Valuation rings of mixed characteristic as limits of complete intersection rings
aa r X i v : . [ m a t h . A C ] J a n VALUATION RINGS OF MIXED CHARACTERISTIC AS LIMITSOF COMPLETE INTERSECTION RINGS
DORIN POPESCU
Abstract.
We show that a mixed characteristic valuation ring with a value group Γ and a residue field k of characteristic p > , is a filtered direct limit of completeintersection Z -algebras if Γ / Z val( p ) has no torsion, k is separably generated over F p and either V is Henselian, or dim V = 1 . Key words : Valuation Rings, Immediate Extensions, Complete IntersectionAlgebras, Cross-sections, Ultrapowers, Henselian Rings
Introduction
In [7] we gave a different proof to a weak form of Zariski’s Uniformization Theorem[11] using the following result (see [7, Theorem 21]).
Theorem 1.
Let V ⊂ V ′ be an immediate extension of valuation rings containing Q . Then V ′ is a filtered direct limit of smooth V -algebras. If the characteristic of the residue field of V is > then this is not true as showsfor example [6, Example 3.13] (see also [6, Remark 6.10]) inspired by [5, Sect 9, No57] and so in this case it is useful to have the following result [8, Theorem 14]. Theorem 2.
Let V ′ be an immediate extension of a valuation ring V . Then V ′ isa filtered union of its complete intersection V -subalgebras. A complete intersection V -algebra is a local V -algebra of type C/ ( P ) , where C isa localization of a polynomial V -algebra and P is a regular system of elements of C . Thus the above theorem says that V ′ is a filtered union of its V -subalgebras oftype C/ ( P ) . Since V ′ is local it is enough to say that V ′ is a filtered union of its V -subalgebras of type T h / ( P ) , T being a polynomial V -algebra, = h ∈ T and P isa regular system of elements of T . Clearly, T h is a smooth V -algebra and in fact it isenough to say that V ′ is a filtered union of its V -subalgebras of type G/ ( P ) , where G is a smooth V -algebra and P is a regular system of elements of G . Conversely,a V -algebra of such type G/ ( P ) has the form T h / ( P ) for some T, h, P using [9,Theorem 2.5]. By abuse we understand by a complete intersection V -algebra a V -algebra of such type G/ ( P ) , or T h / ( P ) . More general, we understand by a completeintersection algebra over a ring A an A -algebra of type G/ ( P ) , or T h / ( P ) , where G is a smooth A -algebra, T is a polynomial A -algebra, = h ∈ T and P is a regularsystem of elements of G , respectively T .Using the above theorem we obtained in [8, Theorem 1] the following result. heorem 3. Let V be a valuation ring containing a perfect field F of characteristic p > , k its residue field, Γ its value group and K its fraction field. Then thefollowing statements hold (1) if Γ is finitely generated, k ⊂ V then V is a filtered union of its completeintersection F -subalgebras, (2) if Γ is finitely generated, k/F is separably generated (that is there exists atranscendental basis x of k over F such that the field extension F ( x ) ⊂ k is algebraic separable) and V is Henselian, then V is a filtered union of itscomplete intersection F -subalgebras, (3) if k ⊂ V then V is a filtered direct limit of complete intersection F -algebras, (4) if k/F is separably generated and either dim V = 1 , or V is Henselian if dim V > , then V is a filtered direct limit of complete intersection F -algebras. The goal of this paper is to extend the above result for the mixed valuation rings(see Theorem 18, Corollary 19).
Theorem 4.
Let V be a mixed characteristic valuation ring, k its residue field, p = char k , Γ its value group. Assume k is separably generated over F p . Then V isa filtered direct limit of Z ( p ) -algebras if one of the following statements hold: (1) V is Henselian and Γ / Z val( p ) has no torsion, (2) V is Henselian and Γ / Z val( p ) ∼ = Z /t Z ⊕ P for some t ∈ N \ p N with k = k t and some Z -module P without torsion, (3) dim V = 1 and Γ / Z val( p ) has no torsion, (4) dim V = 1 and Γ / Z val( p ) ∼ = Z /t Z ⊕ P for some t ∈ N \ p N with k = k t andsome Z -module P without torsion. Valuation rings of mixed characteristic and cross-sections
Lemma 5.
