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Equivariant cycles and cancellation for motivic cohomology. (English) Zbl 1336.14019

In this foundational paper the authors establish the computational basis for one form of equivariant motivic cohomology. More precisely, they give a definition of motivic Bredon cohomology modeled in a natural way on the non-equivariant definition, and establish “the usual results”.
In somewhat more detail, given a group \(G\), the authors define the category of presheaves with \(G\)-equivariant transfers over a field \(k\). These are presheaves on the category of smooth \(k\)-schemes with a \(G\)-action, but where morphisms are the equivariant finite correspondences. On the category of smooth \(G\)-schemes there is the equivariant Nisnevich topology (which is to be distinguished from the fixed-point Nisnevich topology). Given a \(k\)-representation \(V\) of \(G\), the authors make the expected definition \(H^*_G(X, \mathbb{Z}(V)) = H^*_{GNis}(X, C_* \mathbb{Z}(V))\), where \(C_*\) denotes the \(\mathbb{A}^1\)-chain complex and \(\mathbb{Z}(V) = \mathbb{Z}_{tr} \mathbb{P}(V \oplus 1)/\mathbb{P}(V)[-2\dim{V}]\).
Under good conditions, this definition satisfies all the usual properties, in particular homotopy invariance and cancellation. (Here “good conditions” is fairly restrictive compared to the setup above, in particular the authors need to assume that \(G = (\mathbb{Z}/2)^n\) for the strongest results.) The proofs are adaptations of those of Suslin-Voevodsky. Those adaptations are by no means trivial, and the entire paper is highly technical.

MSC:

14F42 Motivic cohomology; motivic homotopy theory
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
14C15 (Equivariant) Chow groups and rings; motives
55N91 Equivariant homology and cohomology in algebraic topology
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