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Cyclic homology, Serre’s local factors and \(\lambda\)-operations. (English) Zbl 1327.14109

Summary: We show that for a smooth, projective variety \(X\) defined over a number field \(K\), cyclic homology with coefficients in the ring \(\mathbb A_\infty = \prod_{\nu|\infty }K_\nu\), provides the right theory to obtain, using {\(\lambda\)}-operations, Serre’s archimedean local factors of the complex \(\mathrm L\)-function of \(X\) as regularized determinants.

MSC:

14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G09 Drinfel’d modules; higher-dimensional motives, etc.
14F25 Classical real and complex (co)homology in algebraic geometry
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
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