A coproduct obstruction for derived unramified cohomology

David Kumallagov

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arXiv · 2606.15451

A coproduct obstruction for derived unramified cohomology

David Kumallagov

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Abstract

Let $k$ be a perfect field of exponential characteristic $p$, and let $R$ be a commutative $\Z[1/p]$-algebra. We prove that the first derived unramified functor \[ F\longmapsto R^1_{\mathrm{nr},R}F \] from homotopy invariant Nisnevich sheaves with transfers of $R$-modules to birational sheaves commutes with arbitrary small direct sums. This gives a positive answer, after inverting the exponential characteristic, to a question of Kahn and Sujatha; on smooth projective varieties no inversion is needed. We also describe an obstruction to this for the functor $R^2_{\mathrm{nr},R}$ in categorical terms, which includes the familiar Griffiths group obstruction. As applications of the motivic nature of the functors $R^q_{\mathrm{nr}}$, we prove torsion-order bounds and a correspondence-detection statement for surfaces.

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David Kumallagov. 2026-06-13. A coproduct obstruction for derived unramified cohomology. https://arxiv.org/abs/2606.15451

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