Section 9. (0NSF) · ScalingStacks

9. Non-connective $K$ -theory

Bass introduced the negative $K$ -groups in order to measure the failure of $K_{0}$ and $K_{1}$ to satisfy localization; this perspective was studied in detail in Thomason-Trobaugh and led to the definition of the Bass-Thomason non-connective $K$ -theory spectrum of rings and schemes. In fact, any nontrivial theory which is “like $K$ -theory” and satisfies localization must be non-connective; there is a nice discussion of this in [48]. In this section we introduce the non-connective algebraic $K$ -theory of $\infty$ -categories and show that it becomes co-representable in ${\mathcal{M}}_{\mathrm{loc}}$ ; see theorem 9.8. This result depends critically on the multistage construction of Section 8, which again follows the general pattern of the argument for dg-categories in [21]. For a connective ring spectrum $R$ , we give a slightly different definition of the non-connective $K$ -theory in terms of a “suspension ring spectrum” of $R$ , and use this show that the negative $K$ -groups of $R$ are isomorphic to those of $\pi_{0}R$ .

9.1. Non-connective $K$ -theory of $\infty$ -categories

In order to construct the non-connective $K$ -theory spectrum associated to a small stable $\infty$ -category, we use a generalization of the axiomatic framework due to Schlichting [70]. For an uncountable regular cardinal $\kappa$ , we will produce functors ${\mathcal{F}}_{\kappa}$ and $\Sigma_{\kappa}$ from $\Cat_{\infty}^{\ex}$ to $\Cat_{\infty}^{\ex}$ such that for any small stable $\infty$ category ${\mathcal{A}}$ :

(i)

$K({\mathcal{F}}_{\kappa}{\mathcal{A}})$ is contractible,
(ii)

there are natural transformations

$\Id\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{F}}_{\kappa}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Sigma_{\kappa}$

such that ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{F}}_{\kappa}{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Sigma_{\kappa}{\mathcal{A}}$ is exact,
(iii)

the functors ${\mathcal{F}}_{\kappa}$ and $\Sigma_{\kappa}$ preserve exact sequences,
(iv)

and ${\mathcal{F}}_{\kappa}$ and $\Sigma_{\kappa}$ preserve $\kappa$ -filtered colimits in $\Cat_{\infty}^{\ex}$ .

The idea is that ${\mathcal{F}}_{\kappa}{\mathcal{A}}$ is a “ $K$ -theoretic cone” and so $\Sigma_{\kappa}{\mathcal{A}}$ is a “suspension” of ${\mathcal{A}}$ . Fix an uncountable regular cardinal $\kappa$ , and for a stable $\infty$ -category ${\mathcal{C}}$ recall from Section 2.4 that ${\mathcal{C}}^{\kappa}$ denotes the $\kappa$ -compact objects in ${\mathcal{C}}$ .

0NP3

Definition 9.1. Using proposition 2.18, we define ${\mathcal{F}}_{\kappa}{\mathcal{A}}=(\Ind_{\omega}({\mathcal{A}}))^{\kappa}$ and $\Sigma_{\kappa}{\mathcal{A}}$ to be the cofiber $(\Ind_{\omega}({\mathcal{A}}))^{\kappa}/{\mathcal{A}}$ .

0NP4

Remark 9.2. One might wish to simply use $\Ind_{\omega}{\mathcal{A}}$ as the cone construction; however, this will rarely turn out to be a small $\infty$ -category, whereas passing to the $\kappa$ -compact objects yields an (essentially) small $\infty$ -category by construction.

Observe that ${\mathcal{F}}_{\kappa}$ is a composite functor

(9.3)

\Cat_{\infty}^{\ex}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St}}}_{\omega}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\ex(\!\kappa)}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\ex}.

By construction and Propositions 5.6 and 5.9, we have an exact sequence

{\mathcal{A}}\longrightarrow{\mathcal{F}}_{\kappa}{\mathcal{A}}\longrightarrow\Sigma_{\kappa}{\mathcal{A}},

which is natural in small stable $\infty$ -categories ${\mathcal{A}}$ . Next, we check that ${\mathcal{F}}_{\kappa}{\mathcal{A}}$ satisfies property (i) above.

