{P})) × ( ∏_{U∈𝒰(X̂)} W(F_{U})) → ∏_{℘∈ℬ(X̂)} W(F_{℘})

Proof. The proof is very similar to the last one. Suppose we have Witt classes α_{P} , α_{U} such that (α_{U})_{F_{℘}} ≅ (α_{P})_{F_{℘}} whenever ℘ is a branch at P on U. Since there are only a finite number of points and components, we may choose an integer n such that each of the Witt classes α_{U}, α_{P} may be represented by quadratic forms q_{U}, q_{P} of the same dimension n. Now, by hypothesis, for each branch ℘ as above, we may find an isometry ϕ_{℘} : (q_{P})_{F_{℘}} → (q_{U})_{F_{w}p}. But this gives the data of a patching problem for quadratic forms, and therefore we may find a form q over F such that q_{F_{P}} ≅ q_{P} and q_{F_{U}} ≅ q_{U} as desired. □

We note that exactness on the left is discussed in Theorem 2.2.2.

3.4. Properties of R̂_{P} , R̂_{U}, F_{P} , F_{U}. Let us now gather together some fundamental facts which we will need in the sequel.

Lemma 3.4.1 [HH], Lemma 6.2 Suppose Ŷ → X̂ is a finite morphisms of projective, normal, finite ℙ_{T}^{1} -schemes. Then the natural inclusions of fields yield isomorphisms:

F_{P} ⊗_{F (X̂)}F (Ŷ) ≅ ∏ F_{P′} F_{U}⊗_{F (X̂)}F (Ŷ) ≅ ∏ F_{U′} F_{℘}⊗_{F (X̂)}F (Ŷ) ≅ ∏ F_{℘′}

where P′ (resp. U′, ℘′) range over all the points (resp. components, branches) lying over P (resp. U, ℘).

Lemma 3.4.2 [HH], Lemma 6.3 Let X̂ be a projective, normal, finite ℙ_{T}^{1} -scheme. Then the natural inclusions of fields yield an exact sequence of F = F (X̂)-vector spaces:

0 → F → ( ∏_{P ∈S(X̂)} F_{P} ) × ( ∏_{U∈𝒰(X̂)} F_{U} ) → ∏_{℘∈ℬ(X̂)} F_{℘}

Lemma 3.4.3 Suppose 𝔙, 𝔚 ⊂ 𝔘 are R̂_{0} modules, where t ∈ R̂_{0} and 𝔙, 𝔚, 𝔘 are all t-adically complete. Suppose further that 𝔙/t𝔙 + 𝔚/t𝔚 = 𝔘/t𝔘. Then 𝔙 + 𝔚 = 𝔘.

Proof. Suppose u ∈ 𝔘. Let v_{0} = w_{0} = 0. We will inductively construct a sequence of elements v_{i} ∈ 𝔙, w_{i} ∈ 𝔚 such that v_{i} − v_{i+1} ∈ t^{i}𝔙, w_{i} − w_{i+1} ∈ t^{i}𝔚, v_{i} + w_{i} − u ∈ t^{i}𝔘. By completeness, these will converge to elements v ∈ 𝔙, w ∈ 𝔚 such that v + w = u.

Suppose we have constructed v_{i}, w_{i} satisfying the above hypotheses. Since u − v_{i} − w_{i} ∈ t^{i}𝔘, we may write u − v_{i} − w_{i} = t^{i}r. By hypothesis, we may write r = v′ + w′ + tr′ for some v′ ∈ 𝔙, w′ ∈ 𝔚, r′ ∈ 𝔘. Setting v_{i+1} = v_{i} + t^{i}v′, w_{i+1} = w_{i} + t^{i}w′, completes the inductive step. □

Lemma 3.4.4 Considering ℙ_{T}^{1} , we have R̂_{𝔸^{1}}+R̂_{∞} = R̂_{℘}, where ℘ is the unique branch at ∞.

Proof. Using Lemma 3.4.3, we need only check that \overline{R}_{𝔸^{1}} + \overline{R}_{∞} = \overline{R}_{℘}, where

\overline{R}_{𝔸^{1}} ≅ R̂_{𝔸^{1}}/tR̂_{𝔸^{1}}, \overline{R}_{∞} ≅ R̂_{∞}/tR̂_{∞}, \overline{R}_{℘} ≅ R̂_{℘}/tR̂_{℘}

But, we may compute \overline{R}_{𝔸^{1}} = k[𝔸_{k}^{1}], \overline{R}_{∞} = { \atop 𝒪_{ℙ_{k,∞}^{1}}}, \overline{R}_{℘} = frac({ \atop 𝒪_{ℙ_{k,∞}^{1}}}). Writing x for the coordinate function on the affine part of the k-line, we may explicitly identify

\overline{R}_{𝔸^{1}} = k[x], \overline{R}_{∞} = k[[x^{−1}]], \overline{R}_{℘} = k((x^{−1})),

and the result follows. □

4. Local-global principles, factorization and patching

Let X̂ → ℙ^{1} be a model for X/K, and let G be an algebraic group defined over F.

4.1. Local-global principles for rational points.

Definition 4.1.1 We say that factorization holds for G, with respect to X̂, if for every tuple (g_{℘})_{℘∈ℬ(X̂)} , there exist collections of elements g_{P} for each P ∈ S(X̂) and g_{U} for each U ∈ 𝒰(X̂) such that whenever ℘ is a branch at P on U we have

g_{℘} = g_{P} g_{U}

with respect to the natural embeddings F_{P} , F_{U} → F_{℘}.

Definition 4.1.2 We say that the local-global principle holds for an F scheme V, with respect to a model X̂ if X(F) ≠ ∅ holds if and only if X(F_{P}), X(F_{U}) ≠ ∅ for every P ∈ S(X̂) and U ∈ 𝒰(X̂).

Definition 4.1.3 Let G be an algebraic group over F and H a scheme over F. We say that H is a transitive G-scheme if G acts transitively on H (see Definition 2.2.3).

Proposition 4.1.4 If factorization holds for a group G, then the local-global principle holds for all transitive schemes over G.

Proof. See the proof of Theorem 3.7 in [HHKa]. □

4.2. Local-global principles for algebraic objects and torsors.

Definition 4.2.1 We say that the local-global principle holds for an algebraic object A (of some given type) if for any algebraic object B (of the same type), we have A ≅ B if and only if A_{F_{P}} ≅ B_{F_{P}} and A_{F_{U}} ≅ B_{F_{U}} for all P, U. We say that the local-global principle holds for a type of algebraic object if it holds for every algebraic object of the type.

