Valued function field

An (algebraic) function field $ F \mid K $( that is, a finitely generated field extension of transcendence degree at least one; cf. also Extension of a field) together with a valuation $ v $, or place $ P $, on $ F $.

The collection of all places $ P $ on $ F $ which are the identity on $ K $ is called the Riemann space or Zariski–Riemann manifold of $ F \mid K $. Such a $ P $ is called a place of the function field $ F \mid K $ and the transcendence degree of its residue field $ FP $ over $ K $ is called the dimension of $ P $. If $ FP = K $, then $ P $ is called a rational place of $ F \mid K $; this is an analogue of the notion of a $ K $- rational point of an algebraic variety defined over $ K $.

Let $ v $ be an arbitrary valuation on $ F $. Then its restriction to $ K $ is a valuation on $ K $; the respective value groups are denoted by $ vF $ and $ vK $ and the respective residue fields are denoted by $ Fv $ and $ Kv $. The transcendence degree of $ F \mid K $ is greater than or equal to the sum of the transcendence degree of the residue field extension $ Fv \mid Kv $ and the $ \mathbf Q $- dimension of $ ( vF/vK ) \otimes \mathbf Q $( which is equal to the maximal number of elements in $ vF $ that are rationally independent over $ vK $; it may be viewed as the "transcendence degree" of the group extension $ vF \mid vK $). If equality holds, one says that $ ( F \mid K,v ) $ is without transcendence defect; in this case, the extensions $ Fv \mid Kv $ and $ vF \mid vK $ are finitely generated. An important special case is when $ v $ is a constant reduction of $ F \mid K $, that is, the transcendence degree of $ F \mid K $ is equal to that of $ Fv \mid Kv $( which is then again a function field).

Stability theorem.

The stability theorem gives criteria for a valued function field to be a defectless field (cf. Defect); a defectless field is also called a stable field. It was first proved by H. Grauert and R. Remmert (1966) for a special case; their proof was later generalized by several authors to cover the case of constant reduction in general (cf. [a1]). A further generalization (with an alternative proof) was given in [a7]: If $ ( F \mid K,v ) $ is a valued function field without transcendence defect and if $ ( K,v ) $ is a defectless field, then so is $ ( F,v ) $. This theorem has applications in the model theory of valued fields via the structure theory of Henselizations of valued function fields, sketched below. As an application to rigid analytic spaces (cf. Rigid analytic space), the stability theorem is used to prove that the quotient field of the free Tate algebra $ T _ {n} ( K ) $ is a defectless field, provided that $ K $ is. This, in turn, is used to deduce the Grauert–Remmert finiteness theorem (cf. Finiteness theorems), in a generalized version due to L. Gruson (1968; see [a1]).

Independence theorem.

If $ F $ contains a set $ {\mathcal T} = \{ x _ {1} \dots x _ {r} ,y _ {1} \dots y _ {s} \} $ such that the values of $ x _ {1} \dots x _ {r} $ form a maximal set of elements in $ vF $ rationally independent over $ vK $, and the residues of $ y _ {1} \dots y _ {s} $ form a transcendence basis of $ Fv \mid Kv $, then the elements of $ {\mathcal T} $ are algebraically independent. Hence, by the initial remarks, $ {\mathcal T} $ is a transcendence basis of $ F \mid K $ and $ ( F \mid K,v ) $ is without transcendence defect. In this case, the stability theorem can be used to prove the independence theorem, which states that the Henselian defect of the finite extension $ F \mid K ( {\mathcal T} ) $ is independent of the choice of such a set $ {\mathcal T} $. This makes it possible to define a Henselian defect for all valued function fields without transcendence defect; in particular, in the constant reduction case. A different notion of defect, the vector space defect, was considered in [a4].

Constant reduction of function fields of transcendence degree one.