Let V be a Henselian mixed characteristic valuation ring, k its residuefield and p = char k . Assume k is separably generated over F p the finite field with p elements. Then there exists a DVR subring R ⊂ V such that (1) pR is the maximal ideal of R , (2) R ⊂ V is an extension of valuation rings with the trivial residue field exten-sion.Proof. By hypothesis there exists a system of elements x of V inducing a separabletranscendental basis of k over F p . Then V ′ = ( Z ( p ) [ x ]) p Z ( p ) [ x ] is a DVR, pV ′ is itsmaximal ideal and the residue field extension of V ′ ⊂ V is algebraic separable. Let ¯ y ∈ k which is not in the residue field k ′ of V ′ and ¯ f = Irr (¯ y, k ′ ) ∈ k ′ [ Y ] . Let f ∈ V ′ [ Y ] be a monic polynomial lifting ¯ f . As V is Henselian we may lift ¯ y to asolution of f in V and V = ( V ′ [ Y ] / ( f )) pV ′ [ Y ] is a DVR such that ¯ y is contained inthe residue field of V . Using this trick by transfinite induction or by Zorn’s Lemmawe find such R . (cid:3) Proposition 6.
Let V be a Henselian mixed characteristic valuation ring, k itsresidue field, p = char k and Γ its value group. Assume Γ is finitely generated, eight ( pV ) = 1 , Γ / Z val( p ) has no torsion and k is separably generated over F p .Then V is a filtered union of its complete intersection algebras over a DVR R withits maximal ideal pR and its residue field k . In particular V is a filtered union ofits complete intersection algebras over Z ( p ) .Proof. By Lemma 5 there exists a DVR subring R ⊂ V such that pR is the maximalideal of R and the extension R ⊂ V has the trivial residue field extension. Then Γ / Z val( p ) is free and choose a Z basis val( x ) , x ∈ V e in it. Thus x is algebraicallyindependent over R by [2, Theorem 1 in VI (10.3)]. Let T be the fraction field of R and set W = V ∩ T ( x ) . Note that Γ is the value group of W . By [7, Lemma 26 (2)], W is a filtered union of its complete intersection R -subalgebras, even regular localrings. Using Theorem 2 note that V is a filtered union of its complete intersection W -subalgebras since W ⊂ V is immediate. But R is a filtered union of its smooth Z ( p ) -subalgebras by Néron desingularization [4] (see also [3]) and we are done. (cid:3) Proposition 7.
Let V be a Henselian mixed characteristic valuation ring, k itsresidue field, p = char k and Γ its value group. Assume k is separably generated over F p , height ( pV ) = 1 and Γ / Z val( p ) ∼ = ( Z /t Z ) ⊕ P for some free finitely generated Z -module P and a t ∈ N \ p N with k = k t . Then V is a filtered union of itscomplete intersection algebras over a DVR A ⊂ V with its residue field k and itslocal parameter π such that Γ / Z val( π ) ∼ = P and A ⊂ V is an extension of valuationrings. Moreover, V is a filtered union of its complete intersection algebras over Z ( p ) .Proof. By Lemma 5 there exists a DVR subring R ⊂ V such that pR is the maximalideal of R and R ⊂ V is an extension with the trivial residue field extension. Let T be the fraction field of R . By hypothesis there exists γ ∈ Γ such that tγ = val( p ) and Γ / Z γ ∼ = P has no torsion. Let π ∈ V be such that val( x ) = γ . Thus π t = pu forsome unit u ∈ V . The equation X t − u = 0 has a solution modulo m the maximalideal of V because k = k t , which can be lifted to a solution u ′ in V by the ImplicitFunction Theorem. Changing π by π/u ′ we may assume π t = p . Note that the DVR A = V ∩ T ( π ) ∼ = ( R [ X ] / ( X t − p )) ( X ) is a complete intersection algebra over R and Γ / Z val( π ) is without torsion.As in the above proposition we see that V is a filtered union of its completeintersection A -subalgebras and R is a filtered union of its smooth Z ( p ) -subalgebrasby Néron desingularization. Since A is a complete intersection R -algebra we see that V is a filtered union of its complete intersection algebras over Z ( p ) . (cid:3) Let q be the minimal prime ideal of pV and L the fraction field of L/q . Theproof of Proposition 6 goes also when Γ is not finitely generated but there exists across-section s : Γ / Z val( p ) → L ∗ = L \ { } , that is a section of the map val : L ∗ → Γ / Z val( p ) . Proposition 8.
