# On the motive of a quotient variety

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- Universitat de Barcelona
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Collect. Math. 49, 2–3 (1998), 203–226 c© 1998 Universitat de Barcelona On the motive of a quotient variety∗ Sebastian del Ban˜o Rollin Departament de Matema`tica Aplicada I, Universitat Polite`cnica de Catalunya, Avinguda Diagonal 647, Barcelona 08028, Spain E-mail: [email protected] Vicente Navarro Aznar Departament d’A`lgebra i Geometria, Facultat de Matema`tiques, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, Barcelona 08007, Spain E-mail: [email protected] En memoria de nuestro compan˜ero y amigo Ferran Serrano Abstract We show that the motive of the quotient of a scheme by a finite group coincides with the invariant submotive. Introduction Let k be a field and Vk the category of smooth projective varieties over k. In the 60’s Grothendieck proved there exists a category Mk called the category of motives and a functor h : Vk −→Mk that factorises the different Weil cohomology theories, such as the �-adic, singular or de Rham cohomologies. If X is a smooth projective variety over k acted on by a finite group G, the quotient variety X/G is no longer necessarily smooth so, a priori, the Grothendieck motive of X/G is not defined, however one can reasonably define h(X/G) to be the G-invariant part of h(X), which is an object of Mk, this definition is consistent ∗ Partially supported by DGCYT grant PB93-0790. 203 Administrador 204 del Ban˜o and Navarro with the realisation functors and Chow groups. Recently, in the case char k = 0, Guille´n and Navarro Aznar have given in [4] an extension of the functor h to arbitrary schemes taking values in the homotopy category of complexes of motives, HoCbMk, in particular it provides with another possible definition of h(X/G). The main result of this note is that these two definitions coincide. In particular if we call pure the objects in the essential image of the functor Mk −→ HoCbMk, then h(X/G) is pure as expected. Contents We start in Section 1 by extending the theory of Chow motives to

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