0NP5

Lemma 9.4. Let ${\mathcal{A}}$ be a small stable $\infty$ -category. Then $K({\mathcal{F}}_{\kappa}{\mathcal{A}})$ is trivial.

0NP6

Proof. Since $\kappa$ is uncountable, ${\mathcal{F}}_{\kappa}{\mathcal{A}}$ has countable coproducts, and so the usual Eilenberg swindle argument implies that the identity map is null-homotopic on $K$ -theory and so its $K$ -theory vanishes. Specifically, the functor $F\colon{\mathcal{F}}_{\kappa}{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{F}}_{\kappa}{\mathcal{A}}$ defined by $X\mapsto\coprod_{{\mathbb{N}}}X$ is exact. Moreover, there is a natural equivalence of exact functors $\id\coprod F\simeq F$ induced by the equivalence $X\coprod(\coprod_{{\mathbb{N}}}X)\simeq\coprod_{{\mathbb{N}}}X$ . Applying $K$ -theory, we can split off the $F$ component of the resulting equivalence of spectra and deduce that the identity of ${\mathcal{F}}_{\kappa}{\mathcal{A}}$ is null-homotopic. ∎

We must check that ${\mathcal{F}}_{\kappa}$ and $\Sigma_{\kappa}$ preserve exact sequences of small stable $\infty$ -categories.

0NP7

Proposition 9.5. Let ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ be an exact sequence of small stable $\infty$ -categories. Then the induced sequences

are exact.

0NP8

Proof. It suffices to show the result for ${\mathcal{F}}_{\kappa}$ , as the statement for $\Sigma_{\kappa}$ follows because colimits commute. Thus, we need to verify that

(\Ind_{\omega}({\mathcal{A}}))^{\kappa}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\Ind_{\omega}({\mathcal{B}}))^{\kappa}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\Ind_{\omega}({\mathcal{C}}))^{\kappa}

is exact. The sequence

\Ind_{\omega}{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind_{\omega}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind_{\omega}{\mathcal{C}}

is exact by Definition 5.12 and Proposition 5.15. Now the result follows from Proposition 5.17. ∎

Passing to the triangulated homotopy category by composing with the functor $\Ho$ , we get a series of functors which satisfies Schlichting’s setup of [70, §2.2] and so produces negative $K$ -groups. Furthermore, we can define the non-connective $K$ -theory spectrum as follows, following [70, §12].

0NP9

Definition 9.6. Let ${\mathcal{A}}$ be a small stable $\infty$ -category. Its non-connective $K$ -theory spectrum $I\mspace{-6.mu}K({\mathcal{A}})$ is given by

I\mspace{-6.mu}K({\mathcal{A}}):=\colim_{n}\Omega^{n}K(\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\,.

Here, $K$ stands for the $K$ -theory spectrum of §7.1, and the structure maps are induced from the exact sequences

\Sigma_{\kappa}^{(n)}({\mathcal{A}})\longrightarrow{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}})\longrightarrow\Sigma_{\kappa}^{(n+1)}({\mathcal{A}})\qquad n\geq 0\,.

Schlichting’s axiomatic framework implies that this construction agrees with his when both are defined, and therefore we deduce from his comparison results [70, §8] that the non-connective $K$ -theory spectrum of Definition 9.6 agrees with the various classical constructions of non-connective $K$ -theory spectra.

Finally, we establish the final technical condition; this will be needed in the following sections.

0NPA

Lemma 9.7. The functors ${\mathcal{F}}_{\kappa}$ and $\Sigma_{\kappa}$ preserve $\kappa$ -filtered colimits.

