Artin approximation

Artin Approximation Dorin Popescu Institute of Mathematics, University of Bucharest, P.O. Box 1-764, Bucharest 70700, Romania E-mail: dorin @stailow, imar. ro

Contents 1. Henselian tings and the property of Artin approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Ultraproducts and the strong approximation property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. l~tale maps and approximation in nested subrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Cohen Algebras and General N6ron Desingulatization in Artinian local tings . . . . . . . . . . . . . . . 5. Jacobi-Zariski sequence and the smooth locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. General N6ron desingularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Proof of the Main Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H A N D B O O K OF ALGEBRA, VOL. 2 Edited by M. Hazewinkel 9 2000 Elsevier Science B.V. All tights reserved 321

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Artin approximation theory has many applications in algebraic geometry (for example to the algebraization of versal deformation and constructions of algebraic spaces (see [5-7])), in algebraic number theory, and in commutative algebra concerning questions of factoriality (see [5,23]). Here we present the main results of this theory with some applications in commutative algebra (see, e.g., 2.10, 2.13). Most of the proofs are based on the so-called General Nrron Desingularization. Its proof is given here only for tings containing Q with the idea of avoiding the difficulties of the nonseparable case; the interested reader is invited to examine the general case in a very nice exposition of Swan [38]. Another consequence of General Nrron Desingularization (we omit this here since it has not too much in common with Artin approximation theory) says that a regular local ring containing a field is a filtered inductive limit of regular local rings essentially of finite type over Z. This is a partial positive answer to a conjecture by Swan and proves the Bass-Quillen Conjecture in the equicharacteristic case using Lindel's result [24] (see also [32,38]). We require here a lot of results from the theory of Henselian tings, 6tale maps and Andrr-Quillen homology which we briefly present without proofs, since the reader can find them in some excellent books, such as, e.g., [34,21,1], or in the short exposition [38].

1. Henselian rings and the property of Artin approximation Let (A, m, k) be a local ring. A is a Henselian ring if it satisfies the Hensel Lemma, that is given a monic polynomial f in a variable Z over A and a factorization f -- f mod m = ~,/Tt of f in two monic polynomials ~, h 6 k[Z] having no common factors, there exist two monic polynomials g, h E A[Z] such that f -- gh and ~ -- g mod m,/t -- h mod m. If A is a Henselian local ring then every finite A-algebra is a product of Henselian local rings. For details concerning this theory see [ 19,34,21,22]. THEOREM 1.1 (Implicit function theorem). Let f = (fl . . . . . f r ) be a system o f polynomials in Y = (Y1 . . . . . Yn) over A, r ~ n, and A f Q A[Y] the ideal generated by the r • r-minors o f t h e Jacobian matrix ( S f i / S Y j ) . Suppose A is Henselian. I f ~ c A n satisfies f ( ~ ) ----0 mod m and A f ( f ) ~ m then there exists a solution y E A n o f f in A such that y ----~ mod m (y is unique if r = n). This theorem is the algebraic version of the well known Implicit Function Theorem from Differential or Analytic Geometry: THEOREM 1.2. Let gi(X1 . . . . . Xs, Y1 . . . . . Yn) = O, 1 <<,i <~ n, be some analytic equations, where gi :C s+n __+ C are analytic maps in a neighborhood o f (0, O) ~ C s+n with the property that gi(O, O) = O, 1 <<,i <~ n, and det(Sgi/OYj(O, 0)) ~ O. Then there exist some unique maps yj : C s ~ C, 1 <~ j <~n, which are analytic in a neighborhood o f O ~ C s such that yj(O) = 0 and g ( X , y) = 0 in a neighborhood o f O. In particular Theorem 1.2 says clearly that in the local ring A of all analytic germs of maps defined in neighborhood of 0 ~ C s, Theorem 1.1 holds. It is well known that A is Henselian if and only if for every monic polynomial h c A[Z] in a variable Z such

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that h(0) 6 m and Oh/OZ(O) q~ m there exists an unique solution z E m of h in A (thus Theorem 1.1 also characterizes Henselian tings). The completion A of A in the m-adic topology is Henselian. There exists also another variant of Theorem 1.1, which says that if f (~) = 0 mod m e for a c c H then the solution y can be taken such that y = ~ mod me. LEMMA 1.3. Let ( A , m , k ) be a Henselian local ring, f = (fl . . . . . fr) a system of polynomials^in Y - (Y1,..., Yn) over A, r <<.n, and ~ a solution o f f in A such that A f (y) ~ m A. Then f has a solution y in A such that y =---~ mod m A. PROOF. Choose ~ 6 A n such that ~ -- ~ mod m,4 (Note that A l m A ~ A / m ! ) . We have f (~) _--__f (~) -- 0 mod m ?] and A f (y) ~ A f (y) ~ 0 mod m/]. By Theorem 1.1 there exists y E A n such that f ( y ) = 0 and y ___~ mod m. [3 REMARK 1.4. Lemma 1.3 says in particular that if a system of polynomial equations f over A has a special solution ~ in ,~ (that is A f (~) ~ 0 mod m/~) then it has also a solution in A. A Noetherian local ring (A, m) has the Artin approximation property (in brief, A has AP) if every finite system of polynomial equations over A has a solution in A if it has one in the completion A of A. The study of the Artin approximation started with the famous papers of M. Artin [4,5], which state that the convergent power series rings over a nontrivial valued field of characteristic zero, the Henselization of a local ring essentially of finite type over a field, and an excellent Dedekind ring all have AP. Note that Remark 1.4 shows a weak form of AP. Moreover, if A has AP then A is necessarily Henselian. Indeed, if f is a polynomial in Z over A such that f ( 0 ) 6 m and (3f/OZ)(O) ~ m then f has an unique root in m A (A is Henselian!), which must be in m since A has AE ^

LEMMA 1.5. A Noetherian local ring (A, m) has AP if and only if every finite system of polynomial equations over A in Y = (YI . . . . . Yn), n E 1~, has its set of solutionsin A dense with respect to the m-adic topology in the set of its solutions in the completion A of A; that is, for every solution ~ of f in A and every positive integer c there exists a solution y of f in A such that y ----~ mod m c,4. PROOF. The sufficiency is trivial. If 33 is a solution of f in ,4 and c 6 H, c h o o s e a system of elements ~ in A such that ~ _= 33 mod mC/~ ( A / m c ~ ,4/mC,4!). We have

i:1 for some elements ~.i E m c and Zi E /~n, 1 ~ i ~< S. Thus (~, (Zi)i) is a solution of

f--O,

g := ~ -

Y-

~,kiZi i=1

--O

inA.

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If A has AP there exists a solution (y, ^ (Zi)i) of f = 0, g - - 0 in A. We have f ( y ) = 0, y -- ~ mod m c and so y -- ~ mod m cA. [] PROPOSITION 1.6. Let (A, m) be a Noetherian local ring which has A P and f a system o f polynomial equations in Y - (I11 . . . . . Yn) which has just a finite set o f solutions in m c (possible n o n e ) f o r a certain positive integer c. Then f has no other solutions in mCA, J being the completion o f A. PROOF. Let y(1) . . . . . y(S) be the solutions of f in m c and suppose that there exists a solution ~ of f in mC,3, which is different from the previous ones. Then there exists t > c such that y(i) ~ f~ mod m t.4 for 1 ~< i ~< s, the m-adic topology of A being separate. As A has AP we find a solution y of f in A such that y = ~ mod m t fi~ (see L e m m a 1.5). Then y ~ y(i) mod m t.4, 1 <~ i <~ s, and so y is a solution of f in m e different from all y(i). Contradiction! D COROLLARY 1.7. Let (A, m) be a Noetherian local ring which has A P and let J be its completion. Then (i) A is reduced if and only if,4 is reduced, (ii) I f A is an integral domain then it is algebraically closed in A. ^

PROOF. (i) A is reduced if ,4 is reduced, because the completion map A --> ,4 is injective. If A is reduced then the polynomial f -- Z n has in A only the solution z - 0. By the previous proposition f cannot have nonzero solutions in ,4,that is ,4 is also reduced. (ii) If f is a polynomial in Z over A then it has at most deg f roots in A. By the previous proposition f cannot have other roots in A. Thus A contains all roots of f in A. U] PROPOSITION 1.8. Let (A, m) be a Noetherian local ring which has A P and B a finite local A-algebra. Then B has A P too. PROOF. Let Wl . . . . . Ws be a system of generators of B as A-module, q~" A s ~ B, the map given by (al, . . . , as) ~ ~ s = 1 aiwi and uj -- (u jl . . . . . Ujs) ~ A s, 1 ~ j ~< t, a system of generators of Ker~b. Let f = (fl . . . . . fr) be a system of A polynomials in Y -- (Y1 . . . . . Yn) over B, B the completion of B and 33 = @l . . . . . Yn) ~ B n. Then

yj -'- ~ yj)~W). ~=1 for some yjx 6 , 4 (,4

or--1

@A B ~ '

B!) and j5 (~) has the form

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for some polynomials fi~ in Yj~ over A. Clearly ~ is a solution of f in B if and only if there exist (~:i~), 1 ~< i ~< r, 1 ~< r ~< t, in A such that t

f/c~ ((Yj)~)) -- ~ Zi/~tt/~ot, /~=l

l<~i~
((Ui) generate also Ker(,4 t~)A t~)!). Suppose ~ is a solution of f in B'. Then (~jz), (Zi/3) is a solution i n / ] of the polynomial equations t

fiot((Yj~.)) -- ~-~ Zi[3gflot,

l <~ i <~ r, l <~u <~ s,

in (Yjz), (Zi~) over A, which must have a solution (yj~), (Zi[3) in A because A has AP. Clearly (yj), yj - ~ = 1 yj~wz, forms a solution of f in B. D PROPOSITION 1.9. Let (A, m) be a Noetherian local ring which has AP and ,J, its completion. Then (i) A is an integral domain if and only if A is an integral domain, (ii) /f p ~ Spec A then p,4 ~ Spec A, (iii) For every field K which is a finite A-algebra, the ring K ~A "~ is an integral domain. PROOF. (i) Suppose ,4 is not an integral domain; that is, there exist two nonzero elements J,~ 6,4 such that J~ = 0. Choose a positive integer c such that ~,~ ~ mC/t. By Lemma 1.3 there exist x , y ~ A such that x y = 0 and x -----~, y -- ~ mod mc. It follows x q~m c, y ~ m e and so A is not an integral domain. (ii) If p 6 Spec A then A / p is an integral domain which has AP by the previous proposition. Thus A l p , 4 ~ (A/p-~ is an integral domain by (i) which is enough. (iii) We may reduce to the case A C K by (ii) and the previous proposition. Choose a basis b = (bl . . . . . bn) of K over the fraction field Q ( A ) of A. Multiplying b with a nonzero element of A, we may suppose all bi integral over A. Thus B = A[b] is a finite A-algebra and so a finite product of local rings (A is Henselian!). Since B is an integral domain, it must be local. Then B has AP and its completion/~ is an integral domain by (i) and the previous proposition. Hence K |

A -- Q ( B ) @A ,~ ~" Q ( B ) (~B (g t~ A A) "~ Q ( B ) ~ B

is an integral domain. REMARK 1.10. A convergent power series in X - (Xl . . . . . Xs) over C is irreducible in C[[X]] if it is irreducible in A "--C{X}. This is a consequence of Proposition 1.9 (ii) because A has AP by the Artin's result from [4].