Let V be a Henselian mixed characteristic valuation ring, k itsresidue field, p = char k , Γ its value group and q the minimal prime ideal of pV .Assume height ( q ) = 1 , Γ ′ = Γ / Z val( p ) has no torsion, k is separably generated over F p and there exists a cross-section s : Γ ′ → L ∗ , L being the fraction field of V /q . hen V is a filtered union of its complete intersection algebras over a DVR R withits maximal ideal pR and its residue field k . Moreover, V is a filtered union of itscomplete intersection algebras over Z ( p ) .Proof. Let Γ be a finitely generated subgroup of Γ containing val( p ) . We mayassume that Γ / Z val( p ) is free of basis val( x ) for some x ∈ V e . Let R be given byLemma 5, T its fraction field and set W = V ∩ T ( s (Γ / Z val( p ))) = V ∩ T ( x ) . Thevalue group of W is Γ . As in Proposition 6 we see that W is a filtered union of itscomplete intersection R -subalgebras, in particular a filtered union of its completeintersection Z ( p ) -subalgebras. Set W = V ∩ T ( s (Γ / Z val( p ))) . Then W is a filteredunion of such W as above and so of its complete intersection R -subalgebras andin particular a filtered union of its complete intersection Z ( p ) -subalgebras. As theextension W ⊂ V is immediate we may apply Theorem 2 as in Proposition 6. (cid:3) Also the proof of Proposition 8 could be extended when Γ / Z val( p ) has torsionbut no p -torsion. Proposition 9.
Let V be a Henselian mixed characteristic valuation ring, k itsresidue field, p = char k , Γ its value group and q the minimal prime ideal of pV .Assume height ( q ) = 1 , Γ ′ = Γ / Z val( p ) has no p -torsion, k is separably generatedover F p , k = k t for all t ∈ N \ p N and there exists a cross-section s : Γ ′′ → L ∗ , L being the fraction field of V /q and Γ ′′ is the factor of Γ / Z val( p ) by its torsion. Then V is a filtered union of its complete intersection algebras over Z ( p ) .Proof. Let R be given by Lemma 5 and T its fraction field. Let Γ ⊂ Γ be afinitely generated subgroup containing val( p ) . Note that Γ is free and we may finda basis of it such that val( p ) is a multiple of one element from the basis. Then Γ / Z val( p ) ∼ = ( Z /t Z ) ⊕ P for some free Z -module P and for some t ∈ N \ p N . Let A be given as in Proposition 7 with residue field k and π a local parameter of A such that val( π ) ∈ Γ and Γ / Z val( π ) has no torsion. Let s : Γ ′′ → L ∗ bethe given cross-section. Let G be the set of all z ∈ V such that z t = p for some t ∈ N \ p N and set W = V ∩ T ( G, s (Γ ′′ )) . Let T be the fraction field of A . Set W = V ∩ T ( s (Γ / Z val( π ))) . Note that Γ is the value group of W . The set ofsuch valuation rings W is filtered and their union is W . The extension W ⊂ V is immediate and V is a filtered union of its complete intersection algebras over W by Theorem 2. But each W is a filtered union of its complete intersection algebrasover Z ( p ) by Proposition 7. (cid:3) Some applications of the Zariski Uniformization Theorem
To extend the above propositions for the case height ( pV ) > we need to applythe Zariski Uniformization Theorem and the following two lemmas, the first being[7, Lemma 7] and the second one being very closed to [8, Lemma 19]. We need topresent firstly some notions.For a finitely presented ring map A → B , an element b ∈ B is standard over A if there exists a presentation B ∼ = A [ Y , . . . , Y m ] /I and f , . . . , f r ∈ I with r ≤ m such that b = b ′ b ′′ with b ′ = det(( ∂f i /∂Y j )) ≤ i, j ≤ r ∈ A [ Y , . . . , Y m ] and a b ′′ ∈ A [ Y , . . . , Y m ] that kills I/ ( f , . . . , f r ) (our standard element is a special power of he standard element from [9, Definition, page 9] given in the particular case of thevaluation rings). The loci of vanishing of standard elements of B over A cut out thelocus of non-smoothness of Spec( B ) → Spec( A ) . The radical of the ideal generatedby the elements of B standard over A is H B/A . Lemma 10.