Proposition 4.2.2 The local-global principle holds for an algebraic object A′ if and only for every patching problem (A, ϕ) of algebraic objects such that A_{P} ≅ (A′)_{F_{P}} , A_{U} ≅ (A′)_{F_{U}} for all P, U, the isomorphism class of (A, ϕ) is independent of ϕ.

Proof. Suppose that the local-global principle holds for A′, and let (A, ϕ), (A, ψ) be two patching problems, such that A_{P} ≅ (A′)_{F_{P}} and A_{U} ≅ (A′)_{F_{U}} for each P, U. Since we may patch algebraic objects, we may find algebraic objects B_{1}, B_{2} over F whose patching problems are equivalent to (A, ϕ), (A, ψ) respectively. Since (B_{1})_{F_{U}} ≅ A_{F_{U}} ≅ (B_{2})_{F_{U}} and similarly for F_{P}, we find that by the local-global principle, B_{1} ≅ B_{2}. But therefore their associated patching problems are isomorphic, implying (A, ϕ) ≅ (A, ψ) as desired.

Conversely, suppose that (A, ϕ)’s isomorphism class is independent of ϕ for every patching problem. Suppose we are given A′, B′ be algebraic objects over F with associated patching problems (A, ϕ) and (B, ψ) respectively. Suppose further that (A′)_{F_{U}} ≅ (B′)_{F_{U}} and similarly for F_{P}. Since A_{U} ≅ (A′)_{F_{U}} ≅ (B′)_{F_{U}} ≅ B_{U} and A_{P} ≅ (A′)_{F_{P}} ≅ (B′)_{F_{P}} ≅ B_{P} for all U, P by definition, we may change ψ via these isomorphisms to find (B, ψ) ≅ (A, ψ′) for some ψ′. But therefore by hypothesis, (A, ϕ) ≅ (A, ψ′) ≅ (B, ψ). Since patching gives an equivalence of categories, we further conclude A′ ≅ B′, completing the proof. □

Remark 4.2.3 Let 𝒯 be a type of algebraic object, and A is a particular object of type 𝒯 . Let 𝒯_{A} denote the subclass of objects which are isomorphic to A (more precisely, 𝒯_{A} is the sub-pseudofunctor of 𝒯 which associates to every field extension L/F the category of algebraic objects of type 𝒯 over L which are isomorphic to the object A_{L}). Then 𝒯_{A} satisfies the hypotheses of patching — i.e. we have an equivalence of categories between the category 𝒫𝒫_{𝒯_{A}} (X̂) and 𝒯_{A}(F) — if and only if the localglobal principle holds for A. Note that in general 𝒯_{A} is not a “type of algebraic object,” described by some monoidal category in the sense described above.

Definition 4.2.4 Let G be an algebraic group over F. We say that the local-global principle holds for G if for α ∈ H^{1}(F, G), with α_{F_{P}} , α_{F_{U}} trivial for each P, U we have α trivial.

Note that since elements of H^{1}(F, G) correspond to torsors for G, we see immediately that the local-global principle will hold for G if and only if the local-global principle holds for all G-torsors, in the sense of Definition 4.1.2. From this observation and from Proposition 4.1.4, we immediately obtain:

Proposition 4.2.5 Suppose G is a linear algebraic group defined over F, and suppose that factorization holds for G with respect to X̂. Then the local global principle holds for G as well.

Proposition 4.2.6 Suppose A is an algebraic object of some type 𝒯 , whose automorphism group is the linear algebraic group G. Then the following are equivalent:

(1) the local-global principle holds for A,

(2) the local-global principle holds for G.

(3) factorization holds for G,

Proof. Since G is the automorphism group of A, by descent (see [Ser79], X.§2, Proposition 4), we may identify H^{1}(L, G_{L}) = Forms(A_{L}), the pointed set of twisted forms of A_{L}. In particular, it is immediate from the definition that the local global principle for A is equivalent to the local global principle for G.

Suppose we have a local-global principle for A, and consider a collection of elements g_{℘} ∈ G(F_{℘}). Consider the patching problem (B, ϕ) where B_{P} = A_{F_{P}} , B_{U} = A_{F_{U}}, and ϕ_{℘} = g_{℘}. By the local-global principle, this is isomorphic to the patching problem (Ã, 𝕀). By definition, we may find an isomorphism h : (B, 𝕀) → (B, ϕ). Let g_{P} = h_{P}^{−1} and g_{U} = h_{U}. By definition of a morphism of patching problems, we find that g_{℘} = g_{P} g_{U}, and that g_{P} ∈ Aut(B_{P}) = G(F_{P}), g_{U} ∈ Aut(B_{U}) = G(F_{U}) as desired.

Conversely, suppose we have factorization for G. In this case it is immediate from Proposition 4.2.5 that the local global principle must hold for G, completing the proof. □

Remark 4.2.7 Theorem 4.2.6 raises the question of whether it would be possible to show the equivalence of the local global principle for a group G and factorization for this group without the presence of an algebraic object with G as its automorphism group. This would give a converse to Proposition 4.2.5. In turn since G-torsors are, in particular, transitive G-schemes, one would then also obtain a converse to Proposition 4.1.4.

5. Factorization for retract rational groups

5.1. Overview and preliminaries. The goal of this section will be to prove the following theorem:

Theorem 5.1.1 Suppose X̂ is a connected normal finite ℙ_{T}^{1} -scheme, with function field F and let G be a connected retract rational algebraic group over F. Then factorization holds for G with respect to X̂.

Using this theorem, we may easily proceed to the proof of the local global principle for schemes with transitive action stated earlier in Theorem 2.2.4: If G is a connected retract rational group over F, then by the theorem, factorization holds for G with respect to X̂. But then by Proposition 4.1.4, the local global principle must hold for transitive G schemes, as desired.

The proof of this theorem will occupy the remainder of the section. Our strategy will be to reduce this to a more abstract factorization problem, arising from the case when X̂ = ℙ_{T}^{1} . Overall, the proof stategy is roughly parallel to that followed in [HHKa], where retractions of open subsets of affine space take the place of open subsets of affine space.

