Artin motives

General Introduction
See Motives, and notes from AndrĂ©.
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Search results
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Online References
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Paper References
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
arXiv:1002.2771 Relative Artin motives and the reductive BorelSerre 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 $\omega^0X(M)$ as the universal Artin motive mapping to M. We use this to define a motive EX 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 $\omega^0X$ and the computation of the motives EX.
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 BorelSerre compactification W onto its BailyBorel Satake compactification Z. We show that $Rp*(\QW)$ is naturally isomorphic to the Betti realization of the motive EZ, where Z is viewed as a scheme. In particular, the direct image of EZ 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 ArtinTate motives over number rings from arXiv Front: math.AG by Jakob Scholbach This paper studies ArtinTate motives over number rings. As a subcategory of geometric motives, the triangulated category of ArtinTate 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 tstructure is constructed. Exactness properties of these functors familiar from perverse sheaves are shown to hold in this context. The cohomological dimension of mixed ArtinTate motives is two, and there is an equivalence of the triangulated category of ArtinTate motives with the derived category of mixed ArtinTate motives.
Update in second version: a functorial and strict weight filtration for mixed ArtinTate motives is established. Moreover, two minor corrections have been performed: first, the category of ArtinTate 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 $f$.
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