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Rational points in henselian discrete valuation rings

Rational points in henselian discrete valuation rings RATIONAL POINTS IN HENSELIAN DISCRETE VALUATION RINGS by MARVIN J. GREENBERG Io Let R be a Henselian discrete valuation ring, with t a generator of the maximal ideal, k the residue field, and K the field of fractions. Let R* be the completion of R, K ~ its field of fractions. If F=(FI,..., Fr) is a system of r polynomials in n variables with coefficients in R, and x is an n-tuple with coordinates in R, set F(x)= (Fl(X), ..., Fr(x)). If F' is another system of r' polynomials, let FF' denote the system of rr' products. By the ideal FR[X] generated by F is meant the ideal in R[X] generated by F1, . .., F r. Theorem 1. -- Assume, in case K has characteristic p>o, that K* is separable over K. Then there are integers N~I, c~i, s~o depending on FR[X] such that for any v>N and any x in R such that F(x) - o (mod e) there exists y in R such that y--x (mod t [~/cl-8) F (y) -= o Corollary 1. -- Let Z be a prescheme of finite type over R. Then there are integers N> I, c>I, s>o depending on Z such that for ,J>_N and for any pointx of Z in R/t ~, the image of x in Z(R/t [~/cl-~) lifts to a point of Z in R. Proof. -- We can take a finite covering of Z by affine opens Zi. We have Z(S) =yZ~(S) for any local R-algebra S, hence the maxima of the integers for the Z~ will do for Z. Corollary 2. -- Z has a point in R if and only if Z has a point in R/t~ for all ,;> I. Let V be the algebraic set in affine n-space over K which is the locus of zeros of F. In the special case that R is complete and V is K-irreducible, non-singular, with a sepa- rably generated function field over K, N6ron [4; Prop. 2% p. 38] has proved this theorem, showing that in this case one can take c= I. However, in the general case we may have c> i (consider the polynomial Y~--X 3 and for any even integer 2v the point x = (t ~, t~)). Theorem I implies that the hypothesis of non-singularity in [4; Prop. 22] can be dropped, so that the sets in that proposition are always constructible. Theorem I is proved by induction on the dimension m of V. If m=--I, i.e., the ideal FR[X] contains a non-zero constant, it is clear. Suppose m>o. We may assume the ideal FR[X] is equal to its own radical (i.e., the scheme over R defined by F is reduced): For let E generate its radical. Then some power E q is in FR[X]. From F(x)=o (modt ~) 563 60 MARVIN J. GREENBERG we conclude t ~ divides Eq(x), so that E(x) - o (rood t [~lql) If N', c', s' are integers for E, we see that N = qN', c = qc', s = s' are integers for F. We may further assume V is K-irreducible: For if V--= W u W', where W, W' are algebraic sets defined respectively by systems of polynomials G, G' with coefficients in R, let N', c', s' (resp. N", c", s") be integers for G (resp. for G'). If x in R satisfies F(x)-o (modt ~) then either G(x) -- o or G'(x) =o (mod t E~I21) since GG' is in the ideal FR[X]. Thus N = 2max(N', N") c = 2max(c', c") s = max(s', s") will work for F. Then there are two cases: Case 1. -- V is separable over K. Let J be the Jacobian matrix of F, and let D be the system of minors of order n--m taken from detJ. The the locus of common zeros of D and F is a proper K-closed W in V. By inductive hypothesis there are integers N', c', s' for the system (D, F). For each system F(i I of n--m polynomials out of F, (i) a system of n--m indices, let Vll I be the locus over K of zeros of F(~I, and let V(i~ be the union of the K-irreducible components of V(0 which have dimension m and are different from V; let Gci/ be a system of generators for the ideal of V(i+~ in R iX]. By inductive assumption there are integers N(0 , c(0, s(0 for the system (G(~), F). If x is a point of V(~) in some extension of K such that for some (j) D(~),(i) (x) + o then the tangent hyperplanes of F,1, ..., F~._= at x are transversal, and x ties on exactly one component of V(~), that component having dimension m. We now invoke (see Lemma ~, n ~ 3) Newton's