Definition 5.1.2 Suppose we have commutative rings F ⊂ F_{1}, F_{2} ⊂ F_{0}, and an algebraic group G over F. We will say that factorization holds with respect to G, F, F_{1}, F_{2}, F_{0} if for every g_{0} ∈ G(F_{0}) there exist g_{1} ∈ G(F_{1}) and g_{2} ∈ G(F_{2}) such that g_{0} = g_{1}g_{2}.

Note that here we are omitting from the notation the homomorphism G(F_{i}) → G(F_{0}) for i = 1, 2. We are particularly interested in this in the following case. Suppose X̂ is a connected, normal, finite ℙ_{T}^{1} -scheme. In this case, we set F = F (X̂), and we let

F_{1} = ∏_{P ∈S(X̂)} F_{P} , F_{2} = ∏_{U∈𝒰(X̂)} F_{U}, F_{0} = ∏_{℘∈ℬ(X̂)} F_{℘}

Remark 5.1.3 It follows immediately from the definitions that factorization holds for the group G with respect to X̂ in the sense of Definition 4.1.1 if and only if factorization holds for G, F, F_{1}, F_{2}, F_{0} in the sense of Definition 5.1.2 where F, F_{1}, F_{2}, F_{0} are as above.

Back to the somewhat more abstract setting, suppose that F is some field, and let L/F be a finite algebra. Recall that if G is a linear algebraic group scheme, we may define its Weil restriction, also referred to as its corestriction or transfer, as the linear algebraic group with the functor of points defined by:

R_{L/F} G(R) = G(R ⊗_{F} L)

where R ranges through all F -algebras ([Gro62], Exp. 195, p. 13 for the definition and Exp. 221, p. 19 for proof of existence). We note that the corestriction in fact comes from a Weil restriction functor from the category of quasi-projective L-schemes to the category of quasiprojective F -schemes, and that this functor takes open inclusions to open inclusions, and takes affine space to affine space (of a different dimension). In particular, it follows that the corestriction of a rational (or retract rational) variety is itself rational (resp. retract rational).

We note the following Lemma, which is an consequence of the definition of the corestriction in terms of the functor of points given above.

Lemma 5.1.4 Let F be a field, and suppose we are given rings F ⊂ F_{1}, F_{2} ⊂ F_{0}, and an finite algebra L/F. Let G be a linear algebraic group over L. Then factorization holds for G, L, L ⊗_{F} F_{1}, L ⊗_{F} F_{2}, L ⊗ F_{0} if and only if it holds for R_{L/F} G, F, F_{1}, F_{2}, F_{0}.

Lemma 5.1.5 Suppose that we are given a morphism of connected projective normal finite ℙ_{T}^{1} -schemes f : Ŷ → X̂. Let L be the function field of Ŷ and F the function field of X̂. Then factorization holds for G, Ŷ if and only if it holds for R_{L/F} G, X̂.

Proof. This follows immediately from the universal property of the Weil restriction, together with Lemma 3.4.1. □

Lemma 5.1.6 Let F be the function field of ℙ_{T}^{1} , which we consider as a ℙ_{T}^{1} -scheme via the identity map. Suppose that for every connected retract rational group G over F, factorization holds for G with respect to ℙ_{T}^{1} (as in Definition 4.1.1). Then for every normal finite ℙ_{T}^{1} -scheme X̂ with function field L, and every connected retract rational group H over L, factorization holds for H with respect to X̂.

Proof. This follows immediately from Lemma 5.1.5. □

As a consequence of this, in order to prove Theorem 5.1.1, we may restrict to the setting where F is the function field of ℙ_{T}^{1} , and where F_{1} = F_{∞}, F_{2} = F_{𝔸_{k}^{1}}, and where F_{0} = F_{℘} is the field associated to the unique branch ℘ along 𝔸_{k}^{1} at ∞. We let R̂_{0} = R̂_{℘}, 𝔙 = R̂_{∞}, 𝔚 = R̂_{𝔸_{k}^{1}}. For convenience, in the sequel we will often refer to the following hypothesis for factorization.

Hypothesis 5.1.7 see [HHKa], Hypothesis 2.4 We assume that the complete discrete valuation ring R̂_{0} contains a subring T which is also a complete discrete valuation ring having uniformizer t, and that F_{1}, F_{2} are subfields of F_{0} containing T. We further assume that 𝔙 ⊂ F_{1} ∩ R̂_{0}, 𝔚 ⊂ F_{2} ∩ R̂_{0} are t-adically complete T -submodules satisfying 𝔙 + 𝔚 = R̂_{0}.

Lemma 5.1.8 With respect to the scheme ℙ_{T}^{1} consider F = F (ℙ_{T}^{1} ), F_{0} = F_{℘}, F_{1} = F_{∞}, F_{2} = F_{𝔸_{k}^{1}}, R̂_{0} = R̂_{℘}, 𝔙 = R̂_{∞}, 𝔚 = R̂_{𝔸_{k}^{1}}. Then these rings and modules satisfy the Hypothesis 5.1.7.

Proof. The completeness of 𝔙, 𝔚 is satisfied by definition. The fact that 𝔙 + 𝔚 = R̂_{0} follows from Lemma 3.4.4. □

5.2. Retractions – basic definitions and properties. Before attacking the problem of factorization directly, it is necessary to collect some facts concerning retractions and retract rational varieties.

Definition 5.2.1 We say that a variety Y is a retraction of a variety U if there exist morphisms i : Y → U and p : U → Y such that pi = id_{Y}. We say that it is a closed retraction if i is a closed embedding.

Remark 5.2.2 In the case of a closed retraction, we will occasionally abuse notation by simply regarding i as an inclusion.

Lemma 5.2.3 Retractions are open Suppose we are given topological spaces Y, U and continuous maps i : Y → U and p : U → Y such that pi = id_{Y}. Then p is an open map (i.e. for V ⊂ U open, p(U) is open).

Proof. This follows immediately from the fact that p(V) = i^{−1}(V) and the continuity of i. □

Definition 5.2.4 We say that Y is a rational retraction of U if there are rational maps i : Y −−→ U and p : U −−→ Y such that pi = id_{Y} on some open set on which pi is defined.

Definition 5.2.5 We say a variety Y is retract rational if it is a rational retraction of 𝔸^{n} for some n.

Lemma 5.2.6 Rational retractions shrink to retractions Suppose Y is a rational retraction of U via rational maps i, p. Then we may find dense open subsets Y_{0} ⊂ Y and U_{0} ⊂ U such that i, p make Y_{0} a retraction of U_{0}.

