Immediate extensions of valuation rings and ultrapowers
aa r X i v : . [ m a t h . A C ] N ov IMMEDIATE EXTENSIONS OF VALUATION RINGS ANDULTRAPOWERS
DORIN POPESCU
Abstract.
We describe the immediate extensions of a one dimensional valua-tion ring V which could be embedded in some separation of a ultrapower of V with respect to a certain ultrafilter. For such extensions holds a kind of Artin’sapproximation. Key words : immediate extensions,pseudo convergent sequences, pseudo limits,ultrapowers, smooth morphisms, Artin approximation.
Let ( R, m ) be a Noetherian local ring and ˜ R the ultrapower of R with respect ofa non principal ultrafilter on N . Then ¯ R = ˜ R/ ∩ n ∈ N m n ˜ R is a Noetherian completelocal ring which is flat over R (see [10, Proposition 2.9], or [13, Theorem 2.5]). Herewe try to find an analogue result in the frame of valuation rings.Let V be a valuation ring with value group Γ containing its residue field k , K itsfraction field and ˜ V = Π U V the ultrapower of V with respect to an ultrafilter U ona set U (see [4], [18], [2]). Then ¯ V = ˜ V / ∩ z ∈ V,z =0 z ˜ V is a valuation ring extending V , a kind of separation of ˜ V . Indeed, q = ∩ z ∈ V,z =0 z ˜ V is a prime ideal because if x x ∈ q for some x i ∈ ˜ V then val( x x ) ≥ γ for all γ ∈ Γ and so one of val( x i ) i = 1 , must be bigger than all γ ∈ Γ , that is one of x i belongsto q .The goal of this paper is to describe the valuation subrings of ¯ V (given for somespecial ultrafilters), which are immediate extensions of V when dim V = 1 . If thecharacteristic of V is > then there exist some immediate extensions which cannotbe embedded in ¯ V (see Remark 10).We owe thanks to the referee who hinted us some mistakes in a preliminary versionof this paper.An inclusion V ⊂ V ′ of valuation rings is an immediate extension if it is local asa map of local rings and induces isomorphisms between the value groups and theresidue fields of V and V ′ . V has some maximal immediate extensions (see [6]).If the characteristic of the residue field of V is zero then there exists an uniquemaximal immediate extension of V .Let λ be a fixed limit ordinal and v = { v i } i<λ a sequence of elements in V indexedby the ordinals i less than λ . Then v is pseudo convergent if val( v i − v i ′′ ) < val( v i ′ − v i ′′ ) (that is, val( v i − v i ′ ) < val( v i ′ − v i ′′ )) for i < i ′ < i ′′ < λ (see [6], [17]). A pseudo limit of v is an element z ∈ V with al( z − v i ) < val( z − v i ′ ) (that is, val( z − v i ) = val( v i − v i ′ )) for i < i ′ < λ . Wesay that v is(1) algebraic if some f ∈ V [ T ] satisfies val( f ( v i )) < val( f ( v i ′ )) for large enough i < i ′ < λ ;(2) transcendental if each f ∈ V [ T ] satisfies val( f ( v i )) = val( f ( v i ′ )) for largeenough i < i ′ < λ ,(3) fundamental if for any γ ∈ Γ there exist i < i ′ large enough such that val( v i − v i ′ ) > γ , Γ being the value group of V .We need [14, Proposition A.6], which is obtained using [4, Theorem 6.1.4] andsays in particular the following: Proposition 1.
Let U be an infinite set with card U = τ . Then there exists anultrafilter U on U such that for any valuation ring V any system of polynomialequations ( g i (( X j ) j ∈ J ) i ∈ I with card I ≤ τ in variables ( X j ) j ∈ J with coefficients inthe ultrapower ˜ V = Π U V has a solution in ˜ V if and only if all its finite subsystemshave. The above proposition is trivial when U = N . In general the ultrafilter U is veryspecial given by [4, Theorem 6.1.4]. Lemma 2.
