e LIBRARY °l ' Michigan State University Ill“! m mm mm “l l l 11m l NIL“! my I 1293 100 T'HECI' This is to certify that the thesis entitled SATURATIONS OF AN ANALYTIC RING OVER i AN ALGEBRAICALLY CLOSED FIELD presented by Ulrich Daepp has been accepted towards fulfillment of the requirements for Ph.D. Mathematics degree in Weaken.“ Major professor Date July 132 1979 0-7 639 OVERDUE FINES ARE 25C pER DAY PER ITEM Return to book drop to remove this checkout from your record. SATURATIONS OF AN ANALYTIC RING OVER AN ALGEBRAICALLY CLOSED FIELD By Ulrich Daepp A DISSERIATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1979 ABSTRACT SATURATIONS OF AN ANALYTIC RING OVER AN ALGEBRAICALLY CLOSED FIELD By Ulrich Daepp The objective of this thesis is to adapt the theory of saturation as developed by Oscar Zariski to the case of analytic rings. We show that some of the necessary conditions for an adequate description of equisingularity ‘with the help of saturation are fulfilled in this particular case. Let k be an algebraically closed, complete and? non-trivially valued field. Let A be an equidimensional and reduced analytic ring over k. A = k[{x1,...,xd}][yl,....ym] where x1....,xd is a system of parameters of A and all yi are integral over the convergent power series ring k[[xl....,xd}]. x1....,xd are called strongly separating if there exist m monic polynomials Pi(Z), l g_i g_m, which are separable over the quotient field k({xl....,xd}) and such that Pi(yi) = 0. Systems of strongly separating parameters always exist. The saturation of A with respect to a strongly separating system of Ulrich Daepp parameters is defined and is again an analytic ring over k; we denote it by' Ax. We can associate analytic set germs V and V; with A and fix respectively. A AX We show that VA and V: are tepologically equivalent, x that is, there are representatives for each of them.which are homeomorphic. The total ring of quotients is §(A) = §(A/pl) @...® @(A/ps) where the pi's are the minimal primes of A. Denote by F; the least Galois extension of k({xl.....xd]) which contains §(A/pj). If we assume further that. [F; :k({xl,...,xd})] and char(k) are relatively prime for all j, l g_j g_s, and that (x1....,xd)A is a reduction of the maximal ideal in A then the multiplicities of A and Xx are the same. We denote the relative Lipschitz-saturation of A by A;. If k and A are as above, except that no * separability conditions are needed, then Ax is again an analytic ring over k and VA and V * are topologically A equivalent. x Meinen Eltern Frieda und Fritz Dapp-Beutler gewidmet ii AC KN OW LEDGMENTS I wish to thank Professor W.E. Kuan for introducing me to interesting tapics in algebraic geometry. His suggestions and encouragement in the preparation of this thesis were very helpful. I would also like to thank my friends P. Gorkin, A. Evans and G. Cross for several useful conversations. Finally, I am very appreciative of the careful and efficient typing of Mary Reynolds. iii TABLE OF CONTENTS Chapter 0 INTRODUCTION I PRELIMINARIES II III IV 1.1 Analytic Rings 1.2 Analytic Set Germs 1.3 Saturation SATURATION OF AN ANALYTIC RING TOPOLOGICAL RELATION BETWEEN RING AND SATURATION THE MULTIPLICITY OF SATURATIONS RELATIVE LIPSCHITZ-SATURATION BIBLIOGRAPHY iv Page 12 15 28 41 51 54 CHAPTER 0 INTRODUCTION In a series of three papers published between 1965 and 1968, [23,24 and 25], Zariski started a theory with the aim of classifying the singularities of alge- braic and algebroid varieties. In the last paper [25] he introduced the algebraic concept of the saturation of a local ring. His intention was to show that two points are equisingular if and only if their local rings have isomorphic saturations. In [25], p.985, Theorem 2.1 he succeeded in doing so for plane algebroid curves over algebraically closed fields of characteristic zero. He also obtained good results in the hypersurface case, Corollary 7.5 of [25], p.1019. In general, however, the question of equisingularity is still open. The definition itself is not agreed upon and several suggestions are competing with each other, see [29]. But the concept of saturation is also interesting from a purely ring theoretic point of view: it is a way to construct a new local ring from a given one. Zariski's second series of three papers, [26,27 and 28], which he published between 1971 and 1975, puts algebraic questions in the foreground. The relation between saturation and other ring theoretic concepts, like localization and completion, are explored. In 1972, A. Seidenberg published a paper in Which he applied the saturation to the complex analytic case. In particular, he proved that two complex analytic varieties Whose saturations of the local rings at the origin are isomorphic, are locally homeomorphic at the origin, [22], Corollary, p.430. Similar results were already obtained by Zariski in [25]. §5 and 6. Our objective is to adapt the theory of saturation to the case of analytic rings of positive characteristic. The geometric objects which are associated with these rings are the analytic set germs. ‘We show that the saturation of this class of rings is well defined and yields other local rings which are again analytic; Theorem (2.12). As in the complex case, we can show that analytic set germs which belong to two analytic rings with isomorphic satura- tions are tOpologically equivalent, Corollary (3.5). under some restrictions the multiplicity of a ring is unchanged if we pass to the saturation, Theorem (4.3). These results can be considered as a minimum requirement for a success- ful algebraic tool Which can help in classifying singu- larities of analytic set germs. This study is therefore a preliminary test for saturations of analytic rings. However, if it will eventually prove adequate has still to be seen. While pursuing this main goal some results concerning analytic rings and separability questions have been obtained which may be of interest on their own, see Chapter III. Also at the end, Chapter V, we include a brief discussion of the situation, if the relative Lipschitz- saturation of Pham-Teissier as defined in [12] is taken instead of the one by Zariski. Terminoloqy and notation not defined explicitly in the text follow that used by Zariski and Samuel in their books [30]. CHAPTER I PRELIMINARIES l.l Analytic Rings A field k is said to be valued if there is a map H :k.» IU- satisfying the following three conditions: 1) 1a] = 0 if and only if a = O 2) |a+b| g |a|+|b| for all a,b 6 k 3) lab] = |a[|b| for all a,b e k. This value defines a metric on k by setting d(a,b) = la-b| if a,b E k. The valuation is said to be non- trivial if there is an element a E k such that |a| #’O,l. It is complete if the induced topology is complete. If char(k) > O and the valuation is non-trivial then it is non-archimedean and we can replace condition 2) by 2*) |aa+b|‘g_max{|a|,lb|] for all a,b 6 k. k[[xl,...,xn]] denotes the ring of formal power series in n .variables over the field k. f(Xl,...,Xn) = J. 1 23f. x l n 11'...'ln 1 ...Xn E k[[X1,...,Xn]] is said to be convergent if there exists a neighborhood U of O in kn such that to every (al,...,an) E U there are elements (A An) 6 (11+)“ and D e m” such that 1100.] j'n OOOAn gD i l [al| < Al....,lan[ < An and Ifil....,i [Al n for all (i1....,in) E (Z+) In All convergent power series form a subring of k[[X1,...,Xn]]. We denote it by k[{xl,...,xn]], it is called the convergent power series ring in n variables. By an analytic ring over k we mean a k-algebra which is the k-homomorphic image of some convergent