Analytical Irreducibility of Normal Varieties Source ... 996/Zariski_Analytic Normality.pdf ·...

Mathematics Department, Princeton University

Analytical Irreducibility of Normal Varieties

Author(s): Oscar Zariski

Source: Annals of Mathematics, Second Series, Vol. 49, No. 2 (Apr., 1948), pp. 352-361

Published by: Mathematics Department, Princeton University

Stable URL: https://www.jstor.org/stable/1969284

Accessed: 25-01-2020 03:13 UTC

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide

range of content in a trusted digital archive. We use information technology and tools to increase productivity and

facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

https://about.jstor.org/terms

Mathematics Department, Princeton University is collaborating with JSTOR to digitize,

preserve and extend access to Annals of Mathematics

This content downloaded from 129.237.35.237 on Sat, 25 Jan 2020 03:13:25 UTCAll use subject to https://about.jstor.org/terms

ANNALS OF MATHEMATICS

Vol. 49, No. 2, April, 1948

ANALYTICAL IRREDUCIBILITY OF NORMAL VARIETIES

BY OSCAR ZARISKI

(Received August 11, 1947)

1. Introductory concepts

By a local domain we mean an integral domain which is at the same time a local ring in the sense of Krull [4]. If m is the ideal of non-units in a local domain o and if o* denotes the completion of o (with respect to the powers of m), we say that o is analytically unramified if the zero ideal in o* is an intersection of prime ideals. In other words: o is analytically unramified if o* has no nilpotent elements.

If , is a prime ideal in an arbitrary local ring o, we say that p is analytically

unramified if the local domain o/p is analytically unramified. It is well known that if o* is the completion of o then o*/o*p is the completion of o/P (Chevalley rl], Proposition 5). It follows that a prime ideal p in a local ring o is analytically unramified if and only if the extended ideal o*p in the completion o* of o is an intersection of prime ideals.

The following theorem has been conjectured by the author and proved by Chevalley ([2], Lemma 9 on p. 9, last sentence, and Theorem 1 on p. 11): The local ring of a point P of an irreducible algebraic variety V is analytically unramifiedt It follows that any prime ideal , in such a ring is also analytically unramified,

because p defines an irreducible subvariety W of V, and the residue class ring o/p is the local ring of the point P, this point now being regarded as a point of W. Note the following special case: V is the affine n-space over k, and P is the origin. In this case the completion of the local ring of the point P is the ring k (x ) of formal power series in n independent variables xi, X2, ... ,Xn, with coef- ficients in k, and therefore it follows that every prime ideal in the polynomial ring k[x] splits into prime ideals in the power series ring k (x). In informal geometric language this result signifies that an irreducible algebraic variety V can de- compose in the neighborhood of a point P only into "simple" analytical branches (i.e., none of the branches has to be "counted" more than once). At any rate, it is true in the complex domain that the analytical reducibility of V in the neighborhood of a point can be no worse ideal-theoretically than it is set-theoretically.

We say that a local domain is analytically irreducible if its completion has no zero divisors, and that a variety V is analytically irreducible at a point P of V if the local ring of P is analytically irreducible. We recall that V is said to be locally normal at P if the local ring of P is integrally closed. The object of this paper is to prove the following theorem:

If an irreducible algebraic variety V is locally normal at a point P, then it is analytically irreducible at P.

In the course of the proof of this Theorem we shall arrive incidentally at another proof of Chevalley's result.

352


NORMAL VARIETIES 353

Our theorem is to be compared with another result concerning normal varieties and proved by the author elsewhere ([7], Definition 4, Theorem 8(A) and Theorem 10, pp. 512-514). We have shown namely that if V is locally normal at P and if a birational transformation of V into another variety V' sends P into a finite set of points of the variety V', then this set consists necessarily of a single point. We saw in this result a strong indication of the analytical irreducibility of normal varieties. For if a variety V consists of s branches in the neighborhood of a point P, then one would expect that these s branches could be separated by a suitable birational transformation. Such a transformation would then replace P by s distinct points.