Lemma. -- If x in R is such that F(~l(x ) =o (mod t ~+~) D(~,C~(x ) ~o (mod t ~) for some (j) then there exists y in R such that Fcil(y ) = o y-x (mod t ~) Hence D(~l,(i) (Y) 4= o If we knew also G(~(y)+o we could deduce that y is a point of V. Take v so large that = [(~-- i)/2] >max(N', all N(~)) 564 RATIONAL POINTS IN HENSELIAN DISCRETE VALUATION RINGS 6x Let x in R be a zero modt ~ ofF. If D(x)=o (modt ~) our inductive hypothesis gives us y in R such that y is a singular point of V and y - x (mod t [~/c']- 8') If for some (i) G(i)(x) ----o (mod t v) then again by induction there isy in R which is a point of VoVc +) such that y-x (mod Otherwise we use Newton's Lemma to find y in R which is a point of V such that y-= x rood P') Thus as integers for 17 we can take N ---- 2 +2max(N', all N(o ) c = 2 max(c', all c(0 ) s = I +max(s', all s(0 ) Case 2. -- V is inseparable over K. In this case we need two facts. Fact 1. -- If K' # a finite extension Of K, then the integral closure R' Of R in K' is a finite R-module. This follows from our assumption K* separable over K (7; Oiv, ~3. i. 7 (ii)]. For the convenience of the reader, we sketch the proof, valid also when R is a higher dimen- sional local domain: K'| is a finite extension field of K*, because of our assumption. R'QRR* is a subring of this field, integral over the complete local domain R*, hence finite over R*. Since R* is faithfully flat over R, R' is a finite R-module. (The assump- tion that R* is a domain, implicit in this argument, can be eliminated (loc. cit.)). Fact 2. -- There is a functor o~ from the category of affine schemes of finite type over R' to affine schemes of finite type over R such that ~" is right adjoint to the change of base functor from R to R'. Thus we have an isomorphism of bifunctors MorR(Y , o~'Z) --~ Morw(Ya, , Z) (for Y/R, Z/R'). Moreover, o*" preserves closed immersions. This follows from Fact I, and can also be established in greater generality (see [8; p. 195-13] where the notation o~'Z=rcR,/RZ is used). Choose a basis bl, ..., b~ for the R-module R'. Every element of R' has uniquely determined coordinates in R with respect to this basis, and the addition and multipli- cation in R' are given by polynomial functions in these coordinates. Hence there is a commutative ring scheme S over R, whose underlying scheme is affine d-space over R, such that for any R-algebra A, MOrR(Spec A, S) = A| t[~'/~(1)]-8(1)) 62 MARVIN J. GREENBERG Now the same arguments as in [9; PP- 638-9] can be repeated word for word. The point is that by using the basis bl, ..., bd, if P is a polynomial in n variables with coefficients in R', the problem of finding a zero of P in A| is replaced by the problem of finding a common zero in A of d polynomials in nd variables with coefficients in R. Let Y be the affine scheme over R defined by the polynomial system F (Y=Spec R[X]/FR[X]). Since the scheme YK over K obtained from Y by change of base is inseparable over K, there is a purely inseparable finite extension K' of K such that the scheme YK' is not reduced, afortiori YR' is not reduced [5; 4.6-3]- Consider the affine scheme ~'YR' over R. There is a canonical R-morphism 0 :Y~o~z'YR, which corresponds by adjointness to the identity morphism of Ya,. Now ~'((Ya')rea) is a closed subscheme of o~-Y R,; let W be its pre-image under 0. Then W is a proper closed subcheme of Y, otherwise the identity morphism of YR, would factor through (YR')red, i.e., YR' would be reduced, contradicting the choice of R'. By inductive assumption, there are integers N', c', s' for W. Suppose y is a point of Y in R/t ~. Let e be the ramification index of the discrete valuation ring R' over R, u a generator of its maximal ideal. Then y induces a point of YR' in R'/u% By a previous argument, there is an integer q (independent of y) such that the image of this point modu [*~/ql is actually a point of (YR')r.d- By adjointness, the image ofy mod t [~/q] is actually a point of W. Hence N = qN', c = qc', s = s' are integers for F. Remark. -- Theorem i is false without the separability assumption. For there exists a discrete valuation ring R whose completion R* is a purely inseparable integral extension of R [6; o. 207]. R must therefore be its own Henselization. The minimal polynomial of an element of R* not in R gives a counter-example to Corollary 2. 