Analytical Irreducibility of Normal Varieties

Abstract

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 coefficients 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 decompose 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.