2. Some auxiliary lemmas

The first two of the following lemmas refer to an arbitrary local ring o and its completion o*. The ideals of non-units in these two rings are denoted by m and m* respectively.

LEMMA 1. If l is an ideal in o and b is any element of o, then oil: o*b = o*(2l: b).

PROOF. It is sufficient to prove the inclusion o*l: o*b C o*(W: b). Let u be any element of o*i o*b, u = lim ui, ui e o. We have

us eu + m*i+l, uib Eub + bm*i+l, i.e., uib e 0* S + o*bmi'l.

Therefore uib e l + bmi+ll ui e W: b + mi+l, and hence u e o* (2l b) + m* for all i. This implies that u e o*(W: b), as asserted.

LEMMA 2. If l and e3 are ideals in o and if a: 3 = Xl, then oil o*23 = oil. PROOF. Let pi, P2, --- , p, be those prime ideals of l which are not contained

in any other prime ideal of l (i.e., they are maximal with respect to the property of being prime ideals of 2l), and let aii be an element of pi which is not in pi (i, j = 1, 2, ... , g, i # y). From W: e = l it follows that e3 is not contained in any of the prime ideals of W. Hence we can find an element ati which is in e3 but not in pui(i =1, 2, - * * , g). We set

bj =aja23 ... agj, b = bi + b2 + + bg.

Then b e 23, b 4 pi, i = 1, 2, , g, and therefore Wl: ob = Wl. It follows then from Lemma 1 that o*i: o*b = oi2l, and since o*W: o*O C o*i: o*b, the inclusion o*9: o*d C o*l follows, and that completes the proof of the lemma.

In the remaining lemmas we assume that o is a local domain which is integrally closed in its quotient field. Lemmas 3-7 refer to a fixed minimal prime ideal p in o,

where we assume that p is analytically unramified, whence

(1) 0 , = P n p* n n pX each 4* being a prime ideal in o*. The lemmas 3-7 concern certain properties

' Here we are making use of the relation o f n o = e which holds for any ideal e in o (Krull [4], Theorem 15; also Chevalley [1], Proposition 5).


354 OSCAR ZARISKI

of these h prime ideals pi. Let p* be one these prime ideals, say * = P. Let $* be one of the prime ideals of the zero ideal in o* such that $* C ,. We set

(2) U = o*/$**, $ = */1*-

LEMMA 3. The quotient ring R = QV is a discrete valuation ring of rank 1. If X is an element of o such that X e p, Xco (2) and if W is the $*-residue of w, then RCx is the ideal of non-units in R.

PROOF. Let a* be an element of p* nf p* ... n p which does not belong to P1,and let b be an element of ow: p, not in p. If we set c* = a*b, then c* 4 p since b is not in P*.2 We have p*a* C op, p*c* C o*pb C O*w. Hence if a denotes the $*-residue of c*, then $a C Qic, and therefore R$ C Rx, since c 4 $. This shows that R$, the ideal of non-units in R, is the principal ideal Rwo, and this completes the proof of the lemma.

LEMMA 4. For any integer n the ideal $* belongs to the symbolic power *(n). PROOF. Since $* C p, it follows from the preceding proof that $*c* C Ow*.

Therefore if a* is any element of $* then a*c* = al c e $* where a,' is an element of o*. Since co is in o, it is not a zero divisor in o* (see footnote 2), and therefore

S 4$*. Consequently ac e $*, a c E 0*co, and this shows that $ (C*)2 c O*W2. In a similar fashion it can be shown that $* (C*)n C o*,n C p*(n), and this completes the proof, since c* o p*.

LEMMA 5. Every primary ideal of p* is a symbolic power of p*. PROOF. By Lemma 4 every primary ideal of P* contains the ideal $*. Hence

there is (1, 1) correspondence between the primary ideals of p* in 0* and the primary ideals of $ in U. But the latter primary ideals are all symbolic powers of $, since the quotient ring of $ is a discrete valuation ring of rank 1 (Lemma 3). This completes the proof.