2. Applications to C i questions. Recall that a domain R is called C~ if any form with coefficients in R of degree d in n variables with n> d ~ has a non-trivial zero in R. Co means that the field of fractions of R is algebraically closed. Theorem 2. -- If k is a CJield, then the field k( (t) ) of formal power series in one variable t over k is C~+1. This generalizes some results of Lang [3], who did the cases i= I, k finite, and i----o. Note that [k:kP]~_p i (take a basis). It suffices to prove that R=k[[t]] is Ci+ t. By Lang [3], k[t] is Ci+ 1. Hence the hypersurface H in projective (n -- i )-space defined by the given form has a point in the ring R/t ~ for aU v. By Corollary 2, H has a point in R. Note 1. -- The same type of argument yields a short proof of Lang's theorem that if R is a Henselian discrete valuation ring with algebraically closed residue field, such that K* is separable over K, then R is C 1. For by Corollary 2, we may assume R complete, and since C 1 is inherited by finite extensions, we may also assume R unramified. Then the argument given in [3; P. 384] shows H has a point in R/t ~ for all ,~. 566 RATIONAL POINTS IN HENSELIAN DISCRETE VALUATION RINGS 6 3 Note 2. -- In the definition of Ci, replace the word "form " by "polynomial without constant term "; a ring with this property is called strongly C~. For example, finite fields are strongly C 1, A theorem of Lang-Nagata states that an algebraic function field in one variable over a strongly C~ field is strongly Ci+ 1. It is natural to ask whether the same statement holds for the power series field in one variable. Ax-Kochen confirm this in characteristic o by showing that the Henselization of k[t] at the origin is elementarily equivalent to k[[t]]. Note 3. -- In the definition of strongly C~, suppose we take the expression " non- trivial " to mean " some coordinate is a unit in R ", instead of " some coordinate is non-zero ". Call this property strongly C~. If R is a strongly C~ discrete valuation ring, then the completion of R is also strongly C~, by Theorem x. It is therefore natural to ask: If a field k is strongly Ci, is the localization of kit] at the origin strongly C~+ 1 ? 3. Newton's Lemma. In this section, R will be an analytically irreducible Henselian local domain with maximal ideal m, F will be a system of r polynomials in n variables with coefficients in R, I <r<n, J the Jacobian matrix of this system. Lemma 1. -- Assume r = n. Given x in R such that F(x) =o (mod m) detJ(x),o (mod m) Then there is y in R such that (i) y-=x (modm) (ii) F(y) -----o Pro@ --There is y in the completion R* satisfying (i) and (ii), by [2; II.I3.3]. Since r=n and det J(y)+o, the domain R[y] is separably algebraic over R. But R is separably algebraically closed in R*, hence y is in R. Lemma 2. -- Let x in R be such that F(x)--o (mod e~m) where e=D(x), D being a subdeterminant of order r of detJ. Then there is y in R such that y-x (modem) F(y) =o Proof. -- We may assume e + o. We may assume x =- o and that D is the subdeter- minant obtained from the first r variables. If r<n, setting F~.(X)-----X i j= r + I, ..., n shows we can assume r = n, hence D = det J. Let J' be the adjoint matrix to J, so that JJ'=DI=J'J, with I the identity matrix. By Taylor's formula, F(eX) -= F(o)+eJ(o)X -t-e2G(X) 567 6 4 MARVIN J. GREENBERG where G(X) is a vector of polynomials each beginning with terms of degree at least 2. Using e =J(o)J'(o) and the hypothesis on F(o), we can factor out eJ(o): F(eX) = eJ(o) H (X) where H is a system whose Jacobian matrix at o is I, and H(o)~o (modm) By lemma i, there isy' in m such that H(y')=o, whence y=ey' does the trick. Note. -- The following argument (due to M. Artin) should eliminate the assumption that R is analytically irreducible, used in the proof of Lemma i: Let Y= Spec R[X]/FR[X], f: Y ~ Spec R the canonical morphism. The hypothesis of Lemma I gives us a point ~ of Y lying over the closed point of Spec R, such that ~ is isolated in its fibre andfis smooth at ~. Hence the local ring D ofx on Y is dtale over R [5; II, 1.4] with the same residue field. Since R is Henselian, R-+o is an isomor- phism [x], hence we have a section Spec R--*Y passing through ~. 