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Let u ' A ~ A t be a morphism of Noetherian rings. The fibers of u are geometrically reduced (resp. geometrically regular) if for every field K, which is a finite A-algebra, the ring K t~) a A t is reduced (resp. regular). The map u is called regular if it is flat and its fibers are geometrically regular. Let A be a Noetherian local ring and A the completion of A. The formal fibers of A are geometrically reduced (resp. geometrically regular) [25, Section 32] if the fibers of the completion map A --+ ,4 are geometrically reduced (resp. geometrically regular). A Noetherian local ring is quasi-excellent (resp. universally Japanese called also Nagata) if its formal fibers are geometrically regular (resp. reduced). An excellent local ring is an universally catenary, quasi-excellent local ring. A Henselian local ring is excellent if and only if it is quasi-excellent. For example Noetherian complete local tings and the convergent power series ring C{X} from 1.10 are all excellent (see [19]). A

PROPOSITION 1.1 1. A Noetherian local ring which has AP is Henselian and universally Japanese. PROOF. Let A be a Noetherian local ring which has AP. Note that the formal fibers of A are integral domains by Proposition 1.9(iii). In particular A is universally Japanese. M THEOREM 1.12 ([19,21,22]). Let A be a Henselian local reduced ring which is universally Japanese, A its completion and Q(A) the total fraction ring of A. Suppose that Q(A) (~A A is normal. Then A is algebraically closed in A. ^

REMARK 1.13. This theorem says in particular that if A is an excellent Henselian local reduced ring then every polynomial equation over A in one variable which has a solution in ,4 has also one (the same) in A. This suggests the following result conjectured by M. Artin [6]. THEOREM 1.14 ([31,38]). An excellent Henselian local ring has A P . The proof goes like in Lemma 1.3 using the following theorem, its proof being given in Sections 4-7 only for rings containing the rational numbers Q (for the complete proof see [30,31,33]; [3,27,38,37])" THEOREM 1.15 (General Nrron desingularization). Let u : A --+ A' be a regular morphism o f Noetherian rings, B a finite type A-algebra and v : B --+ A t a morphism of Aalgebras. Then v factorises through an A-algebra C = ( A [ Y ] / ( f ) ) g , Y = (Y1 . . . . . Yn), where f = (fl . . . . . fr), r <~n, are polynomials in Y over A and g belongs to the ideal A f generated by the r x r-minors of the Jacobian matrix (Ofi /OYj). Indeed, let (A, m) be an excellent Henselian local ring, h a system of polynomial equations in Z -- (Z1, . . . , Zs) over A and s a solution of h in the completion A of A. Then the A-morphism v ' B := A [ Z ] / ( h ) --+ A, Z ~ ~ factors through an A-algebra C as in Theorem 1.15 (A --+ ,4 is regular since A is excellent!), let us say v -- wq, w ' C ~ A, q ' B - + C. A

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Now note that ~ -- w(I?) is a solution of f such that g@) -- w(~) ~ m/~.

Thus Zlf (y)

m/~ and by L e m m a 1.3 there exists a solution y of f in A such that y _----~ mod m/~. Then g(y) ---- g(~) ~ 0 mod m/~ and so we get an A-morphism u ' C ---> A by Y ---> y. Clearly z = uq (Z) is a solution of h in A. REMARK 1.16. Artin's ideas to prove AP in [4,5] use mainly the Weierstrass Preparation Theorem and this is possible in the most important cases. These ideas do not work for the ring C{X}[[Z]], X = (X1 . . . . . Xs), Z = (Z1 . . . . . Zt), which is excellent Henselian and so it has AP by Theorem 1.14.

2. Ultraproducts and the strong approximation property Let D be a filter on N, that is a family of subsets of N satisfying (i) 0 r D, (ii) if s, t ~ D then s N t ~ D, (iii) i f s a D a n d s c t C N t h e n t ~ D . An ultrafilter on N is a maximal in the set of filters on N with respect to the inclusion. A filter D on N is an ultrafilter if and only if N \ s ~ D for each subset s C N which is not in D. An ultrafilter is nonprincipal if there exists no r a N such that D = {s [ r ~ s C N}. An ultrafilter on N is nonprincipal if and only if it contains the filter of all cofinite subsets of N. The ultrapower A* of a ring A with respect to a nonprincipal ultrafilter D on N is the quotient ring of A N by the ideal ID of all (Xn)neN such that the set {n ~ N lXn = 0} c D. Denote by [(Xn)] the class modulo ID of (Xn). Assigning to a e A the constant sequence [(a, a . . . . )] we get a ring morphism CA : A ---->A*. Similarly we may speak about the ultrapower M* of an A-module M which has also a structure of A*-module. As above we have a canonical map CM : M ~ M*. For details concerning this theory see [10,8,29, 311. PROPOSITION 2.1. Let A be a ring, D a nonprincipal ultrafilter on N and A* the ultrapower of A with respect to D. Then (i) A* is an integral domain (resp. field) if A is an integral domain (resp. field), (ii) If A is a domain, then Q(A)* ~- Q(A*), where Q(A), Q(A*) are the fraction fields o f A, resp. A*, (iii) (A*) q ~ (Aq) *, q ~ N, (iv) If A is a field then A* is a separable field extension of A, (v) If a C A is an ideal and a* is the ultrapower o f a with respect to D, then a* C A* is an ideal and ( A / a ) * ~- A*/a*, (vi) If p C A is a prime (resp. maximal) ideal then p* C A* is a prime (resp. maximal) ideal, (vii) If a C A is a finitely generated ideal, then a* -- CA (a)A*. PROOF. (i) If [(Xn)][(Yn)] = 0 in A* then r = {n E N I XnYn = 0} 6 D. If A is an integral domain then r = s tO t for s = {n 6 N I Xn = 0}, t = {n 6 N I Yn = 0}. As D is an ultrafilter

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it follows either s 6 D, or t 6 D (s U t E D!) and so either [ ( X n ) ] = 0, o r [ ( Y n ) ] = 0. N o w suppose that A is a field and let 0 r x -- [(Xn)] 6 A*. Then c = {n 6 1~ I Xn 5~ 0} ~ D and Yn = Xn 1 for n E c and 0 otherwise define an inverse [(Yn)] of x. (ii) If A is an integral d o m a i n then the inclusion A C Q ( A ) induces the inclusion A* C Q ( A ) * . By (i) Q ( A ) * is a field. If z -- [(Zn)] 6 Q ( A ) * then Zn = u n / v n , Vn ~ O, Un,Vn E A, and clearly z = [(Un)]/[(Vn)] ~ Q ( A * ) . Hence Q ( A ) * = Q ( A * ) . (iii) Let (ei) 1<~i<~q be the canonical basis in A q . We must show that (~aq(ei)) 1<~i<~q is a basis in (Aq) * over A*. If u = [(Un)] 6 (Aq) * then we have

lgn -- Z

blniei ,

blni ~ A.

i=1

Hence q

lg -- Z

[(blni)]q~Aq(ei)

i=1

that is (qbAq(ei))i generate ( A q ) *. N o w , if q

l)i (~Aq (ei ) -- 0 i=1

for s o m e l)i --- [(l)ni)] E A * t h e n

s=

n ENI

vniei--O

ED.

i=1

Thus l)ni "-" 0 for n 6 s, 1 ~< i ~< q, and so l)i = O. Hence (~)Aq (ei))i is a basis. (iv) If A is a field then A* is a field by (i). Let K be a finite field extension of A. Then K ~- Aq for a certain q and we have K* ~ (Aq)* ~-- (A*)q ~-- A* (~a K by (iii) as A*-linear spaces. In particular d i m a , K* -- d i m a , A* (~)a K. Hence the canonical ring surjection A * ( ~ a K ~ K* is an isomorphism. (v) The surjection A --+ A / a induces the surjection A* --+ ( A / a ) * whose kernel is given by all x = [(Xn)] E A* such that s = {n 6 1~11xn ~ a} ~ D. Put Yn = Xn if n 6 s and Yn -- 0 if n ~ s. Clearly x = [ (Yn)] c a*. (vi) follows from (i) and (v). (vii) A surjection A q ~ a induces a surjection (Aq) * ~ a* and it is e n o u g h to apply (iii). [] REMARK 2.2. It is easy to see that the ultrapower construction defines an exact functor M o d A --+ M o d A * , which coincides with A* (~)A - - on finitely g e n e r a t e d A - m o d u l e s w h e n A is Noetherian (this is already suggested by Proposition 2.1 (iii), (v), (vii)). In particular t~a is flat w h e n A is Noetherian.

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PROPOSITION 2.3. Let A be a ring, D a nonprincipal ultrafilter on N and A* the ultrap o w e r o f A with respect to D. Then (i) The Jacobson radical J ( A ) o f A is m a p p e d by C/)A in J ( A * ) . (ii) I f m l . . . . . ms are all maximal ideals o f A then m 1. . . . . m s are all maximal ideals o f A*. In particular A* is local if A is local. (iii) I f A is an Artinian local ring then A* is also Artinian a n d length A A = length A, A*. (iv) I f A is Henselian local ring then A* is also Henselian local ring. PROOF. If u C J ( A ) and x = [(Xn)] 6 A* then 1 + ux = [(1 + UXn)] is invertible in A* because 1 + UXn is invertible in A. (ii) By Proposition 1.1(v), (vii) we see that A * / J ( A ) * ~- ( A / J ( A ) ) * is a product of fields ( A / m i ) * ~ A * / m * . Using (i) it follows that (m*) are all m a x i m a l ideals of A*. (iii) If (A, m, k) is Artinian local then m s = 0 for some s 6 N. Since m is finitely generated, m A * = m* - the unique maximal ideal of A* by (ii) is finitely generated and (m*) s = 0. If p 6 Spec A* then (m*) s C p and so m* C p. Thus m* is the unique prime ideal of A*. By the C o h e n T h e o r e m A* is Noetherian and so even Artinian. If 0 = a0 C a l C -.. C at -- A is a composition series of A then 0 = a 0* C a~ C --- C a t -- A * is a composition series of A* because k* ~ ( a i + l / a i ) * ~ ai+ 1 . /a* (as in 2.1(v), or 2.2), k* being the residue field of A* by Proposition 2.1 (i) and (v). Thus length A A = lengthA, A*. (iv) Suppose that (A, m) is a Henselian local ring. Let d

f --Z[(Uni)]zi i=0

be a p o l y n o m i a l over A* in a variable Z such that f ( 0 ) ~ m* and Of/OZ(O) q~ m* and put d Uni Z i E A [ Z ] .

fn -- Z i=0

Then t -- {n ~ N] fn(O) ~- m, (Ofn/OZ)(O) q~m} ~_ D. I f n ~ t there exists Zn ~- m such that fn (Zn) = O, A being Henselian. Put Zn = 0 if n r t. Clearly z = [ (Zn) ] is a solution of f in m*. [51 L E M M A 2.4. Let (A, m) be a Noetherian local ring, f = (fe)eeN a countable system o f polynomials over A, D a nonprincipal ultrafilter on N and A* the ultrapower o f A with respect to D. The f o l l o w i n g statements are equivalent: (i) f has a solution in A I -- A * / m ec, * moo * :-- A i EN mi A *, (ii) f o r every t ~ N, f ( t ) = ( f l . . . . . f t ) has a solution in A * / m t A*.

