On the existence of geometric models for function fields in several variables
aa r X i v : . [ m a t h . N T ] D ec ON THE EXISTENCE OF GEOMETRIC MODELS FORFUNCTION FIELDS IN SEVERAL VARIABLES
FENG-WEN AN
Abstract.
In this paper we will give an explicit construction ofthe geometric model for a prescribed Galois extension of a functionfield in several variables over a number field. As a by-product,we will also prove the existence of quasi-galois closed covers ofarithmetic schemes, which is a generalization of the pseudo-galoiscovers of arithmetic varieties in the sense of Suslin-Voevodsky.
ContentsIntroduction1. Statement of The Main Theorem2. An Explicit Construction for the Model3. Proof of The Main TheoremReferences
Introduction
Let F be a number field and let E be a finitely generated extensionof F . If tr.deg F E = 1, this is on the theory of function fields of onevariable, especially on the Riemann-Roch Theory.Consider the case that [ E : F ] < ∞ . In recent decades one hasbeen attempted to use the related data of arithmetic varieties X/Y todescribe such a given (Galois) extension
E/F (for example, see [3, 7,8, 9, 11, 12, 13, 15, 16]). The reason is that there is a nice relationshipbetween them:For the case that dim X = dim Y , as it has been seen, under certainconditions the arithmetic varieties X/Y behave like Galois extensions
E/F of number fields; at the same time, their automorphism groups
Aut ( X/Y ) behave like the Galois groups
Gal ( E/F ). In particular, therelated data of varieties, such as the arithmetic fundamental groups,encode plenty of information of the maximal abelian class fields of thenumber fields. It needs to decode them for one to obtain class fields.
Mathematics Subject Classification.
Primary 11G35; Secondary 14J50.
Key words and phrases. arithmetic scheme, automorphism group, function fieldin several variables, Galois group.
Moreover, in such a case, one says that the arithmetic varieties
X/Y are a geometric model for the Galois extensions
E/F if the Galois group
Gal ( E/F ) is isomorphic to the automorphism group
Aut ( X/Y ) (forexample, see [5, 10, 11, 13, 14]).Now let F be a finitely generated extension over a number field.In this paper we will have a try to use the related data of arithmeticvarieties X/Y to describe a prescribed field E , a finitely generatedextension over F , of transcendental degree not less than one. We willgive an explicit construction of such a geometric model for functionfields in several variables (see Main Theorem ).On the other hand, we will also demonstrate the existence of quasi-galois closed covers in [2], which is as a by-product of the procedure forthe proof of the
Main Theorem (see
Theorem 3.4 ). It can be regardedas a generalization of the pseudo-galois covers of arithmetic varieties inthe sense of Suslin-Voevodsky (see [13, 14]).
Acknowledgment.
The author would like to express his sinceregratitude to Professor Li Banghe for his invaluable advice and instruc-tions on algebraic geometry and topology.1.
Statement of the Main Theorem
In the present paper, an arithmetic variety is an integral schemeof finite type over
Spec ( Z ). Let k ( X ) , O X,ξ denote the function fieldof an arithmetic variety X with generic point ξ .Let E be a finitely generated extension of a field F . Here E is notnecessarily algebraic over F . Then E is said to be a Galois extension of F if F is the fixed subfield of the Galois group Gal ( E/F ) in E .The following is the main theorem of the paper. Theorem 1.1. ( The Main Theorem ) Let K be a finitely generatedextensions over a number field. Suppose that Y is an arithmetic varietywith K = k ( Y ) . Take any finitely generated extensions L of K suchthat L is Galois over K .Then there exists an arithmetic variety X and a surjective morphism f : X → Y of finite type such that • L = k ( X ) ; • the morphism f is affine; • there is a group isomorphism Aut ( X/Y ) ∼ = Gal ( L/K ) . Remark 1.2.
Let dim X = dim Y . Then X/Y are said to be a geo-metric model of the field extension
E/F provided that k ( X ) = E and k ( Y ) = F and there is a group isomorphism Aut ( X/Y ) ∼ = Gal ( E/F )(for example, see [10, 11, 13, 14]). In the paper
Theorem 1.1 above
N THE EXISTENCE OF GEOMETRIC MODELS FOR FUNCTION FIELDS IN SEVERAL VARIABLES3 gives us a geometric model for function fields in several variables, whichis an analogue of the case for finite extensions of number fields.