LEMMA 6. Each of the h prime ideals Pi contains one and only one of the prime ideals of the zero ideal in o*.

PROOF. Since n,_$(n) = (0) and since, by Lemma 4, *(n) is the full inverse image of $ under the homomorphism o* , Q, it follows that nln p*(n) = p, and so ?* is uniquely determined by P*. Since p* can be any of the h ideals p,, the lemma follows.

LEMMA 7. The relation o*P(n) = fl=14(i ) holds for any interger n.

PROOF. It is sufficient to prove the inclusion o*ip(n) D ml1p*1(n) Let a* be any element of the ideal on the right. We have a* e o*p, and hence a* b = aico, where b is the element which was introduced in the proof of Lemma 3. Since X 4 p(2))

2 It has been proved by Chevalley ([1], Proposition 6) that if o is a local ring and o* is the completion of o, then any prime ideal of the zero ideal of o* contracts in 0 to a prime ideal of the zero ideal of o. If we apply this result to the local domain o/p, where p is a prime ideal of o, and if we take into account that o*/o*p is the completion of o/p, we conclude that any prime ideal of o*p contracts in o to p. In particular, we have in our present case: fl no= P.



it follows from this same lemma that al e nf- Ot (np1)i , and hence aib = al w, i.e. a*b2 = ala,2. In this fashion we find, after n steps, the relation a*b=l where an is some element of o*. Hence a* e o*wco: o*bn c O*p(n) o*bf. Since b 4 p, it follows from Lemma 2 that a* e o*p(n), and this completes the proof of the lemma.

LEMMA 8. Let f = fl I qi be an ideal in o, where qj., q2, X , qn are primary ideals belonging to distinct minimal prime ideals pi, , P2 * P in o. If each of the n ideals pi is analytically unramified, then o*2 = nflno*qi.

PROOF. It is sufficient to prove the inclusion o*e D nfn 1o*qi. Let a* be any element of the ideal on the right. For each i we fix an element coi which

is in pi but not in p ,2) and not in p j, for j F i. Moreover, let bi be an element of ocoi pi, not in pj, j = 1,2, * * , n. We know from the proof of Lemma 7 that if pi is the exponent of qi (so that qi is then necessarily the symbolic power (Pi)), then a*bpl is a multiple of otol, say a*bPl = ac %o , a- E 0*. Since woi pj j # 1, it follows that a* E nf2o*qi , and hence, by a similar argument we have a~'b22 = a2*coP2, where aa2 E ftl3o*qi. Ultimately we find a*bplbP2 ... bpn = * Pi P2 Pn E0P H= . atncwli 922 cO7nn 0*2. Hence if we set b = bVbb2 ... bnt then a* e o*f: o*b,

whence, by Lemma 1, a* E o*(I: b), i.e., a* e o*f, since b pi , i = 1, 2, ... , n. This completes the proof.

LEMMA 9. If there exists an element t $ 0 in o such that all the prime ideals of

the principal ideal o * t are analytically unramified, then o* has no nilpotent elements. PROOF. Let pi, , 2, * * *, Pn be.the prime ideals of o * t and let us assume that

all these ideals are analytically unramified. They are also minimal primes in o since o is integrally closed. Let opx = pi, nf i2n f * f n , i = 1, 2, * n, and let 1*, 2 . *, be the prime ideals of the zero ideal in o* which contain at least one of the prime ideals pij (i = 1, 2, ** , n; j = 1, 2, * * , vi). We know from Lemma 6 that each of the prime ideals p* contains one and only one of the s ideals 1 and from the proof of that lemma it follows that

14l*nf2*n fl ... nf * = nflnfl7 n{P l (j) n ... f (pi)}

or, in view of Lemma 7,

1n2 ... nun* i n c? ie w)j or finally, by Lemma 8,

s1 n s* n *.*.* n Ad = fli o* {ln pif ) n ... n f } C 0*t i)X

where lim v) = + a). Hence 91 n fl* n * ... n 13f * C m*1, for all integers 1. Consequently

(3) 3 n w2 n ... n *=(0)) q.e. d.