This was introduced by M. Deuring in [a2] and studied by many authors; for a survey, see [a3]. The main object of investigation is the relation between the function fields $ F \mid K $ and $ Fv \mid Kv $.

Answering a question of M. Nagata, J. Ohm [a8] gave an elementary proof for the ruled residue theorem: If $ v $ is a valuation on $ K ( x ) $ such that the residue field $ K ( x ) v $ is of transcendence degree one over $ Kv $, then $ K ( x ) v $ is a rational function field over a finite extension of $ Kv $.

More generally, one seeks to relate the genus (cf. Algebraic function) of $ F \mid K $ to that of $ Fv \mid Kv $. Several authors proved genus inequalities; one such inequality, proved by B. Green, M. Matignon and F. Pop in [a4], is given below. Let $ F \mid K $ be a function field of transcendence degree one and assume that $ K $ coincides with the constant field of $ F \mid K $( the relative algebraic closure of $ K $ in $ F $). Let $ v _ {1} \dots v _ {s} $ be distinct constant reductions of $ F \mid K $ having a common restriction to $ K $. Then

$$ 1 - g _ {F} \leq 1 - s + \sum _ {i = 1 } ^ { s } \delta _ {i} e _ {i} r _ {i} ( 1 - g _ {i} ) , $$

where $ g _ {F} $ is the genus of $ F \mid K $ and $ g _ {i} $ is the genus of $ Fv _ {i} \mid Kv _ {i} $, $ r _ {i} $ is the degree of the constant field of $ Fv _ {i} \mid Kv _ {i} $ over $ Kv _ {i} $, $ \delta _ {i} $ is the Henselian defect of $ ( F \mid K,v _ {i} ) $, and $ e _ {i} $ is the ramification index $ ( v _ {i} F:v _ {i} K ) $( which is always finite in the constant reduction case). It follows that constant reductions $ v _ {1} ,v _ {2} $ with common restriction to $ K $ and $ g _ {1} = g _ {2} = g _ {F} \geq 1 $ must be equal. In other words, for a fixed valuation on $ K $ there is at most one extension $ v $ to $ F $ which is a good reduction, that is,

i) $ g _ {F} = g _ {Fv } $;

ii) there exists an element $ f \in F $ such that $ v ( f ) = 0 $ and $ [ F:K ( f ) ] = [ Fv:Kv ( fv ) ] $, where $ fv $ denotes the residue of $ f $;

iii) $ Kv $ is the constant field of $ Fv \mid Kv $. An element $ f $ as in ii) is called a regular function.

More generally, $ f $ is said to have the uniqueness property if $ fv $ is transcendental over $ Kv $ and the restriction of $ v $ to $ K ( f ) $ has a unique extension to $ F $. In this case, $ [ F:K ( f ) ] = \delta e [ Fv:Kv ( fv ) ] $, where $ \delta $ is the Henselian defect of $ ( F \mid K,v ) $ and $ e = ( vF:vK ( f ) ) = ( vF:vK ) $. If $ K $ is algebraically closed, then $ e = 1 $, and it follows from the stability theorem that $ \delta = 1 $; hence in this case, every element with the uniqueness property is regular.

It was proved in [a5] that $ F $ has an element with the uniqueness property already if the restriction of $ v $ to $ K $ is Henselian. The proof uses the model completeness of the elementary theory of algebraically closed valued fields (see Model theory of valued fields), and ultraproducts (cf. Ultrafilter) of function fields. Elements with the uniqueness property also exist if $ vF $ is a subgroup of $ \mathbf Q $ and $ Kv $ is algebraic over a finite field. This follows from work in [a6], where the uniqueness property is related to the local Skolem property, which gives a criterion for the existence of algebraic $ v $- adic integral solutions on geometrically integral varieties.

Divisor reduction mappings.

