Rost nilpotence and higher unramified cohomology
type: Item
instance of: scholarly article
title: Rost nilpotence and higher unramified cohomology
zbMATH Open document ID: 1436.14011
author: H. Anthony Diaz
published in: Advances in Mathematics
publication date: 2019-09-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1807.08163
review text: Let \(X\) be a smooth projective scheme of dimension \(d\) over a perfect field \(k\) of characteristic \(p>0\). Let \(\mathcal{M}_k\) denote the category of Chow motives over \(k\) and the corresponding Chow motive of \(X\) by \(M(X)\). The Rost nilpotence principle predicts that for any field extension \(E/k\), the kernel of \(\mathrm{End}_{\mathcal{M}_k}(M(X))\rightarrow\mathrm{End}_{\mathcal{M}_E}(M(X_E))\) consists of nilpotent correspondences. The main result of the present article shows that a correspondence \(\gamma\) in the kernel is nilpotent if the action of \(\gamma\) on certain higher unramifield cohomology groups is nilpotent. The cohomology group mentioned here is defined as follows. For any positive integer \(r\), let \(W_r\Omega_{X,\log}^*\) be the logarithmic Hodge-Witt complex over \(X\) defined in [\textit{L. Illusie}, Ann. Sci. Éc. Norm. Supér. (4) 12, 501--661 (1979; Zbl 0436.14007)]. Let \((\mathbb{Z}/p^r)_X(n)\) denote the étale sheafification of \(W_r\Omega_{X,\log}^*[-n]\) and let \(\mathcal{H}_X^q(\mathbb{Z}/p^r(n))\) denote the Zariski sheaf associated to the presheaf over \(X\) given by \(U\mapsto H_{\text{ét}}^q(U,\mathbb{Z}_p^r(n))\). The corresponding Zariski cohomology groups \(H_{\mathrm{Zar}}^i(X,\mathcal{H}_X^q(\mathbb{Z}/p^r(n)))\) are what the author refers to as the higher unramified cohomology groups. The main theorem shows that, under the assumption (*) \(H_{\mathrm{Zar}}^{d-3}(X\times X,\mathcal{H}_{X\times X}^{d+1}(\mathbb{Z}/p^r(d)))=0\), if \(\gamma\in \text{CH}^d(X\times X)[p^{\infty}]\) satisfies \(\gamma_{\overline{k}}=0\), then \(\gamma^n=0\) in \(Cor^0(X,X)\) for \(n\gg 0\). Since the assumption (*) is automatically satisfied for surfaces, the theorem shows the validity of the Rost nilpotence principle for any surfaces. Furthermore, using a Bloch-Srinivas-type argument, he proves the following: If \(X\) is a smooth projective scheme of dimension \(\leq 3\) over a perfect field whose Chow group \(\text{CH}_0\) is univ…
reviewed by: Fumio Hazama
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 19D45
zbMATH DE Number: 7107217
zbMATH Keywords: algebraic cycles
zbMATH Keywords: unramified cohomology
zbMATH Keywords: Rost nilpotence
MaRDI profile type: Publication
OpenAlex ID: W2969808942
arXiv ID: 1807.08163
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Wikidata QID: Q127334199
DOI: 10.1016/J.AIM.2019.106770
description: scientific article