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Lefschetz-Riemann-Roch theorem and coherent trace formula. (English) Zbl 0653.14005

Pour une groupe G agissant sur un schéma X sous des conditions convenables, on considère la catègorie abélienne des faisceaux cohérents sur X munis d’une action compatible avec celle de G sur X; la K-théorie algébrique qui s’en déduit est noteé G(G,X). L’objet de ce papier est de définie une K-homologie topologique équivariante \(G/\ell^{\nu^{Top}}(G,X)\) et de la comparer à \(G/\ell^{\nu}(G,X)\) (\(\ell\) premier fixé). La comparaison se fait à l’aide d’un théorème de Lefschetz-Riemann-Roch qui généralise celui de P. Baum, W. Fulton and G. Quart [Acta Math. 143, 193-211 (1979; Zbl 0454.14009)]. Combinant ce dernier avec le thèorème de concentration de G. Segal [Publ. Math., Inst. Hautes Étud. Sci. 34, 129-151 (1968; Zbl 0199.262)], l’A. obtient une formule des traces qui généralise la formule du point fixe de M. F. Atiyah et R. Bott [cf. par exemple, Ann. Math., II. Ser. 86, 374-407 (1967; Zbl 0161.432) et 88, 451-491 (1968; Zbl 0167.217)].
Le point essentiel est la comparaison donneé par le théorème de l’auteur lui-même [cf. Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 437-552 (1985; Zbl 0596.14012); 4.16] entre géométrie algébrique et topologie algébrique selon lequel \(G/\ell^{\nu^{Top}}(X)\simeq G/\ell^{\nu}(X)[\beta^{-1}]\) où \(\beta\) est l’élément de Bott rendu inversible par localisation.
Reviewer: J.C.Douai

MSC:

14C40 Riemann-Roch theorems
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
55N15 Topological \(K\)-theory
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
55N25 Homology with local coefficients, equivariant cohomology
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References:

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