We construct a relative version of the Ainf -cohomology theory developed by Bhatt-Morrow-Scholze and relate it to the prismatic theory of Bhatt-Scholze. The construction relies on the fiber product of topoi. As an application we show that there is an & eacute;tale comparison of the q-crystalline pushforward after inverting mu is an element of Ainf .
Abstract Given a quasi-reductive group G over a local field k , using Berkovich geometry, we exhibit a family of G ( k ) $G(k)$ upper G left parenthesis k right parenthesis -equivariant compactifications of the Bruhat-Tits building B ( G , k ) $\mathcal B(G, k)$ script upper B left parenthesis upper G comma k right parenthesis , constructed and investigated by Solleveld and Lourenço. The compactification procedure consists of mapping the building into the analytification G an $G^{\operatorname {\mathrm {an}}}$ upper G Superscript an of G , then composing this map with the projections from G an $G^{\operatorname {\mathrm {an}}}$ upper G Superscript an to its (in general non-compact) pseudo-flag varieties ( G / P ) an $(G/P)^{\operatorname {\mathrm {an}}}$ left parenthesis upper G divided by upper P right parenthesis Superscript an , for P ranging among the pseudo-parabolic subgroups of G . This generalises previous constructions of Berkovich, then Rémy, Thuillier and Werner. To define the embedding, we are led to giving a partial extension to the quasi-reductive context of results due to Rousseau on the functoriality of Bruhat-Tits buildings with respect to field extensions, which are of independent interest. Finally, we conclude by investigating the geometry at infinity of these compactifications. The boundaries are shown to be stratified, each stratum being equivariantly homeomorphic to the Bruhat-Tits building of the maximal quasi-reductive quotient of a pseudo-parabolic subgroup.
Abstract In this paper we are going to compute the KW $ \mathrm {KW} $ upper K upper W -Euler classes for rank 2 vector bundles on the classifying stack B N $ \mathrm {B} \mathrm{N} $ normal upper B normal upper N , where N is the normaliser of the standard torus in SL 2 $\mathrm{SL}_2$ upper S upper L 2 and KW $\mathrm {KW}$ upper K upper W represents Balmer’s derived Witt groups. Using these computations we will recover, through a new and different strategy, the formulas previously obtained by Levine in Witt-sheaf cohomology. In order to obtain our results, we will prove Künneth formulas for products of GL n $\mathrm{GL}_n$ upper G upper L Subscript n ’s and SL n $\mathrm{SL}_n$ upper S upper L Subscript n ’s classifying spaces and we will develop from scratch the basic theory of twisted symplectic bundles with their associated twisted Borel classes in SL $\mathrm{SL}$ upper S upper L -oriented theories.
Given a smooth variety X over C $\mathbb {C}$ double struck upper C , a smooth divisor i : Y hooked right arrow X $i:Y\hookrightarrow X$ i colon upper Y right arrow with hook upper X and a global function f on X which vanishes on Y and on its critical locus, we compute the map induced on Hochschild homology by the pushforward functor i & lowast; : D b ( Y ) -> D a b s ( M F ( X , f ) ) $i_{\ast }:D<^>b(Y) o D<^>{abs}(MF(X,f))$ i Subscript asterisk Baseline colon upper D Superscript b Baseline left parenthesis upper Y right parenthesis right arrow upper D Superscript a b s Baseline left parenthesis upper M upper F left parenthesis upper X comma f right parenthesis right parenthesis in terms of the Hochschild-Kostant-Rosenberg isomorphisms.
Abstract Let E / F $E/F$ upper E divided by upper F be a quadratic extension of totally real number fields. We show that the generalized Hirzebruch–Zagier cycles arising from the associated Hilbert modular varieties can be put in p -adic families. As an application, using the theory of base change, we give a geometric construction of the multivariable p -adic adjoint L -function twisted by the Hecke character of E / F $E/F$ upper E divided by upper F , attached to Hida families of Hilbert modular forms over F .
