Formal Chow groups, \(p\)-divisible groups and syntomic cohomology.
In the paper under review, the author studies the infinitesimal behavior of the Chow groups \(\text{CH}^r(V)\) of a smooth projective variety \(V\) defined over an absolutely unramified complete discrete valuation field \(K\) of characteristic \(p> 0\) with perfect residue field \(k\). This topic is of particular interest in the context of higher algebraic \(K\)-theories and various \(p\)-adic cohomology theories, where both its significance and a first methodological approach to tackling the involved problems were pointed out by \textit{S. Bloch} in the early 1980s [in: Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. I: Arithmetic, Prog. Math. 35, 13--26 (1983; Zbl 0584.14009)]. In the present work, using the theory of Dieudonné modules over rings of Witt vectors, the theory of \(p\)-divisible groups, \(K\)-cohomological functors, and the framework of the syntomic cohomology theory of Fontaine-Messing and of \textit{K. Kato} [Bull. Soc. Math. Fr. 119, 397--441 (1991; Zbl 0752.14015)], the author develops S. Bloch's approach towards the infinitesimal study of \(p\)-adic Chow groups essentially further, with several deep functorial comparison theorems as breaking new results. More precisely, denote by \({\mathcal K}^M_{r,V}\) the Zariski sheaf of Milnor \(K_r\)-groups on the variety \(V\). Then the so-called Bloch-Quillen formula establishes an isomorphism \[ \text{CH}^r(V)\otimes \mathbb{Z}\Biggl[{1\over (r- 1)!}\Biggr]\cong H^r(V,{\mathcal K}^M_{r,V})\otimes \mathbb{Z}\Biggl[{1\over (r- 1)!}\Biggr], \] which is the starting point ofe the author's refined investigations. Assuming that the variety \(V\) admits a smooth projective model over the valuation ring \(W\) of \(K\), that is, over the ring \(W= W(k)\) of Witt vectors of the residue field \(k\), the category \({\mathcal A}\) of Artinian local \(W\)-algebras with \(k\) as their residue field becomes an appropriate tool for studying the deformation behavior of \(V\) and its Milnor \(K_r\)-sheaves. Namely, the author introduces, for any projective smooth scheme \(X\) over \(W\) and for any integers \(r>0\) and \(q\geq 0\), the \(K\)-cohomological functors \(\widehat H^q({\mathcal K}^M_{r,X})\) on \({\mathcal A}\), which are defined by \[ \widehat H^q({\mathcal K}^M_{r,X})(A):= H^q(X\bigotimes_W A,\ker({\mathcal K}^M_{r,X\bigotimes_W A}\to{\mathcal K}^M_{r,X_k})) \] for \(A\in \text{obj}({\mathcal A})\), andd which are regarded as formal completions of the respective Chow groups \(\text{CH}^r(V)\) in the case \(X= V\). The philosophy behind this construction is that these functors should reflect the infinitesimal behavior of \(\text{CH}^r(V)\) at the origin in the deformation space, and the main goal of the present paper is therefore to describe these functors more explicitely, in particular when \(q= \dim V\). This is achieved by relating the functors \(\widehat H^q({\mathcal K}^M_{r,X})\) to functors of \(p\)-divisible groups over \(W\), to suitable relative syntomic cohomology functors (with respect to Dieudonné modules), and by establishing certain spectral sequences for the latter. The author's fine analysis leads to various new aspects in the \(p\)-adic deformation theory of higher Chow groups of special projective varieties, in particular with a view toward an earlier, rather different approach by \textit{J. Stienstra} [J. Reine Angew. Math. 355, 1--66 (1985; Zbl 0545.14013)] using the theory of the formal de Rham-Witt complex.
scientific article; zbMATH DE number 1436094
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