# Some results
of algebraic geometry

over Henselian rank one valued fields

###### Abstract.

We develop geometry of affine algebraic varieties in over Henselian rank one valued fields of equicharacteristic zero. Several results are provided including: the projection and blow-ups of the -rational points of smooth -varieties are definably closed maps; a descent property for blow-ups; curve selection for definable sets; a general version of the Łojasiewicz inequality for continuous definable functions on subsets locally closed in the -topology and extending continuous hereditarily rational functions, established for the real and -adic varieties in our joint paper with J. Kollár. The descent property enables application of resolution of singularities and transformation to a normal crossing by blowing up in much the same way as over the locally compact ground field. Our approach relies on quantifier elimination due to Pas and a concept of fiber shrinking for definable sets, which is a relaxed version of curve selection. The last three sections are devoted to the theory of regulous functions and sets over such valued fields. Regulous geometry over the real ground field was developed by Fichou–Huisman–Mangolte–Monnier. The main results here are regulous versions of Nullstellensatz and Cartan’s Theorems A and B.

###### Key words and phrases:

Closedness theorem, descent property for blow-ups, curve selection, Łojasiewicz inequality, hereditarily rational functions, regulous functions and sets, Nullstellensatz, Cartan’s Theorems A and B###### 2000 Mathematics Subject Classification:

Primary: 12J25, 03C10; Secondary: 14G27, 14P10.msbm10 \newfont\matbcmbx10 \newfont\goteufm10

## 1. Introduction

In this paper, we develop geometry of affine algebraic varieties in over Henselian rank one valued fields of equicharacteristic zero with valuation , value group , valuation ring and residue field . Every rank one valued field has a metric topology induced by its absolute value. Examples of such fields are the quotient fields of the rings of formal power series and of Puiseux series with coefficients from a field of characteristic zero as well as the fields of Hahn series (maximally complete valued fields also called Malcev–Neumann fields; cf. [24]):

Let be a -algebraic variety. We always assume that is reduced but we allow it to be reducible. The set of its -rational points (-points for short) inherits from a topology, called the -topology. In this paper, we are going to investigate continuous and differentiable functions that come from algebraic geometry and their zero sets. Therefore, we shall (and may) most often assume that is an affine -variety such that is Zariski dense in . Throughout the paper, by ”definable” we shall mean ”definable with parameters”.

Several results concerning algebraic geometry over such ground fields are established. Let be the 3-sorted language of Denef–Pas. We prove that the projection

is an -definably closed map (Theorem 3.1). Further, we shall draw several conclusions, including the theorem that blow-ups of the -points of smooth -varieties are definably closed maps (Corollary 3.5), a descent property for such blow-ups (Corollary 3.6), curve selection for -definable sets (Proposition 8.2) and for valuative semialgebraic sets (Proposition 8.1) as well as a general version of the Łojasiewicz inequality for continuous -definable functions on subsets locally closed in the -topology (Proposition 9.2). Also given is a theorem on extending continuous hereditarily rational functions over such ground fields (Theorem 10.2), established for the real and -adic varieties in our joint paper [27] with J. Kollár. The proof makes use of the descent property and the Łojasiewicz inequality. The descent property enables application of resolution of singularities and transformation to a normal crossing by blowing up (see [25, Chap. III] for references and relatively short proofs) in much the same way as over the locally compact ground field. Our approach relies on quantifier elimination due to Pas and on a certain concept of fiber shrinking for definable sets, which is a relaxed version of curve selection. Note that this paper comprises our two earlier preprints [39, 40].

###### Remark 1.1.

This paper is principally devoted to geometry over rank one valued fields (in other words, fields with non-archimedean absolute value). Therefore, from Section 3 on, we shall most often assume that so is the ground field . Nevertheless, it is plausible that the closedness theorem (Theorem 2.6) and curve selection (Propositions 8.1 and 8.2) hold over arbitrary Henselian valued fields.

We should emphasize that our approach to the subject of this paper is possible just because the language in which we investigate valued fields is not too rich; in particular, it does not contain the inclusion language on the auxiliary sorts and the only symbols of connecting the sorts are two functions from the main -sort to the auxiliary -sort and -sort. Hence and by elimination of -quantifiers, the -definable subsets of the products of the two auxiliary sorts are precisely finite unions of the Cartesian products of sets definable in those two sorts. This allows us to reduce our reasonings to an analysis of ordinary cells (i.e. fibers of a cell in the sense of Pas).

