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Notes on motivic cohomology. (English) Zbl 0632.14010
Let X be a topological space. Then the rational cohomology groups \(H^*(X,{\mathbb{Q}})\) are isomorphic to the associated graded ones of the Atiyah-Hirzebruch filtration of \(K^*(X)\otimes {\mathbb{Q}}\). In algebraic geometry D. Quillen [in Algebr. K-theory I, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 341, 85-147 (1973; Zbl 0292.18004)] has defined algebraic K-groups \(K_ i(X)\) for an algebraic variety X. One would like to define a cohomology theory for algebraic varieties which is “rationally isomorphic” to Quillen’s K-theory in a manner similar to the relation between singular cohomology and topological K-theory described above.
The authors discuss properties that this hoped-for “motivic cohomology” should have, and several approaches to define it. They suggest that the \(\gamma\)-filtration on \(K_ i\) can be used as an analogue for the Atiyah-Hirzebruch filtration on topological K-theory. Denote the associated graded of \(K_ j\) under the \(\gamma\)-filtration by \(gr^ pK_ j\) and the hoped-for motivic cohomology by \(H^ i_{{\mathcal M}}(X,{\mathbb{Z}}(p))\). The authors would then like to have \(H^ i_{{\mathcal M}}(X,{\mathbb{Z}}(p))\otimes {\mathbb{Q}}\cong gr^ pK_{2p-i}(X)\otimes {\mathbb{Q}}.\)
Some properties that motivic cohomology should have are that it should be a universal cohomology theory in some sense, that it should vanish (at least rationally) in certain indices, and that it should be defined in some way by cochains. It is conjectured that \(H^ i_{{\mathcal M}}(X,{\mathbb{Z}}(p))\) should be the hypercohomology groups of X with coefficients in a complex of sheaves \({\mathbb{Z}}(p)_{{\mathcal M}}\). The first approach that the authors take is to construct candidates for the complexes \({\mathbb{Z}}(p)_{{\mathcal M}}\) by representing elements of the complexes as maps into Grassmann manifolds. The second approach is to construct a candidate for \(H^ i_{{\mathcal M}}(X,{\mathbb{Z}}(p))\) as an ext group in a category of Tate motives. The bulk of the paper is taken up in pursuing these two approaches. Complete proofs of the results will appear elsewhere.
Reviewer: Leslie G.Roberts

14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14A20 Generalizations (algebraic spaces, stacks)
14F99 (Co)homology theory in algebraic geometry
Full Text: DOI
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