0NPB

Proof. Recall that ${\mathcal{F}}_{\kappa}$ is the composite (9.3). Hence, the claim follows from the fact that the passage to $\Ind_{\kappa}$ and to $\kappa$ -compact objects preserves $\kappa$ -filtered colimits [52, 5.5.7.8, 5.5.7.10, 5.5.7.11]. Since $\Sigma_{\kappa}$ is the cofiber of the inclusion ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{F}}_{\kappa}$ and colimits commute, we deduce that $\Sigma_{\kappa}$ preserves $\kappa$ -filtered colimits if ${\mathcal{F}}_{\kappa}$ does. ∎

9.2. Co-representability

This subsection is entirely devoted to the proof of the following co-representability result.

0NPC

Theorem 9.8. Let ${\mathcal{A}}$ be a small stable $\infty$ -category. Then there is a natural equivalence of spectra

\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}({\mathcal{S}}_{\infty}^{\omega}),\,{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}))\simeq I\mspace{-6.mu}K({\mathcal{A}})\,.

In particular, for each integer $n$ , we have isomorphisms of abelian groups

\Hom({\mathcal{U}}_{\mathrm{loc}}({\mathcal{S}}_{\infty}^{\omega}),\,\Sigma^{-n}{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}))\simeq I\mspace{-6.mu}K_{n}({\mathcal{A}})

in the triangulated category $\Ho({\mathcal{M}}_{\mathrm{loc}})$ .

The proof of theorem 9.8 will follow from theorems 9.9 and 9.10, and from propositions 9.17 through 9.26.

0NPD

Theorem 9.9. Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be small stable $\infty$ -categories such that ${\mathcal{B}}$ is $\kappa$ -compact. Then there is a natural equivalence of spectra

\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{B}}),\,\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}}))\simeq K(\mathrm{Fun}^{ex}({\mathcal{B}},{\mathcal{A}}))\,.

If ${\mathcal{B}}={\mathcal{S}}_{\infty}^{\omega}$ is the $\infty$ -category of compact spectra, this reduces to an equivalence

\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\,\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}}))\simeq K({\mathcal{A}})\,.

0NPE

Proof. The proof is analogous to the argument for theorem 7.13; instead of the idempotent-complete stable $\infty$ -category $\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}}))$ we consider the small stable $\infty$ -category $\mathrm{Fun}^{\ex}({\mathcal{B}},{\mathcal{A}})$ . Note that since $\kappa>\omega$ , ${\mathcal{S}}_{\infty}^{\omega}$ belongs to $(\Cat_{\infty}^{\ex})^{\kappa}$ . ∎

0NPF

Theorem 9.10. Let ${\mathcal{A}}$ be a small stable $\infty$ -category. Then there is a natural equivalence of spectra

\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\,\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{A}}))\simeq K({\mathcal{A}})\,.

0NPG

Proof. By construction, the object $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega})$ is compact in $\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}$ . Let $S$ denote the set of maps in (8.4), $\overline{S}$ the strongly saturated collection of arrows generated by $S$ [52, 5.5.4.5], and let $X$ be an $S$ -local object such that the map $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$ is an $S$ -local equivalence (i.e., $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$ is in $\overline{S}$ ). Then by definition,

\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{A}}))\simeq\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),X),

so it suffices to show that the functor

(9.11)

R:=\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),-):\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}\longrightarrow{\mathcal{S}}_{\infty}

sends the maps in $\overline{S}$ to equivalences of spectra. Since $\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}$ is a stable $\infty$ -category and $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega})$ is compact, $R$ preserves small colimits, so the two-out-of-three property allows us to reduce to checking that $R$ sends the elements of $S$ to equivalences.

Consider the following diagram

(9.12)

By applying the functor (9.11) to the above diagram (9.12) we obtain by theorem 9.9 a diagram in $\mathcal{S}$

(9.13)

where the upper row is a homotopy cofiber sequence. Now, an argument analogous to the one used in the proof of proposition 7.19 (where we make use of Waldhausen’s fibration theorem) allow us to conclude that the lower row in the above diagram (9.13) is also a homotopy cofiber sequence. This completes the argument. ∎

Let ${\bf V}$ be the partially ordered set $\{(i,j):|i-j|\leq 1,\,i,j\geq 0\}\subset\mathbb{N}\times\mathbb{N}$ . Given a small stable $\infty$ -category ${\mathcal{A}}$ , we denote by $\mathrm{Dia}({\mathcal{A}})$ the $\mathrm{N}({\bf V})$ -diagram

(9.14)

where ${\mathcal{F}}_{\kappa}$ and $\Sigma_{\kappa}$ are as in Definition 9.1.