The organization of the paper is as follows. In Section 2, we set up notation and terminology including, in particular, the language of Denef–Pas and the concept of a cell. We recall the theorems on quantifier elimination and on preparation cell decomposition, due to Pas [41]. Next we draw some conclusions as, for instance, Corollary 2.3 on definable functions and Corollary 2.7 on certain decompositions of definable sets. The former will be applied in Section 5 and the latter is crucial for our proof of the closedness theorem (Theorem 3.1), which is stated in Section 3 together with several direct corollaries, including the descent property. Section 4 gives a proof (being of algorithmic character) of this theorem for the case where the value group is discrete.

In Section 5, we study -definable functions of one variable. A result playing an important role in the sequel is the theorem on existence of the limit (Proposition 5.2). Its proof makes use of Puiseux’s theorem for the local ring of convergent power series. In Section 6, we introduce a certain concept of fiber shrinking for -definable sets (Proposition 6.1), which is a relaxed version of curve selection. Section 7 provides a proof of the closedness theorem (Theorem 3.1) for the general case. This proof makes use of fiber shrinking and existence of the limit for functions of one variable.

In the subsequent three sections, some further conclusions from the closedness theorems are drawn. Section 8 provides some versions of curve selection: for arbitrary -definable sets and for valuative semialgebraic sets. The next section is devoted to a general version of the Łojasiewicz inequality for continuous -definable functions on subsets locally closed in the -topology (Proposition 9.2). In Section 10, the theorem on extending continuous hereditarily rational functions (established for the real and -adic varieties in [27]) is carried over to the case where the ground field is a Henseliam rank one valued field of equicharacteristic zero. Let us mention that in real algebraic geometry applications of continuous hereditarily rational functions and the extension theorem, in particular, are given in the papers [28, 29, 30] and [31], which discuss rational maps into spheres and stratified-algebraic vector bundles on real algebraic varieties.

The last three sections are devoted to the theory of regulous functions and sets over Henselian rank one valued fields of equicharacteristic zero. Regulous geometry over the real ground field was developed by Fichou–Huisman–Mangolte–Monnier [16]. In Section 11 we set up notation and terminology as well as provide basic results about regulous functions and sets, including the noetherianity of the constructible and regulous topologies. Those results are valid over arbitrary fields with the density property. The next section establishes a regulous version of Nullstellensatz (Theorem 12.4), valid over Henselian rank one valued fields of equicharacteristic zero. The proof relies on the Łojasiewicz inequality (Proposition 9.2). Also drawn are several conclusions, including the existence of a one-to-one correspondence between the radical ideals of the ring of regulous functions and the closed regulous subsets, or one-to-one correspondences between the prime ideals of that ring, the irreducible regulous subsets and the irreducible Zariski closed subsets (Corollaries 12.5 and 12.10).

Section 13 provides an exposition of the theory of quasi-coherent regulous sheaves, which generally follows the approach given in the real algebraic case by Fichou–Huisman–Mangolte–Monnier [16]. It is based on the equivalence of categories between the category of -modules on the affine scheme and the category of -modules on which, in turn, is a direct consequence of the one-to-one correspondences mentioned above. The main results here are the regulous versions of Cartan’s Theorems A and B. We also establish a criterion for a continuous function on an affine regulous subvariety to be regulous (Proposition 13.10), which relies on our theorem on extending continuous hereditarily rational functions (Theorem 10.2).

Note finally that the metric topology of a non-archimedean field with a rank one valuation is totally disconnected. Rigid analytic geometry (see e.g. [6] for its comprehensive foundations), developed by Tate, compensates for this defect by introducing sheaves of analytic functions in a Grothendieck topology. Another approach is due to Berkovich [3], who filled in the gaps between the points of , producing a locally compact Hausdorff space (the analytification of ), which contains the metric space as a dense subspace whenever the ground field is algebraically closed. His construction consists in replacing each point of a given -variety with the space of all rank one valuations on the residue field that extend . Further, the theory of stably dominated types, developed by Hrushovski–Loeser [23], deals with non-archimedean fields with valuation of arbitrary rank and generalizes that of tame topology for Berkovich spaces. Currently, various analytic structures over Henselian rank one valued fields are intensively investigated (see e.g. [11, 12] for more information, and [34] for the case of algebraically closed valued fields).

