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Duality and the topological filtration

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Abstract

We investigate some relations between the duality and the topological filtration in algebraic \(K\)-theory. As a result, we obtain a construction of the first Steenrod square for Chow groups modulo two of varieties over a field of arbitrary characteristic. This improves previously obtained results, in the sense that it is not anymore needed to mod out the image modulo two of torsion integral cycles. Along the way we construct a lifting of the first Steenrod square to algebraic connective \(K\)-theory with integral coefficients, and homological Adams operations in this theory. Finally we provide some applications to the Chow groups of quadrics.

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Acknowledgments

Making the Adams operations act on Quillen spectral sequence is certainly very classical. A recent illustration can be found in [15], whence we borrowed some of the arguments and references used in this text. The idea of constructing the first Steenrod square using the duality theory for schemes was inspired by [21]. I am very grateful to Nikita Karpenko, who drew my attention to connective \(K\)-theory and Adams operations while I was trying to construct Steenrod operations for Chow groups. Finally I would like to thank Baptiste Calmès and the anonymous referee, who made suggestions yielding to improvements in the exposition. The support of EPSRC Responsive Mode grant EP/G032556/1 is gratefully acknowledged.

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  1. Mathematisches Institut der Universität München, Theresienstr. 39, 80333 , Munich, Germany

    Olivier Haution

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  1. Olivier Haution
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Haution, O. Duality and the topological filtration. Math. Ann. 357, 1425–1454 (2013). https://doi.org/10.1007/s00208-013-0956-8

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