PROOF. We have to show only (ii) =~ (i). For every t E N, let ~(t) = [@nU))] be in A* such that f ( t ) @ ( t ) ) _ 0 m o d m t A*. Then st = {n ~ N I f(t)@(nt)) -- 0 m o d m t } e D. Note that

,(n)

Sr

S t --

l <~r <~t

-- {0 . . . . . t} 6 D,

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D being nonprincipal. We have S '1 ~ S 2' ~ . - . ~ S t' ~ ' ' . and N r E I ~ Sr' "-- 13. Put Yn - - (~(tn)) if n 6 s it and tn is m a x i m u m such that n 6 s ttn and Yn = 0 if n r s '1. Clearly y = [(yn)] *. [] satisfies f (y) -- 0 mod m ~ THEOREM 2.5 ([29,31]). Let (A, m) be a Noetherian local ring, D a nonprincipal ultrafilter on 1~, A* the ultrapower o f A with respect to D, and ~rA the composite map A ~ A* ~ A1 :-- A* / m ~*,

m ~* = O m i A

*

i 6l~

Then (A1, mA1) is a Noetherian complete local ring, dimA1 = d i m A and lPA is flat. PROOF. Let

B = limA*/m iA* be the completion of A* and p : A* ~ B the canonical map. We claim that p is surjective. Indeed, if zi ~ A*, i ~ N, has the property that Zi+l - - z i mod miA* for all i 6 N, then by L e m m a 2.4 the system of congruences Z - zi -- 0 mod m i A*, i ~ l~I, has a solution z ~ A* because it has solutions in A * / m t A * for all t ~ 1N. Clearly, p maps z on the system ( Z i ) i . As Ker p -- moo we see that A1 is complete local. A1 is Noetherian because its maximal ideal mA1 is finitely generated (see [25, 29.4]). Now A / m s is Artinian and by Proposition 2.3(iii) we have A 1 / m S A 1 ~- A * / m S A * Artinian and length a A / m s = length A, A* / m s A* = length a ~ A 1/ m s A 1. Thus the HilbertSamuel functions associated to A, resp. A 1, coincide. Hence dim A = dim A 1. As m generates the maximal ideal of A1 we see that l P a must be flat by [25, Section 23]. D COROLLARY 2.6. In the notations and hypothesis o f Theorem 2.5 suppose that A is quasiexcellent. Then ~[rA is regular. PROOF. Note that ~r A is a flat morphism of Noetherian local rings by Theorem 2.5 and induces a separable residue field extension k := A / m -+ k* by Proposition 2.1(iv). Since m A1 is the maximal ideal of A1 it is enough to apply the following Lemma, which is proved in Section 4. D LEMMA 2.7 ([2,11,25]). Let ~ :A ~ B be a flat morphism o f Noetherian local rings. Suppose that (i) A is quasi-excellent, (ii) the residue field extension induced by q5 is separable, (iii) the maximal ideal o f A generates the maximal ideal o f B. Then dp is regular. Let (A, m) be a Noetherian local ring. A has the strong Artin approximation property (in brief, A has SAP) if for every finite system of equations f in Y -- (Y1 . . . . . Ys) over A there exists a map v :N --+ 1N with the following property:

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If ~ 6 A s satisfies f ( ~ ) ------_0 mod m vCc), c 6 l~l, then there exists a solution y c A s of f with y -- ~ m o d m c. M. Greenberg [ 18] proved that excellent Henselian discrete valuation rings have SAP and M. Artin [5] showed that the Henselization (see Section 3) of a local ring which is essentially of finite type over a field has SAP. If (A, m) is a Noetherian local ring which has SAP then it has also AP. Indeed, let f be a finite system of polynomials over A in some variables Y, ~ a solution of f in the completion ,4 of A and v the SAP function associated to f . Choose ~ in A such that ~ - ~ mod m v(1)~. We have f (~) = f (~) = 0 mod m v(1)~. Thus f (~) -- 0 mod m uCl) and so there exists a solution y of f in A such that y - ~ mod m. The following lemma shows that the SAP is more easy handled in the framework of ultrapowers. LEMMA 2.8. Let (A, m) be a Noetherian local ring, D a nonprincipal ultrafilter on 1%I, A* the ultrapower o f A with respect to D, and l[rA the composite map A ~ A* ~ A1 "= A * / moo, *

* -- r ) m iA* . moo i6N

The following statements are equivalent: (i) A has SAP. (ii) For every finite system o f polynomials f over A, f o r every positive integer c and f o r * there exists a solution y o f f in A* such every solution ~ o f f in A* modulo moo that y = ~ mod mCA*.

PROOF. (i) :=~ (ii) Let f , ~ = [(Yn)] be like in (ii) and v:l~ ~ 1~1 the SAP function associated to f . We have in particular f (~) -- 0 mod m v(c)A*. Thus, the set s - {n E N I f (Yn) -- 0 mod m v(c)} 6 D. Then for every n ~ s there exists a solution Yn of f in A such that Yn ----Yn mod m c, v being the SAP function of f . Put Yn = 0 for n r s. Then y = [(Yn)] is a solution of f in A* such that y - ~ mod mCA*. (ii) =~ (i) Suppose that there exists a finite system of polynomials f in some variables Y over A which has no SAP function; that is there exists a positive integer c such that ( , ) For every n 6 N there exists Yn in A such that f (Yn) = 0 mod m n but there exists no solution Y'n of f in a such that yn~ -= ~,~ mod m c. Clearly, ~ -- [(Yn)] is a solution of f in A* modulo mr A* for all r 6 1~. Thus f ( ~ ) -0 modm~*. By (ii) there exists a solution y -- [(Yn)] of f in A* such that y -------~ mod m cA* Then the set s = {n ~ 1~ I f (Yn) = O, Yn -- Yn mod m~ 6 D is nonempty. Take n 6 s. Clearly, Yn contradicts (,). Part of the following theorem appeared in [28], extending some results from [40,41]. But the proof there has a gap in the nonseparable case, which was repaired in [23, Ch. 2]. Using ultrapowers easier proofs were given in [29] and [ 14]. The easiest one is given in [31, (4.5)] and it is presented here. E] THEOREM 2.9. An excellent Henselian local ring has SAP. In particular a Noetherian local ring has A P if and only if it has SAP.

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PROOF. We try to imitate the proof of Theorem 1.14. Let (A, m) be an excellent Henselian local ring, D a nonprincipal ultrafilter on H, A* the ultrapower of A with respect to D, and lpA the composite map A ~ A* ~ A1 :-- A*/moo*. By L e m m a 2.8 it is enough to show that given a system of polynomials h in Z = (Z1 . . . . . Zs) over A, a positive integer c 6 H and ~, a solution of h in A* modulo m ~* there exists a solution z of h in A* such that z -* factorises mod mCA *. Then the A-morphism v 9B := A [ Z ] / ( h ) --+ A 1, Z ~ ~ mod moo, through an A-algebra C as in Theorem 1.15 ( ~ a is regular by Corollary 2.6!), let us say v -- wq, w : C --+ A1, q : B ~ C. Thus, we have C = ( A [ Y ] / ( f ) ) g , Y = ( Y 1 , . . . , Yn), f = ( f l . . . . . f r ) , r <~n, g ~ A f and ~ -- w(I~) is a solution of f in A1 such that g(~) -w(~) r mA1. Hence A f (~) ~ reAl. Let ~ be a lifting of ~ to A*. In particular we have f ( ~ ) - 0 mod mCA *, A f ( ~ ) (s mA* and by a variant of the Implicit Function Theorem (A* is Henselian by Proposition 2.3(iv)!) there exists a solution y of f in A* such that y -mod mCA *. Then g ( y ) -- g(~) ~ 0 mod mA* and so we get an A-morphism u " C --+ A* by Y ~ y. Clearly z := u q ( Z ) is a solution of h in A* such that z ---- w q ( Z ) - v ( Z ) = mod mCA *. Now, if (A, m) is a Noetherian local ring, which has AP, then given a finite system of polynomials f over A, let v be the SAP function associated to f considered over the completion A of A (the complete local rings are excellent Henselian and so they have SAP as above!). We claim that the same function v works for f over A. Indeed, if f ( ~ ) -0 mod m v(c) for some elements ~ of A and a c ~ H then there exists a solution ~ of f i n / ] such t h a t ) -- ~ mod mCA. By L e m m a 1.5 f has also a solution y in A such that y -- ~ mod m c A. It follows y -- ~ mod m c,4 and so y _= ~ mod m c. [2 PROPOSITION 2.10 ([23]). Let (A, m) be an excellent Henselian local ring, which is an integral domain and (Xn)n~N a sequence o f elements, which converges in the m-adic topology to an element x E A. I f x is irreducible in A then there exists a positive integer t >> 0 such that Xn is irreducible f o r all n >/t. PROOF. A has SAP by Theorem 2.9. Let v be the SAP function associated to the polynomial f := Y Z - x over A. Then t = v(1) works. Indeed, if Xn is reducible for a certain n ~> t then there exist ~, ~ ~ m such that ~ = Xn -- x mod m t. In particular, f ( ~ , ~,) -- 0 mod m v(1) and so there exist y, z ~ A such that f ( y , z) = 0 and y -----~, z -- ~' mod m. Thus x = y z and y,z ~ m which is impossible since x is irreducible. D LEMMA 2.11. Let (A, m) be a Noetherian local ring which has AP, A its completion, f a system o f polynomials over A in Y = (Y1 . . . . . Yn) and g l , . . . , gr some systems o f polynomials in Y and Z -- (Z1 . . . . . Zs). Then the following statements are equivalent: (i) There exists a solution ~ o f f in A such that all systems gi (y, Z ) -- O, 1 <<.i <~ r, have no solutions in A, (ii) There exists a solution y o f f in A such that all systems gi (y, Z ) = O, 1 <<.i <<.r, have no solutions in A. ^

PROOF. (i) =~ (ii) Suppose there exists ~ i n / ] such that f ( ~ ) = 0 and gi(Y, Z) = 0 has no solutions in A for all 1 ~< i ~< r. Let vi be the SAP functions associated to gi(y, Z), 1 ~< i ~< r (,4 has SAP by Theorem 2.9). Then gi();, Z) has no solutions in A / m c ~- A/mC,4 ^

