Extended powers and Steenrod operations in algebraic geometry (Q1004504)

The authors provide a setting for the construction of Steenrod operations for a generalized cohomology theory (defined over smooth quasi-projective schemes), whose formal group law has order 2. This unified setting gives a better understanding of these and other related constructions which have appeared many times in algebraic geometry for different generalized cohomology theories: see [\textit{M. Karoubi}, The Arnold-Gelfand mathematical seminars: geometry and singularity theory. (Boston), MA: Birkhäuser. 281--323 (1997; Zbl 0877.55012)], \textit{F. Morel} and \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007), Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. I, 579--604 (1998; Zbl 0907.19002)], \textit{Z. Nie}, [Am. J. Math. 130, 3, 713--762 (2008; Zbl 1147.14007)], \textit{P. Brosnan} [Trans. Am. Math. Soc. 355, No. 5, 1869--1903 (2003; Zbl 1045.55005)] and \textit{A. Merkurjev} [J. Reine Angew. Math. 565, 13--26 (2003; Zbl 1091.14006)]. The authors list a set of conditions on a generalized cohomology theory which allow the definition of Steenrod operations. They adapt the methods used by Bisson-Joyal in studying Steenrod and Dyer-Lashof operations in unoriented cobordism and mod 2 cohomology. The article is an enjoyable exposition presenting important examples and references.