## 2. Quantifier elimination and cell decomposition

We begin with quantifier elimination due to Pas in the language of Denef–Pas with three sorts: the valued field -sort, the value group -sort and the residue field -sort. The language of the -sort is the language of rings; that of the -sort is any augmentation of the language of ordered abelian groups (and ); finally, that of the -sort is any augmentation of the language of rings. We denote -sort variables by , -sort variables by , and -sort variables by .

In the case of non-algebraically closed fields, passing to the three sorts with additional two maps: the valuation and the residue map, is not sufficient. Quantifier elimination due to Pas holds for Henselian valued fields of equicharacteristic zero in the above 3-sorted language with additional two maps: the valuation map from the field sort to the value group, and a map from the field sort to the residue field (angular component map) which is multiplicative, sends to and coincides with the residue map on units of the valuation ring of .

Not all valued fields have an angular component map, but it exists if has a cross section, which happens whenever is -saturated (cf. [9, Chap. II]). Moreover, a valued field has an angular component map whenever its residue field is -saturated (cf. [42, Corollary 1.6]). In general, unlike for -adic fields and their finite extensions, adding an angular component map does strengthen the family of definable sets. For both -adic fields (Denef [13]) and Henselian equicharacteristic zero valued fields (Pas [41]), quantifier elimination was established by means of cell decomposition and a certain preparation theorem (for polynomials in one variable with definable coefficients) combined with each other. In the latter case, however, cells are no longer finite in number, but parametrized by residue field variables. In the proof of the closedness theorem, which is a fundamental tool for many results of this paper, we may use an angular component map because a given valued field can always be replaced with an -saturated elementary extension.

Finally, let us mention that quantifier elimination based on the sort (where and is the maximal ideal of the valuation ring ) was introduced by Besarab [4]. This new sort binds together the value group and residue field into one structure. In the paper [22, Section 12], quantifier elimination for Henselian valued fields of equicharacteristic zero, based on this sort, was derived directly from that by A. Robinson [43] for algebraically closed valued fields. Yet another, more general result, including Henselian valued fields of mixed characteristic, was achieved by Cluckers–Loeser [10] for so-called b-minimal structures (from ”ball minimal”); in the case of valued fields, however, countably many sorts , , are needed.

Below we state the theorem on quantifier elimination due to Pas [41, Theorem 4.1].

###### Theorem 2.1.

Let be a structure for the 3-sorted language of Denef–Pas. Assume that the valued field is Henselian and of equicharacteristic zero. Then admits elimination of -quantifiers in the language .

We immediately obtain the following

###### Corollary 2.2.

The 3-sorted structure admits full elimination of quantifiers whenever the theories of the value group and the residue field admit quantifier elimination in the languages of their sorts.

Below we prove another consequence of elimination of -quantifiers, which will be applied to the study of definable functions of one variable in Section 5.

###### Corollary 2.3.

Let be an -definable function on a subset of . Then there is a finite partition of into -definable sets and irreducible polynomials , , such that for each the polynomial in does not vanish and

###### Proof.

By elimination of -quantifiers, the graph of is a finite union of sets , , defined by conditions of the form

where are polynomials, and and are -definable subsets of and , respectively. Each set is the graph of the restriction of to an -definable subset . Since, for each point , the fibre of over consists of one point, the above condition imposed on angular components includes one of the form or, equivalently, , for some , which may depend on , where the polynomial in does not vanish. This means that the set

is contained in the union of hyperplanes and, furthermore, that for each point there is an index such that the polynomial in does not vanish and . Clearly, for any , this property of points is -definable. Therefore we can partition the set into subsets each of which fulfils the condition required in the conclusion with some irreducible factors of the polynomial . ∎

Recall now some notation concerning cell decomposition. Consider an -definable subset of , three -definable functions

and a positive integer . For each set

where stand for or no condition in any occurrence. If the sets , , are pairwise disjoint, the union

is called a cell in with parameters and center ; is called a fiber of the cell .