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for c -- max(vi(1))i. As A has AP we find a solution y of f in A such that y -- ~ mod mCA (see L e m m a 1.5). Then gi (y, Z) has no solutions in A / m c for all i and so they cannot have solutions in A. (ii) = , (i) Suppose there exists y in A such that f (y) = 0 and gi (y, Z) has no solutions in A for all 1 ~< i ~< r. Let v i/ be the SAP functions associated to gi (y, Z), 1 <~ i <<.r (A has also SAP by Theorem 2.9!). Then gi(y, Z), 1 <<.i <~ r have no solutions in A i m t ~ f i / m t A for t -- max(v~(1))i and so they cannot have solutions in ,4. D LEMMA 2.12. Let ( A , m ) be a Noetherian local integral domain, al . . . . . aq a system o f generators o f m, f = X 1 X 2 - X3X4, gl -- XI Z1 - X3, g2 = X1 Z 2 - X4 and g3 = q Zi,j--1 aiaj Ti Vj - Xl polynomials in some variables X, Z, T, V. Then A is not factorial if and only if there exists a solution x = (x l, x2, x3, x4) o f f such that each of the three polynomials gl (x, Z), gz(x, Z), g3(x, T, V) has no solutions in A. PROOF. A is not factorial if and only if there exists an irreducible element x l 6 A, which is not prime; that is, there exists x2, x3, x4 such that x ix2 = x 3 x 4 and x3 ~ x lA, x4 ~ x l A in other words x is a solution of f such that each polynomial g l (x, Z), g2(x, Z) has no solutions in A. Since Xl is also irreducible, it is not a product of two elements from m, that is g3 (x, T, V) has no solutions in A. Conversely, let x = (x l, x2, x3, x4) be a solution of f such that each polynomial gl (x, Z), g2(x, Z), g3(x, T, V) has no solutions in A. Then Xl --P-0 (otherwise T = V = 0 is a solution of g3 (x, T, V)) and Xl ~ m because otherwise gl (x, Z) has clearly a solution in A. As above note that xl is irreducible but not prime. D THEOREM 2.13 ([23,8,31]). Let (A, m) be an excellent Henselian local ring and .4 its completion. Then A is factorial if and only if.4 is factorial. The proof follows using Lemmas 2.11, 2.12, Proposition 1.9 and Theorem 1.14. THEOREM 2.14 ([12,13,8,31]). Let (A, m) be a Noetherian local ring, which has AP and fi its completion. Then (i) A is a normal domain if and only if,4 is a normal domain, (ii) The formal fibers o f A are geometrically normal domains (that is, f o r every field K which is a finite A-algebra, the ring K ~ a A is a normal domain). PROOF. If A is an integral domain, which is not normal then there exist n 6 1~, n n n--i_ i x l , x2, ul , . . . , Un ~ A with x2 ~ 0 such that Xl r x2A and x I + Y~i=l uix I x 2 - 0. Applying L e m m a 2.11 for n

U=X~+

Z

n-i

UiX I

i

X2,

g l = X2,

g2 = X1 - X 2 Z

i=1

we get (i) (gl (X) has no solutions in A means that X2 5~ 0!). (ii) Let K be a field, which is a finite A-algebra and ,4 the integral closure of A in K. A is finite over A because A is universally Japanese by Proposition 1.11 (see [25,33]).

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Then A is a product of Henselian local rings. Moreover A is local because it is an integral domain and it has AP by Proposition 1.8. Then the completion of A is A ~)A A and it must be a normal domain by (i). Thus K (~)A /~ is a normal domain because it is a fraction ring of,4 ~A A. [] REMARK 2.15. C. Rotthaus [35] proved that the formal fibers of a Noetherian local ring, which has AP are even geometrically regular. Thus the converse of Theorem 1.14 holds. On the other hand, we may speak about approximation properties for couples. The variant of Theorem 1.14 for couples goes through (see [31 ]), but the variant of Theorem 2.9 for couples does not hold as Spivakovsky noticed in [36].

3. l~tale maps and approximation in nested subrings A ring morphism f ' A ~ B is called quasi-smooth after [38] if for any R-algebra C and ideal I C C with 12 -- 0 any A-algebra morphism B ~ C / I lifts to an R algebra morphism B ~ C. If such a lifting is unique then f is called quasi-dtale. We call f smooth (resp. dtale) if it is finitely presented and quasi-smooth (resp. quasi-6tale). If f is essentially finitely presented (i.e. a localization of a finitely presented morphism) and quasi-smooth (resp. quasi-6tale) then f is essentially smooth (resp. essentially dtale). A separable field extension and a polynomial extension A ~ A[X1 . . . . . Xn] are smooth. A localization A ~ S - 1 A is essentially 6tale, S being a multiplicative system from A. Let C -- ( A [ Y ] / ( f ) ) g be the A-algebra from Theorem 1.15, Y = (Y1 . . . . . Yn), f = (fl . . . . . fr). Then C is a smooth A-algebra. If r -- n then C is even 6tale over A. Composition of smooth (resp. 6tale) morphisms are still smooth (resp. 6tale) and tensoring a smooth (resp. 6tale) morphism A ~ B by an A-algebra we get still a smooth (resp. 6tale) morphism. Details can be found in [21,34,19 ]. THEOREM 3.1 (Grothendieck [20]). If B is a local algebra essentially smooth over A, then B - (A[Y, T ] / ( f ) ) p , Y = (Y1 . . . . . Yn), where f is a monic polynomial in T over A[Y], P C A[Y, T] is a prime ideal containing f and Of/OT q~ P. Moreover if B is essentially dtale over A then n = 0 above. It follows from this theorem that smooth algebras are flat. Moreover it is easy to see that smooth algebras over fields are regular rings and so smooth maps are regular morphisms. An 6tale algebra over a field k is a product of finite separable field extensions of k. Let (A, m, k) be a local ring. An essentially &ale A-algebra B must have m B as maximal ideal because B / m B is an essentially &ale local k-algebra, that is a finite separable field extension of k. Hence m B is maximal. By [ 19,34] an essentially of finite presentation local A-algebra (B, b) is essentially 6tale if and only if it is flat, b = m B and B / b is a finite separable extension of k. If (B, b) is an essentially of finite type flat-local A,algebra such that b -- m B and B / b is a separable extension of k then B is essentially smooth over A, providing A is Noetherian. Indeed, choose some elements x -- (xl . . . . . Xr) in B lifting a separable transcendence basis of B / b over k. Then the map f :A[X]mA[X] ~ B given by X ~ x is flat by the local flatness criterion [25]. Since B / b is separable over k(x) we get by above that f is essentially 6tale, which is enough.

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An essentially 6tale local A-algebra B is an dtale neighbourhood of A if its structure morphism induces an isomorphism on the residue fields. COROLLARY 3.2 ([34]). A local A-algebra B is an (tale neighbourhood of A if and only

if B ~ (A[T]/(f))(m,T), f being a monic polynomial in T over A such that f (O) 9 m,

af/aT (0) r PROOF. If B = (A[T]/(f))(m,T) with f as above then Of/OT q~ (m, T) and so B is &ale over A as we have seen. Since k (~a B ~- (k[T]/T.-g)(T), where ~ 9 k[T] is a polynomial with ~(0) = Of/OT(O) mod m r 0 we see that k (~a B ~ k[T]/(T) ~- k. Hence B is an 6tale neighbourhood of A. Conversely, if B is an 6tale neighbourhood of A then by Theorem 3.1 B ~- ( A [ T ] / ( f ) ) p , where f is a monic polynomial in T and P C A[T] is a prime ideal such that f 9 P but Of/OT r P. Since B / m B ~ A / m we see that A + mB = B and so T mod f = a + x for an a 9 A and x 9 mB. Changing T by T + a we may reduce to the case when t = T m o d f 9 m B = P B. Then P contains the maximal ideal (m, T) of A[T] and so P = (m, T). By Taylor's formula 0 = f ( t ) = f(O) + t(Of/OT(O) + tfl (t)) for a polynomial fl 9 A [ T ] . It follows f ( 0 ) 9 (t) C mB and by faithful flatness f ( 0 ) 9 m. Similarly Of /OT(t) = Of /OT(O) + tf2(t) for some f2 9 A[T] and so

Of /OT(O) -- Of /OT(t) = Of /OT mod f q~mB. Hence Of / OT (O) q~m. Let (A, m) be a local ring. Then there exists a Henselian local A-algebra A such that for every Henselian local Azalgebra B there exists an unique local A-morphism A ~ B (for details see [34,21,19]). A is unique up to an isomorphism and it is called the Henselization of A. By construction A is the filtered inductive limit of all 6tale neighbourhoods of A. In particular A is a flat A-algebra. LEMMA 3.3. Let ( A , m ) be a local ring, and B an dtale neighbourhood of A. Suppose (A, m) is the filtered inductive limit of an inductive system of local rings ( C i , m i ) , qgij : C i ~ Cj. Then there exists j 9 I and an dtale neighbourhood Bj of Cj such that A (~Cj Bj "~ B. PROOF. By Corollary 3.2, B ~- (A[T]/(f))(m,T) for a monic polynomial f in T over A such that f (0) 9 m, 0 f / 0 T (0) it m. Then

A[T]--limj~IBj[T] and there exists i 9 I, fi 9 Bi(T) which is monic and such that ft is mapped by qgi :Ci[T] ~ A[T] in f . Since f ( 0 ) 9 m and Of/OT(O) q~ m we may find j 9 I, j > i, such that the polynomial f j = qgij(ft') satisfies f j (0) 9 m j, Ofj / 0 T (0) q~ mj. Clearly Bj -- ( C j [ T ] / ( f j))(mj,T) works. [2]

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LEMMA 3.4. Let (A, m) be an excellent Henselian local ring, A its completion, .,4 the category of local A-algebras and F" fit ---> Sets a covariant functor such that the canonical map

limi~lF(Bi) ~ F(limi~lBi) is surjective f o r every filtered inductive system o f local A-algebras (Bi, ~ij )i el, ~ij being local A-morphisms. Then f o r every ~ E F ( A ) and every positive integer c there exists a z E F ( A ) such that z -- ~ mod mC; that is, the canonical morphisms F ( A ) ---> F(A/mC,4) F ( A / m c) and F ( A ) ---> F ( A / m c) map ~, resp. z, in the same element of F ( A / m C ) . PROOF. The proof follows an idea of M. Artin [5]. We may express ,4 as a filtered inductive union of finite type sub-A-algebras of A, let us say A - U i 6 i D i . We have

-- ~ i ~ I Bi for Bi "-- (Di)mAnDi. Fix Z and c. By hypothesis we may find i ~ I such that s is in the image o f the canonical morphism F(goi)'F(Bi) --+ F(A), where q9i denotes the inclusion Bi ~ A. Let us say ~ -- F(qgi(zi)) for a zi E F ( B i ) . Since A has AP by Theorem 1.14, we find an A-morphism 7ti" Di --+ A which coincides with qgi ]Di modulo m c, that is the following diagram commutes

Di

A

>

Bi

99i

> Aim c

~--

> ~i

A/mCA

The left map is ~ i which extends clearly to a local A-morphism oti'Bi --+ A and z = F (Oli) (Zi) works. Vq PROPOSITION 3.5. Let ( A , m ) be an excellent Henselian local ring, A its completion, X - - ( X 1 . . . . , X r ) some indeterminates, A ( X ) (resp. A ( X ) ) the Henselization of A[X](m,X) (resp. A[X](m~,X)), f a system of polynomials over A[X] in Y = (Y1 . . . . . Yn), c a positive integer and y - ('Yl . . . . . ~n) a solution o f f in A ( X ) such that ~i E A(X1 . . . . , Xs), 1 <<.i <<.n, f o r some integers si, 0 <<.si <~ ri. Then there exists a solution y -- (Yl . . . . . Yn) of f in A (X) such that yi E A (X1 . . . . . Xsi) and Yi =y~/mod mC.4 (Xs . . . . . Xsi) f o r each 1 <<.i <<.n. ^

PROOF. Let (B, b) be a local A-algebra. Let F ( B ) be the set of all tuples (C1, . . . , Cn, y), where for each i, 1 <~i <~n, Ci C B (X) is an 6tale neighbourhood of

B[X1 . . . .