power series ring with coefficients in k. A local ring A is called k-analytic if it contains a subring B such that B is an analytic ring over k and A is a finite B-module. If k is algebraically closed then a ring A is k-analytic if and only if it is an analytic ring over k, Corollary 1.5, p.30 of [2]. We will need the following, later on: Lemma [1.1). Let k be an algebraically closed, complete valued field. Let A be an integral domain con- taining k[{xl,...,xd]] as subring where the xi's are analytically independent. If A is a finite module over k[[x1,...,xd]], then A is an analytic ring over k. ‘ggggfz In view of the above remark it is enough to show that A is local. By Hensel's Lemma as stated in (12.2), p.95 of [l], k[{xl,...,xd]] is Henselian. Theorem (43.12), p.183 of [14] gives now the required conclusion. We will need a slightly stronger form of the normalization theorem for convergent power series rings than can usually be found. (Compare e.g., Theorem (45.5). on p.193 of [14].) Theorem (1.2). Let k be an algebraically closed, complete and non-trivially valued field and A a local ring of dimension d. If A is an analytic ring over k and xl,...,xd is any system of parameters of A then k[{xl,...,xd]] EiAo k[[xl,...,xd]] is k-isomorphic to a convergent power series ring in d variables k[{X1,...,Xd]] and A is a finite k[[xl,...,xd]]-module. Conversely, if k[[x1,...,xd]] ggA, A is a finite k[{x1,...,xd]]-module and dim(k[[xl,...,xd}]) = d, then A is an analytic ring over k and xl,...,xd is a system of parameters of A. gzggf: If x1....,xd is a system of parameters then they are analytically independent. Otherwise there exists F(Zl,...,Zd) E k[{Zl,...,Zd]], a convergent power series ring in d variables, such that F(x1,...,xd) = 0. ‘We can apply a k-automorphism m to k[[Zl,...,Zd]] such that CF is regular in 2d“ See Lemma 3, p.147 of [30]. (NOtice that the automorphism given there on the formal power series ring restricts to one on the convergent power series ring). By the Weierstrass Preparation Theorem 3—1 we have CD]? = E(zl....,zd)(z§+ Rs-l(zl""'zd—1)Zd +...) Where E is a unit and the Ri's are non-units. Hence _ . . s s-l _ wF(xl,...,xd) — 0 implies xd-I-Rs_1(xl,...,xd_l)xd -+...— . . . s . This implies that xd 6 (Xl""'xd-1)A making x1,...,x O. d-l into a system of parameters and thus leading to a contra- diction. It is now clear that k[{xl,...,xd]] is k-isomorphic to k[{xl....,xd}]. Next we show that A is a finite module over k[{x1,...,xd]]. We can write A = k[{xl,...,xd,zl,...,zm}]. Since x1,...,xd is a system of parameters we get 2; 6 (xl,...,xd)A for some n. Hence f(xl,...,xd,zl,...,zm) = z: - £51 fi(xl,...,xd,zl,...,zm)xi = 0. If we replace in the above series xj by Xj and zj by Zj' where capital letters genote indeterminates and set F(X1,...,Xd,zl,...,Zm) = z; - 1:: fi(xl,....xd,zl,...,zm)xi then F is Zm—regular. By the Weierstrass Preparation Theorem we can write an arbitrary G 6 k[{xl,...,xd,zl,...,zm]] as G = tI-F-+ n-l . . . 1 . Z) Ri(xl,...,Xd,Zl,...,Zm_l)Zm. Therefore, if i=1 n-l G(xl,...,xd,zl,...,zm) 6 A then G = l2; Ri(xl"°"xd' i . . . . 21,...,zm_l)zm, showing that A is a finite module over k[{x1,...,xd,zl,...,zm_l]]. Inductively we see that A is finite over k[{xl,...,xd]]. For the converse we need only show that x1,...,xd is a system of parameters of A. Let p be an associated prime of (xl,...,xd)A. Then p 0 k[[xl,...,xd]] = (x1,...,xd)k[{xl,...,Xd}]. From Corollary 5.8, p.61 of [3] we conclude that p is the maximal ideal of A. This implies that (xl,...,xd)A is p-primary and hence is generated by a system of parameters. Whenever we have a complete, non-trivial and alge- braically closed field k and a ring A which is an analytic ring over k then the above theorem allows us to write A = k[{x1,...,xd}][yl,...,ys] where x1,...,xd is any system of parameters of A. This is the standard representation of such a ring. 1.2 Analytic Set Germs Again, let k be a valued and algebraically closed field. A set v _c_:_ kn is called analytic at a 6 kn if there is a neighborhood U of a such that U n V is the set of zeroes of finitely many functions which are analytic on V. If Uh is an Open neighborhood of zero in kn and Va is a set which is analytic at each point of U6 then an equivalence relation is defined as follows: (Vfi,Ufi) ~ (VB'UB) if and only if there is a third pair EUanU and vanu =v nu =v. Y B Y B Y Y The equivalence class (Va'Uc) is called an analytic (VY'UY) 'With U set germ and is often only denoted by V or V. Two germs V1 and V2 are topologically equivalent if there are representatives V1 and V2 and continuous maps m :V1 4 V2 and w :Vé 4 V1 such that the compositions to :V1 4 V1 and my :V2 4 V2 are the identity maps. In addition, if the two maps m and w are analytic, then we say V1 and V2 are analytically equivalent. For more details compare [7], where these concepts are develoPed in the case k = C. We associate an analytic set germ V with an analytic ring over k, A, as follows: If A = k[{X1....,Xn}]/m and Fl....,Fs generate m, then there is a neighborhood U of 0 in kn on which Fl,...,F converge. Let W s be the set of the common zeroes of F1....,Fs in U. V is the equivalence class to which (W,U) belongs. As in the algebraic case, V does not depend on the parti- cular generators we took. Also, the radical of u gives rise to the same set germ. We will therefore usually assume that U is a radical ideal. we will show that the analytic set germ does not depend on the particular representation of A. Theorem gl.3). Let k be an algebraically closed, complete and non-trivially valued field . If k[{x Xn]]/m and k[{Yl....,Ym]]/B are k-isomorphic 10...] then their associated analytic set germs are analytically equivalent. Proof:- Choose generators, U = (ol,...,as) and B = (51,...,3t). We set A = k[[Xl,...,Xn]]/fl and B = k[[Yl,...,Ym]]/E. We denote the k-isomorphisms by f :A.4 B and g = f-l. By xi we denote Xi modulo u 10 and by yi, Yi modulo 58. Let Oj(x) E k[{Xl,...,Xn]] such that §j(x) = 9(yj) for 1 g j g m, where fij is Qj modulo 91. Let Ak(Y) E k[{Yl,...,Ym]] such that Kk(y) = fuck) for lgkgn. Here X is A modulo $5. k k For some set UO 5 km we define f# : U0 4 kn by f*(hl,...,hm) = (Al(b),...,An(b)) and for an appropriate V Ekn,g#:V 4km by 9* O O (alto-opan) = (01(3).....Qm(a)). Further define Hk(X) = Ak(fll(X),...,Qm(X)) -Xk for l g k g n and Qj(Y) = Qj(A1(Y),...,An(Y)) -Yj for l g j g m. We claim that Hk(X) 6 9.1 and Qj(Y) 6 58 for all j. For, we have Hk(x) = Kk(§l(x),...,§m(x)) —xk = Kk(g(y1).....g(ym))--xk = 9(Kk(y))-xk) = gf(xk)--xk = 0. Hence Hk(X) 6 91. In the same way we prove that s E oktal. where Qj(Y) E 8. Hence we can write Hk(X) = i=1 t URL 6 k[[X1,...,Xn]] and Qj(Y) = 23 e . B where sz e k[{Yl,....Ym]]. Let U1 be a neighborhood of zero in kn such that 0]”,ch and Qj converge on it for lgkgn, lgzgs, and l g j g m. Let V1 be a neighborhood of zero in k"1 such that 931,51! and Ak converge on it for 1 g j g m, l g L g t, and l g k g n. Let VA be the analytic set defined in U1 by the functions al,...,at. Let VB be the analytic set defined in V1 by the functions 51,...,Bs. Let U2 = g#-1Vl. Then U3 = U D U is a neighborhood 1 2 of zero in kn and g# : VA n U3 4 kn is now well defined. 11 We have: 9 (a) = f#g# a for all a = (a1....,an) 6 VA n U3. (a)-a = f# For, (01(a),...,Qm(a))-a = (A1m(a)).....An(ma))) -a = (H1(a).....Hn(a)) = 0. (Subtraction is meant as a vector Space Operation in kn.) Consequently g* is one-to-one on VA 0 U3. Let _ #-1 _ . . . V2 - f U1. V3 - V1 0 V2 is a neighborhood of zero in km. # We claim.that 9 :‘U n‘v '4 V n V is a homeomorphism. 