Abstract: Let K be the field of fractions of a Henselian discrete valuation ring  ${{\mathcal {O}}_{K}}$ . Let X K /K be a smooth proper geometrically connected scheme admitting a regular model $X/{{\mathcal {O}}_{K}}$ . 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 $\operatorname {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 $(A, {\mathfrak {m}})$ : the greatest common divisor of all the Hilbert-Samuel multiplicities e(Q,A), over all ${\mathfrak {m}}$ -primary ideals Q in ${\mathfrak {m}}$ . We relate this invariant γ(A) to the index of the exceptional divisor in a resolution of the singularity of $\operatorname {Spec}A$ , and we give a new way of computing the index of a smooth subvariety X/K of ${\mathbb{P}}^{n}_{K}$ over any field K, using the invariant γ of the local ring at the vertex of a cone over X.

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{{Explor lien
   |wiki=    Wicri/Mathematiques
   |area=    BourbakiV1
   |flux=    Main
   |étape=   Exploration
   |type=    RBID
   |clé=     ISTEX:4D812E17AD8C6CDC1BE39C4A8BC9F348214223D0
   |texte=   The index of an algebraic variety
}}