, Xsi

](b,X1

.....

Xs i )

such that C1 C C2 C " ' " C Cn and y = (Yl . . . . . Yn) is a solution of f in Cn such that yi E Ci for each 1 ~< i ~< n. Let c~" B ---> B I be a local morphism of local A-algebras and

D. Popescu

338

b ~ the maximal ideal of B ~. Then Di -- B t [ X l . . . . . Xsi](bt,X 1..... Xsi) @B[X1 ..... Xsi ] Ci is an essentially 6tale B~[X1 . . . . . Xsi ](b',Xl ..... Xsi )-algebra and

D i / ( b r, X l . . . . Xsi)Di "~ B ' / b ' |

B i b "~ B ' / b '

is a field. Thus C~ -- D(b',Xl ..... Xsi) is an 6tale neighborhood of B~[X1 Xsi](bt,X 1 Xsi) and clearly C~1 C C~ C - . . C C~ C B' (X). Since the canonical morphism Cn --+ C~n maps a solution y of f in Cn to a solution y~ -- 1 | y of f in C~n we see that we may define a function F ( a ) ' F ( B ) ~ F ( B ' ) by (C1 . . . . . Cn, y) --+ (Crl . . . . . C~, y') and so a covariant functor F : ~A ~ Sets, .A being the category of local A-algebras. Now suppose that B is the filtered inductive limit of a filtered inductive system ( ( B j , b j ) , qgjt)j6j of local A-algebras and let (C1 . . . . . Cn, y) ~ F ( B ) . We have . . . . .

.....

B[X](b,X) '~ lirn Bj[X](bj,X). By Lemma 3.3 there exists j 6 J and an 6tale neighborhood C{ j) Bj[X1

.....

of

Xsi](bj,Xl ..... Xs i )

such that B [ X I . . . . . Xsi](b,Xl ..... X,~i) ( ~ n j [ x !

.....

Xsi l C~ j) ~ Ci.

Clearly for a t 6 J, t > j , we may suppose that f has a solution y(t) in

c(')-(n,[x~ , ' ' ' , n

x~.] . . |

.

.

,

x,.] c(.J))(b,,x,, . . . , x,.) ,

which is mapped to y by C(nt) --+ Cn. Thus (Cl t), . . . . (-"n --,(t) , y(t)) is mapped by F(tpt) in (C1 ..... Cn, y), qgt'Bt ~ B being the limit map and so F satisfies the hypothesis of Lemma 3.4. Next, let ~" be as in the hypothesis. Then there exists an 6tale neighborhood C1 of /~[Xi

.....

Xsl](m,4,Xl ..... X s i )

containing y~. By recurrence we find in this way (Ci . . . . . Cn) such that ( C 1 . . . . . Cn, ~) E F ( A ) . By Lemma 3.4. there exists (Dl . . . . . Dn, y) ~ F ( A ) such that (D1. . . . . Dn, y) --(C1 . . . . . Cn, ~) m o d m c. Thus Yi E Di C A ( X I . . . . . Xsi), Yi -- "Yi m o d m C A ( X 1 , . . . , X s i ) for all 1 <<.i <~Si and f ( y ) = O. D THEOREM 3.6. Let (A, m) be an excellent Henselian local ring, A its completion, X (X1 . . . . . Xr) some indeterminantes, A ( X ) the Henselization o f A[X](m,X), f a system o f polynomials over A[X] in Y -- (Y1 . . . . . Yn), c a positive integer and ~ -- (yl . . . . . Yn) a solution o f f in ,4[[X]] such that Yi E A [ [ X 1 . . . . . Xsi]], 1 <~ i <~ n, f o r some integers si, 0 <<.si <~ r. Then there exists a solution y - ( y l , . . . . Yn) of f in A (X) such that yi E A(X1 . . . . . Xsi) and yi ----Yi mod (m, X1 . . . . . Xsi)CA[[X1 . . . . . Xsi]], 1 <<.i <~n.

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3 39

PROOF. First we see that it is enough to study the case when A is a Noetherian complete local ring. Indeed, if our statement holds for such rings, then there exists a solution ~" = (~l . . . . . Y'n) of f in A(X) such that ~. ~ A(X1 . . . . . Xsi) and ~. ~Yi mod (mft, X1 . . . . , Xsi)C.4[[X1 . . . . . Xsi]], 1 <~ i <<.n. Then it is enough to apply Proposition 3.5. From now on we may suppose that A is a Noetherian complete local ring. After a renumbering we may suppose 0 ~ S1 m . . . __. Sil < Sil + 1 ~ . . . .

Si2 < " " " < Sit_ 1+ 1 ~ " 9 9 --- Sit ~ 1"

and it = n. Set rj -- sij, 1 ~< j ~< t, so = r0 -- i0 = 0 and D j - - A[[X1 . . . . . Xrj_l ]], for 1 ~< j ~< t. Apply decreasing induction on j , 1 ~< j ~< t, to find for each e, i j _ 1 < e <~ n , an element

y(ej) in D j ( X r j _ I + I

X s e ) such that

.....

Y~ej) -~ Ye m o d (m, X l . . . . . X s e ) C A [ [ X l

. . . . .

Xse]]

and . . Y. f ( Y l . . , .Yij-1

+1 . . . . y (nj) ) = 0.

If j -- t this follows because Dt (Xrt_l+l . . . . . Xrt) has A P , being excellent Henselian (see Theorem 19 Indeed, apply L e m m a 1.5 to the system of polynomials f(Yl

.....

Yit-1, Y i t _ l + l . . . . .

Yit).

9

Suppose 1 ~< j < t By induction hypothesis, we have D j + I ( X r j + l . . . . . X r ) such that ....

,,

9 Yij

f(Yl

, y(j+l) ij+l

(,,(j+l)~ ,,~re

)ij
, y(j+l) ....

)--0.

If j -- 1 and Sil = 0 then there exists nothing to show. Otherwise, apply Proposition 3.5 to the case A

= Dj(Xrj_l+l

f(Yl

.....

.....

Xrj), A = Dj+I

Yij_l,Yij_l+l

.....

We obtain the wanted solution y(J).

and to the system of equations

Yn)--'O.

IS]

COROLLARY 3.7 9 Let K be a field, X (X1 . . . . . Xr) some indeterminantes, K ( X ) the algebraic p o w e r series in X over K (that is, the algebraic closure o f K [ X ] in K[[X]] - the Henselization o f K[X](x)), f a system o f polynomials over K ( X ) in Y -- (Y1 . . . . . Yn), c a positive integer and ~ = (yl . . . . . Yn) a solution o f f in K[[X]] such that Yi E K [ [ X 1 . . . . X s i ]], 1 <~ i <<.n, f o r some positive integers si <<.r. Then there exists a solution y = (Yl . . . . . Yn) o f f in K ( X ) such that Yi E K ( X 1 . . . . . Xsi ) and Yi =-- Yi mod (X1, . . . , X s i ) C g [ [ x 1 . . . . . S s i ] ] , 1 <~ i <~n.

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PROOF. Using Theorem 3.6 it is enough to reduce the problem to the case when f is from K [ X , Y]. Let p be the kernel of the map K (X)[Y] ~ K[[X]] given by Y --+ fi (so p is prime!) and q := p (q K[[X, Y]]. We may enlarge f in order to generate p. Since the map K [ X , Y] --+ K (X)[Y] is a filtered inductive limit of 6tale neighborhoods we see that q K (X) [Y] is a reduced ideal and p is a minimal ideal associated to q K (X) [Y]. Thus there exists g ~ K ( X ) [ Y ] \ p such that gp C q K (X)[Y]. Then g@) # 0 and there exists u ~ N, u >~ c such that g@) ~ 0 rood Xu. Apply Theorem 3.6 for a system of generators h of q and u. Then there exists a solution y = (yl . . . . . Yn) of h in K(X) such that Yi E K (X1 . . . . . Xsi) and yi =-- Yi mod (X1 . . . . . Xsi)UK[[X1 . . . . . Xsi]], 1 <~ i <~ n. It remains to show that y is a solution of f too. But g (y) = g (~) ~ 0 mod X u and so g (y) # 0. Thus g 6 a := (Y1 - y! . . . . . Yn - y n ) K (X)[Y] and gp C q K (X)[Y] C a because h(y) = 0. Hence p C a, that is f ( y ) = O.

El

REMARK 3.8. If the sets of variables X involved in the Yi are not "nested", that is they are not totally ordered by inclusion (as above!) then Corollary 3.7 does not hold as Becker [9] noticed. If in this corollary we consider the convergent power series ring over K = C instead of the algebraic power series ring over C then again the result does not hold as Gabrielov noticed in [ 16]. However a special approximation in nested subrings holds also in C{X1 . . . . . Xn } as Grauert showed in [ 17]. The above corollary appeared when char K = 0 in [23, Ch. III] with some nice applications in [23, Ch. IV], the proof being wrong. The idea of Proposition 3.5 comes from H. Kurke and G. Pfister, who actually noticed that the above corollary holds if rings of type K[[XI . . . . . X s ] ] ( X s + l . . . . . X r ) have A P . This follows from Theorem 1.14 and our presentation here follows essentially [31, (3.6), (3.7)]. Extensions of Theorem 3.6 are given in [39,37].

4. Cohen Algebras and General N~ron Desingularization in Artinian local rings LEMMA 4.1. Let (A, m, k) be a Noetherian local ring and L D k a simple field extension. Then there exists a flat local A-algebra (B, b), essentially of finite type such that b - m B and B / b ~ L. I f L / k is separable then B is essentially smooth. PROOF. Let L -- k(x). If x is transcendental over k then B - - A [ X ] m A [ X ] is clearly essentially smooth. If x is algebraic over k then let P ~ A[X] be a monic polynomial lifting of P "-- Irr(k, x) and C = A [ X ] / ( P ) . Then m C is prime in C because C / m C ~ L. Thus B "-- Cmc works. If L Ik is separable then P is separable and so 0 P / O X r m C, that is B is essentially smooth. [3 PROPOSITION 4.2. Let (A, m, k) be a Noetherian local ring and L D k a field extension. Then there exists a flat Noetherian local A-algebra (B, b) such that (i) B / b ~- L, b - - mB,

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(ii) (B, b) is a filtered inductive limit of flat local A-algebras (Ci, ci)itI essentially of finite type such that ci -- m Ci, i E I. Moreover if L~ k is separable then Ci are essentially smooth over A. PROOF. By transfinite induction we construct a family of subfields (Ki)l<~i<~Oof L such that Ko -- k, Ko = L, Ki+l -" Ki(xi+l) for an element xi+l ~ L \ K i , i < 0, and Ki -U j
Bi = lira j
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we have seen. By recurrence the surjective map R --+ L can be lifted to an A-morphism gi "R --+ R ~/q~i for all positive integers i. The maps (gi) defines an A-morphism

g ' R --+ lim R'/q fi "~ R' <---

such that R / q |162 g coincides with 1L modulo some isomorphisms. Note that g is a surjection by [25, 8.4]. Similarly there exists a surjection h" R' ~ R. Then gh and hj are isomorphisms since R, R' are Noetherian tings which is enough. [] The unique (up to an isomorphism) Noetherian complete local A-algebra (R, q) given by Theorem 4.3 is called the Cohen A-algebra of residue field L. REMARK 4.4. The A-algebra A I defined in Section 2 is in fact the Cohen A-algebra of residue field k*. PROPOSITION 4.5. General N~ron Desingularization holds in Artinian local rings. PROOF. Let u : A --+ A' be a regular morphism of Artinian local rings and m the maximal ideal of A. Then A I / m A ' must be a local regular ring which is also Artinian. Thus A~/mA ~ is a field U and m A ~ is the maximal ideal of A'. By Proposition 4.2 there exists a flat Noetherian local A-algebra (B, b) such that (i) B / b ~- k', b = mB, (ii) (B, b) is a filtered inductive limit of essentially smooth A-algebras. Since m is nilpotent, b is too and so B is Artinian too. Thus A', B are complete local tings and by Theorem 4.4 they are A-isomorphic. Hence A' is a filtered inductive limit of essentially smooth A-algebras. By Theorem 3.1 A' is a filtered inductive limit of Aalgebras of type (A[Y, T ] / f ) g , Y = (Yl . . . . . Yn), where f is a monic polynomial in T over A[Y] and g is a multiple of Of/O Y. 7q