3 A 3 A The only thing left to show is that g#(U3 n VA) §3V3 n VB. We have §£(§1(x),...,fim(x)) = 53(9(y1)....,g(ym)) = 9(EL(Y1""'ym)) = O for l g_£ g_t. Hence B£(01(X),....Qm(X)) 6 a for l g_£ g_t. If a 6 U3 0 VA then Bi(g#(a)) = Bi(Ql(a),...,Qm(a)) = O for l g.£ g.t, # ## hence g (a) 6 VB. From above we have a = f g (a), hence 9* 6 V2. Since a 6 U2 = g#'lvl, also g#(a) 6 V1. This establishes the claim. # g is analytic on U3 and f # is analytic on V3. This shows that the germs VA and VB are analytically equivalent. The following lemma and its corollary give some information about the dimension of the ambient space of an analytic set germ associated with a given analytic ring over k. 12 Lemma (1.4). Let k be a complete, non-trivially valued and algebraically closed field. Let A be an analytic ring over k and y1,...,yS a set of generators for the maximal ideal in A. Then A = k[{yl,...,ys]]. Proof: Clearly k[[yl,...,ys]] EgA and therefore A = k[{yl,...,ys,xl,...,xn]] with maximal ideal A(yl,...,ys,xl,...,xn) = A(y1,...,ys). Hence 5 x1 = 2231 AL(Y'X)XL where A£(y,x) 6 A. Set 8 F = x - Z) A£(Y,X)X£. If G(Y,X) is any convergent i=1 power series then by Weierstrass' Preparation Theorem there 1 are convergent series U(Y,X) and RO(Y,X2,...,Xn) such that G(Y,X) = U(Y,X)F(Y,X)4-RO(Y,X2,...,Xn). Hence G(y.x) = Ro(y.x2.....xn) showing that A = klly.x2.....xn}]. After applying this reduction n times we see that A = k[{yl,...,ys}]. Corollary(l.5). Let k be a complete, non-trivially valued and algebraically closed field. Let A be an analytic ring over k of embedding dimension n. we have associated analytic set germs in kl for all L zun. 1.3 Saturation In [25] Zariski gives a definition of saturation which we are going to use and which we will recall here, adapted to the special case of analytic rings over a field k. 13 k is an algebraically closed, complete and non- trivially valued field. A = k[{xl,...,xd]][y1,...,yn] is an analytic ring over k with x1,...,xd a system of parameters. §(A) denotes the total ring of fractions of A, k({xl,...,xd]) is the field of fractions of k[[x1,...,xd]]. The saturation Xx of A with respect to xl,...,xd will only be defined if the following five conditions hold (b,c and e are trivially satisfied): a) A is reduced. b) @(A) is Noetherian and hence @(A) = F @...@ F l s is a direct sum of fields. c) k({xl,...,xd}) contains the element 1 of d(A). d) Let 6i be the identity of Pi in @(A). Then Pi is a finite separable extension of eik([xl,...,xd]) for all i, l g_i g_s. e) A is integral over R = A n k({xl,...,xd]). We will need some notation and a preliminary definition. We denote by Q the algebraic closure of k({x1,...,xd]). If y and z are two elements of §(A) then we say that y dominates 2 if for any two k([xl,...,xd])-homomorphisms $1 and $2 of §(A) into 0 the following is true: If ¢1(Z) i'W2(z) then [W1(Y)-¢2(Y)]/[¢l(2)-¢2(Z)] is integral over R and if ¢l(z) = w2(z) then wl(y) = w2(y). A ring A is said to be saturated if every element of its integral closure A (in its total ring of fractions) which dominates an element of A is contained in A. The l4 intersection of all saturated rings lying between A and A is called the saturation of A and is denoted by ~ Ax‘ Apparently it depends on the system of parameters we have chosen. Recall that if we have two rings A §_B then the morphism f : Spec(B) 4 Spec(A) defined by f(p) = p n A is called radicial if the following two conditions are satisfied: (1) for every prime ideal p in A there is at most one prime in B which lies over p. (2) If p 6 Spec(A), q 6 Spec(B) and q lies over p then §(B/q) is a purely inseparable extension of §(A/p). Zariski's Theorem.4.l [25], p.997 shows that the morphism Spec(Ax) 4 Spec(A) is radicial. ‘We will make extensive use of this fact. CHAPTER II SATURATION OF.AN ANALYTIC RING In this section we will show that we can find good systems of parameters so that it is possible to define the saturation. The saturation is then again an analytic ring over the same field as the original ring. Lemma (2.1). Let k be a valued field which is perfect and let A be a reduced analytic ring over k. The integral closure A of A in its total ring of quotients §(A) is a finite A-module. §(A) is the direct sum of fields §(A) = F @...@ F3 and if ei is the identity of Pi as an 1 element in @(A), then A is the direct sum of the integral closures of the Aei's in the Fi's. lggggf: Since A is a finite module over a convergent power series ring we have by [14], (45.6) on p.194 that A is a Weierstrass ring and hence is in particular pseudo- geometric. That is, if p 6 Spec(A) then the integral closure of. A/p in its field of quotients is a finite A/p-module. The lemma follows now from (19.23), p.167 of [l]. 15 16 The total ring of quotients of a Neetherian ring, and hence of a ring A which is reduced and analytic over k can be described more precisely. Namely §(A) = @(A/pl) @...® @(A/ps) where p1,...,ps denote the minimal primes of A. We introduce the following definition: Definition (2.2). Suppose A = k[{xl.....xd]][Y1:---.ym] where xl,...,xd is a system of parameters for A and the yl,...,ym are integral over k[[x1,...,xd]]. The system of parameters x1....,xd is said to be strongly separating if there exist m monic polynomials Pi(z) in k[{x1,...,xd]][Z] such that Pi(yi) = O for l g_i g.m and which are separable considered as polynomials over the field k({x1,...,xd]). An analytic ring over k which has a strongly separating system of parameters is called strongly separable. A ring A is called equidimensional if dim(A) = dim(A/p) for all associated primes p of the zero-ideal in A. Lemma (2.3). Let k be an algebraically closed, complete and non-trivially valued field. Let A be a reduced and equidimensional analytic ring over k and @(A) = Fl @...@ Fs its total ring of quotients. e. i denotes the unit of Fi in §(A). If x1....,xd is a 1? strongly separating system of parameters of A then Fi is a finite algebraic and separable extension of eik({xl,...,xd]) for 1 g_i g s. Proof: We first consider the case where A is a domain. We have then the following commutative diagram where all maps are the obvious inclusions. k({xl,...,xd])(yl....,ym) / \_ k({xl,...,xd]) k[(xl....,xd]][y1....,ym] \ / k[{x1.....xd}] Clearly k({le...de})(yloooOIYm) = §(k[{x100000xd}] [y1’0001ym]) = §(A). The monic irreducible polynomial of y1 over the field k({x1,...,xd]) divides Pi which is separable. Hence, n is separable. We can now look at the general case and denote the minimal primes of A by p1,...,ps. We claim that k[[x1,...,xd]] 0 pi = (O) for l g,i g,s. To see this 18 we notice that for each pi we have a chain of prime ideals pi C ql C...C qd of length di-l in A. Contracting this chain to k[{x1,...,xd]] we get another proper chain in this ring of length di-l, hence pi n k[[xl,...,xd]] = (O). This establishes the claim. We define the maps in the diagram beneath as follows: 11 1m -11 -1m fl(Z ail"'im Y1 ...Ym +pi)=2ai1...imY1 ...Ym . Since k[{xl,...,xd]] 0 pi = (0) this is an isomorphism. f2 is defined analogously. gl(a) = (a4-p1,...,a4-ps). 92 is defined in the same way. e and h are the natural embeddings. It is clear that all subdiagrams commute, except for diagram D. 19 @(A) h k({X))(Y) 3 F1 92 I ll [K e D 91 ‘\ \ r1 ‘ k({Xl) k[{x}] [5?] a A/Pi k({Xl) A\ I: A 4 HBO] x stands for x1....,xd and y for y1,...,ym. y denotes y 4- pi . 