\(K\)-theory as an Eilenberg-Mac Lane spectrum (Q273865)

Let \(k\) be a commutative ring. The algebraic \(K\)-theory spectrum of a \(k\)-linear Waldhausen category is a module spectrum over the \(K\)-theory spectrum of \(k\); after suitable localisation, it may become an Eilenberg-MacLane spectrum and can thus be described in terms of chain complexes rather than spectra.NEWLINENEWLINEFor \(R\) a localisation of the integers in a set of primes, the commutative ring \(k\) is called \textit{\(R\)-adapted} if \(K_q(k) \otimes R = 0\) for \(q \geq 1\), and if there is a specified ring isomorphism \(K_0(k) \otimes R \cong R\).NEWLINENEWLINEGiven an \(R\)-adapted commutative ring \(k\) and a \(k\)-linear Waldhausen category \(\mathcal{C}\), the author constructs a chain complex \(L\) that models the algebraic \(K\)-theory of \(\mathcal{C}\) up to \(R\)-homology. More precisely, with \(\mathcal{K}^R(\mathcal{C},k)\) denoting the Eilenberg-MacLane spectrum associated to \(L\), the author constructs a natural map \(K(\mathcal{C}) \rightarrow \mathcal{K}^R(\mathcal{C},k)\) from the \(K\)-theory spectrum of \(\mathcal{C}\) which induces an isomorphism on homology with coefficients in \(R\).NEWLINENEWLINEThe main example is that of \(k\) being a finite field of characteristic \(p\) and \(R = \mathbb{Z}_{(p)}\).NEWLINENEWLINEAbout six pages of the paper are devoted to the introduction and a presentation of ``heuristics'', illuminating basic ideas and putting the constructions into context. The remainder of the paper deals with precise statements of the results, and carefully executed proofs thereof.