5. Jacobi-Zariski sequence and the smooth locus Let B be an algebra over a ring A and A [ Y ] / I a presentation of B over A, where Y (Yi)i is a set of indeterminantes not necessarily finite. Let ~ ) BdYi be the free B-module on a basis d Y = (dYi)i in bijection with Y and d" I --> ~ ) BdYi the map =

f --+ Z ( O f / O Yi)dYi. Clearly d~(l2) = 0 and so d" induces a map d" I / I 2 --+ ~ ) BdYi. Let FB/A and ff2B/Abe the kernel and cokernel of d./-'8/A and S-2B/A does not depend of the choice of the presentation and are functorial in B / A . If B -- A / a then FS/A "~ a/a 2 and 12e/a - - 0 . If S C B is a multiplicative system then Fs-~ 8/a ~ S-1 (FS/A) and S2s-~ s/a "~ S-1S28/A.

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THEOREM 5.1 (Jacobi-Zariski sequence). If B ---> C is a morphism of A-algebras then

there exists a natural exact sequence of C-modules I'C/A ~ FC/B ~ C @B ,-QB/A ~ ,.(2C/A ~ ,.~"2C/B "+ 0

which can be extended on the left with the term C |

1-'B/A providing $-2B/A is flat over B.

THEOREM 5.2. Let B = A[Y]/I be an algebra over a ring A. Then the following conditions are equivalent: (1) B is quasi-smooth over A, (2) d : I / I 2 ---> ~ AdYi is a split monomorphism, (3) 1-'B/A "- 0 and ~(2B/A is projective over A. The proofs of these theorems can be found, together with many other details, in [1] (an easy exposition of this subject is given in [38]). Let B be a finitely presented algebra over a ring A and B = A[Y]/I such a finite presentation, Y = (Y1,..., Yn). Let HB/A be the radical in B of ~ A f ( ( f ) : I ) B , the sum being taken over all finite systems of polynomials f = ( f l , . . . , fr) of I, A f being the ideal generated by all r • r-minors of the Jacobian matrix (Ofi/OXj). The ideal HB/A does not depend on the presentation as shows the following: PROPOSITION 5.3. Let q ~ Spec B. Then Bq is essentially smooth over A if and only if q TJ HB/A. PROOF. Suppose that q 75 HB/A and let p C A[Y] be the inverse image of q. Then there exists a finite system of polynomials f of I such that Ip = (f)A[Y]p and Af ~ p, let us say M =det(Ofi/OYj)l<
n ot "~ i=1

+ BqdYi ~

BqdYi i--1

be the projection on the first r summands and

t" +

BqdYi ---> (I/I2)q

i=1

the surjective map given by dYi --+ fi mod I 2. Then adq fl is an isomorphism (its determinant is M!) and so t , c~dq are also isomorphisms. Thus dq is a split monomorphism and so Bq is essentially smooth by Theorem 5.2. Conversely, if dq is a split monomorphism then (I/I2)q is free of a certain rank r ~< n. Let f be a system of r polynomials from I which lifts a basis in (I/I2)q. Then

(f)A[Y]q --qA[Y]q

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by Nakayama's Lemma and the matrix associated to dq in the basis (j5 mod 12) resp. dYj is (8fi/8 Yj). As dq splits we get A f ~ q. E] COROLLARY 5.4. Let s ~ B. Then Bs is smooth over A if and only if s E H B / A . PROOF. Bs is smooth over A if and only if (I-'B/A) s - - 0 and (~(2B/A) s is projective by Theorem 5.2. The last conditions hold exactly when (FB/A)q = 0 and (~QB/A)q is projective for all q 6 Spec B with s ~t q, that is when Bq is essentially smooth (in other words q 73 HB/A) for all q 6 Spec B with s r q. Hence Bs is smooth over A if and only if S E MqDHB/A q = H B / A .

[3

LEMMA 5.5. Let v: B --+ C be a morphism of finite presentation algebras over a ring A.

Then V(HB/A)C fq HC/B C HC/A. PROOF. If q 6 SpecC, q 75 v(HB/A)C 0 HC/B then v - l q 75 HB/A and q 75 Hc/8 and so Bw-1 q is essentially smooth over A and Cq is essentially smooth over B. Thus Cq is essentially smooth over A and so q 75 HC/A. [-1 LEMMA 5.6. Let A be a Noetherian ring u : B ~ A t a morphism of A-algebras and x ~ B an element. Suppose that B is of finite type over A and u (x) ~ v/u (Hs/A) A'. Then there exists a finite type A-algebra C such that u factors through C, let us say u = wv, v(x) E HC/A and v(HB/A)C C HC/B (in particular v(HB/A)C C nc/a). PROOF. Let t = (tl . . . . . ts) be a system of generators of HB/A. By hypothesis we have Y~i~=lU(ti)zi for some elements zi ~ C and a certain r 6 N. Put

u ( x ) r "-

ti)i Z -- ( Z 1 . . . . . Z r ) , and let w : C --+ A' be the map given by Z o z. Clearly Cti is a polynomial B-algebra and so v(ti) ~ Hc/B, v being the structure morphism of the B-algebra C. Thus V(HB/A) C HC/B and v(x) ~ o(HB/A)C. By Lemma 5.5 O(HB/A) C HC/A and

v(x) E HC/A. Let B be a finitely presented A-algebra and b ~ HB/A an element. We say that b is standard (resp. strictly standard) with respect to the presentation B = A [ Y ] / I if b lies in v/A f ( ( f ) : I ) B (resp. A f ( ( f ) : I ) B) for a finite system of polynomials f = (fl . . . . fr) of I.

[3

LEMMA 5.7 (Elkik [15,38]). If b ~ HB/A and (I/I2)b is free over B then b is standard

for the above presentation. PROOF. Let fl . . . . . fr be a base for (l/12)d and h ~ A[Y] representing b. Then

Ih - ( f ) A [ Y l h -k- I 2

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345

and we have (1 + c~)Ih C (f)A[Y]h for a certain a = g~ h s, g ~ I. Thus

h e(h s + g) E ( ( f ) ' I ) for a certain e 6 N and so b s+e E ((f)" I ) . B. Since the map

n i=l is a split m o n o m o r p h i s m the r • r-minors of its matrix (Ofi /O Yj) generate whole Bb. Thus a power of b lies in A f . [-I LEMMA 5.8. If b ~ HB/A and (S'2B/A)b is free then b is standard with respect to the presentation B -- A[Y, Z]/(I, Z) where Z -- (Z1 . . . . . Zn). PROOF. As Bb is smooth over A we have

BbdYi -- (I/I2)b G (,-C2B/A)b

i=1 by Theorem 5.2. Since ~)Bb/A = (~)B/A)b is free of rank ~< n we get (I/I2)b (9 B~ free. Now, let J -- (I, Z). We have

( + ) 0--~ ( J / J 2 ) b - ~

BbdYi

( n 9

i=1 and a surjective map

qg" (I/Ie)b @

(n

)

( ~ BbdZj

-+ Y2Bb/A

0

j=l

)

( ~ BbZj m o d Z 2 ~ (J/Je)b.

j=l But dj maps isomorphically q~((~]=l Bb(Zj m o d Z2)) on ( ~ = 1 BbdZj. After splitting this off from dj~o it remains di which is injective since Bb is smooth over A. Thus ~o must be an isomorphism and so ( J / J 2 ) b "~ (I/I2)b @ B~ which is free. Hence b is standard for the presentation B - A[Y, Z]/(I, Z) by L e m m a 5.7. U PROPOSITION 5.9. Let B = A[Y]/I, Y -- (Y1 . . . . . Yn) be a finitely presented A-algebra and C = S B ( I / I 2) be the symmetric algebra over A associated to 1/12. Then HB/AC C HC/A and $2Cb/A is free for each b ~ HB/A. Therefore C has a presentation such that the image in C of any b ~ HB/A is standard.

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PROOF. Let b ~ HB/A then n

db " (I/I2)b -+ ~

BbdYi

j=l is a split monomorphism and so (I/12)b is projective. Then Cb is locally isomorphic with a polynomial algebra over Bb and so it is smooth. Thus b ~ Hc/B, even b ~ HC/A by Lemma 5.5 and so HB/AC C HC/A. The Jacobi-Zariski sequence written for the A-morphism Bb ~ Cb:

0 = l~Cb/Bb -'->Cb (~Bb IQBb/A ~ ~"2Cb/A~ at-2Cb/Bb~ 0 splits because ff~)Cb/Bb ~ Cb I~)Bb ( l / I 2 ) b is projective. Then

ff2Cb/A ~ Cb ~Bb (ff2Bb/A (~) (l/12)b) ~" Cb (~Bb ( + BbdYi) i=l

which is free, the last isomorphism holds because db is a split monomorphism. Now it is enough to apply Lemma 5.8. FO COROLLARY 5 . 1 0 . Let A be a Noetherian ring, u : B ~ A' a morphism of A-algebras and b ~ B an element. Suppose that B is of finite type over A and u(b) ~ v/HB/AA ~. Then then exists a finite type A-algebra C such that (i) u factors through C, let us say u = wv, v: B ~ C, w : C --~ A', (ii) v(b) is standard f o r a certain presentation of C over A, (iii) V(HB/A) C HC/A.

