Local varieties and asymptotic equivalence

Abstract

1. Let o be a local domain and let a be an ideal of o that is primary for the maximal ideal m of o. If a is an element that is superficial of degree s for a (in the sense of [6, p. -i2 ]) and a,, * * *, at is a basis for c8, let o(a, a)=o[a,/a, * * *, at/a]. If m' is a prime ideal of o(a, a) such that m'no=m, then the quotient ring of o(a, a) at m' is called an a-spot, and the totality of a-spots is called the variety V(a) of a. Let Q be the set of all valuations of the quotient field F of o that dominate o. It is shown that V(a) is a complete variety relative to Q in the sense that each element v of Q dominates a unique a-spot. If o is analytically normal it is shown that for each m-primary ideal a there is an integer g such that the integral closure (a0)a of av is such that all of its powers are integrally closed. Ideals with this property are called normal and (av)a is called a derived normal ideal of a and denoted by a0. If an ideal is normal, then its variety is normal and V(ag) is a normalization of V(a) in the sense that the integral closure in F of any spot P of V(a) is the intersection of a finite number of spots in V(a,) that dominate P. In ?3 it is shown that there is a 1: 1 correspondence between the algebraic points of the variety of the null-form ideal of a (defined over the residue field k of o) and the set of a-spots of maximal rank. Hence it is not surprising that V(a) should have properties analogous to those of varieties over a field. In ?4 some of these properties are elaborated. In particular, the usual relation of dominance between varieties V(a) and V(b) yields a partial order on the set of m-primary ideals that directs the set. If P and P' are a-spots such that P is a quotient ring of P' then we say that P' is a specialization of P. For each m-primary ideal a there is a finite set { Pl, P2, * * * , P8 } of a-spots such that each a-spot is a specialization of one of the Pi. An irreducible ideal is one whose spots are all specializations of a single spot P which is then called a general spot for the ideal. Finally, we show that the notion of asymptotic equivalence in the projective sense introduced by Samuel in [7] can be characterized in terms of local varieties. In fact, if a and b are asymptotically equivalent in the projective sense (in symbols: a T b) then V(a) = V(b), while if a and b are irreducible normal ideals such that V(a) = V(b) then it is shown that a T b.