Reduced power operations in motivic cohomology.

*(English)*Zbl 1057.14027Cohomology operations are very familiar from algebraic topology, where they form a basic toolkit of great power and applicability. Their theory has been extensively developed by N. Steenrod, J. Milnor, and others some decades ago [N. E. Steenrod, Cohomology. Ann. Math. Stud. 50 (1962; Zbl 0102.38104)], later on also in the framework of \(K\)-theory.

In the paper under review, the author develops a theory of cohomology operations in the motivic cohomology of smooth simplicial schemes over a perfect field \(k\) with coefficients in \(\mathbb{Z}/\ell\), where \(\ell\) is a prime number different from the characteristic of the field \(k\). In this context, he constructs the so-called reduced power operations in motivic cohomology, establishes the motivic analogues of the classical Cartan formulas and the Adem relations, describes explicitely the subalgebra generated by these reduced power operations \(P^i\) in the algebra of all (bistable) operations in the motivic cohomology, characterizes an analogue of the Bockstein operator and analyzes the multiplication by the motivic cohomology classes of \(\text{Spec}(k)\). This includes a thorough comparison of the usual topological Steenrod algebra and the motivic Steenrod algebra, the construction of the Thom isomorphism in motivic cohomology, the description of Euler classes of vector bundles in this framework, the analysis of the dual of the motivic Steenrod algebra, and the interrelation between motivic cohomology operations and characteristic classes.

This highly original and pioneering program is carried out in the fourteen sections of the present paper, with a wealth of ingenious new ideas and novel, extremely subtle and fundamental constructions. Whereas the introduction of the motivic reduced power operations via the so-called total power operation follows the classical approach in topology (à la Steenrod-Epstein [loc. cit.]), the author’s construction of the motivic total power operation is completely new and different from any of the standard topological constructions. However, as the author points out, there is a way to obtain the topological analogue of the motivic total operation in terms of Eilenberg-MacLane spectra.

Apart from providing a good deal of this novel theory of motivic cohomology operations, whose completion is still in progress and will be published in the sequel, the present comprehensive paper contains all the rigorous proofs of the results on motivic cohomology operations that the author had already used in his earlier, very spectacular proof of the celebrated Milnor conjecture back in 1996 [V. Voevodsky, The Milnor conjecture, Electronic publication: http://www.math.uiuc.edu/K-theory/170]. Together with some corrections and improvements, the present paper completes that foregoing paper ultimately. Despite its highly advanced, conceptually involved and extremely innovating character, the material is exhibited in a very lucid, detailed and well-organized manner, which is very gratifying in view of its fundamental importance for further research in this central area of contemporary algebraic geometry.

In the paper under review, the author develops a theory of cohomology operations in the motivic cohomology of smooth simplicial schemes over a perfect field \(k\) with coefficients in \(\mathbb{Z}/\ell\), where \(\ell\) is a prime number different from the characteristic of the field \(k\). In this context, he constructs the so-called reduced power operations in motivic cohomology, establishes the motivic analogues of the classical Cartan formulas and the Adem relations, describes explicitely the subalgebra generated by these reduced power operations \(P^i\) in the algebra of all (bistable) operations in the motivic cohomology, characterizes an analogue of the Bockstein operator and analyzes the multiplication by the motivic cohomology classes of \(\text{Spec}(k)\). This includes a thorough comparison of the usual topological Steenrod algebra and the motivic Steenrod algebra, the construction of the Thom isomorphism in motivic cohomology, the description of Euler classes of vector bundles in this framework, the analysis of the dual of the motivic Steenrod algebra, and the interrelation between motivic cohomology operations and characteristic classes.

This highly original and pioneering program is carried out in the fourteen sections of the present paper, with a wealth of ingenious new ideas and novel, extremely subtle and fundamental constructions. Whereas the introduction of the motivic reduced power operations via the so-called total power operation follows the classical approach in topology (à la Steenrod-Epstein [loc. cit.]), the author’s construction of the motivic total power operation is completely new and different from any of the standard topological constructions. However, as the author points out, there is a way to obtain the topological analogue of the motivic total operation in terms of Eilenberg-MacLane spectra.

Apart from providing a good deal of this novel theory of motivic cohomology operations, whose completion is still in progress and will be published in the sequel, the present comprehensive paper contains all the rigorous proofs of the results on motivic cohomology operations that the author had already used in his earlier, very spectacular proof of the celebrated Milnor conjecture back in 1996 [V. Voevodsky, The Milnor conjecture, Electronic publication: http://www.math.uiuc.edu/K-theory/170]. Together with some corrections and improvements, the present paper completes that foregoing paper ultimately. Despite its highly advanced, conceptually involved and extremely innovating character, the material is exhibited in a very lucid, detailed and well-organized manner, which is very gratifying in view of its fundamental importance for further research in this central area of contemporary algebraic geometry.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

14F42 | Motivic cohomology; motivic homotopy theory |

19E15 | Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) |

55S10 | Steenrod algebra |

55S05 | Primary cohomology operations in algebraic topology |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14F45 | Topological properties in algebraic geometry |

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