3 We have alW E pl*(ia) hence passing to the ?*-residues c,, c we find a, o e 13(n). By Lemma 3 it then follows that &a E $3(n-1), whence al e Wn-1). Similarly it is shown that

al = p(fli1, 2, * ,h.


356 OSCAR ZARISKI

3. Application to algebraic varieties

As a first application we shall show that from Lemma 9 it is possible to derive Chevalley's theorem stated in section 1 of this paper. We first observe that it is sufficient to prove that theorem under the assumption that V is locally normal at P. For suppose that the theorem has already been proved under this assumption and suppose that we are dealing with a variety V and a point P of V such that V is not locally normal at P. Then we pass to a derived normal model V' of V, and we consider the points Pi, iN, , IN which correspond on V' to the point P. Let o denote the local ring of P and let 5 denote the intersection of the local rings of the points P'i. Then U is a semilocal ring in the sense of Chev- alley [1] and is a finite o-module (5 is in fact the integral closure of o in the function field 'J(V) of V; see [7], p. 511). The completion of 5 contains the completion o* of o as a subring, in fact is a finite o*-module ([1], Proposition 7, p. 699). To show that o is analytically unramified, it is therefore sufficient to show that the completion of o has no nilpotent elements. Now the completion of 5 is the direct sum of the completions of the s local rings of the points P' ([1], Proposition 8, p. 700), and since V' is locally normal at each of the points P' it follows, from our hypothesis, that the local rings of points P' are all analytically unramified. Hence the completion of 5 has no nilpotent elements, since the rings of which it is a direct sum have no nilpotent elements.

We shall now proceed by induction with respect to the dimension r of V. If r = 1 and if the curve V is locally normal at P, then P is a simple point of V, the local ring of P is a regular ring ([8], p. 19), in fact a valuation ring, and the completion of this ring is itself a regular ring ([4]) which therefore has no zero divisors at all. Having shown this for normal curves, it follows from the preceding observation that Chevalley's theorem is true for algebraic curves. Now let us assume that this theorem is true for algebraic varieties of dimension less than r, and let V be an r-dimensional irreducible variety which is locally normal at a given point P. We have to show that the local ring o of the point P is analytically unramified. By our induction hypothesis we have that every prime ideal in o is analytically unramified (compare with section 1), and in particular every minimal prime ideal in o is analytically unramified. Hence the assumption in Lemma 9 is automatically satisfied for any element t in o, t - 0, and since o is integrally closed, it follows by that lemma that o* has no nilpotent elements.

Before proceeding to the proof of the analytical irreducibility of normal varieties, we shall make a few geometric comments about some of the lemmas proved in the preceding section. The local ring o is now the local ring of the point P of an r-dimensional irreducible variety V, and V is locally normal at P. Let (3) represent the decomposition of the zero ideal of o*. In that case the variety V consists, locally at P, of s analytical branches M1, M2, ***, M, each Mi being an analytical manifold. For each prime ideal $, of the zero ideal we define as in (2) the domain Qi = o*/$!3. Then Qi is the ring of holo- morphic functions on the analytical manifold M,. Now let p be a minimal



prime in o and let (1) be its decomposition in o*. Then p represents an (r - 1)- dimensional subvariety WV of V which contains the point P and which decom-

poses, locally at P, into h analytical manifolds, N1, N2, , , Nh . Lemma 6 signifies that each of the h analytical (r - 1)-dimensional branches Nj of W lies on (one and) only one of the analytical branches Mi of V. This result was to be expected from an intuitive geometric viewpoint, since the intersection of two

distinct analytical branches M1i and Mj of V is part of the singular manifold of V and since, on the other hand, the singular manifold of V cannot pos'sss an (r - 1)-dimensional component at the point P where V is locally normal. Naturally this entire geometric picture which we are painting has real sig- nificance only in the classical case, because in the abstract case an analytical manifold is not a point set at all (there is no conceivable incidence relation be-

