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Steenrod operations in Chow theory. (English) Zbl 1045.55005

Let \(X\) be a complete complex algebraic variety and let \(S_{*}\: H_{*}^{\text{BM}}(X ;{\mathbb Z}/p) \rightarrow H_{*}^{\text{BM}}(X ;{\mathbb Z}/p)\) denote the total Steenrod operation on mod \(p\) Borel-Moore homology. If \(V\) is a subvariety of \(X\) let \(\pi : M \rightarrow V\) be a resolution of singularities of \(V\). Let \(\mu_{M}\) denote the orientation class of \(M\) and let \(\psi : H^{*}(M ;{\mathbb F}/p)@>\cong>> H^*(TM, TM - 0; {\mathbb Z}/p)\) in mod \(p\) singular cohomology. Setting \(cl(V) = \pi_{*}(\mu_{M}) \) and \(w(T_{M}) = \psi^{-1}(S^{*}( \psi(1)))\) a formula in the book by W. Fulton [Intersection theory (1998; Zbl 0885.14002)] states that \(S_{*}( cl(V)) = \pi_{*}( w(T_{M})^{-1}\cap\mu_{M})\).
The author constructs the Steenrod operation \(S_{*}\) on mod \(p\) Chow groups \(A_{*}(X) \otimes {\mathbb Z}/p\). One cannot use the above formula as a definition, because of its possible dependence on the resolution \(\pi\). After the author’s construction is complete one consequence is independence of \(\pi\), which answers a question of Fulton.
The operations are constructed by using an adaptation of the original equivariant cohomology construction of N. E. Steenrod [“Cohomology operations”, Ann. Math. Stud. 50 (1962; Zbl 0102.38104)] to the equivariant intersection theory of D. Edidin and W. Graham [Invent. Math. 131, 595–634 (1998; Zbl 0940.14003)] and B. Totaro [in: Algebraic \(K\)-theory, Proc. AMS-IMS-SIAM Summer Res. Conf., Seattle 1997, Proc. Symp. Pure Math. 67, 249–281 (1999; Zbl 0967.14005)]. The article concludes with a proof of the Adem relations for \(S_{*}\). These operations are related to Voevodsky’s Steenrod operations in motivic cohomology.

MSC:

55N91 Equivariant homology and cohomology in algebraic topology
14C15 (Equivariant) Chow groups and rings; motives
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
55S10 Steenrod algebra
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References:

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