Remark 1.3.
In the course of the proof of
Theorem 1.1 , as a by-product, we will also demonstrate the existence of quasi-galois closedcovers in [2] (see
Theorem 3.4 in the paper), which can be regarded asa generalization of the pseudo-galois covers of arithmetic varieties inthe sense of Suslin-Voevodsky (see [13, 14]).2.
An Explicit Construction for the Model
Notation.
Let us fix some notation and definitions before we givethe procedure for the construction (for details, see [1, 2]). Given anintegral domain D . Let F r ( D ) denote the field of fractions on D . Inparticular, if D is a subring of a field Ω, the field F r ( D ) will alwaysassumed to be contained in Ω.Let ( X, O X ) be a scheme. As usual, an affine covering of the scheme( X, O X ) is a family C X = { ( U α , φ α ; A α ) } α ∈ ∆ such that for each α ∈ ∆, φ α is an isomorphism from an open set U α of X onto the spectrum SpecA α of a commutative ring A α . Each ( U α , φ α ; A α ) ∈ C X is called a local chart . An affine covering C X of ( X, O X ) is said to be reduced if U α = U β holds for any α = β in ∆.Let Comm be the category of commutative rings with identity. Fixeda subcategory
Comm of Comm . An affine covering { ( U α , φ α ; A α ) } α ∈ ∆ of ( X, O X ) is said to be with values in Comm if for each α ∈ ∆ thereare O X ( U α ) = A α and U α = Spec ( A α ), where A α is a ring contained in Comm .Let Ω be a field and let Comm (Ω) be the category consisting of thesubrings of Ω and their isomorphisms. An affine covering C X of ( X, O X )with values in Comm (Ω) is said to be with values in the field
Ω.2.2.
Process of the Construction.
The following is the procedurefor the construction of the geometric model.Let K be a finitely generated extensions over a number field and let Y be an arithmetic variety such that K = k ( Y ) . Take any finitelygenerated extensions L of K such that L/K is a Galois extension.We will proceed in several steps to construct an arithmetic variety X and a surjective morphism f : X → Y satisfying the desired propertyin the Main Theorem of the paper, which will be proved in next section.
Step 1.
Fixed an algebraic closure Ω L of L. PutΩ K = Ω L ∩ K, i.e., an algebraic closure of K. FENG-WEN AN
Without loss of generality, assume that the ring O Y ( V ) is containedin Ω K for each affine open set V of the scheme Y. Otherwise, if that property does not hold, by discussion in [1] we canchoose a scheme ( Y ′ , O Y ′ ) which has that property and is isomorphicto ( Y, O Y ).Evidently, that property holds automatically if Y is an affine scheme.Choose the elements t , t , · · · , t n ∈ L \ K to be a nice basis of L over K (see [2]), that is, they satisfy the followingconditions:( i ) L = K ( t , t , · · · , t n );( ii ) t , t , · · · , t r constitute a transcendental basis of L over K ;( iii ) t r +1 , t r +2 , · · · , t n are linearly independent over K ( w , w , · · · , w r ),where 0 ≤ r ≤ n .Let C Y be the maximal element (by set inclusion) in the collectionof the reduced affine coverings of the scheme Y with values in Ω K . Step 2.
Take any local chart (
V, ψ V , B V ) ∈ C Y . Then V is an affineopen subset of Y and we have F r ( B V ) = K and O Y ( V ) = B V ⊆ Ω K . Define A V to be the subring of L generated over B V by the set ofelements in L ∆ V , { σ ( t j ) ∈ L : σ ∈ Gal ( L/K ) , ≤ j ≤ n } . That is, we have A V = B V [∆ V ] . Put ∆ ′ V = ∆ V \ { t , t , · · · , t r } . We have
F r ( A V ) = L ; A V = B V [ t , t , · · · , t r ] [∆ ′ V ] . Then ∆ ′ V is a nonvoid set. It is seen that B V is exactly the invariantsubring of the natural action of the Galois group Gal ( L/K ) on A V . Set i V : B V → A V to be the inclusion. Step 3.
