ScalingStacks

In this section, we show that for a connective ring spectrum $R$, the non-connective $K$-theory spectrum we associate to the category of perfect $R$-modules has negative homotopy groups determined by the classical non-connective $K$-theory spectrum of the ring $\pi_{0}R$.

We give a proof using a model of non-connective $K$-theory for connective ring spectra based on the construction of a “suspension ring spectrum” coupled with Quillen’s plus construction. We begin by recalling Wagoner’s construction [83] of the non-connective $K$-theory of an ordinary ring $R$. Given a ring $R$, we let $\ell R$ denote the ring of locally finite (countably) infinite matrices in $R$ — i.e., ${\mathbb{N}}\times{\mathbb{N}}$ matrices such that each row and column only has finitely many nonzero elements. We let $mR$ denote the finite matrices, regarded as a 2-sided ideal of $\ell R$ — these are the matrices with only finitely many nonzero elements. Then we can form the quotient ring $\mu R=\ell R/mR$, and Wagoner defines the non-connective $K$-theory spectrum to have $n$th space

It is known that this construction agrees with other possible constructions of the non-connective algebraic $K$-theory spectrum of $R$ (e.g., see [63, §6]).

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

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

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].

Our approach involves constructing a variant of the “suspension ring” construction that allows a construction of a non-connective $K$-theory spectrum which agrees with the non-connective $K$-theory of the ring space $\Omega^{\infty}R$ as defined in equation 9.38 on $\pi_{i}$ for $i<0$ and is equivalent to our version of the the non-connective $K$-theory $I\mspace{-6.mu}K(R):=I\mspace{-6.mu}K(\widehat{R}_{\perf})$ spectrum constructed in definition 9.6. Since $\pi_{0}\Omega^{\infty}R\cong\pi_{0}R$, this equivalence implies the desired comparison.

We begin by recalling the definition of the plus construction introduced in [86], extended to $A_{\infty}$ ring spectra. For convenience, we model $A_{\infty}$ ring spectra as EKMM $S$-algebras. We write $M_{n}R=\map_{R}(R^{\lor n},R^{\lor n})$ for the space of $R$-module endomorphisms of (a cofibrant replacement of) $R^{\lor n}$, and write $GL_{n}(R)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M_{n}(R)$ for the full subspace of $R$-module automorphisms of $R^{\lor n}$; that is, we have a (homotopy) pullback of spaces

Since $GL_{n}(R)$ is a topological monoid, after replacing to ensure the inclusion of the unit is a cofibration, we can form its classifying space $BGL_{n}(R)$. Moreover, there are natural inclusions $GL_{n}(R)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}GL_{n+1}(R)$ which induce maps $BGL_{n}(R)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}BGL_{n+1}(R)$. We can form

Since $\pi_{1}BGL(R)\cong GL(\pi_{0}R)$, we can form the plus construction $BGL(R)^{+}$, and one could define the $K$-theory space to be the infinite loop space $K_{0}(\pi_{0}R)\times BGL(R)^{+}$. The consistency of this definition is proved in [32, 7.1], which we restate below:

This is consistent in the sense that a check of the definition of the plus construction for an $A_{\infty}$ space [55, §7] now yields the following proposition:

We now set up analogues of the constructions of [33]. In order to ensure that our mapping spaces and spectra have the correct homotopy type, we continue to work with the category of EKMM algebra and module spectra. Since all objects are fibrant, it then suffices to work with cofibrant modules. For a connective ring spectrum $R$, in the following we let $\mathrm{Map}_{R}(x,y)$ denote the mapping spectrum between objects $x$ and $y$ and $\map_{R}(x,y)$ the mapping space (which can be computed as $\Omega^{\infty}\mathrm{Map}(x,y)$) in the category of $R$-modules. Moreover, when we write $R^{\lor n}$ inside a mapping object, we will tacitly mean the wedge of a cofibrant replacement of $R$ as an $R$-module.

We now begin to prove the comparison theorem, theorem 9.53 below. As explained in [10, §15], without loss of generality we can work with categories enriched in EKMM $S$-modules as a model for spectral categories, and we tacitly move between categories enriched in EKMM $S$-modules and categories enriched in symmetric spectra in the following discussion.

Let $F_{R}$ denote the spectral category of finitely generated free $R$-modules. The theorem follows from proposition 9.51, which depends on the existence of a spectral category $\tilde{F}^{\infty}_{R}$, equipped with a homotopically fully faithful spectral functor $F_{R}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\tilde{F}^{\infty}_{R}$, whose $\infty$-category of modules has a generator $G$ such that $\pi_{0}$ of the endomorphism ring spectrum of the image of G in the quotient category $\Psi(\tilde{F}^{\infty}_{R})/\Psi(F_{R})$ is $\mu(\pi_{0}\Omega^{\infty}R)$. This approach to constructing analogues of $\mu(\Omega^{\infty}R)$ is motivated by the explicit description of mapping spectra in the stable quotient (see [23, 1.3] for the dg-case and [10, §6] for the spectral analogue) and an idea from [63, 6.1].

We begin by giving a particular construction of such a spectral category. Roughly speaking, the idea is to adjoin the object $R^{\lor\infty}$ to $F_{R}$ in such as way that the inclusion $F_{R}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\tilde{F}^{\infty}_{R}$ is fully faithful and $\Psi(\tilde{F}^{\infty}_{R})$ is generated by an object $G=R^{\lor\infty}$ such that $\pi_{0}\End_{\tilde{F}^{\infty}}(G)\cong\ell(\pi_{0}\Omega^{\infty}R)$.