Let U be an infinite set with card U = τ , V a valuation ring withvalue group Γ and card Γ ≤ τ and λ be an ordinal with card λ ≤ τ . Let U be theultrafilter on U given by the above proposition, ˜ V = Π U V the ultrafilter of V withrespect to U and ¯ V its separation introduced above. Then any pseudo convergentsequence ¯ v = (¯ v i ) i<λ over V has a pseudo limit in ¯ V .Proof. Let S be the system of polynomial equations over ¯ VS i := X − ¯ v i − Y i (¯ v i +1 − ¯ v i ); Y i Y ′ i − , i < λ. For each γ ∈ Γ + choose an element z γ ∈ V with val( z γ ) = γ and lift ¯ v i to someelements ˜ v i ∈ ˜ V . Let S ′ be the system of polynomial equations S ′ iγ := X − ˜ v i − Y i (˜ v i +1 − ˜ v i ) − z γ Z γ ; Y i Y ′ i − , for i < λ, γ ∈ Γ + , and some variables X, Y i , Y ′ i , Z γ .Then S ′ has a solution in ˜ V if and only if S has a solution modulo z γ ˜ V for all γ ∈ Γ , that is if S has a solution in ¯ V , which happens if and only if (˜ v i ) i<λ has apseudo limit in ˜ V . Note that the cardinal of the system S ′ is ≤ τ . By the aboveproposition, S ′ has solutions in ˜ V if and only if every finite subsystem T of S ′ hasa solution in ˜ V . We may enlarge T such that it has the form ( S ′ iγ ) i = i ,...,i e ; γ = γ ,...,γ e for some i < . . . < i e < λ and γ , . . . , γ e ∈ Γ + . But then x = ˜ v i e +1 induces asolution of T in ˜ V because val(˜ v i e +1 − ˜ v i j ) = val(˜ v i j +1 − ˜ v i j ) or ≤ j ≤ e and so there exist some units y j ∈ ˜ V such that ˜ v i e +1 − ˜ v i j − y j (˜ v i j +1 − ˜ v i j ) ∈ ∩ z ∈ V,z =0 z ˜ V , for ≤ j ≤ e . Thus (¯ v i ) i<λ has a pseudo limit x in ¯ V . (cid:3) Remark 3.
Let K be the fraction field of V . If in the above proof v is transcendentalthen val( x ) ∈ Γ and even the extension K ⊂ K ( x ) is immediate (see [6, Theorem2]). If v is algebraic then val( x ) could be in ˜Γ \ Γ , ˜Γ being the value group of ˜ V . Lemma 4.
Let U , τ , U , V , Γ be as in Lemma 2. Then the extension V ⊂ ¯ V factorsthrough the completion of V .Proof. By the above lemma any fundamental sequence over V has a limit in ¯ V . Thelimits of the fundamental sequences over V form a valuation subring ˆ V which mustbe separate because ∩ z ∈ V,z =0 z ¯ V = 0 . Hence ˆ V is the completion of V . (cid:3) Lemma 5.
Let V, Γ , U , U, τ ˜ V , ¯ V be as in Lemma 2, a an element of V with val( a ) > and B a finitely presented V -algebra. Assume V is Henselian and the completioninclusion V ⊂ ˆ V is separable. Then any V -morphism B → ¯ V could be lifted modulo a ¯ V to a V -morphism B → ˜ V .Proof. The proof is similar to the proof of [10, Corollary 2.7] or part of the proofof [13, Theorem 2.9] (see also [3]). Let B ∼ = V [ Y ] / ( f ) , Y = ( Y , . . . , Y n ) , f =( f , . . . , f m ) and ¯ w : B → ¯ V given by Y → ¯ y ∈ ˆ V n , let us say that ¯ y is induced by ˜ y = [( y u ) u ∈ U ] ∈ ˜ V . Set γ = val( a ) . By [8, Theorem 1.2] applied to V , there exista positive integer N and ν ∈ Γ + such that if z ∈ V and val( f ( z )) ≥ N γ + ν thenthere exists z ′ ∈ V such that f ( z ′ ) = 0 and val( z − z ′ ) ≥ γ .By construction we have in particular val( f (( y u )) ≥ N γ + ν for all u from a set δ ∈ U . So there exists y ′ u ∈ V such that f ( y ′ u ) = 0 and val( y u − y ′ u ) > γ . Define y ′ t = 0 if t δ and let ˜ y ′ = [( y ′ u ) u ∈ ˜ V ] . Then f (˜ y ′ ) = 0 in ˜ V and the V -morphism B → ˜ V given by Y → ˜ y ′ lifts ¯ w modulo a ˆ V . (cid:3) Lemma 6.