Define the disjoint unionΣ = a ( V,ψ V ,B V ) ∈C Y Spec ( A V ) . N THE EXISTENCE OF GEOMETRIC MODELS FOR FUNCTION FIELDS IN SEVERAL VARIABLES5
Let π Y : Σ → Y be the projection.Σ is a topological space, where the topology τ Σ on Σ is naturallydetermined by the Zariski topologies on all Spec ( A V ) . Step 4.
Define an equivalence relation R Σ in Σ in such a manner:Take any x , x ∈ Σ. We say x ∼ x if and only if j x = j x holds in L .Here, j x denotes the corresponding prime ideal of A V to a point x ∈ Spec ( A V ) (see [4]).Define X = Σ / ∼ . Let π X : Σ → X be the projection.Hence, X is a topological space as a quotient of Σ . Step 5.
Define a map f : X → Y by π X ( z ) π Y ( z )for each z ∈ Σ. Step 6.
Put C X = { ( U V , ϕ V , A V ) } ( V,ψ V ,B V ) ∈C Y where U V = π − Y ( V ) holds and ϕ V : U V → Spec ( A V ) is the identitymap for each ( V, ψ V , B V ) ∈ C Y . Then C X is a reduced affine coveringon the space X with values in Ω L .Define the scheme ( X, O X )to be obtained by gluing the affine schemes Spec ( A V ) for all localcharts ( V, ψ V , B V ) ∈ C Y with respect to the equivalence relation R Σ (see [4, 6]).Then C X is admissible and the sheaf O X is an extension of C X onthe space X (see [1]).Finally, ( X, O X ) is the desired scheme and f : X → Y is the de-sired morphism of schemes. (Note that the proof will be given in thefollowing section.) FENG-WEN AN
This completes the construction.3.
Proof of the Main Theorem
Definitions.
Assume that O X and O ′ X are two structure sheaveson the underlying space of an integral scheme X . The integral schemes( X, O X ) and ( X, O ′ X ) are said to be essentially equal provided thatfor any open set U in X , we have U is affine open in ( X, O X ) ⇐⇒ so is U in ( X, O ′ X )and in such a case, D = D holds or there is F r ( D ) = F r ( D ) suchthat for any nonzero x ∈ F r ( D ), either x ∈ D \ D or x ∈ D \ D ⇐⇒ x − ∈ D \ D holds, where D = O X ( U ) and D = O ′ X ( U ).Two schemes ( X, O X ) and ( Z, O Z ) are said to be essentially equal if the underlying spaces of X and Z are equal and the schemes ( X, O X )and ( X, O Z ) are essentially equal.Let X and Y be two arithmetic varieties and let f : X → Y be asurjective morphism of finite type. By a conjugate Z of X over Y weunderstand an arithmetic variety Z that is isomorphic to X over Y .Let Aut ( X/Y ) denote the group of automorphisms of X over Z .Then X is said to be quasi-galois closed over Y by f if there isan algebraically closed field Ω and a reduced affine covering C X of X with values in Ω such that for any conjugate Z of X over Y the twoconditions are satisfied: • ( X, O X ) and ( Z, O Z ) are essentially equal if Z has a reducedaffine covering with values in Ω. • C Z ⊆ C X holds if C Z is a reduced affine covering of Z with valuesin Ω.Let K be an extension of a field k . Here K/k is not necessarily alge-braic. Recall that K is said to be quasi-galois over k if each irreduciblepolynomial f ( X ) ∈ F [ X ] that has a root in K factors completely in K [ X ] into linear factors for any intermediate field k ⊆ F ⊆ K (see[2]).The elements t , t , · · · , t n ∈ K \ k to be a nice basis of K over k if they satisfy the following conditions:( i ) L = K ( t , t , · · · , t n );( ii ) t , t , · · · , t r constitute a transcendental basis of L over K ; N THE EXISTENCE OF GEOMETRIC MODELS FOR FUNCTION FIELDS IN SEVERAL VARIABLES7 ( iii ) t r +1 , t r +2 , · · · , t n are linearly independent over K ( t , t , · · · , t r ),where 0 ≤ r ≤ n .Now let D ⊆ D ∩ D be three integral domains. The ring D issaid to be quasi-galois over D if the field F r ( D ) is a quasi-galoisextension of F r ( D ).The ring D is said to be a conjugation of D over D if thereis a ( r, n ) − nice k − basis w , w , · · · , w n of the field F r ( D ) and an F − isomorphism τ ( r,n ) : F r ( D ) → F r ( D ) of fields such that τ ( r,n ) ( D ) = D , where k = F r ( D ) and F , k ( w , w , · · · , w r ) is assumed to be con-tained in the intersection F r ( D ) ∩ F r ( D ).3.2. Criterion for Quasi-gaois Closed.