20 If Pi(Z) is an integral relation for yi over k[{xl,...,xd]] then it is also one for §i over the same ring. The special case treated first shows now that Pi is a finite separable algebraic extension of k({x1,...,xd]). The lemma will be proven if we show that hf2e(k({x})) = eigz(k({x])) =hfze(a) = h(a4-pi) = (opooo'a+Pipooo'O) = €i(a+Pl'...'a+ps) = 6192(a)o Our next goal is to show that all analytic rings over k are strongly separable. To achieve this we will have to introduce some notation and to quote some theorems from [21]. Suppose A is an analytic ring over k. A k-derivation 6 :A.4 M is called finite if M is a finite A-module. The pair (Dk(A)'dk)' where Dk(A) is a finite A-module and dk;:A’4 Dk(A). is a derivation, is called the univer- sally finite derivation of A if the following holds: For every finite module M and every k-derivation. 5 :A.4 M there exists a unique A-homomorphism. h which makes the following diagram commutative: Dk(A) In case d.k :A.4 Dk(A) exists, it is determined uniquely up to A-isomorphism. Dk(A) is called the universally finite module of k-differentials of A. 21 Theorem(2.4) (Sheja-Storch) [21], p.146, (2.6). If A is an analytic ring over k then Dk(A) exists. Definition (2.5) [21], p.149. Suppose A is an analytic integral domain of dimension d. A system of parameters x1....,xd in A is called separating if the quotient field of A is separable algebraic over k({x1a...pxd)). Note that separating and strongly separating agree for analytic domains over k. If A is a domain and M is an A-module then rankA(M) = dim§(A)(M.Qk i(A)). Theorem (2.6) (Scheja-Storch) [21], p.149, (4.1). Suppose A is an analytic integral domain of dimension d. Then 1) rankA(Dk(A)) 2Dd. 2) rankA(Dk(A)) = d if and only if A contains a separating system of parameters. 3) A system of parameters x1,...,xd of A is separating if and only if Dk(A)/(Adxl+...+ Adxd) is a torsion module. Definition (2.1) [21], p.150. Suppose A is an analytic ring over k and p 6 Spec(A). p is said to be separable if and only if rankA/P(Dk(A/p)) = dim(A/p). A is said to be separable if A is reduced and each minimal prime of A is separable. 22 Theorem (2.8) (Scheja-Storch) [21]. (4.2), p.150. If k is perfect then every reduced analytic ring over k is separable. For our purposes we need strong separability. However, the following theorem shows that the two conditions imply each other. Theorem (2.9). Let k be an algebraically closed complete and non-trivially valued field. Let A be an equidimensional and reduced analytic ring over k. Then A is separable if and only if it is strongly separable. Proof: Of course we can assume that char(k) = p > 0. It follows easily from Lemma (2.3) and Theorem (2.6) that strongly separable implies separable. NOw we assume that A is separable. Let p1,...,pS be the minimal primes, Wi.:A'* A/pi the projections and di:A./'pi 4 Dk(A/pi) the universally finite derivations. By [21], p.149, Zusatz zu (4.1) we get d = dim(A) = dim(A/pi) = rankA/Pi(di(wim)). We can now apply Hilfssatz (7.2) in [21], p.157, to find xl,...,xd 6 m such that for all i {wi(x1),...,wi(xd)] is a system of parameters and a di-free set, that means {divi(xl),...,diri(xd)] is linearly independent over A/pi. Let Y1’°'°'Yd be any system of parameters of A. Then y§,...,y§ is a system of parameters, too. Now let 21 = yEKlntxi); then 23 21,...,zd is a system of parameters since lwtxi are . = p units. dj[wj(y§(l-txi))] (wjyj) djrj(xi) and wj(yi) # 0 for otherwise dim(A/pj) g dim(A/yiA) = d-1 by [6], (10.5), p.73. Hence [wj(zl),...,wj(zd)] is dj-free for l g_j g,s. Consider the following sequence: a. . o 4 [i (A/pj)dj7rj(zj) 4 Dk(A/pj) 4 nkm/pjviz (A/pj)djvrj(zi) 4 0 Since 4(A/pj) is flat over A/pj ‘we get: 0‘4 (§:(A/pj)djfij(zi)) G @(A/pj) 4 Dk(A/pj) ® é(A/pj) 4 (ka/PjV‘E’ (A/pj)dj7rj(zi)) 0A MA/Pj) 4 O /pj Since the ranks of the first two terms are equal, the rank fth lat ' ,that an A.ZA.d.1r.. o e 3 one is zero me 8 Dk( /pJ)/ti( /p3) 3 3(21) is torsion. It follows now from (2.6) that wj(zl),...,wj(zd) are separating systems of parameters for all j, l g.j g_s. We write now' A = k[{zl,...,zd]][y1,...,ym] where 21,...,zd is the system of parameters defined above. For clarity we set y = y1 where i is any of the indices 1,...,m. ‘We denote by AssA(O) the set of all associated primes of the ideal (0) in A. Let p 6 AssA(0) with n-l yip and let P(Z)=Zn+a z +...+a e O k[{zl,...,zd}][Z] such that P(y) = 0. Since n-l k[{zl,...,zd]] n p = (O) ‘we have a canonical inclusion k[{zl,...,zd]] E A/p and we denote the image of a 6 k[{zl,...,zd]] in A/p by a. Let 24 n-l - 5(2) =zn+ar1 z +...+a =I>l(2)...15k(2) where 51(2) -1 0 denote the irreducible factors of P(Z) as polynomials over wk[(zl,...,zd]] which is a unique factorization domain. By a corollary to Gauss Lemma, see [8], p.147, Lemma 3, the Pi(Z)'s are irreducible over vk({zl,...,zd]). Since also P(y) = 0 (here y = w(y) where w :A.4 A/p is the canonical surjection), we get 51(2) = Irr(y,wk([zl,...,zd])), the irreducible monic polynomial, for some i, 1 g_i g_k. By choice of notation we may assume that i = 1. Since §(A/p) is separable over #k[{zl,...,zd}] we have that P1(Z) is separable. If m-l .. m _ - 131(2) -z +bm_z +...+b l m m-l Z +bm-lz +...+ bO of Bi under the map F. Since Pi is separable so is 0 then we set QP(Z) = where bi denotes the unique preimage d 6 . QP an Qp(y) P If t is the number of primes belonging to zero and containing y then set t z = . Fy( ) z 11 Q (2) p6AssA(0) P Yip Fy is a monic polynomial over k[[zl,...,zd]] which is separable since each irreducible factor is. Also Fy(0) = 0 since F(y)=ytHQ(y)€ n pg 0 p=(o). Y P p6AssA(O) p6AssA(O) This shows that 21,...,zd is a strongly separating system of parameters. 25 Corollar 2.10 . If k and A are as in Theorem (2.9) then A is strongly separable. Proof: Immediate consequence of (2.8) and (2.9). The following example serves two purposes. First, it shows that not every system of parameters is strongly separating. It also shows that a system of parameters may be separating for one summand of the total ring of fractions but not for another one. Example (gylll, Let k be an algebraically closed, complete and non-trivially valued field of characteristic p > 2. Then k[{xl,x2}] is a regular local unique factorization domain of dimension 2. Xi-—X2 and X§-X2 are two prime elements in this ring. Let P = (Xi-szllxlmzll. Q = (XE-x2)k[{xl,x2}] and I = P n Q. The example we want to consider is A = k[[X1,X2]]/I. If we set p = P/I and q = 0/1 then AssA(0) = [p,q] and A is reduced. A/p a k[[xl]] and A/q a k[[xl]]. Hence dim(A/p) dim(A/q) = dim(A) = l and A is equi- dimensional. m = (xl,x2)A is the maximal ideal of A . . 2 . and x1 is a parameter since m ‘EEXIA' We can now write 1 Z2- (xi+x§)z+x§+2 = (Z-xiHZ-XE) is separable over A = k[{xl]][Y]/(Y2- (xi+x§)Y+xp+2). Since Py(Z) = k([xl}). x1 is a strongly separating parameter. x2 is a parameter, too: ml»2 5; sz. Therefore A = k[{x2]] [YJ/(Ypil'Z-szp-szz-I-xg), A/p # k[{x2]] [SH/(Y2 -x2) = k[{x2]] and A/q = k[[x2]][Y]/KYp-x2). We see that 26 @(A/p) is separable and §(A/q) is not separable over k({x2}). We can now state and prove our desired result: Theorem (2.12). Let k be an algebraically closed, complete and non-trivially valued field. Let A be an equidimensional and reduced analytic ring over k. Then there exists a system of parameters xl,...,xd of A such that the saturation of A with respect to this system exists. For each system of parameters for which it exists the saturation is again an analytic ring over k of dimension d. ‘ggggf: ‘We first have to check conditions a through e mentioned in Section 1.3: a) is part of the assumptions; b) follows from (2.1); c) is obvious. To satisfy d) we have to choose a strongly separating system of parameters. That we can find such a system follows from Corollary (2.10). Lemma (2.3) shows now that d) is fulfilled. For e): k[{xl,...,xd]] 93A n k({xl,...,xd]) = R and that A is integral over R follows from Theorem (1.2). Now suppose that x1....,xd is a system of parameters for which the saturation is defined. we denote it by Ax and A stands for the integral closure of A in §(A). From Lemma (2.1) and the fact that A is Noetherian we conclude that Ax is a finite A module. From Theorem (1.2) it follows that Ax is finite over k[{xl,...,xd]]. 