4. Acknowledgements. The argument in Case 1 has been developed from ideas of P. Cohen and A. N~ron. My original argument in Case 2 required the extra assumption [k : k-~]<oo; the present argument is essentially due to M. Raynaud. Newton's lemma for Hense!ian local rings was suggested by M. Artin. BIBLIOGRAPHY [i] M. AaroN, Grothendie& Topologies, pp. 86-9i , Harvard Notes, I962. [2l A. GROr~-mr~I~.CK, SOninaire Ggom~trie alg~brique, ,96o, I.H.E.S., Paris. [3] S. I.uu~o, On Quasi-Algebraic Closure, Annals of Math., vol. 55, 1952, 373-39o- [4] A. N~RON, ModUles minimaux des varifies ab~lienaes sur les corps locaux et global, Publ. I.H.E.S., n ~ 21, Paris, I964. [5l A. GRort-m~mmcK et J. DmtmoN~, 1210r~nts de g~orr~trie alg~brique, IV (a e pattie), Publ. I.H.E.S., n ~ 24, Paris, [6] M. NAOATA, Local Rings, Interseience, x962. [7l A. GROT~m~mmCK et J. D1~.tmo~., l~lOnents de g~orndtrie alg~brique, IV (premiere partie), Publ. I.H.E.S., n ~ 2o, Paris, x964. [8] A. Gao~m~,mmcK, Technique de descente, II, Sdminaire Bourbaki, t959-6o , expos~ x95. [9] M. GR~.~.~m~.RO, Schemata Over Local Rings, Annals of Math., vol. 73, I961, 624-648. Northeastern University, Boston, Mass. Manuscrit refu le 24 avril 1966. 1967. -- Imprimerie des Presses Universitaires de France. -- Vend6me (France) t~DIT, N ~ 29 469 ~PR~ E~r FRJu'~C~ IMP. ]go 19 965 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Rational points in henselian discrete valuation rings

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Publisher
Springer Journals
Copyright
Copyright © 1966 by Publications mathématiques de l’I.H.É.S
Subject
Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN
0073-8301
eISSN
1618-1913
DOI
10.1007/BF02684802
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Abstract

RATIONAL POINTS IN HENSELIAN DISCRETE VALUATION RINGS by MARVIN J. GREENBERG Io Let R be a Henselian discrete valuation ring, with t a generator of the maximal ideal, k the residue field, and K the field of fractions. Let R* be the completion of R, K ~ its field of fractions. If F=(FI,..., Fr) is a system of r polynomials in n variables with coefficients in R, and x is an n-tuple with coordinates in R, set F(x)= (Fl(X), ..., Fr(x)). If F' is another system of r' polynomials, let FF' denote the system of rr' products. By the ideal FR[X] generated by F is meant the ideal in R[X] generated by F1, . .., F r. Theorem 1. -- Assume, in case K has characteristic p>o, that K* is separable over K. Then there are integers N~I, c~i, s~o depending on FR[X] such that for any v>N and any x in R such that F(x) - o (mod e) there exists y in R such that y--x (mod t [~/cl-8) F (y) -= o Corollary 1. -- Let Z be a prescheme of finite type over R. Then there are integers N> I, c>I, s>o depending on Z such that for ,J>_N and for any pointx of Z in R/t ~, the image of x in Z(R/t [~/cl-~) lifts to a point of Z in R. Proof. -- We can take a finite covering of Z by affine opens Zi. We have Z(S) =yZ~(S) for any local R-algebra S, hence the maxima of the integers for the Z~ will do for Z. Corollary 2. -- Z has a point in R if and only if Z has a point in R/t~ for all ,;> I. Let V be the algebraic set in affine n-space over K which is the locus of zeros of F. In the special case that R is complete and V is K-irreducible, non-singular, with a sepa- rably generated function field over K, N6ron [4; Prop. 2% p. 38] has proved this theorem, showing that in this case one can take c= I. However, in the general case we may have c> i (consider the polynomial Y~--X 3 and for any even integer 2v the point x = (t ~, t~)). Theorem I