Rigid cohomology and de Rham-Witt complexes (Q1947826)

Let \(k\) be a perfect field of characteristic \(p>0\), \(W\) the ring of Witt vectors of \(k\) and \(K\) the fraction field of \(W\). Let \(X\) be a proper and smooth scheme over \(k\). There is a canonical comparison isomorphism between the crystalline cohomology and the cohomology of the de Rham-Witt complex: \[ H^*_{\mathrm{crys}}(X/W)_{K} \simeq H^*(X,W\Omega^{\bullet}_{X,K}), \] where the subscript \(K\) denotes tensorisation with \(K\) (see [\textit{S. Bloch}, Publ. Math., Inst. Hautes Étud. Sci. 47, 187--268 (1977; Zbl 0388.14010)] and [\textit{L. Illusie}, Ann. Sci. Éc. Norm. Supér. (4) 12, 501--661 (1979; Zbl 0436.14007)]). The aim of the article is to generalize this result to the case where \(X\) is only assumed to be separated of finite type over \(k\), replacing the groups \(H^*_{\mathrm{crys}}(X/W)_{K}\) by the groups of rigid cohomology with compact supports \(H^*_{\mathrm{rig},c}(X/K)\) (which coincide with the former when \(X\) is proper and smooth). Let us assume that \(X\) is proper over \(k\) (the general case reduces to this one by standard arguments) and consider a closed embedding of \(X\) into a smooth scheme \(Y\) of finite type over \(k\). The author proves that there exists a \(W \mathcal{O}_{Y,K}\)-algebra \(\mathcal{A}^W_{X,Y}\) on \(Y\) supported in \(X\) endowed with a de Rham-Witt connection and a canonical isomorphism \[ H^*_{\mathrm{rig}}(X/K) \simeq H^*(Y,\mathcal{A}^W_{X,Y} \hat{\otimes}_{W \mathcal{O}_{Y}} W\Omega^{\bullet}_{Y}). \] The paper also provides an integral version of this isomorphism. One of the key points is a comparison isomorphism similar to the first one mentioned here with \(X\) smooth and coefficients in a crystal \(\mathcal{E}\) in \(\mathcal{O}_{X/W}\)-modules that is flat and quasi-coherent over~\(W\). This generalizes previous results where \(\mathcal{E}\) was assumed to be flat over \(\mathcal{O}_{X/W}\) (see [\textit{J.-Y. Etesse}, Compos. Math. 66, No. 1, 57--120 (1988; Zbl 0708.14013)] and [\textit{A. Langer} and \textit{T. Zink}, J. Inst. Math. Jussieu 3, No. 2, 231--314 (2004; Zbl 1100.14506)]).