27 Since k[{xl,...,xd]] is integrally closed in its quotient field we have R = A n k({x1,...,xd]) and it follows from [25], (4.1), p.997 that Spec(Ax) .. Spec(A) is radicial. Hence Ax is a local ring and dim(Ax) = dim(A) = d. By the second part of Theorem (1.2) we get that Ax is an analytic ring over k. Corollagy (2.13). Let k and A be as in Theorem (2.12). If xl,...,xd is a system of parameters for which the saturation exists then it is strongly separating. Proof: The second half of the proof to Theorem (2.9) shows that Zariski's condition d) implies that the system of parameters is strongly separating. CHAPTER III TOPOLOGICAL RELATION BETWEEN RING AND SATURATION As we have shown the saturation of an analytic ring over k is again an analytic ring over k, provided one takes a strongly separating system of parameters. As explained in Section 1.2 one can associate analytic set germs with both rings. The requirement that the two analytic set germs are t0pologically equivalent can be considered as a minimum requirement for an adequate definition of equisingularity. The purpose of this sec- tion is to show that this requirement is fulfilled. Suppose we have two analytic rings over k, A and A’, AEA’ and A’ is finite over A, say A’ = A[y1,...,ym]. If A = k[{xl,...,xn]] then there is an associated analytic set germ VA in kn. A’ = k[{x1,...,xn,yl,...,ym]] gives then rise to a set germ VA, in kn+m. In this situation we say that VA, lies over VA. If D 6 A then we will write D(xl,...,xn) for a representation of D in k[{xl,...,xn]]. D(X1,...,Xn) is then the power series which has the same coefficients 28 29 as D(xl....,xn) but has the ring elements xi replaced by the indeterminates xi. If a = (al,...,an) 6 kn then D(a) simply means D(Xl,...,Xn) evaluated at Lemma (3.1). Let k be an algebraically closed, complete and non-trivially valued field of characteristic p > 0. Let A ggA’ be two analytic rings over k such that A’ is a finite A-module. Further assume that there are D 6 A, D y'o and a 6 It such that Dapa 6 A for all a 6 A’. Then the analytic set germs VA and VA. , where VA, lies over VA. have representatives (VA,U) and (VA,,U’) sudh that above every a 6 VA with D(a) #’0 there lies one and only one point of YA” Proof: We first prove the uniqueness: If a A’ = A[yl,...,ym] then we‘have Dy? = 9i 6 A for l g.i g_m. We take a set of defining functions for VA, and include a among them the m functions D(X)Y§ -—gi(x). D(X) and gi(X) are defined as explained previous to the statement of the theorem. Let a = (a1,...,an) 6 VA with D(a) y'o. Let b and c be two different points above a in YA” b = (all-colanlbIIOOOlbm) and c = (alIOOOIanIclIOOOICm)O G Since b and c are in V I we have D(a)b§ -gi(a) = 0 A or bpa - gi(a) - 0 for l ' I th 1 D(a) - g_i g,m. n e same way pa 91(a) . ci — D(a) = O for l g_i g.m. Hence b1 and c1 are 30 both solutions of the equation ZpG-r = 0. Since k is algebraically closed there is s 6 k such that spa = r and therefore Zpa-r = (Z-s)pa = 0. Hence the equation has only one solution and we conclude that bi = ci for l g.i g.m, ‘which shows that b = c. It remains to show the existence. If A =k[{xl,...,xn,yl,...,Ym]]/$ let F1(X,Y),...,FS(X,Y) be a set of generators for 8. Since A’ is a finite A-module we may assume that Fi(X,Y)6 k[{X]][Y]. Take p big enough such that -a —G G [D(X)lP[Fi 0 such that -a m g e, [gi(a)/D(a)]p < e for all i, 1 g i g m when— ' n ever a 6 ow = [(al,...,an) 6 k llail < w] and such that all Gi(X), l g_i g_s and D(X) are convergent on Uw' We can now include the Gi(X) among the generators for an analytic set (V ,UE). If a 6 VA and D(a) #'0 then G G we let b= (a,[gl(a)/D(a)]P .....[gmtr=1)/D(zi)]p ). a b 6 U8 and [Fi(b)]p = Gi(a)/D(a)p = o for l g i g s. Hence Fi(b) = 0 which implies that b 6 (VA,,U€). We now state two lemmas which are well known in the complex case. We will point out at the end of this section 31 (see Theorem (3.6)) why the usual short proofs will not work in our case. We think it is justified to repeat the proofs given by Kneser [10] here because they are not well known and are given only for the case k = C. But the reader will see that the particular field is not of importance. Lemma (3.2). Let k be a valued field. Let 1 g_m g n and 01,...,on 6 k ordered such that [all g,..g [on . Then there is a non-negative real valued function cpm on (11+)m such that the following two conditions are satisfied: (1) |am|‘$-mmitl""'tm) where ti = Isn_i+1(ol,...,on)| and sj denotes the j-th elementary symmetric polynomial. (2) cpm(x100001xk) 4 O as x1 ‘0 Op...,Xk 4 0. Proof: Kneser [10], pp.102-104. We do induction on m. If m= 1 then we set Cp1(x) = :35. (2) is clearly satisfied. To see that (1) holds, consider cpl(tl) = Mal...on| 2 n./lcl|n= loll. We assume now that we have functions $1,...,@k each of them satisfying (1) and (2). Suppose that ak+l = 0. We set mk+l = 0. we can therefore assume that ak+l # 0. Then we have oi #‘O for all i 2_k4-1. Setting r = mdn{k,n-k] we get 32 (3) Sn_k(allooolan) = (a](+l ocean) [1+ 1' -1 -1 .Z} si(cxl,...,ak)si(ak+1,...,dn )]. i=1 We notice the following inequalities: (4) [si(ol,...,o.k)| 3 [all [oi] +...+ ‘Gk-il lakl S cpl-(t1) coo cpi(tllooooti) +ooo+cpk_i(tllooootk_i) oo- wk(t1.....tk) = si(ml(tl).....mk(tl.....tk)) for i g_k. Also: n—k (5) IO‘k+1| 31Qk+1"’ani (e) (si(a;il.....a;1>1 g (“gknakfll'i for all i g n-k. USing (4) through (6) in (3) we get: -k tk+l = Isn_k(allooopan)! 2 |01C+lin [1... I}: (a )s ('1 a‘l) i=1 Si 1'..°'ak i ak+l’°"' n i] r . 2 Well“ k” '?3 '(nik>|°k+ll 1 si( 0 there is a 6 > 0 such that if [ai-bi| < 6, O g.i g_n-l, then |x1-c| < e,...,|xm-c| < c after appropriate enumeration of the roots of f2. Proof: If c 6 k we define _ _ n n-l 91(y) - fl(y+ C) — y +an_ly +...+aly and - _ n n-l n n n , L E-k k g(y)=(y+c) +...+a = 2(2) ( _ )ac ]y n n 6‘ fl t-k k = 2.:[21 (k)aLc ]y . Ik=0 £=k (NOte that if m 6 It and a 6 k then we write ma to m abbreviate the field Operation. 2) a. In particular, ex— i=1 pressions of the form ($)a have to be understood in this 34 sense, that is, with (E) 6 nu.) Similar 92(y) = n. n r L-k k 21[ Z} ( )b c ]y . Hence the new coefficients are k L k=0 £=k n n c:k = 23h (fi)c"ka£ and Bk = ‘3‘ (fi)c“kbfi. lok-Bkl = I; <fi>c""l _<. £2: (fiHCIL’kIay-byl _<. n(n!)c*[a£*-b£*l where c* = max{l,[c|n-k] and E* is such that [a£*-b£*l = max {lai-bil]. Qgign-l Suppose now that c is a root of multiplicity m of f1(x), then do =...= a roots of 92(y) ordered such that [Y1] g,..g,|vn = O. Denote by Yl,...,y the m—l n . Using the notation from Lemma (3.2) we have ti = Is (Y1....,Yn)l = [Bi-l . By Lemma (3.2) there is a n-i+1 function wm. such that |Yl| g,..g_|le g_¢m(tl,...,tm). By the above inequalities we have ti = IBi-ll S n.-(n!)c*|a£*-b£*|, l g_i g_m. Since mm(tl,...,tm) 4 O as tl,...,tm4 O and [xi-cl = IYi+C-C| = [Yil g wm(tl,...,tm) for l g_i g_m, the lemma is proven. We will now combine the lemmas to prove the following theorem: Theorem (3.4). Let A = k[{x1,...,xn]] and I A = k[{x1,...,xn,y1,...,ym]] where k is an algebraically closed complete and non-trivially valued field. Suppose that l) A’ is a finite A-module and A E:A’- 2) A' is reduced. 