implies that the hypothesis of non-singularity in [4; Prop. 22] can be dropped, so that the sets in that proposition are always constructible. Theorem I is proved by induction on the dimension m of V. If m=--I, i.e., the ideal FR[X] contains a non-zero constant, it is clear. Suppose m>o. We may assume the ideal FR[X] is equal to its own radical (i.e., the scheme over R defined by F is reduced): For let E generate its radical. Then some power E q is in FR[X]. From F(x)=o (modt ~) 563 60 MARVIN J. GREENBERG we conclude t ~ divides Eq(x), so that E(x) - o (rood t [~lql) If N', c', s' are integers for E, we see that N = qN', c = qc', s = s' are integers for F. We may further assume V is K-irreducible: For if V--= W u W', where W, W' are algebraic sets defined respectively by systems of polynomials G, G' with coefficients in R, let N', c', s' (resp. N", c", s") be integers for G (resp. for G'). If x in R satisfies F(x)-o (modt ~) then either G(x) -- o or G'(x) =o (mod t E~I21) since GG' is in the ideal FR[X]. Thus N = 2max(N', N") c = 2max(c', c") s = max(s', s") will work for F. Then there are two cases: Case 1. -- V is separable over K. Let J be the Jacobian matrix of F, and let D be the system of minors of order n--m taken from detJ. The the locus of common zeros of D and F is a proper K-closed W in V. By inductive hypothesis there are integers N', c', s' for the system (D, F). For each system F(i I of n--m polynomials out of F, (i) a system of n--m indices, let Vll I be the locus over K of zeros of F(~I, and let V(i~ be the union of the K-irreducible components of V(0 which have dimension m and are different from V; let Gci/ be a system of generators for the ideal of V(i+~ in R iX]. By inductive assumption there are integers N(0 , c(0, s(0 for the system (G(~), F). If x is a point of V(~) in some extension of K such that for some (j) D(~),(i) (x) + o then the tangent hyperplanes of F,1, ..., F~._= at x are transversal, and x ties on exactly one component of V(~), that component having dimension m. We now invoke (see Lemma ~, n ~ 3) Newton's Lemma. -- If x in R is such that F(~l(x ) =o (mod t ~+~) D(~,C~(x ) ~o (mod t ~) for some (j) then there exists y in R such that Fcil(y ) = o y-x (mod t ~) Hence D(~l,(i) (Y) 4= o If we knew also G(~(y)+o we could deduce that y is a point of V. Take v so large that = [(~-- i)/2] >max(N', all N(~)) 564 RATIONAL POINTS IN HENSELIAN DISCRETE VALUATION RINGS 6x Let x in R be a zero modt ~ ofF. If D(x)=o (modt ~) our inductive hypothesis gives us y in R such that y is a singular point of V and y - x (mod t [~/c']- 8') If for some (i) G(i)(x) ----o (mod t v) then again by induction there isy in R which is a point of VoVc +) such that y-x (mod Otherwise we use Newton's Lemma to find y in R which is a point of V such that y-= x rood P') Thus as integers for 17 we can take N ---- 2 +2max(N', all N(o ) c = 2 max(c', all c(0 ) s = I +max(s', all s(0 ) Case 2. -- V is inseparable over K. In this case we need two facts. Fact 1. -- If K' # a finite extension Of K, then the integral closure R' Of R in K' is a finite R-module. This follows from our assumption K* separable over K (7; Oiv, ~3. i. 7 (ii)]. For the convenience of the reader, we sketch the proof, valid also when R is a higher dimen- sional local domain: K'| is a finite extension field of K*, because of our assumption. R'QRR* is a subring of this field, integral over the complete local domain R*, hence finite over R*. Since R* is faithfully flat over R, R' is a finite R-module. (The assump- tion that R* is a domain, implicit in this argument, can be eliminated (loc. cit.)). Fact 2. -- There is a functor o~ from the category of affine schemes of finite type over R' to affine schemes of finite type over R such that ~" is right adjoint to the change of base functor from R to R'. Thus we have an isomorphism of bifunctors MorR(Y , o~'Z) --~ Morw(Ya, , Z) (for Y/R, Z/R'). Moreover, o*" preserves closed immersions. This follows from Fact I, and can also be established in greater generality (see [8; p. 195-13] where