Proof. We may find open subsets Ỹ ⊂ Y and Ũ ⊂ U such that i, p restrict to morphisms on these sets, i.e. we have:

\xymatrix{Y \ar@{-->}[r] & 𝔸^{n} \ar@{-->}[r] & Y \\ Ỹ \ar[u] \ar[ur]^{i} & Ũ \ar[u] \ar[ur]^{p} & Ỹ \ar[u]}

We choose Y′ = i^{−1}p^{−1}Ỹ ⊂ Ỹ. We note that pi is defined on Y′ and so by definition, we may find Y_{0} ⊂ Y′ such that pi|_{Y_{0}} = id_{Y_{0}}. Let U = p^{−1}(Y_{0}). Then we have pi(Y_{0}) ⊂ Y_{0} and so i(Y_{0}) ⊂ p^{−1}(Y_{0}) = U. Since p(U) ⊂ Y_{0} by definition, we have constructed the desired morphisms. □

Lemma 5.2.7 Retractions shrink to closed retractions Suppose Y is a retraction of U via morphisms i, p. Then we may find dense open subvarieties Y_{0} ⊂ Y and U_{0} ⊂ U such that Y_{0} is a closed retraction of U_{0} via the restrictions of i, p.

Proof. Since we may identify Y with the image of p it follows that Y is locally constructible in U [EGA 4-1, p. 239 (Chevalley’s thm)]. By [EGA 3-1, p. 12], it follows that Y is the intersection of a closed and an open set in U. By setting U_{0} to be this open set, and Y_{0} = Y ∩ U_{0} it follows that Y_{0} is closed in U_{0}. Now it is easy to see that the restrictions of i, p exhibit Y_{0} as a retraction of U_{0}. □

Corollary 5.2.8 Rational retractions shrink to closed retractions Suppose Y is a rational retraction of U via rational maps i, p. Then we may find dense open subsets Y_{0} ⊂ Y and U_{0} ⊂ U such that i, p make Y_{0} a closed retraction of U_{0}.

Proof. This follows immediately from Lemmas 5.2.6 and 5.2.7. □

The following lemma gives us a first hint that retractions inherit some of the geometry of the larger spaces.

Lemma 5.2.9 Retractions of smooth schemes are smooth Suppose Y is a retraction of a smooth scheme U. Then Y is smooth.

Proof. This follows from the formal criterion for smoothness (see for example [Gro67] §17 or [Ill96] §2). From this formulation, in the language of [Ill96], we must show that if S_{0} → S is a thickening, and f : S_{0} → Y is a morphism, then we must be able to find a cover {V_{i}} of S and morphisms g_{i} : V_{i} → Y extending f|_{S_{0}∩V_{i}}. To see this, we first use the smoothness of U to find g̃_{i} : V_{i} → U extending i ∘ f|_{S_{0}∩V_{i}}. Now we set g_{i} = p ∘ g̃_{i}. We then have

g_{i}|_{S_{0}∩V_{i}} = p ∘ g̃_{i}|_{S_{0}∩V_{i}}
= p ∘ i ∘ f|_{S_{0}∩V_{i}}
= f|_{S_{0}∩V_{i}}

as desired. □

Lemma 5.2.10 Standard position for retractions Suppose Y is a smooth d-dimensional variety which is a closed retraction of an open subscheme U ⊂ 𝔸^{n}. We also suppose that with respect to the inclusion of Y in 𝔸^{n}, that 0 ∈ Y. Then we may shrink U and choose coordinates on U so that Y is the zero locus of polynomials f_{1}, …, f_{n−d} with

f_{i} = x_{i} + P_{i}

where the x_{i}’s are the coordinate functions on 𝔸^{n} and P_{i} is a polynomial in the x_{j}’s, each of whose terms is of degree at least 2.

Further, we may alter i and p defining the retraction so that the morphism ip : U → Y → U is given by

(x_{1}, …, x_{n}) ↦ (M_{1} + Q_{1}, …, M_{n} + Q_{n})

where

M_{i} = \begin{cases}0 & if 1 ≤ i ≤ n − d \\ x_{i} & if n − d < i ≤ n \end{cases}

and Q_{i} is a rational function in the variables x_{i}, regular on U, such that \frac{∂}{∂x_{j}} Q_{i}|_{0} = 0 for all i, j.

Proof. For purposes of skimmability, we have placed this proof at the end of the section. □

5.3. Adic convergence of Taylor series. The basic strategy for factorization will be to produce closer and closer approximations to a particular factorization. In order to carry this out, it is necessary to discuss notions of convergence and approximations in the adic setting, paralleling the discussion of [HHKa], Section 2.

Suppose F_{0} is a field complete with respect to a discrete valuation v with uniformizer t, and let |a| = e^{−v(a)} be a corresponding norm. Let A = F_{0}[x_{1}, …, x_{N}], 𝔪 the maximal ideal at 0, A_{𝔪} the local ring at 0 and Â = F_{0}[[x_{1}, …, x_{N}]] the complete local ring at 0. For I = (i_{1}, …, i_{N}) ∈ ℕ^{N}, we let |I| = ∑_{j} i_{j}. Define for r ∈ ℝ, r > 0

Â_{r} = {∑_{I} a_{I}x^{I} | lim_{|I|→∞} |a_{I}|r^{|I|} = 0}

and for f = ∑ a_{I}x^{I} ∈ Â_{r}, we set

|f|_{r} = sup_{I} |a_{I}|r^{|I|}.

We give 𝔸^{n}(F_{0}) a norm via the supremum of the coordinates

|(a_{1}, …, a_{N})| = max_{i}{|a_{i}|}

and we let D(a, r) be the closed disk of radius r about a ∈ 𝔸^{n}(F_{0}) with respect to the induced metric. We note that since the values of the metric are discrete, this disk is in fact both open and closed in the t-adic topology.

We note the following elementary lemma:

Lemma 5.3.1 Suppose a ∈ D(0, r), and f, g ∈ Â_{r}. Then

(1) f + g, fg ∈ Â_{r},

(2) |f + g|_{r} ≤ max{|f|_{r}, |g|_{r}},

(3) the group

{a ∈ Â_{r} | |a|_{r} < M} ⊂ Â_{r}

is complete with respect to the filtered collection of subgroups 𝔪^{i} ∩ Â_{r},

(4) |f|_{r} is finite,

(5) |fg|_{r} ≤ |f|_{r}|g|_{r},

(6) if r′ < r, then |f|_{r′} ≤ max\{|f(0)|, \frac{r′}{r} |f|_{r}\},

(7) f(a) is well defined (i.e. is a convergent series), and

(8) |f(a)| ≤ |f|_{r}, and if f(0) = 0 then |f(a)| ≤ |f|_{r}|a|r^{−1}.

