Rost nilpotence and higher unramified cohomology
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 universally supported in dimension \(\leq 2\) in the sense of [\textit{S. Bloch} and \textit{V. Srinivas}, Am. J. Math. 105, 1235--1253 (1983; Zbl 0525.14003)], then \(X\) satisfies the Rost nilpotence principle.
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