the notation o~'Z=rcR,/RZ is used). Choose a basis bl, ..., b~ for the R-module R'. Every element of R' has uniquely determined coordinates in R with respect to this basis, and the addition and multipli- cation in R' are given by polynomial functions in these coordinates. Hence there is a commutative ring scheme S over R, whose underlying scheme is affine d-space over R, such that for any R-algebra A, MOrR(Spec A, S) = A| t[~'/~(1)]-8(1)) 62 MARVIN J. GREENBERG Now the same arguments as in [9; PP- 638-9] can be repeated word for word. The point is that by using the basis bl, ..., bd, if P is a polynomial in n variables with coefficients in R', the problem of finding a zero of P in A| is replaced by the problem of finding a common zero in A of d polynomials in nd variables with coefficients in R. Let Y be the affine scheme over R defined by the polynomial system F (Y=Spec R[X]/FR[X]). Since the scheme YK over K obtained from Y by change of base is inseparable over K, there is a purely inseparable finite extension K' of K such that the scheme YK' is not reduced, afortiori YR' is not reduced [5; 4.6-3]- Consider the affine scheme ~'YR' over R. There is a canonical R-morphism 0 :Y~o~z'YR, which corresponds by adjointness to the identity morphism of Ya,. Now ~'((Ya')rea) is a closed subscheme of o~-Y R,; let W be its pre-image under 0. Then W is a proper closed subcheme of Y, otherwise the identity morphism of YR, would factor through (YR')red, i.e., YR' would be reduced, contradicting the choice of R'. By inductive assumption, there are integers N', c', s' for W. Suppose y is a point of Y in R/t ~. Let e be the ramification index of the discrete valuation ring R' over R, u a generator of its maximal ideal. Then y induces a point of YR' in R'/u% By a previous argument, there is an integer q (independent of y) such that the image of this point modu [*~/ql is actually a point of (YR')r.d- By adjointness, the image ofy mod t [~/q] is actually a point of W. Hence N = qN', c = qc', s = s' are integers for F. Remark. -- Theorem i is false without the separability assumption. For there exists a discrete valuation ring R whose completion R* is a purely inseparable integral extension of R [6; o. 207]. R must therefore be its own Henselization. The minimal polynomial of an element of R* not in R gives a counter-example to Corollary 2. 2. Applications to C i questions. Recall that a domain R is called C~ if any form with coefficients in R of degree d in n variables with n> d ~ has a non-trivial zero in R. Co means that the field of fractions of R is algebraically closed. Theorem 2. -- If k is a CJield, then the field k( (t) ) of formal power series in one variable t over k is C~+1. This generalizes some results of Lang [3], who did the cases i= I, k finite, and i----o. Note that [k:kP]~_p i (take a basis). It suffices to prove that R=k[[t]] is Ci+ t. By Lang [3], k[t] is Ci+ 1. Hence the hypersurface H in projective (n -- i )-space defined by the given form has a point in the ring R/t ~ for aU v. By Corollary 2, H has a point in R. Note 1. -- The same type of argument yields a short proof of Lang's theorem that if R is a Henselian discrete valuation ring with algebraically closed residue field, such that K* is separable over K, then R is C 1. For by Corollary 2, we may assume R complete, and since C 1 is inherited by finite extensions, we may also assume R unramified. Then the argument given in [3; P. 384] shows H has a point in R/t ~ for all ,~. 566 RATIONAL POINTS IN HENSELIAN DISCRETE VALUATION RINGS 6 3 Note 2. -- In the definition of Ci, replace the word "form " by "polynomial without constant term "; a ring with this property is called strongly C~. For example, finite fields are strongly C 1, A theorem of Lang-Nagata states that an algebraic function field in one variable over a strongly C~ field is strongly Ci+ 1. It is natural to ask whether the same statement holds for the power series field in one variable. Ax-Kochen confirm this in characteristic o by showing that the Henselization of k[t] at the origin is elementarily equivalent to k[[t]]. Note 3. -- In the definition of strongly C~, suppose we take the expression " non- trivial " to mean " some coordinate is a unit in R ", instead of " some coordinate is non-zero ". Call this property strongly C~. If R is a strongly C~ discrete valuation ring, then the completion of R is also strongly C~, by Theorem x. It is therefore natural to ask: If a field k is strongly Ci, is the localization of kit] at the origin strongly C~+ 1 ? 