Holmstrom Artin motives

Artin motives

AG (Algebraic geometry)

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Artin motives

See Motives, and notes from André.

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Artin motives

Pure

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Artin motives

http://mathoverflow.net/questions/11932/artin-motives-references-for

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Artin motives

arXiv:1002.2771 Relative Artin motives and the reductive Borel-Serre compactification of a locally symmetric variety from arXiv Front: math.AG by Joseph Ayoub, Steven Zucker We introduce the notion of Artin motives and cohomological motives over a scheme X. Given a cohomological motive M over X, we construct its punctual weight zero part $\omega^0_X(M)$ as the universal Artin motive mapping to M. We use this to define a motive E_X over X which is an invariant of the singularities of X. The first half of the paper is devoted to the study of the functors $\omega^0_X$ and the computation of the motives E_X

In the second half of the paper, we develop the application to locally symmetric varieties. Specifically, let Y be a locally symmetric variety and denote by p:W–>Z the projection of its reductive Borel-Serre compactification W onto its Baily-Borel Satake compactification Z. We show that $Rp_*(\Q_W)$ is naturally isomorphic to the Betti realization of the motive E_Z, where Z is viewed as a scheme. In particular, the direct image of E_Z along the projection of Z to Spec(C) gives a motive whose Betti realization is naturally isomorphic to the cohomology of W.

arXiv:1003.1267 Mixed Artin-Tate motives over number rings from arXiv Front: math.AG by Jakob Scholbach This paper studies Artin-Tate motives over number rings. As a subcategory of geometric motives, the triangulated category of Artin-Tate motives DATM(S) is generated by motives of schemes that are finite over the base S. After establishing stability of these subcategories under pullback and pushforward along open and closed immersions, a motivic t-structure is constructed. Exactness properties of these functors familiar from perverse sheaves are shown to hold in this context. The cohomological dimension of mixed Artin-Tate motives is two, and there is an equivalence of the triangulated category of Artin-Tate motives with the derived category of mixed Artin-Tate motives

Update in second version: a functorial and strict weight filtration for mixed Artin-Tate motives is established. Moreover, two minor corrections have been performed: first, the category of Artin-Tate motives is now defined to be the triangulated category generated by direct factors of $f_* \mathbf 1(n)$—as opposed to the thick category generated by these generators. Secondly, the exactness of $f^*$ for a finite map is only stated for etale maps $f$.

nLab page on Artin motives