A length formula for the multiplicity of distinguished components of intersections

Abstract

To an arbitrary ideal I in a local ring (A, m) one can associate a multiplicity j( I, A) that generalizes the classical Hilbert–Samuel multiplicity of an m -primary ideal and which plays an important role in intersection theory. If the ideal is strongly Cohen–Macaulay in A and satisfies a suitable Artin–Nagata condition then our main result states that j( I, M) is given by the length of I/( x 1,…, x d−1 )+ x d I where d≔ dim A and x 1,…, x d are sufficiently generic elements of I. This generalizes the classical length formula for m -primary ideals in Cohen–Macaulay rings. Applying this to an hypersurface H in the affine space we show for instance that an irreducible component C of codimension c of the singular set of H appears in the self-intersection cycle H c+1 with multiplicity e( jac H,C, O H,C) , where jac H is the Jacobian ideal generated by the partial derivatives of a defining equation of H.