Let V, Γ , U , U, τ ˜ V , ¯ V be as in Lemma 2 and V ′ ⊂ ¯ V a valuation subring,which is an immediate extension of V . Assume V is Henselian and the completioninclusion V ⊂ ˆ V is separable. Then any algebraic pseudo convergent sequence of V which has a pseudo limit in V ′ has one also in V .Proof. Let v = ( v j ) j<λ be an algebraic pseudo convergent sequence of V which hasa pseudo limit x in V ′ . Let h ∈ V [ X ] be a polynomial of minimal degree amongthe polynomials f ∈ V [ Y ] such that val( f ( v i )) < val( f ( v j )) for large i < j < λ . Set h ( i ) = ∂ i h/∂X i , ≤ i ≤ deg h with h ( i ) = 0 . By [12, Proposition 6.5] there existsan ordinal ν < λ such that an element z ∈ V is a pseudo limit of v if and only if val( h ( i ) ( z )) = val( h ( i ) ( v ν )) where h ( i ) = 0 and val( h ( z )) > val( h ( v ρ )) for ν ≤ ρ < λ .In particular, we have val( h ( i ) ( x )) = val( h ( i ) ( v ν )) where h ( i ) = 0 and val( h ( x )) > val( h ( v ρ )) for ν ≤ ρ < λ . Thus if z ∈ V satisfies val( h ( i ) ( z )) = val( h ( i ) ( x )) for all ≤ i ≤ deg h with h ( i ) = 0 then z is a pseudo limit of v .Let d i ∈ V such that val( d i ) = val( h ( i ) ( x )) for ≤ i ≤ deg h with h ( i ) = 0 , letus say h ( i ) ( x ) = d i t i for some invertible t i ∈ V ′ , and g the system of equations ( i ) ( Z ) − d i U i , U i U ′ i − . If z, ( u i ) i is a solution of g in V then z is a pseudo limitof ( v j ) j<λ . But the map B := V [ Z, ( U i ) i ] / ( g ) → ¯ V given by ( Z, ( U i ) , ( U ′ i )) → ( x, ( t i ) , ( t − i )) could be lifted by Lemma 5 to a map B → ˜ V , that is g has a solutionin ˜ V and so in V as well. This ends the proof. (cid:3) Remark 7. If V is Henselian and V ′ is a filtered direct limit of smooth V -algebraswe get as above that any algebraic pseudo convergent sequence of V which has apseudo limit in V ′ has also one in V . Indeed, let x, ( v j ) j<λ , h, ( d i ) , ( t i ) , g as above.Then the solution ( x, ( t i ) , ( t − i )) of g in V ′ comes from a solution of g in a smooth V -algebra C . But there exists a V -morphism ρ from C to V because V is Henselian.Thus we get a solution of g in V via ρ , so ( v j ) j<λ has a pseudo limit in V . Lemma 8.
Let V, Γ , U , U, τ ˜ V , ¯ V be as in Lemma 2 and V ′′ ⊂ V ′ ⊂ ¯ V some valuationsubrings such that V ⊂ V ′′ , V ′′ ⊂ V ′ are immediate extensions. Assume V ′′ isHenselian and the completion inclusion V ′′ ⊂ ˆ V ′′ is separable. Then any algebraicpseudo convergent sequence of V ′′ which has a pseudo limit in V ′ has one also in V ′′ .Proof. Let ¯ V ′′ be given from V as ¯ V from V and ( v j ) j an algebraic pseudo convergentsequence over V ′′ which has a pseudo limit in V ′ ⊂ ¯ V ⊂ ¯ V ′′ . By Lemma 6 appliedto V ′′ it has one in V ′′ . Note that the construction of ¯ V , ¯ V ′′ are done with the same U and U . (cid:3) Theorem 9.
