Intersection theory in an equicharacteristic regular local ring and the relative intersection theory

Abstract

Using the intersection theory of the notes of Serre, Algebre locale. Multipliciues, the valuation theoretic formula for a hypersurface is given, and it is shown that transversality is equivalent to intersection multiplicity one. The intersection multiplicity of Algebre locale. Multiplicites is computed for two algebraic varieties over an arbitrary field and compared to the intersection number of Weil's Foundations of algebraic geometry. The intersection theory and the notation developed in the notes of Serre, Alge'bre locale. Multiplicites [31, will be used. Let A be a regular noetherian ring which is locally equicharacteristic. Let Z be the group of cycles of A, the free abelian group generated by the prime ideals of A. If p is a prime ideal of A, p also denotes the cycle. If M is a noetherian A module and if Pl, p,p are the isolated prime ideals of M, let Z(M)= _i=l,-*,s tAlpt(M)pi c Z. Ifp and q are two prime ideals of A which intersect properly at another prime ideal m of A, that is p + ? c m and height pAm + height qAm=height (p+q)Am=height mAin, define as in [3, Chapter V], i(p * q, m)=XAm(Arn/pA m, Am/qAm). Extending i bilinearly to cycles which intersect properly, i(Z(M) Z(N), m) =XAm (Mm, Nm). Let Pl, , p, be distinct prime ideals of A, let ql, q, be distinct prime ideals of A, and let ai and bj be nonzero integers for i= 1, *, s and j=1, *, t. The cycles a1p?+ * +a,p, and b1ql+ ? -+b1q, intersect properly if for each pair i,j the prime idealsp, and q, intersect properly at each isolated prime ideal mi,k of pi+qj. If this holds, define (alp, + + asp,) (blql + + btqt) = 2 a,b1i(p< * qj, mijk)riJk ijk which is again a cycle. Associativity and the projection formula are given in [3, Chapter V, C]. Received by the editors June 25, 1971 and, in revised form, December 21, 1971. A MS 1970 subject classifications. Primary 13 H 15, 14C 15.