Let K be the field of fractions of a Henselian discrete valuation ring O_K.
Let X_K/K be a smooth proper geometrically connected scheme admitting a regular
model X/O_K. We show that the index \delta(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 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 \gamma(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 \gamma(A) to the index of
the exceptional divisor in a resolution of the singularity of Spec(A), and we
give a new way of computing the index of a smooth subvariety X_K/K of P^n_K
over any field K, using the invariant \gamma of the local ring at the vertex of
a cone over X.
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