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The index of an algebraic variety

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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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Notes

  1. Recall that a ring A of finite Krull dimension is equidimensional if \(\dim A/{\mathfrak {p}}=\dim A\) for every minimal prime ideal \({\mathfrak {p}}\) of A. A point x∈X is equidimensional if \({\mathcal{O}}_{X,x}\) is.

  2. It should be noted that the definition of e(Q,M) taken in [42], page 107, (or in [53], 11.1.5) is different from the definition taken at the beginning of this section and in [7], VIII.72. Thus the statement of [42], 14.6, cannot be applied directly.

  3. Due to the two different definitions of e(Q,M), the proof of [42], 14.11, needs to be slightly adjusted to prove the second equality.

  4. See [23] IV.13.2.2, with a correction in [23] (Err IV, 34) on pp. 356–357 of no. 32.

  5. The case of inseparable extensions requires careful consideration (see for instance the Notes on page 230 of [17]). We also note that Proposition 5.2 in Chapter 9 of [35], page 240, requires further hypotheses to ensure that the element y in its proof exists.

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Authors and Affiliations

  1. IHÉS, 35 route de Chartres, 91440, Bures-sur-Yvette, France

    Ofer Gabber

  2. Institut de Mathématiques de Bordeaux, Université de Bordeaux 1, 33405, Talence, France

    Qing Liu

  3. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Dino Lorenzini

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  1. Ofer Gabber
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  2. Qing Liu
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  3. Dino Lorenzini
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Correspondence to Dino Lorenzini.

Additional information

D.L. was supported by NSF Grant 0902161.

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Gabber, O., Liu, Q. & Lorenzini, D. The index of an algebraic variety. Invent. math. 192, 567–626 (2013). https://doi.org/10.1007/s00222-012-0418-z

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