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We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for étale cohomology with nontrivial coefficients, as well as in the relative setting, i.e., for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the proétale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the proétale site. Moreover, we need to prove the Poincaré lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. Another ingredient for the proof is the geometric acyclicity of crystalline period sheaves, whose computation is due to Andreatta and Brinon.
Hesselholt and Madsen (2004) define and study the (absolute, -typical) de Rham–Witt complex in mixed characteristic, where is an odd prime. They give as an example an elementary algebraic description of the de Rham–Witt complex over , . The main goal of this paper is to construct, for a perfect ring of characteristic , a Witt complex over with an algebraic description which is completely analogous to Hesselholt and Madsen’s description for . Our Witt complex is not isomorphic to the de Rham–Witt complex; instead we prove that, in each level, the de Rham–Witt complex over surjects onto our Witt complex, and that the kernel consists of all elements which are divisible by arbitrarily high powers of . We deduce an explicit description of for each . We also deduce results concerning the de Rham–Witt complex over certain -torsion-free perfectoid rings.
Let be a quadratic extension of nonarchimedean locally compact fields of residual characteristic and let denote its nontrivial automorphism. Let be an algebraically closed field of characteristic different from . To any cuspidal representation of , with coefficients in , such that (such a representation is said to be -selfdual) we associate a quadratic extension , where is a tamely ramified extension of and is a tamely ramified extension of , together with a quadratic character of . When is supercuspidal, we give a necessary and sufficient condition, in terms of these data, for to be -distinguished. When the characteristic of is not , denoting by the nontrivial -character of trivial on -norms, we prove that any -selfdual supercuspidal -representation is either distinguished or -distinguished, but not both. In the modular case, that is when , we give examples of -selfdual cuspidal nonsupercuspidal representations which are not distinguished nor -distinguished. In the particular case where is the field of complex numbers, in which case all cuspidal representations are supercuspidal, this gives a complete distinction criterion for arbitrary complex cuspidal representations, as well as a purely local proof, for cuspidal representations, of the dichotomy and disjunction theorem due to Kable and Anandavardhanan, Kable and Tandon, when .
In this paper we prove a general vanishing result for Kohlhaase’s higher smooth duality functors . If is any unramified connected reductive -adic group, is a hyperspecial subgroup, and is a Serre weight, we show that for , where is a Borel subgroup and the dimension is over . This is due to Kohlhaase for , in which case it has applications to the calculation of for supersingular representations. Our proof avoids explicit matrix computations by making use of Lazard theory, and we deduce our result from an analogous statement for graded algebras via a spectral sequence argument. The graded case essentially follows from Koszul duality between symmetric and exterior algebras.
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