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Algebraic Morava $$K$$-theories. (English) Zbl 1030.55003
This paper is devoted to the construction of an analogue, $$k(t)$$, of connective Morava $$K$$-theory in the category of motivic spectra. The setting for this construction is the homotopy theory of schemes developed in [F. Morel and V. Voevodsky, Publ. Math., Inst. Hautes Etud. Sci. 90, 45-145 (1999; Zbl 0983.14007)] in which a spectrum is a sequence of simplicial sheaves of sets on the site of smooth schemes over a field $$k$$ with the Nisnevich topology. $$k$$ is assumed to come equipped with an embedding $$k \hookrightarrow {\mathbb C}$$.
The paper gives a summary of the properties of the stable homotopy category of such spectra, and then develops the properties of the motivic spectrum $$MGI$$, the motivic analogue of the complex cobordism spectrum $$MU$$. This is used to construct an $$MGI$$-module spectrum, $$k^\prime(t)$$, by killing appropriate elements of $$\pi_*(MGI)$$ and from this, by cofibre sequences, a tower of spectra whose homotopy limit is $$k(t)$$. The embedding $$k \hookrightarrow {\mathbb C}$$ is used to construct a topological realization functor which is used at several points to identify elements in motivic cohomology. The author suggests that, in analogy with the situation at the prime $$2$$, the existence of $$k(t)$$ may lead to a proof of the Block-Kato conjecture at odd primes.

##### MSC:
 55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology 14F42 Motivic cohomology; motivic homotopy theory 14F35 Homotopy theory and fundamental groups in algebraic geometry 14F99 (Co)homology theory in algebraic geometry 55P42 Stable homotopy theory, spectra
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