PROOF. By Lemma 5.6 we may reduce to the case when b ~ HB/A. Applying Proposition 5.9 we get C as it is necessary except that we must show that there exists w : C ~ A ~ such that u -- wv. Since C is a symmetric algebra over B, the inclusion v : B ~ C has a canonical retraction t given by $ 8 ( I / I 2) ~ $ 8 ( I / 1 2 ) / ( I / I 2) ~- B. Thus w = ut works. []

6. General N~ron desingularization THEOREM 6.1 (N6ron [24]). Let R C R' be an unramified extension of discrete valuation rings which induces separable field extensions on the fraction and residue fields. Then R t is a filtered inductive union of its sub-R-algebras which are essentially smooth. Note that the hypothesis of N6ron's desingularization says in fact that the inclusion map R --+ R' is regular. Thus Theorem 1.15 on discrete valuation rings follows from Theorem 6.1. The following theorem is a stronger variant of Theorem 1.15:

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347

THEOREM 6.2. Let u" A --+ A ~ be a morphism o f Noetherian rings. Then u is regular if and only if A ~ is a filtered inductive limit o f smooth A-algebras. The sufficiency is trivial because a filtered inductive limit of regular morphisms is regular, since a Noetherian ring, which is a filtered inductive limit of regular rings must be regular too. The necessity is equivalent with Theorem 1.15. Indeed, if Theorem 1.15 holds then for every finite A-algebra B and every morphism of A-algebras v: B --+ A ! there exists a smooth A-algebra C such that v factors through C. Thus A ! is a filtered inductive limit of smooth A-algebras by the following elementary lemma (a proof is given in [38]). LEMMA 6.3. Let S be a class o f finitely presented algebras over a ring A and A ! an Aalgebra. The following statements are equivalent: (1) A~ is a filtered inductive limit o f algebras in S, (2) I f B is a finitely presented A-algebra and v : B --+ A' is a morphism o f A-algebras then v factors through some C in S. Conversely, if the necessity in Theorem 6.2 holds then for every finite type A-algebra B and every v: B --+ A ! morphism of A-algebras there exists a smooth A-algebra D such that v factors through D, let us say v = wt, w: D ~ A!, t : B --+ D. Thus it is enough to apply the following LEMMA 6.4. Let v : B ~ A t be a morphism o f algebras over a Noetherian ring A. Suppose that B is o f finite type over A and v ( H B / A ) A ! = A'. Then v factors through an Aalgebra C = ( A [ Y ] / ( f ) ) g , Y -- (}11. . . . . Yn), where f -- ( f l . . . . . fr), r <~ n, are polynomials in Y over A and g belongs to the ideal A f generated by the r • r-minors o f the Jacobian matrix ( Of i / 0 Yj ). PROOF. By Corollary 5.10 there exists a finite type A-algebra C such that (i) v factors through C, let us say v -- wt, t" B ---> C, w" C --+ A !, (ii) 1 is standard for a certain presentation of C over A, let us say C ~ A [ Y ] / I , Y = (Y1 . . . . . yn). Then there exists f ---- ( f l . . . . . f r ) in I, r ~< n such that 1 c A f ( ( f ) ' I ) C . Thus there exists ct 6 I such that g -- 1 + ot E A f A ( ( f ) " I). Hence C ~ ( A [ Y ] / ( f ) ) g as it is required. [] From now on, let u :A --+ A ! be a morphism of Noetherian rings, B a finite type Aalgebra and v : B ---> A ! an A-morphism. We can suppose that hB := v / v ( H B / A ) A ' =7s A t because otherwise we may apply L e m m a 6.4. Let q be a minimal prime over ideal of hB. After [38] we say that A ~ B ~ A ! D q is resolvable if there exists a finite type A-algebra C such that v factors through C, let us say v = wt, t: B --+ C, w : C --+ A t and hB C h c := V / w ( H c / A ) A ! ~ q. !

PROPOSITION 6.5. Suppose that p "-- u - l q is a minimal over ideal o f u - 1 (hB), A --~ Aq is flat and Aq' / p A q is a geometrically regular k ( p ) :-- A p / p A p - a l g e b r a . Then A --+ B --+ A ! D q is resolvable.

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For proving the necessity in Theorem 6.2 and so Theorem 1.15 it is enough to show that the above proposition holds. Indeed if h 8 -~ A t then we may choose a minimal prime over ideal q of hB such that p = u - l q is a minimal prime over ideal of a = u -1 (hs) (if h B = ~ = 1 qi, qi being minimal primes over ideals of h B, then a minimal prime over ideal p of a = ~ i s= 1 u - - 1 (qi) must contain one of u - 1qi and so p = u - 1qi ). Thus the hypothesis of Proposition 6.5 are fulfilled (u is regular in Theorem 6.2!). Hence A ~ B ~ A t 3 q is resolvable (roughly speaking we may increase h 8). By Noetherian induction we arrive to the case h B = A t which is solved by Lemma 6.4. If A contains Q then Proposition 6.5 follows from the following two lemmas: LEMMA6.6. Supposeht q - - O a n d A p , - - ~ A p | able. Then A ~ B ~ A t D q is resolvable too.

B---~ Aqt D q A qt , p - - u

--1

(q) isresolv-

LEMMA 6.7 (Main Lemma). Let a ~ A,,4 :-- A / a 8 A , B :-- B/a8B, ~t := At/aSA t and Cl = q~ a8At. Suppose that (i) Anna (a 2) = Anna (a), Anna,(u(a) 2) = Anna,(u(a)), (ii) a is strictly standard f o r a certain presentation of B over A, (iii) ,4 ~ / ) ~ ,~t D ~ is resolvable. Then A ~ B ~ A t D q is resolvable too. PROOF OF PROPOSITION 6.5 WHEN A D Q. Apply induction on ht q. If ht q - 0 by Lemma 6.6 it is enough to show that t

then

t

Ap ~ Ap @A B ~ Aq D qAq is resolvable. It is easy to see that HAp|

D HB/A" (Ap @A B),

since the smoothness preserves by base change. If q A qt ~ HAp| t, (otherwise we have nothing to show) then we may apply the Proposition 4.6 (the Artinian local case) because the morphism a p ~ Aq is regular by hypothesis (it is fiat and Aq/pAq is regular, Artinian local k(p)-algebra thus a separable field extension of k(p)!). Suppose now ht q > 0. If ht p = 0 then choose x in h8 which is mapped by v in a regular parameter of A qt / p A q . Consider u t ' A [ X ] ~ A t given by X ~ x B t - B[X] and v t" B t ~ A t extending v by X ~ x. The map A[X](p,x) ~ Aqt is flat by the local flatness criterion [25] since the map k(p)[X]~x~ ~ A qt / p A q is flat by [25, Th. 23.1]. By construction A qt / ( p , x ) A qt is still a regular local ring, even a geometrically regular k(p)algebra since chark(p) = 0 (A contains Q!). Clearly HB[X]/A[X] D HB/A " B[X] as above and it is enough to show that A[X] ~ B[X] ~ A' D q

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349

is resolvable. Thus we may suppose ht p > 0 too. Choose a in u - l ( h s ) such that a lies in no height 0 prime contained in p. We have ht p / a A < ht p and so ht q / a A t < ht q because ht q = ht p + d i m A q /l p A q by flatness of Ap --+ Aql (see [25, 15.1]). By Corollary 5.10 there exists a finite type A-algebra C such that (i) v factors through C, let us say v = wt, t" B --+ C, w ' C ~ A t, (ii) a is standard for a certain presentation of C over A, (iii) t(HB/A) C HC/A. Changing a to one of its powers we may suppose that a is strictly standard for the same presentation of C over A. Since the chain Anna (a) C A n n a (a 2) C --. stops by Noetherianity we may suppose A n n a (a 2) = A n n a (a) and similarly Anna,(u(a)) 2 = Anna,(u(a)), changing a to a power o f a . L e t / l -- A / ( a 8 ) , ~ t = A t / a 8 A t, ~; = C / a 8 C a n d ~ -- q / a 8 A t. By induction hypothesis (ht ~ < ht q !) we may suppose that ,4 --+ t~ ~ ~t D ~ is resolvable. Then by the Main L e m m a A ~ C ~ A t D q is resolvable too. [2 PROOF OF LEMMA 6.6. Let D be a finite type A p-algebra such that

vp" Bp :-- Ap QA B --+ Aqt factors through D, let us say Vp = ~ p O l p , Olp " Bp ~ D, tip" D --+ Aq' a n d flp(nD/Ap) -Aqt (by hypothesis Ap --+ Bp --+ Aqt 53 q Aqt is resolvable). Using L e m m a 6.4 we may suppose even D smooth over Ap. We claim that we can change D such that there exists a finite type B-algebra C such that v factors through C, let us say v = fla, fi :C --+ A t, ~ being canonically, D ~- Bp | C as B-algebras and Vp is induced by v. Indeed, let D ~- Bp[Z]/(g), Z -- (Z1 . . . . . Zs), g = (gl . . . . . ge), gi being supposed with coefficients in B and tip given by Z --+ y / t for y E A ts, t ~ A t \ q . Indeed, for large integer c ~> 1 we get homogeneous polynomials Gi (Y, T) = T cgi ( Y / T ) , Y = (Y1 . . . . , Ys), of positive degree such that G i ( y , t ) = 0 in Aq, t that is r G i ( y t ) - 0 for a certain r ~ A t \ q . Changing y to ry and t by rt we may suppose G i ( y , t ) = 0 in A t. Let C = B[Y, T ] / ( G ) and f l : C --+ A t be the extension of v by Y --+ y, T ~ t. The Bpmorphism y : D[T, T -1 ] ~ (Bp @B C)T given by Z ~ Y~ T, T --+ T is an isomorphism and the map fl" (Bp @B C)T ~ Aq induced by fl is such that fly extends tip. Thus we may change D to the smooth Ap-algebra D[T, T - l ] . Clearly fl(Hc/A) ~ q let us say fl(c) r q for a certain c ~ HC/A. Suppose C ~- A [ X ] / ( f ) , X -- (X1 . . . . . Xn), f = ( f l . . . . , fm). If hB C hc then C works. Otherwise, choose w ~ A t \ q and an integer k > 0 such that w ( h s ) k = 0 (ht q = 0!). Let bl, . . . , bz be a system of generators of HB/A,

E-

C[X, U, V, W]/

Ui Vij, WUi

fj i--1

i,j

where U = (U1 . . . . . Uz), V -" ( V / j ) , W are new indeterminantes and / ~ ' E --+ A' the map extending fl by Ui --~ v(bki ), V ~ O, W ~ w. Since EUi is a polynomial ring over

B[Ui, U/-1] it is smooth over B. Then Ebigi is smooth over A, thus biUi E HE/A and

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350

so hB C hE. But E w ~- C[V, W, W -1 ] is smooth over C and so Ecw is smooth over A. Hence/3 (c) w ~ h E \ q . []

7. Proof of the Main Lemma The main Lemma will follow from the following two lemmas: LEMMA 7.1 (Lifting Lemma). Let A --+ A' be a morphism of Noetherian rings, d c A, 5, .-A/(d2), j ' - A ' / ( d 2 ) , X : = A/(d), X' "= A'/(d), ?~ a finite type A-algebra and ~ " C; --+ A' a morphism of A-algebras. Suppose that Anna (d 2) -- Anna (d), Anna, (d 2) Anna,(d). Then there exist a finite type A-algebra D and a morphism w" D --+ A ~ of Aalgebras such that (i) A | fl factors through A @a W, (ii) rr -1 (hd) C ho, Jr being the surjection A' ~ A' and h d = v'/~(Hd/J)J'" LEMMA 7.2 (Desingularization Lemma). Let A --+ A' be a mor~ghism of Noetherian rings, B afinite type A-algebra, v" B ~ A' an A-morphism, d E A, A := A/(d4), A' "- A ' / ( d 4) and B "-- B / d 4 B. Suppose that d is strictly standard for a presentation of B over A, Anna (d e) = Anna (d) and Anna, (d 2) ...--AnnA,(d). Let D be a finite type A-algebra and w" D --+ A ~an A-morphism such that A (~A 1) factors through A @A 110. Then there exist a finite type A-algebra E and an A-morphism y" E --+ A' such that (i) v and w factor through y, (ii) HD/AE C HE/A (so hD C hE). Indeed, let a ~ A, B be as in the Main Lemma and set d -- a 4. Since

