Duality and the topological filtration (Q383592)

Let \(\mathrm{Ch}\) be the Chow group modulo two and \(\widetilde{\mathrm{Ch}}\) its quotient group modulo the image of torsion integral cycles. In the paper the author constructs an operation \(Sq_{1}: \mathrm{Ch}_{p}\rightarrow \mathrm{Ch}_{p-1}\). This is in fact a generalization of the result in [Am. J. Math. 135, No. 1, 53--63 (2013; Zbl 1267.14012)], where the author constructed \(Sq_{1}: \mathrm{Ch}_{p}\rightarrow {\widetilde{\mathrm{Ch}}_{p-1}}\). The author proves several properties of this construction such as the descent of the \(Sq_{1}\) to a map \({\mathbb Z}/2 {\otimes} gr_{p} K_{0}^{\prime} \rightarrow {\mathbb Z}/2 {\otimes} gr_{p-1} K_{0}^{\prime}.\) The group \(\, K_{0}^{\prime}\) (for a noetherian scheme \(X\)) is the \(0\)-th \(K\)-theory group of the category \({\mathcal M}(X)\) of coherent \({\mathcal O}_{X}\)-modules and the filtration is the topological filtration modulo two. The application of \(Sq_{1}\) yields {2}-torsion elements in a large class of quadrics. Lifting the involution \({\psi}_{-1}\) induced by the duality on \(K\)-theory to algebraic connective \(K\)-theory allows the author to define another operation \({\tau}_{-1}\) on connective \(K\)-theory and establish its properties.