3) A’ is a radicial extension of A. 35 Let VA and VA, be associated analytic set germs in n and kn+m respectively. Then there are repre- k sentatives (V ,U) and (VA, ,U’) such that the projection w':kn+m‘4 kn induces a homeomorphism on the analytic sets. Proof: If char(k) = 0 then R = c and the theorem is identical to Theorem 9 of [22], p.429. Hence we assume throughout the proof that char(k) > 0. Let P1,...,PS be the minimal primes of A’. Since A’ is reduced P1 n...n P8 = (0) is an irredundant primary decomposition. Let pi = Pi 0 A, then (0) = pl n...n p8. Suppose we could leave out one of the primes, say p1. Then pl 3 p2 n...n ps and we have p1 _:_:>_ pi for some i, 2 g_i g_s, say pl 2,p2. By the going up theorem, [3], (5.11), p.62, there is a prime Q in A’ such that P2.E:Q and Q n A = p1: Since the extension is radicial we have Q = P1 and hence PZEPl which is a contradiction. This shows that p1,...,ps are exactly the minimal primes of A and A is therefore also reduced. we have now 6(A) = 6(A/pl) @...® 6(A/ps) and §(A') = @(A’/P1) @...@ HIV/P8). The map (a+pi)/(b+pi) 4 (a+Pi)/(b+Pi) determines a natural embedding of MA) in @(A'). 36 Next we show that there are analytic sets (VA,U) and (VA’ ,U’) such that above every point of VA there lies exactly one point of VA,. Hence the projection is a bijection between (VA,U) and (VA, n (U> O and that the existence of two sets lying above each other in the required way is established for all smaller dimensions. Since A g A’ g 6(A’) we can write A’ = 2%Aai where a. 6 6(A’). Since the extension is radiciil, each J. §(A’/Pi) is purely inseparable over 6(A/pi) and hence a there is an a 6 nt such that a? = r./s. 6 @(A), i i i t r.,s. 6 A, for all i, l g_i g.t. Let D = n s. 6 A. i l a 1 1 Then D 7! o and Dap e A for all a e A’. By Lemma (3.1) we can find (VA,U) and (VA, ,U’) such that there is exactly one element in VA, above each element a 6 VA if D(a) #'0. We consider the analytic subset of V on whidh D vanishes. Let I’ = rad(D -A7), A 1 = 1’ n A, A = A/I and A" = A’/i’. Clearly, (I), 37' is a finite A-module and, (2), A7' is reduced. But also (3) holds, the extension A EEK? is radicial. For, suppose p 6 Spec(A), then it corresponds to some '5 6 Spec(A) with I555. If P and Q are primes in A7} P n A = Q n A = p then consider the corresponding primes P and 5 in A’. NOW ‘3 n A = 5 n A = p. 37 This contradicts the radiciality of A’ over A, hence there is at most one prime above p in A7} Further we have A/p a A/E and A7]? 2.A’/P. Since A/E 4 A’/P is purely inseparable so is A/p 4 Af/P. In conclusion we have that Spec(Av) 4 Spec(A) is radicial. Since D is not a zero divisor in A’ ‘we have dim A7.< dim.A’. By the induction hypothesis we have two sets (V§,W) and (V;7.,W’) which lie above each other in the required A way and therefore do the sets (VA,UTTW) and (VA, ,UJFWW’). It remains to show that the projection restricted to the analytic sets is a topological map. Clearly it is continuous, so the only thing left to show is that (x1,...,xn) 4 (x1....,xn,yl,...,ym) is continuous. Let (xii),...,xél)) be a sequence in V which A (0) (0),. n converges to (x1 ,...,x Denote by (xii),...,xéi), (1)) Yil).....ym the corresponding sequence in V ,. Let A m be a set of generating functions for VA, which includes the integral relations of the generators for q A’ over A. We denote by Fk(x,Z) = Z kkt..utfko(x) the integral polynomial for yk and hence fkj(xl,....xn) 6 k[{xl,...,xn]] for all j and k. (0) (0), ak1,...,aqu are the qk roots of Fk(xl ,...,xn Z) = O and we set .akl = yfio). Let EL be a sequence of positive real numbers such that e£+1 < EL, 2el < Iaki-akj| if aki y'akj and 6‘ 4 0 as L 4 m. ‘We denote by 38 zfii),...,zfié) the solutions of Fk(x(l),Z) = 0. Then k y(i) (i) =zk Qk(i)' all continuous functions we can apply Lemma (3.3) m ( Z, q )- times to find N such that whenever i‘z N k_1 k I. r then we will have Iakj- 1 g_flk(i) g qk. Since the fkj are (1)] < en for l g_k.g_m and l g_j g qk. The above statement requires a proper choice of indexing for the roots of Fk = 0 which can always be made. If we have y(i)- it follows that yfil) 4 kyfio ) as required. Suppose this —zki) from some L on, then is not the case for at least one k. Since we have only a finite number of choices we would have a subsequence ir such that Qk(ir) #’1 for some k but Qk(ir) = uk (ir ) (ir) = constant for all k. Since P(x1,...,ym ) = O for all F 6 N and since all F are continuous we have (0) (0) P(x1,...,xn 'alul""'amu ) = 0 for all F 6 u with m a. y'a. for at least one i, l g_i gim. Thus we aui 11 ‘would have a second element lying above (xio),...,xéo)) which contradicts our previous findings. The main theorem of this section follows now easily. Theorem (3.5). Let k be an algebraically closed, complete and non-trivially valued field. Let A be an equidimensional reduced ring which is analytic over k. Let xl,...,xd be a strongly separating system of para- meters of A and AX the saturation with respect to 39 this system. Then the two associated analytic set germs are tOpologically equivalent. In fact, the homeomorphism can be induced by the natural projection of the ambient spaces if the repre- sentations of the rings are chosen so that the associated set germs lie above each other. [Egggf: By Theorem (2.12) Ax is analytic over k and hence a finite k[{x]]-module. This shows that Ax is finite over A. Since Ax E §(A) and A is reduced, we conclude that A; is reduced. The fact that Ax is radicial is proven in Theorem 4.1, [25], p.997. We can now apply (3.4) and get the second half of our theorem. Theorem (1.3) shows that the particular representation of the ring does not matter and therefore finishes up the proof. Lemma (3.2) and the part in the proof to Theorem (3.4) which establishes the continuity of the map can be proven much more easily in the case when k = C. The shorter proofs are based on the fact that every bounded sequence in C has a convergent subsequence. The following theorem shows that we do not have this fact available in our situ- ation and that we can therefore not h0pe to adapt the usual proofs. Recall that a space is called sequentially compact if and only if every sequence has a convergent subsequence. 40 Theorem (3.6). Let k be an algebraically closed, non-trivially valued field of positive characteristic and let Ac = [x 6 k|1x| g a], where a 6 I91. Then Ad is not sequentially compact. ‘ggggf: Since AG is metric it is paracompact; see [4], p.186, Theorem 5.3. By [9], p.162, E). part (d), Ad is sequentially compact if and only if it is countably compact. The latter is the case if and only if Ad is compact, [4], p.230, Corollary 3.4. Now suppose Ad is sequentially compact and hence compact. Then k is locally compact, since addition is continuous. Since char(k) > O the valuation is non- archimedean and from Theorem 1 of [18], p.245, it follows that the valuation is discrete, that is [k-—[0]| is a cyclic subgroup of the positive real numbers. Say Ix] is a generator of this group. We can assume that [xl > 1. It is easy to see that [x] = min{|y| > lly 6 k]. Since k is algebraically closed there is a 6 k such that 2 a = x and therefore 1 < [a| < [x . This contradiction shows that Ad cannot be sequentially compact. CHAPTER IV MULTIPLICITY In order to be equisingular it is certainly a necessary condition for two analytic set germs that their local rings have the same multiplicity. One of the main theorems in [28], Theorem 4.1, p.455, states that under certain conditions the multiplicity is preserved by passing to the saturation. The proof of our corresponding theorem follows basically the one of Zariski. However, the conditions are somewhat different so that some of the details have to be changed. For this reason we include here the somewhat lengthy proof in full. We will need some definitions and results which are due to NOrthcott and Rees. Definition (4.1). If u and B are ideals of a ring R then B is a reduction of m if Big A and r r+l 8% = T for at least one positive integer r. Theorem (4.2) (NOrthcott-Rees, [17], p.357). Let (A,m) be local with char(A) = char(A/m). If q is an meprimary ideal of A and (x1,...,xd) = u is a parameter ideal which is a reduction of q then their multiplicities are the same: e(fl) = e(q). 