Mathematical Research Letters

Volume 19 (2012)

Number 1

A lemma on nearby cycles and its application to the tame Lubin-Tate space

Pages: 165 – 173

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n1.a13

Author

Jean-François Dat (Université Pierre et Marie Curie, Institut de Mathématiques de Jussieu, 4, place Jussieu, 75005 Paris, France)

Abstract

This note is concerned with a cohomological consequence of ageometric construction due to Yoshida, which relates the tame levelof the Lubin–Tate tower to some Deligne–Lusztig variety of Coxetertype. More precisely, we show that the equivariant morphism incohomology which follows from Yoshida’s construction is an\emph{isomorphism}, whatever the coefficients are. In particular,this gives a conceptual explanation to the observation that$\ell$-adic cohomologies indeed were “the same”, once computedindependently on each side (by Boyer, resp. Lusztig). This alsogives a “simple” proof of the absence of torsion in the integralcohomology of the tame Lubin–Tate space. Our main tool is a generalresult on vanishing cycles for schemes with semi-stable reductionwhich generalizes previous results of Zheng and Illusie. In roughterms, this states that the restriction of the nearby cycles complexto a closed stratum is the push-forward of its restriction to thecorresponding open stratum.

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