###### Theorem 2.4.

(Preparation Cell Decomposition, [41, Theorem 3.2]) Let

be polynomials in one variable with coefficients being -definable functions on . Then admits a finite partition into cells such that on each cell with parameters and center and for all we have:

where are -definable functions, for all , and the map does not depend on .

###### Remark 2.5.

The functions are said to be prepared with respect to the variable .

Every divisible ordered group admits quantifier elimination in the language of ordered groups. Therefore it is not difficult to deduce from Theorems 2.1 and 2.4 the following

###### Corollary 2.6.

(Cell decomposition) If, in addition, the value group is divisible, then every -definable subset of is a finite disjoint union of cells.

Every archimedean ordered group (which of course may be regarded as a subgroup of the additive group of real numbers) admits quantifier elimination in the Presburger language with binary relation symbols for congruences modulo , , where denotes the minimal positive element of if it exists or otherwise. Under the circumstances one can deduce in a similar manner the following

###### Corollary 2.7.

If, in addition, the valuation is of rank , then every -definable subset of is a finite disjoint union of sets each of which is a subset

of a cell

determined by finitely many congruences:

where are -definable functions, for , and , .

## 3. Closedness theorem

In this paper we are interested mainly in geometry over a Henselian rank one valued field of equicharacteristic zero. From now on we shall assume (unless otherwise stated) that the ground field is such a field. Below we state one of the basic theorems, on which many other results of our paper rely.

###### Theorem 3.1.

(Closedness theorem) Let be an -definable subset of . Then the canonical projection

is definably closed in the -topology, i.e. if is an -definable closed subset, so is its image .

Observe that the -topology is -definable whence the above theorem is a first order property. Therefore it can be proven using elementary extensions and thus one may assume that an angular component map exists. We shall provide two different proofs for this theorem. The first, given in Section 4, is valid whenever the value group is discrete, and is based on a procedure of algorithmic character. The other, given in Section 7, is valid for the general case, and makes use of Corollary 2.7 and fiber shrinking from Section 6 which, in turn, relies on some results on -definable functions of one variable from Section 5. When the ground field is locally compact, the closedness theorem holds by a routine topological argument. We immediately obtain five corollaries stated below.

###### Corollary 3.2.

Let be an -definable subset of and stand for the projective space of dimension over . Then the canonical projection

is definably closed.

###### Corollary 3.3.

Let be a closed -definable subset of or . Then every continuous -definable map is definably closed in the -topology.

###### Corollary 3.4.

Let , , be regular functions on , be an -definable subset of and the blow-up of the affine space with respect to the ideal . Then the restriction

is a definably closed quotient map.

###### Proof.

Indeed, can be regarded as a closed algebraic subvariety of and as the canonical projection. ∎

Since the problem is local with respect to the target space, the above corollary immediately generalizes to the case where the -variety is the blow-up of a smooth -variety .

###### Corollary 3.5.

Let be a smooth -variety, , , regular functions on , be an -definable subset of and the blow-up of the ideal . Then the restriction

is a definably closed quotient map.

###### Corollary 3.6.

(Descent property) Under the assumptions of the above corollary, every continuous -definable function

that is constant on the fibers of the blow-up descends to a (unique) continuous -definable function .

## 4. Proof of Theorem 3.1 when the valuation is discrete

The proof given in this section is of algorithmic character. Through the transfer principle of Ax–Kochen–Ershov (see e.g. [9]), it suffices to prove Theorem 3.1 for the case where the ground field is a complete, discretely valued field of equicharacteristic zero. Such fields are, by virtue of Cohen’s structure theorem, the quotient fields of formal power series rings in one variable with coefficients from a field of characteristic zero. The valuation and the angular component of a formal power series are the degree and the coefficient of its initial monomial, respectively.

The additive group is an example of ordered -group, i.e. an ordered abelian group with a (unique) smallest positive element (denoted by ) subject to the following additional axioms:

and

for all integers . The language of the value group sort will be the Presburger language of ordered -groups, i.e. the language of ordered groups augmented by and binary relation symbols for congruence modulo subject to the axioms:

for all integers . This theory of ordered -groups has quantifier elimination and definable Skolem (choice) functions. We can replace the above two countable axiom schemas with universal ones after adding the unary function symbols of one variable for division by with remainder, which fulfil the following postulates:

for all integers . The theory of ordered -groups admits therefore both quantifier elimination and universal axioms in the Presburger language augmented by division with remainder. Thus every definable function is piecewise given by finitely many terms and, consequently, is piecewise linear.