tween Mi and points of the affine ambient space of V, except that we may say that the origin P of the branch Mi is on Mi). At any rate is is now becoming clear why the hypothesis that o is integrally closed was a priori necessary in the proof of Lemma 6. As to Lemma 3, we shall only make the following observation: the fact that the quotient ring R is a valuation ring is to be interpreted

geometrically in the sense that if Nj lies on Mi then Nj is not singular for Mi . This interpretation is again in connection with the fact that the singular manifold

of V must be, locally at P, of dimension < r. The remarks just made disre- gard our final result (which we shall now proceed to prove) that actually V is analytically irreducible at P, whence s = 1, and o* is an integral domain. It is still an open question whether o* is integrally closed. In other words: if V is locally normal at P, is V also normal as an analytical manifold? We now proceed to the proof of the theorem on the analytical irreducibility of normal varieties stated in section 1. Let (3) be the decomposition of the zero ideal of o* into prime ideals. We shall prove in the next section the following relations:

(4) ($* + $) n o (o), if i j. Assuming for the moment relation (4), we now show that the assumption that s > 1 leads to a contradiction. Let us then assume that s > 1 and let uj be an element of( + ) n o, uj O, j = 2, 3, ,s. If we set u= U2U3 **u we can write u in the form

u = v* + w*, where v* Ef$* and w* e 2*=* n $ n .. . Let ou = pP1) nf pP2)n ...n p (Pn) where the pi are minimal prime ideals in o' and let o"pi = Pi*l n pf*l n fl Div (Since we are dealing with the local ring of a point of an algebraic variety V, we know that every prime ideal of 0 is analytically unramified.) All the lemmas of section 2 are applicable. In particular, we find by lemmas 7 and 8 that

0* u-=f1fl~nZ w. (P Each of the ideals , icontains one of the ideals $,. If pWj D $ ,,, then by Lemma


358 OSCAR ZARISKI

4 any symbolic power of p*i{ contains $ and in particular p D $3*. If ,u = 1 then v* e $* Cpp*j("$ ,and since also u e ,{ ; it follows that w* =

u - vi) If 1A / 1, then w* E (P$),and hence we have again v* e p*ii Therefore in either case we find that both v* and w* are in p*,(" , for all i and

j. Hence both v* and w* belong to the ideal 0 u, say v* = valu and w* = wJ*u. From Vi + wi = 1 it follows that v* and w* cannot both belong to m*, and hence at least one of these two elements must be a unit in o*. On the other hand we have yr u e $13, w u e 2*, and u does not belong to any of the prime ideals $ since u $ 0 is an element of o and is therefore not a zero divisor in o*. Hence v* e $* and wr e 9V*, so that neither va nor wi can be a unit in o*. Thus the assumption s > 1 leads to a contradiction, and this proves our theorem.

4. Proof of relation (4)

We first point out the geometric meaning of relation (4). The two prime

ideals $* and $*3 represent two distinct analytical branches Mi and Mj of V through the point P. The sum $ + $*I represents the intersection L of these two branches. The left hand member of (4) is an ideal in the local ring of P whose zero manifold is the least algebraic subvariety W of V which contains the analytical manifold L. Therefore relation (4) is equivalent to the assertion that W is not the entire variety V. The geometric facts which lie behind this assertion are the following: a) as an intersection of two analytical branches the manifold L must belong to the singular manifold of V; b) therefore also W belongs to the singular manifold of V; c) the singular manifold of V is a proper subvariety of V. The formal proof shall now be given.

Let the ambient linear space S of V be of dimension n and let q be the prime

ideal of V in the polynomial ring k[x] of the n variables xl, k2, * ... X n. Let R denote the local ring Q(P/S) of the point P regarded as a point of S, and let R* be the completion of R. The decomposition (3) of the zero ideal in o* implies a corresponding decomposition of the ideal R*q into s prime ideals:

(5) ~~~~R*q = q1 q* n ..n q,*. The ring R* is a complete local domain of dimension n. Each of the prime ideals q, is of dimension r, the same as the dimension of the prime R-ideal R- q ([2], Theorem 1). Relation (4) is equivalent to the following relation:

(6) (q + q!) nR i RRq.

