Algebraic Morava \(K\)-theories.

*(English)*Zbl 1030.55003This 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.

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.

Reviewer: Keith Johnson (Halifax)