Lemma 5.3.2 Suppose f ∈ A_{𝔪}. Then for all ε ≥ |f(0)| with ε > 0, there exists r > 0 such that f ∈ Â_{r} and |f|_{r} < ε. Further, for any δ > 0 we may choose r < δ.

Proof. Write f = g/h, g, h ∈ A with h ∉ 𝔪. Since A/𝔪 is a field, we may find h′ ∈ A with hh′ − 1 = −b ∈ 𝔪. Therefore, in Â, we have f = gh′(∑ b^{i}). Since g, h′, b are polynomials, they are in A_{r} for any r. Further, by Lemma 5.3.1(6), we may reduce r so that |gh′|_{r} ≤ |f(0)|, and since b(0) = 0, we may also ensure |b|_{r} < 1. In doing this, note that we may also ensure that r < δ. We note that by Lemma 5.3.1(5), |b^{i}|_{r} < 1. Now, by Lemma 5.3.1(3), it follows that | ∑ b^{i}|_{r} < 1. But therefore |f|_{r} = |gh′ ∑ b^{i}|_{r} < ε as desired. □

Lemma 5.3.3 The t-adic topology on 𝔸^{N}(F_{0}) is finer than the Zariski topology.

Proof. It suffices to show that if p ∈ 𝔸^{N}(F_{0}) and f is a polynomial not vanishing on p, we may find a disk about p on which f is nonvanishing. Without loss of generality, we may apply a translation and assume that p = 0. Let g = f − f(0). By Lemma 5.3.2, since g(0) = 0, we may find an r > 0 such that f ∈ Â_{r} and such that |g|_{r} < |f(0)| (using ε = |f(0)|). In particular, if a ∈ 𝔸^{N}(F_{0}) with |a| < r, we have |g(a)| ≤ |g|_{r} < |f(0)| by Lemma 5.3.1(8). Therefore, for such an a, f(a) = g(a) + f(0) ≠ 0. Therefore f does not vanish on a disk of radius r about the origin as desired. □

Proposition 5.3.4 Linear approximations and error term Suppose f ∈ Â_{r} for r ≤ 1. Write

f = c_{0} + L + P where P (x⃗) = ∑_{|ν|≥2} c_{ν}x^{ν},

where L is a linear form with coefficients in F_{0} and all c_{ν} ∈ F_{0}. Suppose |L + P |_{r} ≤ 1. Let 0 < ε ≤ |t|r^{2}, and suppose a, h ∈ 𝔸^{N}(F_{0}) with |h|, |a| ≤ ε. Then

|f(a + h) − f(a) − L(h)| ≤ |t||h|

.

Proof. This proof is a very slight modification of Lemma 2.2 in [HHKa]. Choose a real number s so that we may write |h| = ε|t|^{s}. We may rearrange the quantity of interest as:

f(a + h) − f(a) − L(h) = ∑_{|ν|≥2} c_{ν} ((a + h)^{ν} − a^{ν}) .

Since the absolute value is nonarchimedean, it suffices to show that for every term m = c_{ν}x^{ν} with |ν| ≥ 2 we have

|m(a + h) − m(a)| ≤ ε|t|^{s+1}.

For a given ν with |ν| ≥ 2, consider the expression (x + x′)^{ν} − x^{ν}, regarded as a homogeneous element of degree j = |ν| in the polynomial ring F_{0}[x_{1}, …, x_{N}, x′_{1}, …, x′_{N}]. Since the terms of degree j in x_{1}, …, x_{N} cancel, the result is a sum of terms of the form λℓ where λ is an integer and ℓ is a monomial in the variables x, x′ with total degree d in x_{1}, …, x_{N} and total degree d′ in x′_{1}, …, x′_{N}, such that d + d′ = j and d < j. Hence d′ ≥ 1. Consequently, for each term of this form,

|λℓ(a, h)| ≤ |ℓ(a, h)| ≤ ε^{d}(ε|t|^{s})^{d′} = ε^{j}|t|^{sd′} ≤ ε^{j}|t|^{s}.

Since (a + h)^{ν} − a^{ν} is a sum of such terms, and the norm is nonarchimedean, we conclude |(a + h)^{ν} − a^{ν}| ≤ ε^{j}|t|^{s}.

Since m = c_{ν}x^{ν}, it follows that

|m(a + h) − m(a)| ≤ |c_{ν}|ε^{j}|t|^{s} ≤ r^{−j}ε^{j}|t|^{s}.

Now ε ≤ |t|r^{2}, so ε^{j−1} ≤ |t|^{j−1}r^{2j−2}. Since |t| < 1, r ≤ 1, and j ≥ 2, we have

ε^{j−1} ≤ |t|^{j−1}r^{j+j−2} ≤ |t|r^{j}.

Rearranging this gives the inequality (ε/r)^{j} ≤ ε|t| and so (ε/r)^{j}|t|^{s} ≤ ε|t|^{s+1}. Therefore

|m(a + h) − m(a)| ≤ r^{−j}ε^{j}|t|^{s} ≤ ε|t|^{s+1} = |t||h|,

as desired.

□

Lemma 5.3.5 Local bijectivity Suppose f : U → V is a morphism between Zariski open subschemes of 𝔸_{F_{0}}^{d} containing the origins and such that f(0) = 0. Suppose further, that after writing the coordinates of f as elements of Â, we have f = (f_{1}, …, f_{d}) with f_{i} = x_{i} + Q_{i} and Q_{i} consisting of terms of degree higher than 2. Then we may find t-adic neighborhoods U′ ⊂ U(F_{0}) and V′ ⊂ V (F_{0}) of 0 such that f maps U′ bijectively onto V′. Further, we may assume that U′ and V′ are disks about the origin of equal radii.

