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Perversité et variation

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Abstract

Let S be the spectrum of a strictly henselian discrete valuation ring with residue characteristic p and Λ=ℤ/ℓνℤ, where ℓ is a prime number ≠p and ν is an integer ≥1. For a scheme X of finite type over S and smooth over S along the special fiber X s outside a closed point x, we study the vanishing cycles complex RΦ(Λ) and the tame variation \({{\mathop{{\rm{ Var}}}(\sigma) : R\Phi_{{{{\rm{ t}}}}}(\Lambda)_x \rightarrow R\Gamma_{{\{x\}}}(X_s,R\Psi_{{{{\rm{ t}}}}}(\Lambda))}}\), for Σ in the tame inertia group I t . In particular, we show that if X is regular, flat over S of relative dimension n≥1, and Σ is a topological generator of I t , then R qΦ(Λ) x =0 for q≠n and \({{\mathop{{\rm{ Var}}}(\sigma) : R^n\Phi_{{{{\rm{ t}}}}}(\Lambda)_x \rightarrow H^n_{{\{x\}}}(X_s,R\Psi_{{{{\rm{ t}}}}}(\Lambda))}}\) is an isomorphism.

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  1. Université de Paris-Sud, Mathématique - Bât. 425, UMR 8628, 91405, Orsay Cedex, France

    Luc Illusie

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  1. Luc Illusie
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Correspondence to Luc Illusie.

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Mathematics Subject Classification (2000): 14F20, 14D05, 14D06

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Illusie, L. Perversité et variation. manuscripta math. 112, 271–295 (2003). https://doi.org/10.1007/s00229-003-0407-z

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