Let Z* be any prime ideal of q* + q7. To prove (5) we have to show that for any such prime ideal l* we have:

(7) ?2* nR 0 Rq.

4Note that u is not a zero divisor in o*, since u e o. Therefore the relation (v* + w*)u = u implies vl + w4 = 1.



Let p be the dimension of DC*. Since D* is a proper divisor of qs* and qj ,it follows that p < r. We pass to the quotient ring Z = R* . This ring is a local ring of dimension n - p, and since the ring R* is a regular complete ring, it

follows that also Z) is a regular ring (Cohen [3], Theorem 20, p. 97). We consider the ideal Zi) q. By well known properties of quotient rings, the decom-

position of the ideal kD- q is obtained from the decomposition (5) of R*q by

replacing the prime ideals qi by their extensions -S. q '. These extensions are prime ideals. Moreover Z2. qs is the unit ideal in Z' if and only if ?X* does not contain qt, and two distinct prime ideals which are contained in I* give rise to distinct extensions. Since qi and q* are both contained in a*, it follows that Z'q is not a prime ideal. Another property of the ideal Z'q which we shall have to use is that its prime ideals are all of dimension r - p. This follows from the fact that the ideals q* are of dimension r and the ideal I* of which SD is the quotient ring is of dimension p.

Now let {fi(x), f2(x), ... , fN(x) I be a basis of the prime polynomial ideal q. These N polynomials will also form a basis of the ideal Ziq. We denote by 9M the ideal of non-units of the regular (n - p)-dimensional local ring Z. The

additive group /12 can be regarded as a vector space OR*, of dimension it -p over the field Z'/Wf ([8], p. 6). We now use the two properties of the ideal k q which have been derived above. Since this ideal is pure (r - p)-dimensional,

it follows that at most n - r of the polynomials fi(x) map, modulo 9)12, on independent vectors of OR*. Moreover, since the ideal Z?q is not prime, it follows that the set of vectors of Go* which correspond to the polynomials fi(x) contains actually less than n-r independent vectors ([3], corollary on p. 87). On the other hand, if we consider the local vector space OR = YR(V/S) of V in S (see [8], p. 6) we find that the set of vectors of 'Dt which correspond to the polynomials fi(x) contains exactly n-r independent vectors, since any variety V in S is simple for S. We shall now show that relation (7), and hence also relation (4), follows from this discrepancy between the dimensionalities of the vector spaces spanned by the

polynomials fi(x) in the vector spaces OR and IR* respectively. We first assume that the function field 9i(V) is separably generated over k. We

use the results of our paper [8] concerning the vector space O1t = DR(V/S) and the vector space M(V) of local V-differentials in S ([8], p. 25). Let ts be the q-residue of xi and let i be the D-residue of xi. Since the function field of V is separably generated over k , it follows that the Jacobian matrix aj(f, f2, * ,N) 49(X1 2X 2 , x*,n) is of rank n-r at x = t ([8], Theorem 7', p. 31). On the other hand, since the polynomials fi(x) span in 'JR* a space of dimension less than n-r, it follows a fortiori that any n-r rows in the Jacobian matrix a(f1(x),

f2(X) *... * f (X))/3(X1 x2 X ... * X1n) are linearly dependent over the field 9)I/9)2.5

6 A few words will suffice to explain this assertion. The partial derivations a/8x, have obviously the property of transforming into itself the quotient ring of any prime ideal in k[x]. In particular, these derivations transform R into itself. Moreover, if m is the ideal of non-units in R, then the partial derivatives of any element of m' are elements of m''.


360 OSCAR ZARISKI

Hence this matrix is of rank less than n-r. We conclude therefore that while

all the (n-r)-rowed minors of the matrix a(fi(x), f2(x), * - x , , x2, * *2 , xn) belong to A, at least one of these minors is not in q. This establishes (7) and hence also (4).