For a commutative diagram of ring morphisms B ( ( ❘❘❘❘❘❘ B b a (cid:15) (cid:15) + + ❲❲❲❲❲❲❲❲❲ A ' ' PPPPPP ❧❧❧❧❧❧ V that factors as follows A ✐✐✐✐✐✐✐✐✐ ) ) ❚❚❚❚❚❚ V /a VA ′ ♥♥♥♥♥♥ A ′ /a A ′ ❤❤❤❤❤ with B finitely presented over A , a b ∈ B that is standard over A (this means aspecial element from the ideal H B/A defining the non smooth locus of B over A ,for details see for example [7, Lemma 4] ), and a nonzerodivisor a ∈ A ′ that mapsto a nonzerodivisor in V that lies in every maximal ideal of V , there is a smooth A ′ -algebra S such that the original diagram factors as follows: B ●●●●●●● , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨ A ) ) ❙❙❙❙❙❙ ❦❦❦❦❦❦ V.A ′ / / S ❦❦❦❦❦❦ Lemma 11.
Let V ⊂ V ′ be an extension of valuation rings, E a finitely presented V -algebra, w : E → V ′ a V -morphism and b ∈ V inducing a standard elementfor E over V . Assume the map E/b E → V ′ /b V ′ induced by w factors through acomplete intersection V /b V -algebra. Then w factors through a complete intersection V -algebra.Proof. We follow the proof of [8, Lemma 19]. By assumption the composite map E w −→ V ′ ⊂ V ′ /b V ′ factors through a complete intersection V /b V -algebra B , letus say through a map B ρ ′ −→ V ′ /b V ′ . We may assume that B has the form T h / ( P ) for a polynomial V /b V -algebra T , U = ( U , . . . , U l ) , h ∈ T and a regular systemof elements P = ( P , . . . , P t ) from T . Suppose the map ρ ′ is given by U → u ′ forsome u ′ ∈ ( V ′ /b V ′ ) l . Let u ∈ V l be a lifting of u ′ . We have P ( u ) = b z for some z in V ′ l and h ( u ) invertible. Set D = ( V [ U, Z ] / ( P − b Z )) h , Z = ( Z , . . . , Z t ) andlet ρ : D → V be the map ( U, Z ) → ( u, z ) . Note that D is a complete intersection V -algebra. As w factors through ρ modulo b there exists a smooth D -algebra D ′ such that w factors through D ′ using Lemma 10. It follows that D ′ is a completeintersection V -algebra as well. (cid:3) The next proposition is Proposition 8 when height ( pV ) is not necessarily . Proposition 12.