3. Newton's Lemma. In this section, R will be an analytically irreducible Henselian local domain with maximal ideal m, F will be a system of r polynomials in n variables with coefficients in R, I <r<n, J the Jacobian matrix of this system. Lemma 1. -- Assume r = n. Given x in R such that F(x) =o (mod m) detJ(x),o (mod m) Then there is y in R such that (i) y-=x (modm) (ii) F(y) -----o Pro@ --There is y in the completion R* satisfying (i) and (ii), by [2; II.I3.3]. Since r=n and det J(y)+o, the domain R[y] is separably algebraic over R. But R is separably algebraically closed in R*, hence y is in R. Lemma 2. -- Let x in R be such that F(x)--o (mod e~m) where e=D(x), D being a subdeterminant of order r of detJ. Then there is y in R such that y-x (modem) F(y) =o Proof. -- We may assume e + o. We may assume x =- o and that D is the subdeter- minant obtained from the first r variables. If r<n, setting F~.(X)-----X i j= r + I, ..., n shows we can assume r = n, hence D = det J. Let J' be the adjoint matrix to J, so that JJ'=DI=J'J, with I the identity matrix. By Taylor's formula, F(eX) -= F(o)+eJ(o)X -t-e2G(X) 567 6 4 MARVIN J. GREENBERG where G(X) is a vector of polynomials each beginning with terms of degree at least 2. Using e =J(o)J'(o) and the hypothesis on F(o), we can factor out eJ(o): F(eX) = eJ(o) H (X) where H is a system whose Jacobian matrix at o is I, and H(o)~o (modm) By lemma i, there isy' in m such that H(y')=o, whence y=ey' does the trick. Note. -- The following argument (due to M. Artin) should eliminate the assumption that R is analytically irreducible, used in the proof of Lemma i: Let Y= Spec R[X]/FR[X], f: Y ~ Spec R the canonical morphism. The hypothesis of Lemma I gives us a point ~ of Y lying over the closed point of Spec R, such that ~ is isolated in its fibre andfis smooth at ~. Hence the local ring D ofx on Y is dtale over R [5; II, 1.4] with the same residue field. Since R is Henselian, R-+o is an isomor- phism [x], hence we have a section Spec R--*Y passing through ~. 4. Acknowledgements. The argument in Case 1 has been developed from ideas of P. Cohen and A. N~ron. My original argument in Case 2 required the extra assumption [k : k-~]<oo; the present argument is essentially due to M. Raynaud. Newton's lemma for Hense!ian local rings was suggested by M. Artin. BIBLIOGRAPHY [i] M. AaroN, Grothendie& Topologies, pp. 86-9i , Harvard Notes, I962. [2l A. GROr~-mr~I~.CK, SOninaire Ggom~trie alg~brique, ,96o, I.H.E.S., Paris. [3] S. I.uu~o, On Quasi-Algebraic Closure, Annals of Math., vol. 55, 1952, 373-39o- [4] A. N~RON, ModUles minimaux des varifies ab~lienaes sur les corps locaux et global, Publ. I.H.E.S., n ~ 21, Paris, I964. [5l A. GRort-m~mmcK et J. DmtmoN~, 1210r~nts de g~orr~trie alg~brique, IV (a e pattie), Publ. I.H.E.S., n ~ 24, Paris, [6] M. NAOATA, Local Rings, Interseience, x962. [7l A. GROT~m~mmCK et J. D1~.tmo~., l~lOnents de g~orndtrie alg~brique, IV (premiere partie), Publ. I.H.E.S., n ~ 2o, Paris, x964. [8] A. Gao~m~,mmcK, Technique de descente, II, Sdminaire Bourbaki, t959-6o , expos~ x95. [9] M. GR~.~.~m~.RO, Schemata Over Local Rings, Annals of Math., vol. 73, I961, 624-648. Northeastern University, Boston, Mass. Manuscrit refu le 24 avril 1966. 1967. -- Imprimerie des Presses Universitaires de France. -- Vend6me (France) t~DIT, N ~ 29 469 ~PR~ E~r FRJu'~C~ IMP. ]go 19 965

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Publications mathématiques de l'IHÉSSpringer Journals

Published: Dec 1, 1966

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