In the residue field sort, we can add new relation symbols for all definable sets and impose suitable postulates. This enables quantifier elimination for the residue field in the augmented language. In this fashion, we have full quantifier elimination in the 3-sorted structure with .

Now we can readily pass to the proof of Theorem 3.1 which, of course, reduces easily to the case . So let be an -definable closed (in the -topology) subset of . It suffices to prove that if lies in the closure of the projection , then there is a point such that .

Without loss of generality, we may assume that . Put

The set contains points all coordinates of which are arbitrarily large, because the point lies in the closure of . Hence and by definable choice, contains a set of the form

where is an unbounded definable subset and

are increasing unbounded functions given by a term (because a function in one variable given by a term is either increasing or decreasing). We are going to recursively construct a point with by performing the following procedure of algorithmic character.

Step 1. Let

and

If , there is a sequence , , such that

when . Since the set is a closed subset of , we get

and thus is the point we are looking for. Here the process stops. Otherwise

for some infinite definable subset of and . The set

is definable in the language . By full quantifier elimination, it is given by a quantifier-free formula with variables only from the value group -sort and the residue field -sort. Therefore there is a finite partitioning of into definable subsets over each of which the fibres of the above set are constant, because quantifier-free -definable subsets of the product of the two sorts are finite unions of the Cartesian products of definable subsets in and in , respectively. One of those definable subsets, say , must be infinite. Consequently, for some , the set

contains points of the form , where and .

Step 2. Let

If , there is a sequence , , such that

when . Since the set is a closed subset of , we get

and thus is the point we are looking for. Here the process stops. Otherwise

for some infinite definable subset of and . Again, for some , the set

contains points of the form , where , is an infinite definable subset of and .

Step 3 is carried out in the same way as the previous ones; and so on.

In this fashion, the process either stops after a finite number of steps and then yields the desired point (actually, ) such that , or it does not stop and then yields a formal power series

such that for each there exists an element for which

Hence , and thus the sequence tends to the point when tends to . Since the set is a closed subset of , the point belongs to , which completes the proof.

## 5. Definable functions of one variable

Consider first a complete rank one valued field . For every non-negative integer , let be the local ring of all formal power series

in one variable such that tends to when ; coincides with the ring of restricted formal power series. Then the local ring

is Henselian, which can be directly deduced by means of the implicit function theorem for restricted power series in several variables (see [7, Chap. III, § 4], [17] and also [19, Chap. I, § 5]).

We keep the assumption that the ground field is a Henselian rank one valued field of equicharacteristic zero. Let be the completion of the algebraic closure of . Clearly, the Henselian local ring is closed under division by the coordinate and power substitution. Therefore it follows from our paper [38, Section 2] that Puiseux’s theorem holds for . We still need an auxiliary lemma.

###### Lemma 5.1.

The field is a closed subspace of its algebraic closure .

###### Proof.

This follows directly from that the field is algebraically maximal (as it is Henselian and finitely ramified; see e.g. [15, Chap. 4]), but can also be shown as follows. Denote by the closure of a subset in , and let be the completion of . We have

Now, through the transfer principle of Ax-Kochen–Ershov (see e.g. [9]), is an elementary substructure of and, a fortiori, is algebraically closed in . Hence , as asserted. ∎

Now consider an irreducible polynomial

in two variables of -degree . Let be the Zariski closure of its zero locus in . Performing a linear fractional transformation over the ground field of the variable , we can assume that the fiber , , of over does not contain the point at infinity, i.e. . Then and is a unit in . Via Hensel’s lemma, we get the Hensel decomposition

of into polynomials

which are Weierstrass with respect to , , respectively. By Puiseux’s theorem, there is a neighbourhood of such that the trace of on is a finite union of sets of the form

Obviously, for , the fiber of over tends to the point when .

Let us mention that if

then

after perhaps shrinking the neighbourhood . Indeed, let

and be the smallest positive integer with . Since is a closed subspace of , we get

for close enough to , and thus the assertion follows.