Let V be a Henselian mixed characteristic valuation ring, k itsresidue field, p = char k , Γ its value group and q the minimal prime ideal of pV .Assume Γ ′ = Γ / Z val( p ) has no torsion, k is separably generated over F p and thereexists a cross-section s : Γ ′ → L ∗ , L being the fraction field of V /q . Then V is afiltered direct limit of some complete intersection algebras over a DVR R with itsmaximal ideal pR and its residue field k . In particular V is a filtered direct limit ofsome complete intersection Z ( p ) -algebras. roof. Let R be the DVR given by Lemma 5 and T its fraction field. By Propo-sition 8 we may assume that height ( pV ) > and let q ′ ∈ Spec ( V ) the prime idealcorresponding to the maximal ideal of T ⊗ R V . Then height ( q/q ′ ) = 1 and applyingProposition 8 for the extension R ⊂ V /q ′ we see that V /q ′ is a filtered union of itscomplete intersection R -subalgebras.Let E ⊂ V be a finitely generated R -algebra of V and w : E → V its inclusion.Using the Zariski Uniformization Theorem [11] (see also [7, Theorem 35]) we see thatthe map L ⊗ R E → V q ′ factors through a smooth L -algebra and so w factors througha finitely generated R -algebra E ′ , let us say through a map w ′ : E ′ → V such that w ′ ( H E/R ) q ′ applying [7, Lemma 13]. Then a power of p belongs to w ′ ( H E/R ) andwe may assume that p s ′ is standard for E ′ over R . Set s = 3 s ′ . As above we get inparticular that the composite map E ′ w ′ −→ V → V /p s V factors through a completeintersection R/ ( p s ) -algebra and using Lemma 11 we see that w ′ factors through acomplete intersection R -algebra. Now it is enough to apply [9, Lemma 1.5]. (cid:3) Similarly, we get the following proposition from Proposition 9 when height ( pV ) is not necessarily . Proposition 13.
Let V be a Henselian mixed characteristic valuation ring, k itsresidue field, p = char k , Γ its value group and q the minimal prime ideal of pV .Assume Γ ′ = Γ / Z val( p ) has no p -torsion, k is separably generated over F p , k = k t for all t ∈ N \ p N and there exists a cross-section s : Γ ′′ → L ∗ , L being the fractionfield of V /q and Γ ′′ is the factor of Γ / Z val( p ) by its torsion. Then V is a filtereddirect limit of some complete intersection algebras over Z ( p ) . Corollary 14.
Let V be a Henselian mixed characteristic valuation ring, k itsresidue field, p = char k , Γ its value group and q the minimal prime ideal of pV .Assume Γ ′ = Γ / Z val( p ) ∼ = Z /t Z ⊕ P for some t ∈ N \ p N and some Z -module with-out torsion, k is separably generated over F p , k = k t and there exists a cross-section s : Γ ′′ → L ∗ , L being the fraction field of V /q and Γ ′′ is the factor of Γ / Z val( p ) byits torsion. Then V is a filtered direct limit of some complete intersection algebrasover Z ( p ) . We need the following proposition [7, Proposition 22].
Proposition 15.
Let V ⊂ V ′ be an extension of valuation rings. Suppose that (1) V is a discrete valuation ring with a local parameter π (2) πV ′ is the maximal ideal of V ′ , (3) the residue field extension of V ⊂ V ′ is separable.Then V ′ is a filtered direct limit of some smooth V -algebras. The proof of the Proposition 12 could be modified to show the following lemma.
Lemma 16.
Let V be a mixed characteristic valuation ring, k its residue field, p = char k and Γ its value group. Assume there exist (1) a DVR A ⊂ V with π as a local parameter such that A ⊂ V is an extensionand its residue field extension is trivial, a valuation subring B ⊂ V containing A with πB the maximal ideal of B and with value group Γ ′ ⊂ Γ such that B ⊂ V is an extension and Γ ′′ = Γ / Γ ′ has no torsion, (3) there exists a cross-section s : Γ ′′ → L ∗ , L being the fraction field of V /q , q being the minimal prime ideal of πV .Then V is a filtered direct limit of some complete intersection algebras over A .Proof. Let q be the minimal prime ideal of πV and q ′ ∈ Spec V the prime idealcorresponding to the maximal ideal of the fraction ring of V by the multiplicativesystem generated by π . Then height ( q/q ′ ) = 1 , q ∩ B = πB and height ( πB/ ( q ′ ∩ B ) =1 . Using (3) and [7, Lemma 26 (2)] as in Proposition 8 we see that V /q ′ is a filteredunion of its complete intersection B/ ( q ′ ∩ B ) -subalgebras. This shows that V is afiltered union of its complete intersection B -algebras if q ′ = 0 . Otherwise we applythe Zariski Uniformization Theorem as in Proposition 12 to get V as a filtered directlimit of complete intersection algebras over B . Using Proposition 15 we see that B is a filtered direct limit of smooth A -algebras, which is enough. (cid:3) Main results
To extend the Propositions 6, 7 for the case when Γ is not finitely generated weneed to apply some facts from Model Theory. We will need [7, Theorem A10] whichwe present below (we could use also [10, Proposition 5.4], or [1, 3..3.39, 3.3.40]). Theorem 17.
