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'Is Regular.2 We Intend VOL. 41, 1955 MATHEMATICS: P. SAMUEL 647 * This work was carried out at the University of Wisconsin Naval Research Laboratory under Contract No. AT(11-1)-298 with the United States Atomic Energy Commission. 'Some authors, including MacRoberts, Magnus, and Oberhettinger, define Ferrers' associated Legendre function of the first kind as (1)' times the definition given in eqt (2). Our definition, however, agrees with Jahnke and Emde and with Whittaker and Watspn. 2 H. Margenau and G. M. Murphy, Mathematics of Physics and Chenmistry (New York: D. Van Nostrand Co., 1943), p. 103, eq. (3-45). 3A. Erdelyi, "Higher Transcendental Functions" (Bateman Manuscript Project) (McGraw- Hill, 1953) 1, 172, eq. (27). See also T. M. MacRoberts, Quart. J. Math, (Oxford ser.), 11, 95 and 96, 1940; T. M. MacRoberts, Spherical Harmonics (New York: Dover Publications, Inc., 1947), p. 367, ex. 86(ii), together with p. 143, ex. 10, and p. 125, eq. (14). SIMPLE SUBVARIETIES OF ANALYTIC VARIETIES* BY PIERRE SAMUEL HARVARD UNIVERSITY AND UNIVERSITY OF CLERMONT-FERRAND Communicated by Oscar Zariski, June 2, 1955 1. Inttoduction.-Let k be a field. We make the simplifying assumption that its characteristic p is not 0. To every prime ideal p in the power series ring A = kI[[X, . ., X.]] we associate an object V(p) called an analytic variety over U.' The dimension of the local ring A/p is called the dimension of V(p).2 If p c p17 we say that V(p') is a subvariety of V(p). A subvariety V(p') of an analytic variety V(p) is called simple if its local ring (A/A),yi,,= is' regular.2 We intend to show that Zariski's generalized Jacobian criterion for simple subvarieties of algebraic varieties may be extended to analytic varieties. The proof closely follows the proof of Zariski's criterion given in a recent book of ours.3 2. Notions about Derivations and Differentials.-Let A be any, commutative ring (not necessarily with unit). One usually considers derivations of A with values in some overring of A. It is convenient to have here the more general notion of a derivation of A taking its values in an A-module M: such a derivation is an additive mapping of A into M such that D(ab) = a * Db + b * Da. If f is a homomorphism of 31 into another A-module M', then f o D is also a derivation of A. One can construct an A-module Z(A) and a derivation d: A -i Z(A) which ure universal for the derivations of A,in the sense that, given any derivation D: A -M-)-, there exists a homomorphism f: Z(A) --iAI such that D = f o d. In fact, one asso- ciates, with every element x of A, a symbol mx, one constructs the free module F(A) = E A * me, and Z(A) is then the factor module of F(A) by the submodule eA generated by all the elements mX+y - m. - my and M, - X m, -y * mx. If we denote by dx the class of m, in S(A), then d is a derivation of A, which obviously enjoys the universal mapping property. We say that Z(A) is the module of dif- ferentials of A. More generally, given another ring B and a homomorphism h: A -X B, every B- module M is also an A-module, and we can consider derivations of A into B-modules (here we have D(aa')- h(a)Da' + h(a')Da). There still exists a universal B-module B(R, A) for thiese derivations: the factor module of the free module F(B, A) - Downloaded by guest on October 2, 2021 648 MATHEMATICS: P. SAMUEL PROC. N. A. S. E B * mi by the submodule R(B, A) generated by all the elements m+y - m - eA mY, m, .