0NPH

Lemma 9.15. Let ${\mathcal{A}}$ be a small stable $\infty$ -category. Then $\Sigma_{\kappa}^{(n)}({\mathcal{A}})/\Sigma_{\kappa}^{(n)}({\mathcal{A}})$ and ${\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}})$ become trivial after application of $\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}$ .

0NPI

Proof. The object $\Sigma_{\kappa}^{(n)}({\mathcal{A}})/\Sigma_{\kappa}^{(n)}({\mathcal{A}})$ is already trivial in $\Cat_{\infty}^{\ex}$ . Since Proposition 2.18 implies that ${\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}})$ admits all $\kappa$ -small colimits, for any ${\mathcal{B}}$ in $(\Cat_{\infty}^{\ex})^{\kappa}$ the small stable $\infty$ -category $\mathrm{Fun}({\mathcal{B}},{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}}))$ also admits all $\kappa$ -small colimits. Thus, the connective $K$ -theory spectrum $K(\mathrm{Fun}({\mathcal{B}},{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}}))$ is trivial. Finally, theorem 9.9 and the fact that the objects $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{B}})$ , with ${\mathcal{B}}$ in $(\Cat_{\infty}^{\ex})^{\kappa}$ generate the category $\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}$ [52, 5.5.7.3] allow us to conclude that ${\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}})$ becomes trivial after application of $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}$ , and thus after application of $\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}$ . ∎

Let ${\mathcal{A}}$ be a small stable $\infty$ -category. We denote by $V({\mathcal{A}})$ the object

V({\mathcal{A}})=\colim_{n}\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{A}}))

in $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ whose indexing maps are induced from the above diagram (9.14). Note that $V({\mathcal{A}})$ is functorial in ${\mathcal{A}}$ and that we have a natural map $\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}V({\mathcal{A}})$ . We obtain then a well-defined functor $V$ along with a natural transformation:

(9.16)

\displaystyle V(-):\Cat_{\infty}^{\ex}\longrightarrow\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}

\displaystyle\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}\Rightarrow V(-)\,.

0NPJ

Proposition 9.17. Let ${\mathcal{A}}$ be a small stable $\infty$ -category. Then, there is a natural equivalence of spectra

\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\,V({\mathcal{A}}))\simeq I\mspace{-6.mu}K({\mathcal{A}})\,.

0NPK

Proof. This follows from the following equivalences

(9.18)	$\displaystyle I\mspace{-6.mu}K({\mathcal{A}})$	$\displaystyle=$	$\displaystyle\underset{n\geq 0}{\mathrm{hocolim}}\,\Omega^{n}K(\Sigma_{\kappa}^{(n)}({\mathcal{A}}))$
(9.18)		$\displaystyle\simeq$	$\displaystyle\underset{n\geq 0}{\mathrm{hocolim}}\,\Omega^{n}\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\,\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{A}})))$
(9.19)		$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty})^{\kappa}),\,\underset{n\geq 0}{\mathrm{colim}}\,\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{A}})))$
(9.19)		$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\kappa}),\,V({\mathcal{A}}))\,.$

Equivalence (9.18) comes from theorem 9.10 and equivalence (9.19) comes from the compactness of $\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega})$ in $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ . ∎

0NPL

Proposition 9.20. The functor $V$ (9.16) inverts Morita equivalences.

