On the \(K\)-theory spectrum of a smooth curve over a finite field (Q676989)

The authors define a certain Iwasawa module associated to a curve \(X\) over a finite field \(\mathbb{F}\) (3.14). This module appears in the formulation of the main results (theorems 5.1 and 5.2). Here they calculate \(\widehat {\mathcal K}^*(KX)= \widehat {\mathcal K}^* (\widehat KX)\) as a module over the ring of cohomology operations in \(\widehat {\mathcal K}^*\), where \(K X\) denotes an algebraic \(K\)-theory spectrum associated to the category of locally free sheaves on \(X\), \(\widehat KX\) denotes the \(\ell\)-completion of \(KX\) \((\ell\) an odd rational prime different from \(\text{char} \mathbb{F})\), \(\widehat {\mathcal K}\) denotes the \(\ell\)-completion of the periodic topological complex \(K\)-theory spectrum, and \(\widehat {\mathcal K}^*\) is the associated cohomology theory. Calculation of the homotopy type of the Bousfield localisation with respect to \(\widehat {\mathcal K}^*\) (theorem 7.5) is one of the consequences of the main results.