If %f(V) is not separably generated over k, the proof is the same except that

instead of the ordinary Jacobian matrix we must use the mixed Jacobian matrix introduced in our paper [8] on p. 38.

In conclusion we observe that in our proof of the analytical irreducibility of

normal varieties we have made use only of two special properties of local, integrally closed, domains which are true for local rings of points of normal varieties

and which are not known to be true in general. These two properties are the following: 1) the local domain o, and every prime ideal in o, is analytically

unramified; 2) if $, and $ are any two distinct prime ideals of the zero ideal in o*, then relation (4) holds. The first property is certainly false for general local domains, if we drop the condition that o is integrally closed ([6]; also [8], p. 24, where the ring o defined in (8) is easily seen to be analytically ramified), but it is possible that it holds for all integrally closed local domains. Also the extent

to which relation (4) is valid in the non-geometric case, is an unsolved question. On the answer to these questions depends the answer to the following general question: if o is an integrally closed local domain, is it true that the completion of

o is also an integral domain? It is trivial that the answer is affirmative if o is of dimension 1 (for o is then a discrete rank valuation ring).

ADDED IN PROOF. A related question is the following: if ois a local domain such that (a) its integral closure o is a finite o-module, is it true then that (b) o is analytically unramified? It is known (Krull [5]) that if o is of dimension 1 then (a) and (b) are equivalent. An affirmative answer to the first question would imply an affirmative answer to this second question, since it can be easily shown that under assumption (a) o is analytically unramified if and only if i is analytically unramified.

HARVARD UNIVERSITY

REFERENCES

[1] C. CHEVALLEY, On the theory of local rings, Ann. of Math., vol. 44 (1943), pp. 690-708. [2] C. CHEVALLEY, Intersections of algebraic and algebroid varieties, Trans., Amer. Math.

Soc., vol. 57 (1945), pp. 1-85.

From this it follows that each of the derivations i/axf has a unique extension in the complete ring R*. This extension will be denoted by the same symbol /ax, . By the same argument, the extended derivation a/axi in R* can be further extended to the quotient ring Z, and moreover, the partial derivatives of any element of TIP are elements of TP-'. Now consider any n - r of the polynomials f,(x), say f1(x), f2(x), - - - , fn-,(x). Since the corresponding vectors of 9)/9)2 are linearly dependent, there is a relation of the form: A*fi(x) + A f2(x) + + A*-,f7(x) e 9J2, where the AF are elements of ?Z, not all in WZ. Applying the derivation a axi and observing that the polynomials f j(x) belong to 9), we find that Alaf1(x)/axi + A2af2(x)/ ax + + AL, afn-r(x)/axi is in 9J, for i = 1, 2, * , n. Since the Al are not all zero mod5D, the assertion in the text follows.



[3] I. S. COHEN, On the structure and ideal theory of complete local rings, Trans., Amer. Math.

Soc., vol. 59 (1946), pp. 54-106. [4] W. KRULL, Dimensionstheorie in Stellenringen, J. Reine Angew. Math., vol. 179 (1938),

pp. 204-226.

[5] W. KRULL, Ein Satz uber primeire Integritdtsbereiche, Math. Ann., vol. 103 (1930), pp. 540-565.

[6] F. K. SCHMIDT, (Yber die Erhaltung der Kettensatze der Idealtheorie bei beliebigen endlichen

K6rpererweiterungen, Math. Zeit., vol. 41 (1936), pp. 443-450. [7] 0. ZARISKI, Foundations of a general theory of birational correspondences, Trans., Amer.

Math. Soc., vol. 53 (1943), pp. 490-542. [8] A. ZARISKI, The concept of a simple point of an abstract algebraic variety, Trans., Amer.

Math. Soc., vol. 61 (1947), pp. 1-52.


Analytical Irreducibility of Normal Varieties Source ... 996/Zariski_Analytic Normality.pdf ·...

Documents

Transcript of Analytical Irreducibility of Normal Varieties Source ... 996/Zariski_Analytic Normality.pdf ·...