41 42 Now we come to the main theorem in this section. Theorem (4.3). Suppose k is an algebraically closed, complete and non-trivially valued field. Let A be an equidimensional and reduced ring which is analytic over k. Suppose x1....,xd is a strongly separating system of parameters and the ideal it generates is a reduction of the maximal ideal of A. Denote by F; the least Galois extension of k({xl,...,xd]) which contains e(A/pj), where {p1,...,ps] = AssA(O). Suppose that char(A) and [F3 :k({x1,...,xd])] are relatively prime for all j, 1 g_j g_s. Then e(A) = e(Ax). Proof: See [28], §4. We construct a sequence of rings A. = Ai-1[L1]’ where A l O = A and Li contains all elements of A which dominate some element in 'Ai-l' By Lemma (2.1), A is a finitely generated NOetherian A- ~ module and hence so is Ax. Thus, there is an integer n such that An = Ax. Each Ai is finitely generated over A and thus over k[{x1,...,xd]]. Since the latter ring is a Henselian domain, we see that Ai is local ([14], (43.12), p.183). Hence each A1 is analytic over k ~ ‘Wlth maXimal ideal Mi' Clearly Aix = Ax' If m is the maximal ideal in k[{x1,...,xd]] then (1) e(mA) = e(mAi) for all 1 2,0. We will show first that no element of k[[x1,...,xd]] is a zero divisor in Ai' Recall from the proof of Lemma (2.3) that k[[xl,...,xd]] 0 p = (O) for all primes p in A 43 which belong to zero. If r 6 A and r is not a zero divisor in A then r is not a zero divisor in e(A), for, otherwise there is an element a/s with a,s 6 A and (r/l) ~(a/s) = 0. Then rat = O for some non-zero divisor t in A. This is impossible. Con- sequently, if a 6 Ai is a zero divisor and a 6 A, then a 6 p for some p 6 AssA(O) and hence a £’k[{xl,...,xd]]. We can now apply the projection formula of [30], p.299, Corollary 1 and get: [Ai/h&_:k[[x1,...,xd]]/m]e(mAi) = [A1 :k[[x1,...,xd]]e(m) [A.:k[{xl,...,xd]]e(m) [A/M : k[{xl,. . . ,xd]]/m]e(mA) . This implies that e(mAi) = e(mA). we will need a valuation theoretic characterization of a reduction: Let Fj = §(A/pj) where (Pl....ops} = ‘ASSA(O)‘ Then @(A) = F @...® Fs and we denote by ”j l the j-th projection. Let Sj be the set of all non- trivial valuations of Fj which are nonnegative on A/pj. If v 6 Sj and x 6 A then we will write v(x) for v(wj(x)) in order to keep the notation simpler. If v 6 Sj and u is an ideal in A, then v(m) = min(v(x)|x 6 N]. n. denotes the ideal in A/pj which is generated by rjfl. J The derived complete ideal of m is defined as s A 91'= n (n 15191.3 ) where R denotes the valuation j=l v68. 3 3 V V 3 5 ring of v. S = L) S.. 44 Claim: If m and B are ideals of A, T contains some non-zero divisor and fl 9:8, then m is a reduction of B if and only if v(fl) = v(B) for all v 6 S. First, assume that T is a reduction of B. We set A = {x 6 Alx is integral over U]. By [16], p.156, Theorem 3, is 9 it. By [15]. p.167, Theorem 1, we have 8 E m’. For arbitrary v 6 Sj and arbitrary x 6 B ‘we have x 6 fl’ and in particular x 6 #31 ijv. We can n 'write wj(x) = 121 aiwj(ai) where (a1....,an) = m and oi 6 RV. Now v(x) 2_min{v1(ole(al)),...,v(onwj(an))] = v(dfiwj(a£))‘2 v(at) for some t, l g.£ g_n. Therefore v(m) g v(B). The other inequality is obvious and we get v(fl) = v(B). Now suppose that v(m) = v(B) for all v 6 S. Let b 6 B, wj(b) #’O and v 6 S then there is a 6 fl jl such that v(a) g v(b). In case v(a) = v(b) then v(a/b) = v(a)-v(b) = O or wj(a)/hj(b) = u 6 Rv and -1 1 u = Wj(b)/hj(a) 6 Rv' Hence wj(b) = u wj(a) 6 N.Rv. J In case v(b) > v(a), then v(a+b) = v(a). We may assume 1rj(a+b) a! 0, then v(a/a+b) = 0, hence -1 _ wj(a)/wj(a4-b) — u 6 Rv and nj(a)u — rj(a4-b) 6 Nij. Since wj(a) 6 9.1.R.V we get as before wj(b) 6 N.R The 3 JV' same is true_if rj(b) = 0. Since j and v were arbitrary we see that b 6 u’ and hence B E_U'. By Theorem 1 of [15], p.167, 8 g.§ and by Theorem 3 of [16], p.156 we have that B is a reduction of m as required. This finishes the proof of the claim. 45 We will now use induction on the index i to show that (x1....,xd)Ai is a reduction of Mi for all i 2_O. The case i = O is trivial. The fact that x1....,xd is a system of parameters in Ai for all i follows from [30], p.276, Theorem 15(d). We may now assume that (x1,...,xd)Ai is a reduction of Mi or, equivalently, that v(mAi) = v(Mi) for all v 6 S. We have to show that this implies (2) v(mA. 1+1) = v(Mi+l) for all v 6 S. To prove (2) notice first that we clearly have v(M ) g_v(mAi+l). We write Ai+1 = Ai[T] where we can i+l assume that T E-Mi+l’ For, if y 6 T, but y £.Mi+1 then replace it by z = y-c where c 6 k. It suffices now to show the following: For all y 6 T and for all v 6 S- we have v(y) 2 v(xi) for all i, 1 g_i g,d. If this were true and a 6 M1+1 then a = ZLGibi, bi 6 T and ci 6 Ai. v(a) 2_min{v(oibi)] = v(olbl) = v(a£)4- v(bz) 2_v(b£) for some L. Hence v(Mi+1) 2 v(T) 2 v("‘Ai+l) ‘ Let F; be the least Galois extension of K = k({xl,...,xd]) containing Fj' Denote by s; the set of all valuations of F; which are nonnegative on R = k[[xl,...,xd]]. Since ij is integral over R the * elements of Sj are nonnegative on rjA, that means * * v*|F e sj for all v e sj. By [5], Theorem 13.2, p.94 j to every v 6 Sj there is an extension v* 6 8;. Hence 46 * Sj consists exactly of the extensions of v 6 Sj to * at Fj. Hence we can also prove equivalently that v (y) 2 'k 'k 'k v (m) for all v 6 S and all y 6 T. Let S0 be all valuations of K. which are nonnegative on R. we have * * O . * * . v [K 6 S if v 6 Sj and Sj conSlsts of all the extensions of S0 to F3. All extensions belonging to V0 6 So form a complete set of conjugates under * * G = Gal(Fj|K). More precisely, if v extends v0 and * T 6 G then v T extends v0 and if v’ extends v I then v' = v*r’ for some T 6 G; see [30], p.28, 0 Corollary 3. (3) If y 6 T arbitrary and v0 6 SO arbitrary, * * then there exists at least one v 6 Sj which extends v and v*(y) 2.v*(m). O For the proof of this statement, assume that this is * * * not the case, or v (y) < v (m) for all v extending v0. * Let h = [Fj : K], g = [Fj : K]. Note that if p = char(A) then (h.p) = l = (9.1:) and hlg. T * (y) = )3 yr = FjIK rec (g/h)TFj/K(Y)' Let Y0 = (l/ngFf KW) = (l/thFj‘KW) J n = (1/9)Z 171‘. (If n62 then n= 2 1, 16k and TEG i=1 (l/n) = n‘l.) We ShOW'that yO 6 m, the maximal ideal of R. Since y is integral over R there is a minimal polynomial f(X) = Xn4-cn_lxn'1+u... of y over R. By [8], p.147, 47 Lemma 3, f(x) is irreducible over K and hence y0 = cn-l 6 R. Let Rj be the integral closure of R in Fj. By the lying over theorem there is p 6 Spec(Rj) such that p n ‘lTj (A )T is integral by [3] . i+1 i+1 T over R and we have p n (Fin+l) - (iji+1) (5.8), p.61. USing the same theorem, R n p = m. ) = FjMi+l' (WjA T wj(y0) 6 p Since y 6 Mi+l' Hence yO 6 m. * For all extensions v of V0 * * * * 'k v (m), hence v (y) < v (yo), hence v (y-yo) = v (y) we have V*(YO)-2 * T 0' Therefore v ( E (y -y)) = * T6G * v (y) for all extensions v of v0. This implies that there is an element * for all v extending v 'k for every extension v of V0 * T 6 G such that v*(yT-y) g_v (y). Since y 6 Li+1 it * dominates some 2 6 A1 and we have v (yT-y).2 v*(zT-z) for all v* in .S; and for all T 6 G. We may assume that z 6 Mi (if not, replace by z-c where c-I-Mi * * = zani and c 6 k). Hence v (y) 2 v (zT-z) for all 'k * v extending v0 and for a suitable T 6 G. Fix vO * T extending v0 and TO 6 G such that vo(z 0) = . * T w w * min{vo(z )]. Set v1 = vOT0 then we have Vl(2) = T6G T 'k 'k 'k * - vo'ro(2) = vo(z o) = minivotzTH = min[v1Tol(zT)] = _1 TEG T6G 1' 'T min{v:(z O )]. v:(z) = min{v:(zT)]. Hence v:(zT-z) TEG T6G 2_v:(z) for all T 6 G. For some T 6 G, v:(y)‘2 * T w H w ) w ) w M )__ * v1(z -z '2 v1(z). ence v1(y 2_vl(z 2_v1( i - vl(m). The last equality comes from the induction hypothesis. This proves statement (3). 