Recall that we denote by $\widehat{R}$ the category of $R$-modules, which we can regard as a spectral category. Let $F_{R}$ denote the full spectral subcategory of $\widehat{R}$ spanned by the finite free $R$-modules $R^{\lor n}$, $n\in\mathbb{N}$, and let $F^{\infty}_{R}$ denote the full spectral subcategory of $\widehat{R}$ spanned by the finite free $R$-modules as well as the countable wedge $R^{\lor\infty}\simeq\colim_{n}R^{\lor n}$. The inclusion gives a fully faithful spectral functor $i\colon F_{R}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F^{\infty}_{R}$. The spectral category $F^{\infty}_{R}$ is an intermediate construction that we will use to construct $\tilde{F}_{R}^{\infty}$.

Write $\Psi(F_{R})$ and $\Psi(F^{\infty}_{R})$ for the presentable stable $\infty$-categories of $F_{R}$-modules and $F^{\infty}_{R}$-modules, and let $i_{!}\colon\Psi(F_{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi(F^{\infty}_{R})$ denote the left adjoint of the restriction $i^{*}\colon\Psi(F^{\infty}_{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi(F_{R})$. Given an $R$-algebra $A$, we will also write $\Psi(A)$ for the stable $\infty$-category of $A$-modules.

Since $i_{!}$ is fully faithful, we have an exact sequence of presentable stable $\infty$-categories

where ${\mathcal{C}}$ denotes the cofiber of $i_{!}$. We can regard ${\mathcal{C}}$ as the full subcategory of $\Psi(F^{\infty}_{R})$ spanned by the local objects. In mild abuse of notation, for each $0\leq n\leq\infty$, we will write $R^{\lor n}$ for the $F^{\infty}_{R}$-module represented by $R^{\lor n}$.

Since $i^{*}i_{!}$ is equivalent to the identity, the counit map $i_{!}i^{*}R^{\lor\infty}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}R^{\lor\infty}$ restricts to an equivalence of $F_{R}$-modules. However, it is not an equivalence of $F^{\infty}_{R}$-modules, since not all endomorphisms of $R^{\lor\infty}$ (e.g., the identity) factor through $i_{!}i^{*}R^{\lor\infty}$.

The following proposition is standard; we restate it for convenience.

As a consequence, we can identify a compact generator of ${\mathcal{C}}$.

This suggests that we might consider $\End_{{\mathcal{C}}}(G)$, regarded as an $A_{\infty}$ ring spectrum under composition, as an analogue of $\hat{\mu}R$. Note that by corollary 9.46, $\End_{{\mathcal{C}}}(G)\simeq\mathrm{Map}_{F^{\infty}R}(R^{\lor\infty},G)$ is equivalent to the cofiber (in spectra) of the map

However, since $\pi_{0}(\mathrm{Map}_{F^{\infty}_{R}}(R^{\lor\infty},R^{\lor\infty}))$ can be identified as the collection of infinite matrices with values in $\pi_{0}(R)$ that have finitely many elements per row, we need to perform a construction analogous to definition 9.41.

Let

denote the subcategory of $\pi_{0}F_{R}^{\infty}$ consisting of those maps

for $0\leq m,n\leq\infty$, which are locally finite when regarded as elements of the group of $\pi_{0}R$-valued $m\times n$-matrices. Since the composition induces on $\pi_{0}$ the product of matrices and products of locally finite matrices are locally finite, this specification does indeed define a subcategory of $\pi_{0}F_{R}^{\infty}$. Furthermore, $\pi_{0}\tilde{F}_{R}^{\infty}$ inherits an enrichment over abelian groups from that of $\pi_{0}F_{R}^{\infty}$. We now perform a categorical analogue of definition 9.41, using the Eilenberg-Mac Lane functor $H$ from categories enriched in abelian groups to spectral categories [69, 5.1.5].

Using this we obtain a morphism of spectral categories

Note that for $0\leq m,n<\infty$, the induced map of Eilenberg-Mac Lane spectra

is an equivalence, as finite matrices are locally finite.

We now define spectral categories $\tilde{F}^{\infty}_{R}$ and $\tilde{F}_{R}$ as the homotopy pullbacks

Observe that $\tilde{F}_{R}$ and $\tilde{F}^{\infty}_{R}$ have the same objects as $F_{R}$ and $F^{\infty}_{R}$, respectively, but $\End_{\tilde{F}^{\infty}_{R}}(R^{\lor\infty})\simeq LR$.

The spectral functor $\tilde{F}^{\infty}_{R}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F^{\infty}_{R}$ induces a functor (which is not fully faithful)

on $\infty$-categories of modules.

Carrying out the same analysis as above, we see that the quotient

can be described as modules over $\End_{{\mathcal{C}}^{\prime}}(G^{\prime})$, where $G^{\prime}$ is the cofiber of the map

(here $R^{\lor\infty}$ is regarded as an object of $\tilde{F}_{R}^{\infty}$) and hence as a spectrum $\End_{{\mathcal{C}}^{\prime}}(G^{\prime})$ is equivalent to the cofiber in spectra of the map

Based on this, we define

using the setup described above, and we proceed to relate this suspension ring spectrum construction to an $\infty$-categorical delooping. The basic idea is that our constructions of the suspension rings give (smaller) models of the $\infty$-categorical cone ${\mathcal{F}}_{\kappa}$ from definition 9.1 which are more closely related to the suspension ring spectrum $\hat{\mu}R$.

Proposition 9.51 allows us finally to establish the desired result.