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Register a new userLa $mathrm{Z}_l$-cohomologie du mod`ele de Deligne-Carayol est sans torsion
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Authors
Pascal Boyer
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This article is the $mathrm{Z}_l$-version of my paper Monodromie du faisceau pervers des cycles evanescents de quelques varietes de Shimura simples in Invent. Math. 2009 vol 177 pp. 239-280, where we study the vanishing cycles of some unitary Shimura variety. The aim is to prove that the cohomology sheaves of this complexe are free so that, thanks to the main theorem of Berkovich on vanishing cycles, we can deduce that the $mathrm{Z}_l$-cohomology of the model of Deligne-Carayol is free. There will be a second article which will be the $mathrm{Z}_l$ version of my paper Conjecture de monodromie-poids pour quelques varites de Shimura unitaires in Compositio vol 146 part 2, pp. 367-403. The aim of this second article will be to study the torsion of the cohomology groups of these Shimura varieties.
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The principal result of this work is the freeness in the $ overline{mathbb Z}_l$-cohomology of the Lubin-Tate tower. The strategy is of global nature and relies on studying the filtration of stratification of the perverse sheaf of vanishing cycles of some Shimura varieties of Kottwitz-Harris-Taylor types, whose graduates can be explicited as some intermediate extension of some local system constructed in the book of Harris andTaylor. The crucial point relies on the study of the difference between such extension for the two classical $t$-structures $p$ and $p+$. The main ingredients use the theory of derivative for representations of the mirabolic group.
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