Suppose now that an -definable function satisfies the equation

and is an accumulation point of the set . It follows immediately from the foregoing discussion that the set can be partitioned into a finite number of -definable sets , with , such that, after perhaps renumbering of the fiber of the set over , we have

Hence and by Corollary 2.3, we immediately obtain the following

###### Proposition 5.2.

(Existence of the limit) Let be an -definable function on a subset of and suppose is an accumulation point of . Then there is a finite partition of into -definable sets and points such that

Moreover, there is a neighbourhood of such that each definable set

is contained in an affine line with rational slope

with , , , or in .

###### Remark 5.3.

Note that the first conclusion (existence of the limit) could also be established via the lemma on the continuity of roots of a monic polynomial (which can be found in e.g. [6, Chap. 3, § 4]). Yet another approach for the case of tame theories is provided in [18, Lemma 2.20]. The second conclusion relies on Puiseux’s parametrization.

## 6. Fiber shrinking for definable sets

Let be an -definable subset of with accumulation point

and an -definable subset of with accumulation point . We call an -definable family of sets

an -definable -fiber shrinking for the set at if

i.e. for any neighbourhood of , there is a neighbourhood of such that for every , . When , is itself a fiber shrinking for the subset of at an accumulation point . This concept is a relaxed version of curve selection. It is used in Sections 7 and 8 in the proofs of the closedness theorem and a certain version of curve selection.

###### Proposition 6.1.

(Fiber shrinking) Every -definable subset of with accumulation point has, after a permutation of the coordinates, an -definable -fiber shrinking at .

###### Proof.

We proceed with induction with respect to the dimension of the ambient affine space . The case is trivial. So assuming the assertion to hold for , we shall prove it for . We may, of course, assume that . Let be coordinates in .

If is an accumulation point of the intersections

we are done by the induction hypothesis. Thus we may assume that the intersection

is empty. Then the definable (in the -sort) set

has an accumulation point .

Since the -sort admits quantifier elimination in the language of ordered groups augmented by binary relation symbols for congruence modulo , every definable subset of is a finite union of subsets of semi-linear sets contained in that are determined by a finite number of congruences

(6.1) |

here , , , for , .

Consequently, there exists a semi-linear subset of given by finitely many linear equations and inequalities with integer coefficients and with constant terms from such that the subset of determined by congruences of the form 6.1 is contained in and has an accumulation point . Therefore there exists an affine semi-line

where are positive integers, passing through a point

and contained in . It is easy to check that the set

is contained in . Then

is an -definable -fiber shrinking for the set at . This finishes the proof. ∎

## 7. Proof of Theorem 3.1 for the general case

The proof reduces easily to the case . We must show that if is an -definable subset of and a point lies in the closure of , then there is a point in the closure of such that . We may obviously assume that . By Proposition 6.1, there exists, after a permutation of the coordinates, an -definable -fiber shrinking for at :

here is the canonical projection of onto the -axis. Put

it easy to check that if a point lies in the closure of , then the point lies in the closure of . The problem is thus reduced to the case and .

By Corollary 2.7, we can assume that is a subset of a cell

of the form

But the set

is an -definable subset of the product of the two sorts, which is, by elimination of -quantifiers, a finite union of the Cartesian products of definable subsets in and in , respectively. It follows that is an accumulation point of the projection of the fiber for a parameter . We are thus reduced to the case where is the fiber of the set for a parameter . For simplicity, we abbreviate and to and , . Denote by the domain of these functions; then is an accumulation point of .

In the statement of Theorem 3.1, we may equivalently replace with the projective line , because the latter is the union of two open and closed charts biregular to . By Proposition 5.2, we can thus assume that the limits, say , of , () when exist in and, moreover, there is a neighbourhood of such that, each definable set

is contained in an affine line with rational slope

(7.1) |

with , , , or in .

Performing a linear fractional transformation of the coordinate , we get

The role of the center is immaterial. We can assume, without loss of generality, that it vanishes, , for if a point lies in the closure of the cell with zero center, the point lies in the closure of the cell with center .

When occurs and , the set is itself an -fiber shrinking at and the point is an accumulation point of lying over , as desired.

So suppose that either only occurs or occurs and . By elimination of -quantifiers, the set is a definable subset of . The value group admits quantifier elimination in the language of ordered groups augmented by symbols for congruences modulo , , (cf. Section 2). Therefore the set is of the form

(7.2) |

where , for .