The Index of an Algebraic Variety

Let K be the field of fractions of a Henselian discrete valuation ring OK . Let XK/K be a smooth proper geometrically connected scheme admitting a regular model X/OK . We show that the index δ(XK/K) of XK/K can be explicitly computed using data pertaining only to the special fiber Xk/k of the model X. We give two proofs of this theorem, using two moving lemmas. One moving lemma pertains to horizontal 1-cycles on a regular projective scheme X over the spectrum of a semi-local Dedekind domain, and the second moving lemma can be applied to 0-cycles on an FA-scheme X which need not be regular. The study of the local algebra needed to prove these moving lemmas led us to introduce an invariant γ(A) of a singular local ring (A,m): the greatest common divisor of all the Hilbert-Samuel multiplicities e(Q,A), over all m-primary ideals Q in m. We relate this invariant γ(A) to the index of the exceptional divisor in a resolution of the singularity of SpecA, and we give a new way of computing the index of a smooth subvariety X/K of PK over any field K, using the invariant γ of the local ring at the vertex of a cone over X.