Proof. By Lemma 5.3.2, since f(0) = 0, we may find 0 < r ≤ 1 such that f ∈ Â_{r} and |f|_{r} ≤ 1. Choose ε ≤ |t|r^{2} as in the statement of Proposition 5.3.4 and such that D_{0}(ε) ⊂ V (F_{0}) and D_{0}(ε) ⊂ U(F_{0}). Let V′ = D_{0}(ε) ⊂ V (F_{0}) and U′ = D_{0}(ε) ⊂ V (F_{0}). We claim that for b ∈ U′, we have |f(b)| ≤ ε and so f(b) ∈ V′. To see this, we note that Q_{i} ∈ Â_{r} and |Q_{i}|_{r} ≤ 1, and hence we may apply Proposition 5.3.4 (with 0 linear and constant term) to see that |Q_{i}(b)| ≤ |t||b| < |b| = max{|b_{i}|}. By the nonarchimedean property, this gives

|f(b)| = max{|f_{i}(b)|} = max{|b_{i} + Q_{i}(b)|} ≤ max{|b_{i}|, |Q_{i}(b)|} = max{|b_{j}|} = |b|

We consider first surjectivity. Note that both U′ and V′ are both closed and open. Since they are closed in a complete metric space, they contain all limits of their Cauchy sequences. Let a ∈ V′, and let b_{0} = 0. We will inductively construct elements b_{i} ∈ U′ such that |f(b_{i}) − a| ≤ ε|t|^{i}. In particular, since |a| ≤ ε, we have |f(b_{0}) − a| = |a| ≤ ε. Assuming we have constructed b_{i−1}, we let h = a − f(b_{i−1}), and note |h| ≤ ε|t|^{i−1} by hypothesis, and |b_{i−1}| ≤ ε since b_{i−1} ∈ U′. Therefore, by Proposition 5.3.4, we have

|f(b_{i−1} + h) − f(b_{i−1}) − h| ≤ |t||h| ≤ ε|t|^{i}

and so setting b_{i} = b_{i−1} + h, we find that since f(b_{i−1}) + h = a, we have

|f(b_{i}) − a| = |f(b_{i−1} + h) − a|
= |f(b_{i−1} + h) − f(b_{i−1}) − h + f(b_{i−1}) + h − a|

= |(f(b_{i−1} + h) − f(b_{i−1}) − h) + f(b_{i−1}) − a + a − f(b_{i−1})|

= |f(b_{i−1} + h) − f(b_{i−1}) − h| ≤ |t||h| ≤ ε|t|^{i}

as desired. Since b_{i} is a Cauchy sequence, using the completeness of U′, we may set b = lim b_{i} ∈ U′ and we find by continuity that f(b) = a as desired.

Next, we consider injectivity. Suppose a, b ∈ U′, let h = b − a and suppose h ≠ 0. We need to show that f(a) ≠ f(b). Since the valuation is nonarchimedean we have a, h ≤ ε. Let E = f(a+h)−f(a)−h. Then we find |E| ≤ |h||t| by Proposition 5.3.4. But this means in particular that |E+h| = |h| by the nonarchimedean triangle inequality. Therefore

|f(b) − f(a)| = |f(a + h) − f(a)| = |E + h| = |h| ≠ 0

so f(b) ≠ f(a) as desired. □

Lemma 5.3.6 Suppose that Y is a closed retraction of an open subscheme U ⊂ 𝔸^{n} in the standard form of Lemma 5.2.10 with respect to morphisms i, p. Then we may find a t-adic neighborhood V′ of 0 ∈ Y (regarding Y as a subscheme of U via i) such that the composition V′ → 𝔸^{n}(F_{0}) → 𝔸^{d}(F_{0}) is bijective onto a t-adic disk, where the last map is given by projection onto the last d coordinates.

Proof. As in Lemma 5.2.10, we suppose that Y is the zero locus of polynomials f_{1}, …, f_{n−d} with

f_{i} = x_{i} + P_{i}

where P_{i} is a polynomial in the x_{j}’s of degree at least 2. Using Lemma 5.3.2, we may choose 0 < r ≤ 1 such that f_{i} ∈ Â_{r} and |f_{i}|_{r} ≤ 1 for each of the finitely many functions f_{i}. Choose ε ≤ |t|r^{2} as in Proposition 5.3.4. Let U′ ∈ 𝔸^{n}(F_{0}) be the disk about the origin of radius ε, and let V′ be the intersection of the inverse image of U′ in Y (F_{0}) with the disk of radius ε in 𝔸^{n}(F_{0}). Let g : V′ → U′ be the composition map in the statement.

First, suppose a, b ∈ V′ with a ≠ b, and let h = b − a. If g(a) = g(b), then by definition of g, the last d coordinates of a and b must match. Since a ≠ b, we therefore know that x_{i}(h) ≠ 0 for some i = 1, …, n − d where x_{i} is the i’th coordinate function on 𝔸^{d}. We may therefore choose i such that |x_{i}(h)| has the largest possible value, and in particular, we then would have |x_{i}(h)| = |h|. But, estimating |f_{i}(b) − f_{i}(a)| = |f_{i}(a + h) − f_{i}(a)| using Proposition 5.3.4, we find

|f_{i}(a + h) − f_{i}(a) − x_{i}(h)| ≤ |t||h|.

We claim that |f_{i}(a + h) − f_{i}(a)| ≥ |h| and in particular that f_{i}(a) ≠ f_{i}(b). To see this must hold, assume by contradiction that |f_{i}(a + h) − f_{i}(a)| < |h|. In this case, we have

|f_{i}(a + h) − f_{i}(a) − x_{i}(h)| = |x_{i}(h)|

since |x_{i}(h)| = |h|. But therefore we have |h| ≤ |t||h| which is a contradiction since |t| < 1. Therefore, f_{i}(b) ≠ f_{i}(a) as claimed, forcing g to be injective.

For surjectivity, consider a ∈ U′, and consider its image b ∈ 𝔸^{d}. Using the form for the retraction in Lemma 5.2.10, we may apply Lemma 5.3.5 to the composition (shrinking ε if necessary)

\xymatrix{𝔸^{d} ∩ U \ar[r] & U \ar[r]^{p} & Y \ar[r] \ar[rr]^{g} & U \ar[r] & 𝔸^{d}}

By Lemma 5.3.5, we may find an inverse image b′ of b in 𝔸^{d} of norm less than ε. Consequently, by definition, the image of b′ in Y must actually live in V′, and this is an inverse image for a as desired. □

Corollary 5.3.7 In the notation of the previous lemma, we may choose t-adic neighborhoods of the origin U′ ∈ 𝔸^{d} and V′ ∈ Y such that the composition U′ → U → Y takes U′ bijectively to V′ and the composition V′ → Y → U → 𝔸^{n} → 𝔸^{d} takes V′ bijectively to U′.