For a valuation ring V with value group Γ , there is a countablesequence of ultrafilters U , U , . . . on some respective sets U , U , . . . for which thevaluation rings { V n } n ≥ defined inductively by V := V and V n +1 := Q U n +1 V n aresuch that the valuation ring e V := lim −→ n ≥ V n has a cross-section e s : e Γ → e K ∗ , where e K and e Γ are the fraction field and the value group of e V . Theorem 18.
Let V be a Henselian mixed characteristic valuation ring, k its residuefield, p = char k and Γ its value group. Assume k is separably generated over F p andeither (1) Γ / Z val( p ) has no torsion, or (2) Γ / Z val( p ) ∼ = Z /t Z ⊕ P for some t ∈ N \ p N with k = k t and some Z -module P without torsion.Then V is a filtered direct limit of some complete intersection algebras over Z ( p ) .Proof. Let R be given by Lemma 5 and L its fraction field. Let q be the minimalprime ideal of pV . In case (2) as in Proposition 9 we find an element π ∈ V suchthat π t = p and set A = V ∩ L ( π ) . In both cases we have a DVR subring A of V with Γ / Z π without torsion and the residue field extension of A ⊂ V trivial. Apply theabove theorem for V /q . Then there is a countable sequence of ultrafilters U , U , . . . on some respective sets U , U , . . . for which the valuation rings { W n } n ≥ defined nductively by W := V /q and W n +1 := Q U n +1 W n are such that the valuation ring f W := lim −→ n ≥ W n has a cross-section e s : e Γ → e K ∗ , where e K and e Γ are the fraction field and the value group of f W . Set V = V and V n +1 := Q U n +1 V n . Then the valuation ring e V := lim −→ n ≥ V n has the property q e V isa prime ideal and e V /q e V ∼ = f W . Similarly we define e A . Note that the residue fieldextension of e A ⊂ e V is trivial and π e A is the maximal ideal of e A . As Γ / Z val( π ) has notorsion we see that e Γ / Z val( π ) has also no torsion. Thus e V is a filtered direct limit ofcomplete intersection algebras of e A -algebras as in Proposition 13 and Corollary 14.Since e ( A ) is a filtered direct limit of some complete intersection algebras over A byProposition 15 we see that e V is a filtered direct limit of some complete intersectionalgebras over A and even over Z ( p ) (see Lemma 16).Let E ⊂ V be a finitely generated Z ( p ) -algebra and w its inclusion. Then thecomposite map E → V → e V induced by w factors through a complete intersection Z ( p ) -algebra G , let us say through a map ρ : G → e V . As G is finitely generated over Z ( p ) we may consider ρ : G → V n for some n . Set ρ n = ρ .As in the proof of [7, Theorem 35] we may find also such map ρ n − : G → V n − if n > . Indeed, let E = Z ( p ) [ Z ] / ( h ) , Z = ( Z , . . . , Z s ) , where h is a system ofpolynomials from Z [ Z ] . Let G = Z ( p ) [ Y ] / ( P ) , Y = ( Y , . . . , Y n ) and P = ( P , . . . , P l ) a regular system of elements of Z ( p ) [ Y ] . Assume the map E → G be given by Z → g for some polynomials g in Z ( p ) [ Y ] . Suppose that the map ρ n is given by Y → y for some y in V n and w is given by Z → z for some z in V . Then the system ofpolynomial equations P ( Y ) = 0 , g ( Y ) = z over V has a solution y in V n . As V n isthe ultrapower of V n − we see that this system has also a solution in V n − , that isthe composite map E → V → V n − factors also through G .Step by step we see that w factors through G and it is enough to apply [9, Lemma1.5]. (cid:3) Corollary 19.
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