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Remarks on motivic homotopy theory over algebraically closed fields. (English) Zbl 1248.14026
In this interesting article the authors discuss several versions of the Adams and Adams-Novikov spectral sequence in the framework of motivic stable homotopy theory. The spectral sequence constructions are considered in the \(2\)-complete motivic stable homotopy category over an algebraically closed field of characteristic \(0\). The authors establish especially here important convergence results for both spectral sequence types. Besides showing that these spectral sequences observe similar behavior as their classical topological counterparts, new and more algebro-geometric phenomena also show up, like the existence of permanent cycles related to the Tate twist.
Many interesting examples and applications are discussed, which nicely emphasize the interplay of topological and algebro-geometric methods. For example, they perform several interesting calculations on the cohomology of symmetric products of spheres in the motivic setting analogous to classical topological calculations obtained by Nakaoka in the 50s. As an interesting application they also discuss a \(2\)-complete version of the complex motvic \(J\)-homomorphism.
The appendix gives a thorough tour on the construction of motivic Adams spectral sequences and their main properties. The article also discusses nicely the different motivic cobordism spectra used in the construction of the motivic Adams-Novikov spectral sequences.

MSC:
14F42 Motivic cohomology; motivic homotopy theory
19L20 \(J\)-homomorphism, Adams operations
55T15 Adams spectral sequences
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