A case of the deformational Hodge conjecture via a pro Hochschild-Kostant-Rosenberg theorem (Q2445180)

Consider a proper smooth schemes \(X\) over \(K[[t]]\), where \(K\) is an algebraic extension of \(\mathbb Q\). Let \(Y\) be the closed fibre, and let \(Y_r\) be its \(r^{th}\) infinitesimal thickening. The homology class \(\zeta\) of an algebraic cycle of codimension \(p\) on \(Y\) is an element in \(F^pH^{2p}_{dR}(Y/K)\). The deformational Hodge conjecture can be interpreted as the question whether the condition that \(\zeta \) stays infinitesimally in \(F^p\) implies that there is a family of cycles on the system of schemes \(Y_r\) which lift, up to some equivalence relation, the cycle in the central fibre. The author shows that such is the case. More formally, he proves that the Chern character map produces a commutative diagram containing the two exact rows \[ \varprojlim_r K_0(Y_r) \longrightarrow K_0(Y) \longrightarrow \varprojlim_rK_{-1}(Y_r,Y) \] \[ \bigoplus_pF^p_{flat}H^{2p}_{dR}(Y/K) \hookrightarrow \bigoplus_pF^pH^{2p}_{dR}(Y/K) \to \bigoplus_p\varprojlim_r\mathbb H^{2p+1}(Y_r,\Omega_{(Y_r,Y)/K}^{\geq p}) \] Here \(F^p_{flat}H^{2p}_{dR}(Y/K)\) denotes the classes of \(H^{2p}_{dR}(Y/K)\) whose flat lift to \(H^{2p}_{dR}(X/A)\) belongs to \(F^pH^{2p}_{dR}(X/A)\) and \(\Omega_{(Y_r,Y)/K}^\bullet:=\ker(\Omega_{Y_r/K}^\bullet\to \Omega_{Y/K}^\bullet)\). The left square is cocartesian while the groups on the right are isomorphic, this yields: Theorem. Given an element \(z\in K_0(Y)\), the Grothendieck group, the following are equivalent: (i) \(z\) lifts to \(\varprojlim_rK_0(Y_r)\) (ii) The de Rham class \(ch(z)_p \in F^pH_{dR}^{2p}(Y/K)\) belongs to \(F^p_{flat}H_{dR}^{2p}(Y/K)\) for \(p=0,\dots,\dim Y\). The main idea for the construction of the diagram is to appeal to Goodwillie's Chern character from \(K\)-theory to negative cyclic homology, and to use his isomorphism theorem [\textit{T. G. Goodwillie}, Ann. Math. (2) 124, 347--402 (1986; Zbl 0627.18004)]. Working with the couples \((Y_r,Y\)) one finds a diagram of pro abelian group. To complete the proof it is necessary to invoke a pro Hochschild--Kostant--Rosenberg theorem for Hochschild homology, which is contained in a preprint due to the author, \textit{Pro unitality and pro excision in algebraic {\(K\)}-theory and cyclic homology.}