A Special Case of Positivity (ii)

In this note we prove the following special case of Serre’s conjecture on Intersection Multiplicity: Let (R,m) be a regular local ring and let P , Q be two prime ideals such that (R/(P + Q)) < ∞, dimR/P + dimR/Q = dimR and dimension of Gm(R/P ) ⊗Gm(R) Gm(R/Q) < 2. Then χ(R/P, R/Q) ≥ em(R/P )em(R/Q); here em(T ) denotes the Hilbert-Samuel multiplicity for any finitely generated module T with respect to m. Let (R,m) denote a regular local ring of dimension n, essentially of finite type over a field L or a discrete valuation ring V . Let X = SpecR, W1 = Spec(R/P ), W2 = Spec(R/Q); P , Q are prime ideals of R. Assume that (R/(P+Q)) < ∞. Let π : X̃ → X be the blow-up ofX at {m}, E the exceptional divisor and η : E → {m} the induced map, i.e., η = π |π−1{m}. Since R is a regular local ring of dimension n, E = Pn−1 K , where K = R/m. Let W̃1, W̃2 denote the blow-ups of W1 and W2 at {m}. The exceptional divisors for W̃1 and W̃2 are W̃1∩E and W̃2∩E, respectively. For any finitely generated R-module T , let em(T ) denote the Hilbert multiplicity of T and let Gm(T ) denote the associated graded module ⊕∞ n=0 m T/mT . In this note we intend to prove the following (with notations as above): Corollary to the Theorem. Suppose that W̃1 ∩ W̃2 ∩ E is either a finite set of points or empty. Then χ(R/P,R/Q) ≥ em(R/P )em(R/Q). Algebraically, the above condition, i.e. W̃1 ∩ W̃2 ∩ E is either a finite set of points or empty, is equivalent to stating that the dimension (henceforth dim) of Gm(R/P )⊗Gm(R/Q) is less than 2. Serre [S] showed that the above inequality holds in the equacharacteristic case for any proper intersection. Tennison [T] proved that equality holds (in the above, “≥”) if dimGm(R/P ) ⊗ Gm(R/Q) = 0, i.e. W̃1 ∩ W̃2 ∩ E = ∅. Our approach is completely different from those of Serre and Tennison. We use intersection theory as developed by Fulton and MacPherson [Fu] and Gabber’s theorem of non-negativity [B]. The main point is to understand intersection multiplicity as defined by Serre [S] via the blow-up at {m} × {∞} in SpecR × P, where (R,m) is a regular local ring of essentially finite type over a discrete valuation ring or a field. We mostly Received by the editors September 12, 2003 and, in revised form, February 17, 2004. 2000 Mathematics Subject Classification. Primary 13H05, 14C17; Secondary 13D15, 14H15.