Let X and Y be two arith-metic varieties. Let Ω be a fixed algebraically closed closure of thefunction field k ( X ). Definition 3.1.
Let ϕ : X → Y be a surjective morphism of finitetype. A reduced affine covering C X of X with values in Ω is said to be quasi-galois closed over Y by ϕ if the below condition is satisfied:There exists a local chart ( U ′ α , φ ′ α ; A ′ α ) ∈ C X such that U ′ α ⊆ ϕ − ( V α )for any ( U α , φ α ; A α ) ∈ C X , for any affine open set V α in Y with U α ⊆ ϕ − ( V α ), and for any conjugate A ′ α of A α over B α , where B α is thecanonical image of O Y ( V α ) in the function field k ( Y ). Lemma 3.2.
Let ϕ : X → Y be a surjective morphism of finite type.Suppose that the function field k ( Y ) is contained in Ω . Then the scheme X is quasi-galois closed over Y if there is a unique maximal reducedaffine covering C X of X with values in Ω such that C X is quasi-galoisclosed over Y. Proof.
Assume that there is a unique maximal reduced affine covering C X of X with values in Ω such that C X is quasi-galois closed over Y. Fixed any a conjugate Z of X over Y . Let σ : Z → X be anisomorphism of schemes over Y . Suppose that Z has a reduced affinecovering C Z with values in Ω.Take any local chart ( W, δ, C ) ∈ C Z . Put U = σ ( W ); A = O X ( U ); C = O Z ( W ) . Then we have U = Spec ( A ) and W = Spec ( C ) . FENG-WEN AN As C X is quasi-galois closed over Y , it is seen that there is an affineopen subset U ′ in X such that C = O X ( U ′ ) . As U ′ = Spec ( C ) = W, we have σ − ( U ) = U ′ ⊆ X ;hence, Z = σ − ( X ) = X. It follows that we must have ( X, O X ) = ( Z, O Z ) . (cid:3) An affine covering { ( U α , φ α ; A α ) } α ∈ ∆ of ( X, O X ) is said to be an affine patching of ( X, O X ) if φ α is the identity map on U α = SpecA α for each α ∈ ∆ . Evidently, an affine patching is reduced.
Lemma 3.3.
Let ϕ : X → Y be a surjective morphism of finite type.Suppose that the function field k ( Y ) is contained in Ω . Then X isquasi-galois closed over Y if there is a unique maximal affine patching C X of X with values in Ω such that • either C X is quasi-galois closed over Y , • or A α has only one conjugate over B α for any ( U α , φ α ; A α ) ∈ C X and for any affine open set V α in Y with U α ⊆ ϕ − ( V α ) , where B α is the canonical image of O Y ( V α ) in the function field k ( Y ) .Proof. It is immediate from
Lemma 3.2 . (cid:3) Existence of Quasi-gaois Closed Covers.
Now we give theexistence of quasi-galois closed covers which take values in a prescribedextension of the function field in several variables.
Theorem 3.4.
Let K be a finitely generated extensions of a numberfield and let Y be an arithmetic variety with K = k ( Y ) . Fixed anyfinitely generated extensions L of K such that L is Galois over K .Then there exists an arithmetic variety X and a surjective morphism f : X → Y of finite type such that • L = k ( X ) ; • the morphism f is affine; • X is a quasi-galois closed over Y by f .Proof. It is immediate from
Lemma 3.3 and the construction in § (cid:3) N THE EXISTENCE OF GEOMETRIC MODELS FOR FUNCTION FIELDS IN SEVERAL VARIABLES9
Proof of the Main Theorem.
Now we can give the proof of theMain Theorem of the paper.
Proof. (Proof of Theorem 1.1)
It is immediate from
Theorem 3.4 above and the
Main Theorem in [2]. (cid:3)
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