Higher arithmetic Chow groups (Q439889)

To every regular, flat and quasi-projective scheme \(X\) over an arithmetic ring, which is called an arithmetic variety, Gillet-Soulé gave a definition for the arithmetic Chow groups \(\widehat{\text{CH}}^p(X)\) whose elements are classes of \(p\)-cycles and corresponding Green currents. Later, Burgos Gil gave an alternative definition of \(\widehat{\text{CH}}^p(X)\) involving the Deligne complex of differential forms with logarithmic singularities along infinity \(\mathcal{D}_{\log}^*(X,p)\). Burgos Gil's construction naturally isomorphic to Gillet-Soulé's in the proper case. Moreover, Burgos Gil proved that \(\widehat{\text{CH}}^p(X)\) fit into an exact sequence: NEWLINE\[NEWLINE \text{CH}^{p-1,p}(X) \to \mathcal{D}_{\log}^{2p-1}(X,p)/{\text{im} d_\mathcal{D}} \to \widehat{\text{CH}}^p(X) \to \text{CH}^p(X) \to 0NEWLINE\]NEWLINE where \(\rho: \text{CH}^{p-1,p}(X) \to \mathcal{D}_{\log}^{2p-1}(X,p)/{\text{im} d_\mathcal{D}}\) is the Beilinson regulator. Assume that \(X\) is proper and defined over an arithmetic field, the arithmetic Chow groups have been extended to higher degrees by Goncharov, they fit into a long exact sequence of the form NEWLINE\[NEWLINE \cdots \to \widehat{\text{CH}}^p(X,n) \to \text{CH}^p(X,n) \to \text{H}_\mathcal{D}^{2p-n}(X,\mathbb{R}(p)) \to \widehat{\text{CH}}^p(X,n-1) \to \cdotsNEWLINE\]NEWLINE NEWLINE\[NEWLINE \text{CH}^p(X,1) \to \mathcal{D}_{\log}^{2p-1}(X,p)/{\text{im} d_\mathcal{D}} \to \widehat{\text{CH}}^p(X) \to \text{CH}^p(X) \to 0. \quad (*)NEWLINE\]NEWLINENEWLINENEWLINEThe aim of the article under review is to give a new construction of \(\widehat{\text{CH}}^p(X,n)\) for quasi-projective arithmetic varieties over a field. The authors' definition agrees with the higher arithmetic Chow group defined by Goncharov for projective arithmetic varieties over a field and it admits several satisfying properties:NEWLINENEWLINE(i). if \(X\) is proper, \(\widehat{\text{CH}}^p(X,0)\) agrees with Gillet-Soulé's arithmetic Chow group;NEWLINENEWLINE(ii). \(\widehat{\text{CH}}^p(X,n)\) fit into the long exact sequence (*);NEWLINENEWLINE(iii). \(\widehat{\text{CH}}^p(X,n)\) admit contravariant functoriality for any morphism;NEWLINENEWLINE(iv). let \(\pi: X\times \mathbb{A}^m\to X\) be the natural projection, then the pull-back \(\pi^*: \widehat{\text{CH}}^p(X,n)\to \widehat{\text{CH}}^p(X\times \mathbb{A}^m,n)\) is an isomorphism;NEWLINENEWLINE(v). there exists an associative and graded commutative product on NEWLINE\[NEWLINE\widehat{\text{CH}}^*(X,*):=\bigoplus_{p\geq0,n\geq0}\widehat{\text{CH}}^p(X,n).NEWLINE\]