-h(x) my - h(y) * mx. We still denote by dx (or dB, AX, or dAx to avoid confusion) the class of mx in T(B, A). The B-module Z(B, A) is isomorphic to the tensor product B ®AZ(A) (the structure of A-module of B being defined by h). We have Z(A) = TD(A, A). If we have four rings A, A', B, B' with commuting homomorphisms h A---*B lu VIv lh l A MO--B ' the mapping Ebi* ma, v(b) *mu(a,) (ai e A, bi e B) of F(B, A) into F(B', A') i i carries R(B, A) into R(B', A'). We thus have a mapping Z(B, A) -- ZD(B', A'), which is a homomorphism for their structures of B-modules (the structure of B- module of T(B', A') being defined by v). In particular, a homomorphism of A into A' defines a homomorphism of Z(A) into Z(A'). Let p be an ideal in A. From the homomorphisms p -- A -- A/p - 0, we deduce homomorphisms Z(A/p, p) A (A/p, A) Z(A/p, A/p) -O 0 (notice that p operates trivially on A/p). LEMMA 1. The preceding sequence is exact. Actually, the only point which is not absolutely obvious is the fact that every element co of the kernel of t is in the image of s. Since t(w) = 0, we see, by "lifting" w in the free module F(A/p, A), that X is a linear combination of elements like da-da'-da", where a = a' + a" (p), or da-a'da"-a'da', where a = a'a" (p); as all such elements are of the form dz with z in p, w is in the image of s. QED. Simple examples-e.g., the existence of a nontrivial derivation of 2Z into Z/(2)- show that s need not be a monomorphism. In other words, Z is a right exact functor (in its second variable), but it is not left exact. It could be interesting to study its satellites. If, instead of rings, we consider algebras over a fixed ring k, the consideration of the derivations which are k-linear leads, in a similar way, to the definition of rela- tive differential modules Tk(B, A). Analogous property of right exactness. 3. Statement of the Theorem and Reduction to a Lemma.-Every derivation of A = k [[X1, Ia Xn]] is trivial on AP = kP[[X1P, . , XP] ]. If (zj) denotes a p- basis of k (the finite products II zj'(i)I where 0 < n(j) < p - 1, forming a linear basis of k over kP), then A is a "topologically" free module over AP, with the finite products II Xiz'/Z (O < n(i), n(j) < p - 1) as basic elements.4 It follows that t.4 every derivation of A (with values in any A-module) is a linear combination of the derivations Di, D/' defined by Di(Xie) = bii, Di(zj) = 0, Dj'(X1) = 0, Dj'(zj,) = bjj, (consequently, every such derivation is continuous for the local ring topology of A, a circumstance which does not happen in characteristic 0). In other words, O(A, A) is a "topologically" free A-module, freely generated by the elements dXi, dzjC4 n tpt Collsider Llow twro prime ideals p, q of A such that q c p; denote b~y d and c the Downloaded by guest on October 2, 2021 VOL.VMA41, 1955 THEMA TICS: P. SAMUEL 649 dimensions of V(p) and V(q), and by xi thep-residue of Xi. Let (Fa) be any basis of q THEOREM. The following assertions are equivalent: 1. V(p) is a simple subvariety of V(q). 2. The Jacobian matrix ((DiFa) (x), (Dj'F,,) (x)) has rank n - e. If the complete local ring k [ [x]] = A/p is separable over k,5 then assertion 1 is equiva- lent to 3. The matrix ((DiFa) (x)) has rank n - e. By definition, relation 1 means that A8/qA, is a regular local ring (of dimension e - d). Since A. is a regular local ring of dimension n - d,6 this means that qA, is generated by n -e elements of some regular system of parameters of A,.7 If we set R = A8, m = pe7; this is equivalent to saying that, in the (n - d)-dimen- sional vector space rn/min2 over R/m,; the subspace (Rq + M2)/m2 generated by the classes of the Fa's has dimension n -e. (Notice that this subspace has, at any rate, a dimension <n - e, since R/Rq has dimension e - d.) Now let (u1, . ., un-d) be a regular system of parameters of R. Any derivation D of m into R/m is obviously trivial on m2 and is uniquely determined by the knowledge of the values D(ui) (which may be arbitrarily selected in R/m): if x - E ajui e m, then Dx = E aiDui (di = m-residue of ar). Thus Z(R/m, m) is an i i (n - d)-dimensional vector space over R/m, generated by the elements dmui. The formula dm(E ajui) = E id..i shows that Z(R/m, m) is canonically iso- morphic to m/m2. In view of this isomorphism, we have to express the fact that the rank of the sys- tem of vectors (dmFa) in 4(R/m, m) is n - e. We have a homomorphism V: Z(R/m, m) -- )(R/m, R). Since Z(A, A) is freely generated by the.elements dXi, dzj, Z(A/p, A) and Z(R/m, A) are also freely generated by these elements.
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