A further way to compare $ F \mid K $ with $ Fv \mid Kv $ is to construct a relation between their Riemann spaces by divisor reduction mappings. Such morphisms, which preserve arithmetical properties, were introduced by M. Deuring in [a2] for the case of good reduction when the valuations are discrete. This was generalized to non-discrete valuations by P. Roquette in [a9]. A partial reduction mapping not needing the assumption of good reduction was used in [a5] for the construction of elements with the uniqueness property.

Structure of Henselizations of valued function fields.

Valued function fields play a role also in the model theory of valued fields. The question whether an elementary theory is model complete or complete can be reduced to the existence of embeddings of finitely generated extensions of structures (cf. Existentially closed; Robinson test; Prime model). In the case of valued fields, these are just the valued function fields (or the finite extensions, but a field is never existentially closed in a non-trivial finite extension). Since there is no hope for a general classification of valued function fields up to isomorphism, it makes sense to pass to their Henselizations and use the universal property of Henselizations (see Henselization of a valued field). The main results are as follows (cf. [a7]).

1) In the case of valued function fields without transcendence defect, natural criteria can be given for the isomorphism class of their Henselizations to be determined by the isomorphism classes of the value group and the residue field. This makes essential use of the stability theorem.

2) If $ ( F \mid K,v ) $ is a valued function field of transcendence degree one which is an immediate extension, and if $ ( K,v ) $ is a tame field (see Ramification theory of valued fields), then the Henselization of $ ( F,v ) $ is equal to the Henselization of a suitably chosen rational function field contained in this Henselization. This reduces the classification problem to the rational function field, where in turn it can be solved using methods developed by I. Kaplansky (1942; see Kaplansky field). If the residue field of $ K $ has characteristic zero, the above result is a direct consequence of the fact that in this case $ ( K ( x ) ,v ) $ is a defectless field, for every $ x \in F $.

This structure theory, together with the stability theorem, can be used to show the following. Let $ P $ be a place of the algebraic function field $ F\mid K $. Then there is a finite extension $ {\mathcal F}\mid F $ and an extension $ P $ to $ {\mathcal F} $ wich admits local uniformization This result also follows from work of A.J. de Jong (1995). But, in addition, a valuation-theoretical description of the extension $ {\mathcal F}\mid F $ can be given. In particular, if $ v $ is the valuation induced by $ P $ and if $ ( F\mid K,v) $ is without transcendence defect, then $ P $ admits local uniformization without extending the function field. See [a10].

References

[a1]	S. Bosch, U. Güntzer, R. Remmert, "Non–Archimedean analysis" , Springer (1984)
[a2]	M. Deuring, "Reduktion algebraischer Funktionenkörper nach Primdivisoren des Konstantenkörpers" Math. Z. , 47 (1942) pp. 643–654
[a3]	B. Green, "Recent results in the theory of constant reductions" Sém. de Théorie des Nombres, Bordeaux , 3 (1991) pp. 275–310
[a4]	B. Green, M. Matignon, F. Pop, "On valued function fields I" Manuscr. Math. , 65 (1989) pp. 357–376
[a5]	B. Green, M. Matignon, F. Pop, "On valued function fields II" J. Reine Angew. Math. , 412 (1990) pp. 128–149
[a6]	B. Green, M. Matignon, F. Pop, "On the local Skolem property" J. Reine Angew. Math. , 458 (1995) pp. 183–199
[a7]	F.-V. Kuhlmann, "Valuation theory of fields, abelian groups and modules" , Algebra, Logic and Applications , Gordon&Breach (to appear)
[a8]	J. Ohm, "The ruled residue theorem for simple transcendental extensions of valued fields" Proc. Amer. Math. Soc. , 89 (1983) pp. 16–18
[a9]	P. Roquette, "Zur Theorie der Konstantenreduktion algebraischer Mannigfaltigkeiten" J. Reine Angew. Math. , 200 (1958) pp. 1–44
[a10]	F.-V. Kuhlmann, "On local uniformization in arbitrary characteristic" The Fields Institute Preprint Series (1997)