On the structure and ideal theory of complete local rings

Introduction. The concept of a local ring was introduced by Krull 7 (1), who defined such a ring as a commutative ring 9 in which every ideal has a finite basis and in which the set m of all non-units is an ideal, necessarily maximal. He proved that the intersection of all the powers of m is the zero ideal. If the powers of m are introduced as a system of neighborhoods of zero, then 3 thus becomes a topological ring, in which the usual topological notions -such as that of regular sequence (2) -may be defined. The local ring dt is called complete if every regular sequence has a limit. Let m = (mi, 2, , "*) and assume that no element of this basis may be omitted. If the ideals (i, u2, , Ui), i = l, 2, , n, are all prime, 9 is said to be a regular local ring (2) of dimension. It was conjectured by Krull 7, p. 219 that a complete regular local ring 9 of dimension ra whose characteristic equals that of its residue field 9/tn is isomorphic to the ring of formal power series in ra variables with coefficients in this field. If on the other hand the characteristics are different, so that the characteristic of 9 is zero and that of K/m is a prime number p, then 9 cannot have this structure. In this case we note that p must be contained in m. Krull then conjectured that if 9 is unramified (that is, if p is not contained in m2), then 3 is uniquely determined by its residue field and its dimension. He conjectured finally that every complete local ring is a homomorphic image of a complete regular local ring. These conjectures are proved in 4-7, in particular, in Theorem 15 and its corollaries. Actually the basic result, to the proof of which is devoted Part II (4-6), is the one concerning arbitrary (that is, not necessarily regular) complete local rings. It is proved (Theorems 9 and 12) that every complete local ring 9 is a homomorphic image of a complete regular local ring of a specific type, namely, the ring of all power series in a certain number of variables with coefficients taken from a field or from a valuation ring of a certain simple kind. This follows easily as soon as it is shown that a suitable "coefficient domain" can be imbedded in 9? , and the burden of the proof of Presented to the Society, December 27, 1942; received by the editors July 14, 1945. (l) Numbers in brackets refer to the Bibliography at the end of the paper. (!) This is equivalent to Krull's definition of a p-Reihenring, as given in 7 ; the equivalence is proved in Theorem 14 and its corollary. The term "regular local ring" was introduced by Chevalley l.