ScalingStacks

Next, we recall the generalization of this construction to connective ring spectra. Prior to the invention of modern notions of structured ring spectra, May initiated the study of the algebraic $K$-theory of a multiplicative object called an “$A_{\infty}$ ring space”, which is an $E_{\infty}$ space with a suitably compatible $A_{\infty}$ multiplication (for a particular pair of operads) [55, 72]. The prototype example of an $A_{\infty}$ ring space is $\Omega^{\infty}R$ for a connective ring spectrum $R$ [55, 3.1]. Fiedorowicz, Schwänzl, Steiner, and Vogt [33] extended Wagoner’s constructions by defining $mR$ and $\ell R$ for $A_{\infty}$ ring spaces (using the work of [72] to define matrices with entries in $A_{\infty}$ ring spaces), and then defining $\mu R$ to be the homotopy cofiber of the inclusion $mR\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\ell R$. Furthermore, they prove that there is an equivalence of spaces

(9.37)

$\displaystyle K_{0}(\pi_{0}(R))\times BGL^{+}(R)\simeq\Omega(K_{0}(\pi_{0}(\mu R))\times BGL^{+}(\mu R)).$

These constructions then allow a definition of the non-connective algebraic $K$-theory of an $A_{\infty}$ ring space $R$ with spaces

(9.38)

$\displaystyle I\mspace{-6.mu}K(R)_{n}=K_{0}(\mu^{n}\pi_{0}R)\times BGL^{+}(\mu^{n}R).$

This definition implies that for an $A_{\infty}$ ring space $R$, the natural map $R\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\pi_{0}R$ induces an isomorphism on the algebraic $K$-groups $I\mspace{-6.mu}K_{-n}(R):=K_{0}(\mu^{n}\pi_{0}R)$ for $n\geq 0$ [33, 1.1].