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Chow categories. (English) Zbl 0726.14005

Let p: \(X\to S\) be a proper morphism of relative dimension n and \(P(c_ i(E_ j))\) a given polynomial of absolute degree \(n+1\) \((\deg (c_ i)=i)\) in the Chern classes of vector bundles \(E_ 1,...,E_ k\). In connection with metrical versions of the Riemann-Roch-Grothendieck theorem, P. Deligne [in Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata/Calif. 1985, Contemp. Math. 67, 93-117 (1987; Zbl 0629.14008)] asks for a construction of a line bundle \(I_{X| S}P(c_ i(E_ j))\) on S which is a realisation of \(\int_{X| S}P(c_ i(E_ j)) \in CH^ 1(S).\)
The author defines (for noetherian, separated, universally catenary schemes) Chow categories \({\mathbb{C}}{\mathbb{H}}^ i\) and develops basic formalism for such categories. In a further paper [cf. Arithmetic algebraic geometry, Proc. Conf., Texel/Neth. 1989, Prog. Math. 89, 75-152 (1991)] he defines Chern functors \(c_ i(E)\), which take values in the i-th Chow category \({\mathbb{C}}{\mathbb{H}}^ i(X)\), and gives a proposal for the wanted \(I_{X| S}P(c_ i(E_ j))\). - The starting point of the construction of the Chow categories is the equivalence between line bundles and \({\mathcal O}^*\)-principal homogeneous sheaves. \({\mathbb{C}}{\mathbb{H}}^ i(X)\) will be the category of \(G_ i\)-principal homogeneous sheaves on some site \(X_{(k)}\) on X. \(G_ i(X)\) is the sheaf \(E_ 2^{i-1,-i}(X)\) \((E_{\bullet}^{p,q}(X)\) is the spectral sequence in the K-theory of Quillen, associated with the filtration given by codimension).

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
14C40 Riemann-Roch theorems
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry

Citations:

Zbl 0629.14008
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References:

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