--+ B --+ A' D cl "-- q / d A ' is resolvable there exist a finite type ,4-algebra 6" and a morphism/~" 6" ~ ,4' of ,4-algebras such that h b C h~ ~ ~ and ~"/) --+ ,4' factors through/~. By Lemma 7.1 there exist a finite type A-a~ebra D and a morphism w ' D --+ A' of A-algebras such that ,7t | factors through A @A W and rr -! (h~) C hD. Now apply Lemma 7.2 for d -- a. Thus there exist a finite type A-algebra E and an A-morphism V" E --+ A' such that v and w factor through }, and hD C hE. Since HB/AB C Hi~/j by base change we get hB,4' C ht~ C h~ and so h O D h B. We have also h O ~ q because h ~. ~Z q. PROOF OF THE LIFTING LEMMA. Let

where Pi ~ A f(i) ( ( f ( i ) )

. [) for some presentation 6" -- J [ X ] / [ , X -- (X1 . . . . . Xn), where

each f(i) is a finite subset of/7. Let fl . . . . . fs be polynomials in A[X] such that f j

-

-

Artin approximation

351

f j mod d 2, 1 ~< j ~< s, generate [ and {jq . . . . . fs} contains all elements from f ( i ) . Set I = (fl . . . . . fs, d 2) and let/~ be given by X ~ 2 = x mod d 2 for a certain x ~ A 'n. Then f (x) -- dz, z ~ d A 'n. Set g j ( X , Z ) = f j - d Z j , 1 <~ j <~ s, Z -- (Z1 . . . . . Zs) and g--(gj)l<~jNs. To every Pi we associate a system of polynomials F (i) in A [ X , Z] in the following way: Let Pi be a lifting o f / 5 / t o A[X]. Since Pi I C (d 2, f l . . . . . f r ) for a certain r in a certain ordering of ( f j ) depending on i, it follows that

Pi f j --

~-~ /_/!i)

--jr ft 4-

d 2 (i)

Gj ,

t=l for some polynomials

-l--l!i) -jt '

Gj(i) from A [ X ] , r < j <~ s. Set

F(i)--eizj-~n(1)Zt-da~i)

,

r < j <~ s, F (i) - - (FJi))r
t=l and D -- A [ X , Z ] / J , where J -- (g, F), F - - [,.Ji f ( i ) .

d E ) i) = P i ( f j - g j ) _ ~

Note that

H(i) (ft -- gt) _ d 2 a j (i) "'jt

t=l = - P i g j 4-

~H!

"-it

i)

gt E (g),

(1)

r < j <<.s.

t=l By construction J C (g) 4- ((g) "d) (3 (Z, d) (see (1)) and so J ( x , z) C AnnA, d A d A ' = 0. Thus we may define w by (X, Z) ----> (x, z), so (i) holds. Using again (1) we see that Dd ~ A d [ X , Z ] / ( g ) .v Ad[X] is smooth over A. Hence d ~ HD/A. It remains to show that Pi ~ HD/A. Let f stand for the column vector with entries f j o . t u~ j t. and let H (i) be the (s x s)-matrix (H)I)), where we set Hit(i) : 0 if t > r and ,t-t(i) ljt -__- ~t if j ~< r for some fixed ordering of the f depending on i. Then we may write Pi f -H (i) f mod d 2. Set S (ik) = Pk H (i) - H (k) H (i), i # k. We have s ( i k ) f =-- Pi Pk f -- H (k) Pi f = Pi Pk f -- Pi Pk f -- 0 mod

d 2.

Differentiating it follows that s ( i k ) o f / O X e e (d 2, f ) C (d, g).

(2)

But Pi is a multiple of a r • r-minor of ( O f j / O X e ) l < ~ j ~ r , l < ~ e < ~ n , given by the columns, let us say el . . . . . er and S (ik) has the form (S[0), where S denotes the first

352 r

D. Popescu

columns of S (ik). Thus there exists an

r x

r-matrix

T (i)

over

A[X, Z] such that

(Ofj/OXeq)l<~j,q~r T (i) -- Pi Idr. By (2) it follows that Pi S (ik) ~ 0

mod

(d, g)

(3)

and from the definition of F (i) we obtain H (i) Z = Pi Z m o d (d, F(i)). Thus

s(ik)z

= PkH(i)z-

H ( k ) H ( i ) z =-- P k P i Z -

PiH(k)Z

= Pi(PkZ - H(k)Z) = Pi F (k) m o d (d, F(i)).

(4)

Set E (i) -- (A[X, Z ] / ( g , F(i)))pi. By (3) S (ik) = 0 in E ( i ) / d E (i) and it follows that F (k) = 0, k 7~ i in E ( i ) / d E (i) by (4). Thus F (k) 6 d E (i). It is enough to show that E (i) is smooth over A. Indeed, if this holds then E (i) is fiat over A and we have AnnE(i)d = AnnE(i) d 2. Then F (k) 6 AnnE(i) d N d E (i) = 0 (see (1)), that is Dp i "~ E (i) . Thus Dp i is smooth over A and so Pi ~ HD/A. Hence Jr -1 (hal) C h o . It remains to show that E (i) is smooth over A. Note that by (1) (gj)r
dLb~;t', the last one being the structure map. Then there exists a finite type D-algebra E such that: (i) c~factors through E, (ii) the annihilator a in D o f A n n o ( d 2 ) / A n n o ( d ) satisfies a E C HE/D. PROOF. Let C = D [ X ] / I , X = ( X l . . . . . Xn) be a presentation of C over D for which d is strictly standard. Then there exists a finite system of polynomials f = ( f i ) 1~
H1 --

af/ax )

O]Idn-r

"

Artin approximation

353

Similarly define the n square matrices Hu such that det(Hv) -- My. Clearly the first r rows of all Hu coincide. Let Gtv be the adjoint matrix of Hu and Gv = TvG~. We have

E

G v Hv = E

Tv My Idn = P Idn

and so

ds Idn - P ( x ) I d n - E

Gv(x)Hv(x).

Let ot be given by X ~ y ~ A in. By hypothesis we have x t - y ~ d 4A in, where x t is the image o f x by D --+ A t. L e t u s say x t - y = d3s for e ~ d A in. Then v (v) "-- Hv(x)s satisfies

E

Gv(x)v(V) -- P ( x ) s = dse

and so

s(xt - Y) = d2 E

Gv(x)v(V)"

(v (v As the first r rows of Hv coincide we see that v (v) -- (Ul . . . . . Ur, Vr+ 1. . . . . Vn ), where Ul . . . . . Ur are independent of v. Let

h - s ( X - x) - d 3 W - d 2 E

Gv(x) V (v),

where W = (W1, . . . , Wn) and V (v) are variables of the form (U1 . . . . . Ur, Vr~+],... Vn~V)). Clearly h maps to 0 under the map dp'D[X, W, V] ~ v (v). Since

f (X) - f (x) = E

A t given by X --+ y, W --+ O, V (v) --+

Of/ O X j ( X j - x j )

J m o d u l o higher order terms in X j -- x j and

s ( X - x ) =_d3W + d 2 E

Gv(x)V(V) m o d h

we see that for m = m a x deg j~ we have

J + d 4 Qt m o d h with Q' E (W, V)2D[W, V] n. Thus

J

D. Popescu

354

where Q E (~--~D W j + d ( W , V)2)D[W, V] n. But ~ 3 f i / O X j ( x ) ( G v ( x ) V ( V ) ) j is the i-th entry of

H v ( x ) G v ( x ) V (v) = Tv(x)Mv(x) V (v), that is Tv(x)Mv(x)Ui because i ~< r. It follows that

sm f ( x ) - sm f (x) =--smd3U -1- d3 Q m o d h. As f (x) _= 0 m o d d 4 we have f (x) = d3c for a c 6 dAtr. Take g = smc -Jr-s m U -]- Q and note that g lies in D[W, V]. We have

d 3g = s m f m o d h.

(,)

Set E = D[X, W, V ] / ( I , g , h ) ~- C[W, V l / ( g , h ) . We have d3q~(g) = 0 from ( , ) and ~(g) ~ d A 'r because c ~ d A 'r, v (v) ~ d A m and q~(Q) ~ d A 'r. Since A n n z , ( d ) = A n n a , ( d 2) it follows ~b(g) = 0. Thus ~b factors through E and so c~ factors too. Note that Ed ~- Cd[W, V]/(h) because ( . ) shows that g is 0 in Ed. Solving the equations h = 0 for the W we see that Ea ~- Cd[V]. Thus Ea is smooth over C and so over D because d ~ HC/D by hypothesis. Hence d ~ HE/D. Let F = D[X, W, V]/(g, h). As P I C ( f ) we get

P(x)lFs C PIFs + (X-

x ) I F s C ( f ) F s + d21Fs

because s ( X - x) ~ (d 2, h) as above. But P(x) = ds and ( f ) Fs -- 0 because of (,). Thus d l F s - - d Z l F s . Then there exists s' ~ 1 + d D [ X , W, V] such that s ' d l F s = 0 and so d lFss, = 0. It follows that IFdss, = 0 and so Fdss, ~- Edss'. As Ed is smooth over D we see that dss' ~ HF/ D. Next we show that ss' ~ HF/D. Since E~ "~ Ds[W, V]/(g) solving the equations h = 0 in X we see that Ag C HF~/D. But ( 3 g / 3 U ) =-- s m Idr m o d d and so s mr + de ~ HFs/D for a certain e E F. Then ss'(s mr + de) ~ HF/D and we get star+Is ' ~ HF/D because dss t E HF/D. Thus ss' ~ HF/D. Since h = 0 in Fs we note that X ----x m o d d 2 and so I Fs =-- I (x) Fs =--0 m o d d 2. Thus I Fs C d 2 Fs, let us say I Fs = d b for some ideal b of Fs. As above dlFss, = 0 and so dZbFss , -- 0. But F.~s, is smooth over D, in particular flat. It follows that aFs.,., is the annihilator in F.~s, of AnnF, s, (dZ)/AnnF, e (d). Thus dabFss, -- 0 and so a I Fss, = 0. H e n c e Fss,t = Ess,t for all t 6 a. Since Fss, is smooth over D we get sstt ~ HE/D. As ss ~ = 1 m o d d and d ~ HE/O it follows t ~ HE/D for all t ~ a, that is aE C HE/D . I-1 PROOF OF THE DESINGULARIZATION LEMMA. Let C --- B @A D and c~'C --+ A f being given by b | z --+ v(b) | w(z). By hypothesis there exists f " / ~ - - > / ) "-- D / d a D such that /~ | tO "-" (/~ | tO)~. Then the m a p / 5 " C ~ / ) given by/~ | ~, --> f (/)) | ~, is clearly a retraction of D-algebras such that C @ c~ is the composite map

Artin approximation

355

Applying Lemma 7.4 to the case D --+ C --% A',/5 we find a finite type D-algebra E such that (1) ct factors through E (so v and w factor too), (2) the annihilator a in D of AnnD(d2)/AnnD(d) satisfies aE C HE/D. Let t ~ HD/A. Then Dt is smooth over A, in particular flat. Thus AnnD, (d 2) = AnnD, (d) and so a Dt = Dr. By 2) it follows Et smooth over Dr. Hence Et is smooth over A and so

HD/AE C HE/A.

D

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