A LEMMA ON NEARBY CYCLES AND ITS APPLICATION TO THE TAME LUBIN-TATE SPACE

作者:Dat Jean Francois*
来源:Mathematical Research Letters, 2012, 19(1): 165-173.
DOI:10.4310/MRL.2012.v19.n1.a13

摘要

This note is concerned with a cohomological consequence of a geometric construction due to Yoshida, which relates the tame level of the Lubin-Tate tower to some Deligne-Lusztig variety of Coxeter type. More precisely, we show that the equivariant morphism in cohomology which follows from Yoshida's construction is an isomorphism, whatever the coefficients are. In particular, this gives a conceptual explanation to the observation that l-adic cohomologies indeed were "the same", once computed independently on each side (by Boyer, resp. Lusztig). This also gives a "simple" proof of the absence of torsion in the integral cohomology of the tame Lubin-Tate space. Our main tool is a general result on vanishing cycles for schemes with semi-stable reduction which generalizes previous results of Zheng and Illusie. In rough terms, this states that the restriction of the nearby cycles complex to a closed stratum is the push-forward of its restriction to the corresponding open stratum.

  • 出版日期2012