Let V ⊂ V ′ be an immediate extension of one dimensional valuationrings and Γ , U , U, τ, ˜ V , ¯ V be as in Lemma 2. Assume V ′ is complete and card U ≥ card Γ . The following statements are equivalents: (1) the extension V ⊂ ¯ V factors through V ′ , (2) for any valuation subring V ′′ ⊂ V ′ such that V ⊂ V ′′ and V ′′ ⊂ V ′ areimmediate extensions any algebraic pseudo convergent sequence of V ′′ whichis not fundamental and has a pseudo limit in V ′ has one also in V ′′ .Moreover, if V ⊂ V ′ is separable and one from (1) , (2) holds then V ′ is a filtereddirect limit of smooth V -algebras.Proof. Suppose (1) holds and let V ′′ be as in (2) . Then the completion ˆ V ′′ of V ′′ is contained in ¯ V ′′ by Lemma 4. An algebraic pseudo convergent sequence v over V ′′ which has a pseudo limit in V ′ ⊂ ¯ V ⊂ ¯ V ′′ must have a pseudo limit in ˆ V ′′ byLemma 8 because ˆ V ′′ is Henselian since dim V ′′ = 1 . Then v has a pseudo limit in V ′′ too by [12, Lemma 2.5] for example.Assume (2) holds. Let V ′′ ⊂ V ′ be a valuation subring such that V ⊂ V ′′ and V ′′ ⊂ V ′ are immediate and K ′′ its fraction field. Applying Zorn’s Lemma we maysuppose that V ′′ is maximal for inclusion among those immediate extensions W ⊂ V ′ of V such that V ⊂ ¯ V factors through W . Assume that V ′′ = V ′ . Let x ∈ V ′ \ V ′′ and v be a pseudo convergent sequence over V ′′ having x as a pseudo limit butwith no pseudo limit in V ′′ (see [6, Theorem 1]). Then v is either fundamental ortranscendental by (2) . If v is transcendental then K ′′ ⊂ K ′′ ( x ) is the extensionconstructed in [6, Theorem 2] for v . By Lemma 2 we see that v has a pseudo limit z in ¯ V ′′ . Actually, the proof of Lemma 8 gives that z could be taken in the completion V ′′ of V ′′ . Then the unicity given by [6, Theorem 2] shows that K ′′ ( x ) ∼ = K ′′ ( z ) andso the extension V ′′ ⊂ ¯ V factors through V = V ′ ∩ K ′′ ( x ) because z ∈ ˆ V ′′ ⊂ V ′ ⊂ ¯ V since V ′ is complete. If v is fundamental then as above x ∈ ˆ V ′′ ⊂ ¯ V . In both casesthe extension V ′′ ⊂ ¯ V factors through V = V ′ ∩ K ( x ) . These contradict that V ′′ ismaximal by inclusion, that is V ′′ must be V ′ .Now suppose V ⊂ V ′ is separable and (2) holds. We reduce to the case whenthe fraction field extension K ⊂ K ′ of V ⊂ V ′ is of finite type because V ′ is afiltered direct union of V ′ ∩ L for some subfields L ⊂ K ′ , which are finite type fieldextensions of K . By induction on the number of generators of L over K we mayreduce to the case when K ′ = K ′′ ( x ) for some element x ∈ V ′ , K ′′ being the fractionfield of a valuation subring V ′′ ⊂ V ′ as in (2) . As K ′ /K is separable of finite typewe may arrange that K ′ /K ′′ is still separable. Then x is a pseudo limit of a pseudoconvergent sequence v from V ′′ which has no pseudo limit in K ′′ (see [6, Theorem 1]).Assume that v is not a fundamental sequence. Then v is transcendental by (2) andso V ′ is a filtered direct union of localizations of polynomial V ′′ -subalgebras of V ′ in one variable by [11, Theorem 3.2] (see also [14, Lemma 15]). If v is fundamentalthen V ′′ ⊂ V ′ is dense and separable and we may apply a theorem of type Néron-Schappacher (see e.g. [12, Theorem 4.1]). (cid:3) Remark 10. If V ⊂ V ′ is the valuation ring extension given in [12, Example 3.13]then V ′ is not a filtered direct limit of smooth V -algebras (see [12, Remark 6.10])and so cannot be embedded in ¯ V by Theorem 9.The following corollary is a kind of Artin approximation (see [1], [13], [5], [16])in the frame of valuation rings. Its statement extends the idea of [7, Corollary 8,Theorem 11] and [15, Theorem 14] replacing the order by the valuation. Corollary 11.
