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Motivic connective \(k\)-theories and the cohomology of \(A(1)\). (English) Zbl 1266.14015
This article provides a plethora of interesting calculations in stable motivic homotopy theory over the complex numbers completed at the prime \(2\). Among many other things, the authors give a thorough construction of a motivic analogue of the classical real \(K\)-theory spectrum \(KO\) and related motivic connective covers, like \(ko\). The role of classical topological \(K\)-theory in this context here is played by algebraic \(K\)-theory. The authors also compute \(ko\)-homology using an adequate motivic analogue of the Adams spectral sequence, where the \(E_2\)-term is giving as Ext groups \(\mathrm{Ext}_{A(1)}(H^*X, \mathbb{F}_2)\) over the subalgebra \(A(1)\), involving the Steenrod squares \(Sq^1\) and \(Sq^2\) of the motivic Steenrod algebra is a certain motivic space or spectrum. The spectral sequence converges to the \(2\)-adic completion of the motivic \(ko\)-homology \(ko_{*,*}(X)\). The authors also produce some explicit calculations in particular cases.

MSC:
14F42 Motivic cohomology; motivic homotopy theory
19L41 Connective \(K\)-theory, cobordism
55N15 Topological \(K\)-theory
55T15 Adams spectral sequences
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