48 * Now conSider an arbitrary extension v of v . O v*(m) = V*(Mi) and since 2 6 Mi ‘we have for all v* * * * in Sj and for all T in G, v (zT) 2_v (m). Hence * v*(zT-z) 2,v (m). Since y dominates 2 'we get * T * * * . v (y -y) 2_v (m) for all v 6 Sj and T 6 G. USing * T . * T * * . (3) we get v1(y ) 2m1n[v1(z -2),v1(y)] 2v1(m). Since 1' * m = m and VlT ranges over all extensions of v 'we 0 * * have v (y) 2.v (m) for all v* extending v0. This finishes the proof of (2). The theorem follows now easily. Namely, by Theorem (4.2) and the assumption that mA is a reduction of M , 'we have e(A) = e(Mo) = e(mA). By (1), e(mA) = e(mAn). By (2), together with Theorem (4.2): e(mAn) = e(Mh) = e(An). Since An.= A.x ‘we have e(A) = e(Ax) as desired. Definition (4.4). A system of parameters xl,...,xd of a local ring (A,m) is said to be transversal if e((xl,...,xd)A) = e(m). Corollary (4.5). Let k be as in (4.3). Let A be an analytic integral domain over k. Let xl,...,xd be a (strongly) separating and transversal system of para- meters. Denote by F* the least Galois extension of k([xl,...,xd]) which contains 6(A). Suppose that char(k) and. [F* :k({x1,...,xd])] are relatively prime. Then e(A) = e(Ax). 49 ‘ggggf: The statement follows from (4.3) if we can show that (xl,...,xd)A is a reduction Of the maximal ideal in A. This follows from [20], p.16, Theorem 3.2, if we can show that all minimal primes in the completion of A are of dimension d = dim(A). [14], p.188, Theorem (44.1) implies that A is analytically irreducible, that is, the completion is a domain. As shown in (2.10) we can always find a strongly separating system of parameters if k is algebraically closed and A is equidimensional. If k is infinite (which is Of course the case if k is algebraically closed), then there is a system of parameters in A which generates a reduction Of the maximal ideal, [17], p.356, Corollary to Theorem 2. It is not known to us, under what conditions we can find a system of parameters which satisfies both conditions. As in the algebroid case, the question remains Open, whether the multiplicity is preserved if the parameters do not generate a reduction of the maximal ideal (see the remark beneath Corollary 4.2 in [28], p.460). However, there definitely are non-trivial cases to which our theorem applies. To show this, is the purpose of the following example. Example (4.6). This is a ring satisfying all con- ditions of (4.5) and having non-trivial saturation. Let k be an algebraically closed, non-trivially valued field with char(k) > 3. We consider A = k[{X,Y]]/(Y34-Y24-X2 + 2XY). It is easy to check that A is a domain and 50 dim(A) = 1. We write x and y for X and Y modulo the relation. m = (x,y) is the maximal ideal and x is a system Of parameters since m2 E (x). Hence A=k[{x]][y] where y3+y2+2xy+x =0. (x) isa reduction of m since m2(x) = m3. One also checks that it is (strongly) separating. f(Z) = 234-224-2x24-x2 N is irreducible in k[{x]][Z] and hence in k({x])[Z]. Therefore [e(A) :k({x])] = [k({x])(y) :k({x])] = deg(f(Z)) = 3. F* is the splitting field of f(Z) over k({x]), hence char(A) and [F* :k({x])] are relatively prime. That A is not saturated can be seen as follows: If it were saturated then A were an .Arf ring since dim(A) = l and A is Cohen-Macaulay, [11], p.682, Corollary 5.3. Then we would have dimA/m(m/m2) = e(A), by [11], p.661, 2+2xr+x2) g (x,y)2 and Theorem 2.2. Since (Y34VY k[{x,Y]] is a regular local ring of dimension 2, we have dimA/m(m/m2) = 2. To calculate the multiplicity of A we use [30], p.299, Corollary 1 and get e(xA) = 3. Since (x) is a reduction Of m.= (x,y) we have e(A) = 3. This contradiction shows that A is not saturated. CHAPTER V RELATIVE LIPSCHITZ-SATURATION An alternative definition of saturation was develOped by Pham and Teissier [19]. We repeat it here as given in [12], p.792. Definition (Ell). Let R be a ring and let 9 :A.4 B be a homomorphism of R—algebras. The Lipschitz- * saturation AB R Of A in B, relative to R.4 A.4 B I * O O is the set AB,R = {x 6 le ORIJ-l GR}: is integral over the kernal of the canonical map B ®R B44 B GA B]. (This kernal is generated by all the elements g(a) O l-l 3 g(a), a 6 A). A is saturated in B relative to R.4 A 4 B if A = 9(A). For properties of this saturation, see [12]. In the case Of an analytic ring A = k[{xl,...,xd]][y1,...,yn] we have k[{xl,...,xd]] 4 A.4 A, A is the integral closure Of A in e(A). We will just write A; for * A , where x = (xl,...,xd). A.k[{Xl] 51 52 Theorem (Ball. Let k be an algebraically closed, complete and non-trivially valued field, A an equi— dimensional and reduced analytic ring over k, and x1,...,xd a strongly separating system Of parameters Of ~ * C . A. Then, Ax __AX Proof: The existence Of Ax is proven in Theorem (2.12). In the proof of Lemma (2.3) we have shown that under the present hypothesis, [ L) p] n p 6AssA(O) k[{x1,...,xd}] = (O). k[{xl,...,xd]] is integrally closed in its quotient field. Hence the conditions for Corollary (4.2) of [12], p.807 are fulfilled and we get the statement of the theorem. proposition (1.4) of [12], p.797 shows that A .. A; is a radicial extension. For this reason we can translate some of our earlier results directly to the case of the Lipschitz—saturation. Because of Theorem (5.2) these results are actually stronger. Notice also, that no separability conditions are required in this case. Corolla 5.3 . Let k be an algebraically closed, complete and non-trivially valued field. Let A be an equidimensional and reduced analytic ring over k. The Lipschitz—saturation of A with respect to any system Of parameters exists, is again an analytic ring over k and dim(A) = dim(A;). 53 Proof: The same as for Theorem (2.12) except that we use Proposition (1.4) of [12], p.797 instead of [25], (4.1). Corollary (5.4). Let k and A be as in (5.3). If V and w are analytic set germs associated with A and A; respectively, then V and W are topologically equivalent. For apprOpriate representations, the homeomorphism is induced by the natural projection Of the ambient spaces. Proof: The same proof as for (3.5). Again PrOposition (1.4) Of [12], p.797 replaces Theorem (4.1) of [25]. we do however not know if an analogous statement to Theorem (4.3) holds. BIBLIOGRAPHY 10. ll. 12. BIBLIOGRAPHY S.S. Abhyankar, Local Analytic Geometry, Academic Press, New YOrk and London, 1964. 5.8. Abhyankar and M. van der Put, Homomorphisms of analytic rings, J. reine angew. Math. 242 (1970), 26-60. M.F. Atiyah and I.G. 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