Let V ⊂ V ′ be a separable immediate extension such that for anyvaluation subring V ′′ ⊂ V ′ such that V ⊂ V ′′ and V ′′ ⊂ V ′ are immediate extensionsany algebraic pseudo convergent sequence of V ′′ which has a pseudo limit in V ′ has one also in V ′′ . Let f be a finite system of polynomials equations from V [ Y ] , Y = ( Y , . . . , Y n ) , which has a solution in V ′ . Assume V is complete and dim V = 1 .Then f has a solution in V . Moreover, if y ′ = ( y ′ , . . . , y ′ n ) is a solution of f in V ′ then there exists a solution y = ( y , . . . , y n ) of f in V such that val( y i ) = val( y ′ i ) for ≤ i ≤ n .Proof. Let y ′ be a solution of f in V ′ , B = V [ Y ] / ( f ) and w : B → V ′ be the mapgiven by Y → y ′ . Let Γ , U , U, ˜ V , ¯ V be as in Theorem 9. Then the extension V ⊂ ¯ V factors through V ′ and ¯ w the composite map B w −→ V ′ → ¯ V could be lifted to a map ˜ w : B → ˜ V by Lemma 5. Thus f has in ˜ V the solution ˜ w ( Y ) and so it has also asolution in V .Now, let a = ( a , . . . , a n ) ∈ V n be such that val( a i ) = val( y ′ i ) for ≤ i ≤ n .Then there exists an unit z ′ i ∈ V ′ such that y ′ i = a i z ′ i and the system g obtained byadding to f the equations Y i − a i Z i , Z i T i − , ≤ i ≤ n has in V ′ the solution y ′ , z ′ = ( z ′ , . . . , z ′ n ) , t ′ = ( t ′ , . . . , t ′ n ) , the last ones are given by the inverses of ( z ′ i ) . So g has a solution y, z, t in V and it follows that val( y i ) = val( y ′ i ) , ≤ i ≤ n . (cid:3) emark 12. Another proof of the above corollary could be done using that V ′ is afiltered direct limit of smooth V -algebras (see Theorem 9). References [1] M. Artin,
Algebraic approximation of structures over complete local rings . Pub.Math. Inst. Des. Hautes. Scientif. ,(1969), 23–58.[2] M. Aschenbrenner, L. van den Dries, J. van der Hoeven,
Asymptotic differentialalgebra and model theory of transseries , Annals of Mathematics Studies, ,Princeton University Press, Princeton, NJ, 2017.[3] J. Becker, J. Denef, L. Lipshitz, L. van der Dries,
Ultraproducts and approxi-mation in local rings I , Inventiones Math., , (1979), 189-203.[4] C. C. Chang, H. J. Keisler, Model theory , 3rd ed., Studies in Logic and theFoundations of Mathematics, vol. 73, North-Holland Publishing Co., Amster-dam, 1990.[5] H. Hauser,
The classical Artin approximation theorems , Bull. Amer. Math. Soc., , (2017), 595-633;[6] I. Kaplansky, Maximal fields with valuations , Duke Math. J. (1942), 303-321.[7] Z. Kosar, D. Popescu, Nested Artin Strong Approximation Property , J. of PureAlgebra and Applications,
222 (4) , (2018), 818-827.[8] L. Moret-Bailly,
An extension of Greenberg’s theorem to general valuation rings ,Manuscripta Math. (2012), no. 1-2, 153-166.[9] A. Ostrowski,
Untersuchungen zur arithmetischen Theorie der Körper , Math.Z. (1935), no. 1, 321-404.[10] D. Popescu, Algebraically pure morphisms , Rev. Roum. Math. Pures et Appl., , (1979), 947-977.[11] D. Popescu, On Zariski’s uniformization theorem , in Algebraic geometry,Bucharest 1982 (Bucharest, 1982), Lecture Notes in Math., vol. 1056, Springer,Berlin, 1984, 264-296.[12] D. Popescu,
Algebraic extensions of valued fields , J. Algebra , (1987), 513-533.[13] D. Popescu,
Artin approximation , in Handbook of Algebra, vol. , Ed.M.Hazewinkel, (2000), Elsevier Science, 321-356.[14] D. Popescu, Néron desingularization of extensions of valuation rings with anAppendix by Kęstutis Česnavičius , to appear in Proceedings of Transient Tran-scendence in Transylvania 2019, Eds. Alin Bostan, Kilian Raschel (possible inSpringer Collection PROMS), arxiv/AC:1910.09123v4.[15] D. Popescu, G. Rond,
Remarks on Artin Approximation with constraints , OsakaJ. Math. , (2019), 431–440.[16] G. Rond, Artin Approximation , J. of Singularities, , (2018), 108-192.[17] O. F. G. Schilling, The theory of valuations , Mathematical Surveys, NumberIV, American Math. Soc., (1950).[18] H. Schoutens,
The use of Ultraproducts in Commutative Algebra , Lect. Notesin Math., Springer, 1999. imion Stoilow Institute of Mathematics of the Romanian Academy, Researchunit 5, University of Bucharest, P.O. Box 1-764, Bucharest 014700, Romaniaimion Stoilow Institute of Mathematics of the Romanian Academy, Researchunit 5, University of Bucharest, P.O. Box 1-764, Bucharest 014700, Romania