The index of an algebraic variety

Let K be the field of fractions of a Henselian discrete valuation ring . Let X K / K be a smooth proper geometrically connected scheme admitting a regular model . We show that the index δ ( X K / K ) of X K / K can be explicitly computed using data pertaining only to the special fiber X k / 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 -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 : the greatest common divisor of all the Hilbert-Samuel multiplicities e ( Q , A ), over all -primary ideals Q in . We relate this invariant γ ( A ) to the index of the exceptional divisor in a resolution of the singularity of , and we give a new way of computing the index of a smooth subvariety X / K of over any field K , using the invariant γ of the local ring at the vertex of a cone over X .