Proof. By Proposition 5.2.10 and Lemma 5.3.5, we may find U′ ⊂ 𝔸^{d}, V′ ⊂ Y so that the composition U′ → V′ → U′ is bijective. By Lemma 5.3.6, we may find V′′ ⊂ V′ such that V′′ → U′ in bijective onto a t-adic disk U′′ ⊂ U′. But now again the composition U′′ → U′′ is bijective, and since V′′ → U′′ is also bijective, we find U′′ → V′′ is bijective as well. □

5.4. Factorization.

Theorem 5.4.1 Under Hypothesis 5.1.7, let f : 𝔸_{F_{0}}^{d} × 𝔸_{F_{0}}^{d} −−→ 𝔸_{F_{0}}^{d} be an F_{0}-rational map that is defined on a Zariski open set U ⊆ 𝔸_{F_{0}}^{d} × 𝔸_{F_{0}}^{d} containing the origin (0, 0). Suppose further that we may write:

f = (f_{1}, …, f_{d}), f_{i} ∈ k̂[x_{1}, y_{1} …, x_{d}, y_{d}]_{𝔪}

where f_{i} = x_{i} + y_{i} + ∑_{|(ν,ρ)|≥2} c_{ν,ρ,i}x^{ν}y^{ρ}.

Then there is a real number ε > 0 such that for all a ∈ 𝔸^{d}(F_{0}) with |a| ≤ ε, there exist v ∈ 𝔙^{d} and w ∈ 𝔚^{d} such that (v, w) ∈ U(F_{0}) and f(v, w) = a.

Proof. The proof of this theorem is exactly as in [HHKa], Theorem 2.5, wherein in the first paragraph, the problem is reduced to exactly the hypotheses which we assume. □

Theorem 5.4.2 Under Hypothesis 5.1.7, let m : Y × Y → Y be a rational F -morphism defined at (0, 0), and suppose that m(y, 0) = y = m(0, y) where it is defined. Suppose that Y is a closed retraction of an open subscheme of 𝔸^{n}. Then there exists ε > 0 such that for y ∈ Y (F_{0}) ⊂ 𝔸^{n}(F_{0}), |y| ≤ ε, there exist y_{i} ∈ Y (F_{i}), i = 1, 2 such that y = m(y_{1}, y_{2}).

Proof. We consider as in Corollary 5.3.7, t-adic neighborhoods of 0 U′ ⊂ 𝔸^{d}(F_{0}) and V′ ⊂ Y (F_{0}) such that we have bijections U′ → V′ and V′ → U′ defined by algebraic rational morphisms p′ : 𝔸^{d} −−→ Y and i′ : Y −−→ 𝔸^{d}. We consider

\xymatrix{V′ × V′ \ar[r]^{m|_{V′×V′}} & V′ \ar[d]_{i′} \\ U′ × U′ \ar[u]^{p′} \ar[r] & U′}

By hypothesis, the composition in the bottom U′ × U′ → U′ is given as the restriction of an algebraic rational morphism μ : 𝔸^{d} × 𝔸^{d} → 𝔸^{d}. By Corollary 5.3.7, it is sufficient to show that μ is surjective when restricted to a sufficiently small t-adic neighborhood.

We first shrink V′, U′ if necessary to make them contained in Zariski neighborhoods V, U as in Lemma 5.2.10. Now, we note that the rational map μ|_{𝔸^{d}×{0}} is just ip, since m|_{Y ×{0}} = id_{Y}. By Lemma 5.2.10, we find

m|_{𝔸^{d}×{0}}(x_{1}, …, x_{d}) = (x_{1} + Q_{1}, …, x_{d} + Q_{d})

where Q_{i} is a rational function in the variables x_{i}, regular on U, such that \frac{∂}{∂x_{j}} Q_{i}|_{0} = 0 for all i, j. But now we are done, using Theorem 5.4.1.

□

Theorem 5.4.3 Factorization for retract rational groups Under Hypothesis 5.1.7, assume that F = F (ℙ_{T}^{1} ), F_{1} = F_{𝔸_{k}^{1}}, F_{2} = F_{∞}, and F_{0} = F_{℘}, where ℘ is the unique branch at ∞. Let G be a retract rational connected linear algebraic group defined over F. Then for any g_{0} ∈ G(F_{0}) there exist g_{i} ∈ G(F_{i}), i = 1, 2, such that g_{1}g_{2} = g_{0} — that is to say, factorization holds for G with respect to ℙ_{T}^{1} (see Definition 4.1.1).

Proof. Using Lemma 5.2.7, we may find an open subscheme Y ⊂ G which is a retraction of an open subscheme U of affine space. In particular, Y must contain an F -rational point y ∈ Y (F), and after replacing Y by y^{−1}Y if necessary, we may assume Y contains the identity element of G. Using 5.4.2, where m is the multiplication map, we find that there exists ε > 0 such that factorization holds for g_{0} ∈ G(F_{0}) provided that |g_{0}| < ε. Fix such an epsilon, and suppose g_{0} ∈ G(F_{0}) is an arbitrary element. Since G is retract rational, it follows that G(F) is Zariski dense in G(F_{0}). Therefore, we have the existence of an element g′ ∈ G(F) such that g′^{−1}g_{0} ∈ Y. Since Y is a retraction of affine space, it follows that Y (F_{2}) is t-adically dense in Y (F_{0}). Therefore we may find g′′ ∈ Y (F_{2}) such that |g′^{−1}g_{0}g′′^{−1}| < ε. Writing g^{′−1}g_{0}g^{′′−1} = g_{1}g_{2} where g_{i} ∈ G(F_{i}), we conclude that g_{0} = (g′g_{1})(g_{2}g′′). Since g′g_{1} ∈ G(F_{1}) and g_{2}g′′ ∈ G(F_{2}), we are done. □

By Lemma 5.1.5 and the comments just following, we conclude that Theorem 5.1.1 holds.

5.5. Proof of Lemma 5.2.10.

Lemma 5.5.1 Suppose f = g/h for g, h ∈ k[x_{1}, …, x_{n}] with h(0) ≠ 0, g(0) = 0 and (∂f/∂x_{i})|_{0} = 0 for all i. Then if R is a k-algebra with h(0) ∈ R^{∗} and containing and element ϵ ∈ R, ϵ^{2} = 0 then f(ϵv) = 0 for v ∈ k^{n}.

Proof. Since g(0) = 0, we may write g = L + Q where L is a linear polynomial, and Q is a sum of homogeneous terms of degree at least 2. Now we simply note that

\frac{∂f}{∂x_{i}} = \frac{h(∂L/∂x_{i} + ∂Q/∂x_{i}) − (L + Q)(∂h/∂x_{i})}{h^{2}}

and in particular since h(0) ≠ 0, we find (∂f/∂x_{i})|_{0} = 0 implies that h(0)(∂L/∂x_{i}(0)) = 0, which implies that all the coefficients of the linear form L are 0 and so L = 0. Since h(0) ≠ 0, it follows that h(ϵv) is a unit, and we therefore may note that f(ϵv) = Q(ϵv)/h(ϵv) is well defined and ϵ^{2} = 0 implies Q(ϵv) = 0, showing that f(ϵv) = 0 as desired. □

We now proceed with the proof of Lemma 5.2.10. To begin, choose f_{1}, …, f_{r} which are regular on a neighborhood of 0 ∈ U and which cut out Y. Writing f_{i} = g_{i}/h_{i}, for g_{i} and h_{i} with no common factors, we see that since the h_{i} don’t vanish at 0, after shrinking U so that the h_{i} don’t vanish on U, we may ensure that the h_{i} are units, and hence Y is cut out by the g_{i}. Therefore we may assume (after replacing f_{i} by g_{i} and shrinking U), that the f_{i} are polynomials. Next, we write

f_{i} = L_{i} + P_{i}

where L_{i} is a linear polynomial and P_{i} has degree at least 2. Note that f_{i} has no constant term since it must vanish at 0. Since Y is smooth of dimension d, by the Jacobian criterion, the L_{i}’s (which we may identify with the gradient of f_{i} at 0), span a n − d dimensional space. After relabelling, we may assume that L_{1}, …, L_{n−d} give a basis for this space. Let Ỹ be the zero locus of f_{1}, …, f_{n−d}. Since Y ⊂ Ỹ we have the codimension of Ỹ at 0, codim_{0}(Ỹ) ≤ codim(Y) = n − d. By construction, the Jacobian matrix of the defining equations for Ỹ at 0 has rank n − d, and so by [Eis95], page 402, n − d ≤ codim(Ỹ). But then

n − d ≠ codim_{0}(Ỹ) ≤ codim(Y) = n − d

so codim_{0}(Ỹ) = n − d and also by the Jacobian criterion, we conclude that Ỹ is smooth at 0. We may therefore, after shrinking U assume that Ỹ is smooth, irreducible, and of the same dimension as Y. But Y ⊂ Ỹ therefore implies Y = Ỹ, and in particular we may assume r = n − d, and the L_{i} are independent.

After choosing a new basis for 𝔸^{n}, it is clear that we may assume L_{i} = x_{i} while preserving our hypotheses.

Finally, consider the morphism γ = ip : U → U (where i and p are as in the definition of the retraction), and write γ(x⃗) = (γ_{1}(x⃗), …, γ_{n}(x⃗)), where each γ_{i} is a regular function on U. Since γ_{i}(0) = 0, we may write γ_{i} = M_{i} + Q_{i} for the linear function

M_{i} = ∑ \frac{∂}{∂x_{i}} γ_{i}|_{x⃗=0} x_{i}

and have all the partial derivatives of the Q_{i} vanishing. Let 𝕋 = Spec k[ϵ]/(ϵ^{2}), and consider a 𝕋-valued point τ : 𝕋 → U given by a⃗ϵ = (a_{1}ϵ, …, a_{n}ϵ) ∈ 𝔸^{n}(k[ϵ]/(ϵ^{2})). We note that τ maps 𝕋 into Y if and only if f_{i}(a⃗ϵ) = 0 for each i. But we have

f_{i}(a⃗ϵ) = L_{i}(a⃗ϵ) = ϵL_{i}(a⃗).

In particular, this occurs exactly when a_{i} = 0 for 1 ≤ i ≤ n − d. Since γ(a⃗ϵ) ∈ Y, we therefore have M_{i} = 0 for 1 ≤ i ≤ n−d. Since γ|_{Y} = id_{Y}, looking on 𝕋-valued points of Y under γ, we find M_{i} = M′_{i} + x_{i} for n − d < i ≤ n where M′_{i} is a linear function of x_{1}, …, x_{n−d}. Consider the linear function 𝔸^{n} → 𝔸^{n} given by

ϕ : (x_{1}, … x_{n}) ↦ (y_{1}, …, y_{n})

where

y_{i} = \begin{cases}x_{i} & if 1 ≤ i ≤ n − d \\ x_{i} − M_{i} & n − d < i ≤ n \end{cases}

Define rational maps i′ = i ∘ ϕ : Y −−→ 𝔸^{n} and p′ = ϕ^{−1} ∘ p : 𝔸^{n} −−→ Y. We then have p′ ∘ i′ = ϕ^{−1}piϕ = id_{Y} as rational maps, and therefore define a rational retraction. By Lemma 5.2.7 we may shrink U and Y to make this a closed retraction. Note also that i′p′ = ip = γ.

As before, let τ : 𝕋 → U be given by a⃗ϵ = (a_{1}ϵ, …, a_{n}ϵ) ∈ 𝔸^{n}(k[ϵ]/(ϵ^{2})), where a⃗ ∈ 𝔸^{n}(k). We consider the morphism i′p′ : U → Y → U, which we write as

(x_{1}, …, x_{n}) ↦ (M′_{1} + Q′_{1}, …, M′_{n} + Q′_{n})

with M′_{i} linear and the first derivatives of the Q′_{i} vanishing at the origin. Computing using Lemma 5.5.1 applied to functions Q′_{i}, we then find

i′p′(τ) = ϵ(M′_{1}(a⃗), …, M′_{n}(a⃗))

and also, using the linearity of ϕ and the fact that i′p′ = ip = γ = M + Q,

i′p′(τ) = ϵϕ^{−1}(M_{1}(ϕ(a⃗), …, M_{n}(ϕ(a⃗)))
= ϵ(0, …, 0, a_{n−d+1}, …, a_{n}),

and so

M′_{i} = \begin{cases}0 & if 1 ≤ i ≤ n − d \\ x_{i} & if n − d < i ≤ n \end{cases}

Therefore, upon replacing p, i by p′, i′, we obtain the desired conclusion.

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