0NPM

Proof. It suffices to show that $V(-)$ sends maps of shape ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Idem({\mathcal{A}})$ to isomorphisms. Consider the following diagram

Proposition 2.18 implies that ${\mathcal{F}}_{\kappa}(P)$ is an equivalence. Therefore, since both rows are strict-exact sequences and $\Ho({\mathcal{A}})$ and $\Ho(\Idem({\mathcal{A}}))$ differ by direct summands, we conclude that $\Sigma_{\kappa}(P)$ is an equivalence. The definition of the functor $V(-)$ allow us to conclude the proof. ∎

0NPN

Proposition 9.21. The functor

V(-)\colon\Cat_{\infty}^{\ex}\longrightarrow\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}

inverts Morita equivalences, preserves $\kappa$ -filtered colimits, and sends exact sequences to cofiber sequences.

0NPP

Proof. Proposition 9.20 implies that $V(-)$ inverts Morita equivalences. Furthermore, by Lemma 9.7, ${\mathcal{F}}_{\kappa}$ and $\Sigma_{\kappa}$ preserve $\kappa$ -filtered colimits for $\kappa>\omega$ , and so $V(-)$ does as well. Now, let

{\mathcal{A}}\longrightarrow{\mathcal{B}}\longrightarrow{\mathcal{C}}

be an exact sequence. Proposition 9.20 implies that we can assume that $\Ho({\mathcal{A}})$ is a thick triangulated subcategory of $\Ho({\mathcal{B}})$ . Consider the following diagram

(9.22)

where $\mathrm{Dia}({\mathcal{A}},{\mathcal{B}})$ is obtained by passage to the cofiber objectwise. Note that since in the above diagram (9.22) the upper row is objectwise a strict-exact sequence, we obtain a cofiber sequence

V({\mathcal{A}})\longrightarrow V({\mathcal{B}})\longrightarrow V({\mathcal{B}},{\mathcal{A}})\longrightarrow\Sigma V({\mathcal{A}})

in $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ , where

V({\mathcal{B}},{\mathcal{A}}):=\colim_{n}\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{B}})/\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\,.

We now show that the induced map

(9.23)

V({\mathcal{B}},{\mathcal{A}})\longrightarrow V({\mathcal{C}})

is an equivalence. For this, consider the following commutative diagram

Since the induced triangulated functor

\Ho({\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\longrightarrow\Ho({\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{B}}))

preserves $\kappa$ -small colimits, [70, §3.1] implies that the triangulated category

\Ho({\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{B}})/{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}}))

is idempotent complete. Therefore, $\theta_{n}$ is an equivalence, and we obtain maps

\psi_{n}:\Sigma_{\kappa}^{(n)}({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{B}})/{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}})

which induce maps

\Psi_{n}:\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{C}}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Sigma^{-n-1}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n+1)}({\mathcal{B}})/\Sigma^{(n+1)}_{\kappa}({\mathcal{A}})).

It follows that the natural map

\colim_{n}\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{B}})/\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\colim_{n}\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{C}}))

is an equivalence, which implies that the map (9.23) is an equivalence. ∎

0NPQ

Corollary 9.24 (of proposition 9.21). There is a functor

\mathrm{Loc}:{\mathcal{M}}_{\mathrm{loc}}^{\kappa}\longrightarrow\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}

such that $\mathrm{Loc}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}}))\simeq V({\mathcal{A}})$ , for every small stable $\infty$ -category ${\mathcal{A}}$ .

0NPR

Proof. This follows from propositions 9.21 and 8.6. ∎

0NPS

Proposition 9.25. The two functors

\mathrm{Loc},\gamma^{\ast}\colon{\mathcal{M}}_{\mathrm{loc}}^{\kappa}\longrightarrow\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}

are canonically equivalent, where $\gamma^{\ast}$ is the right adjoint of the localization functor.

0NPT

Proof. Let us denote by ${\bf L}$ the endofunctor $\mathrm{Loc}\circ\gamma$ of $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ . Note that we have a natural transformation $\Id\Rightarrow{\bf L}$ . Making use of the definition of $V(-)$ and of the fact that colimits in $\infty$ -categories commute, we observe that ${\bf L}$ is a localization functor on $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ [52, 5.2.7.4]. Therefore, it suffices to show that a map in $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ becomes an equivalence in ${\mathcal{M}}_{\mathrm{loc}}^{\kappa}$ if and only if it becomes an equivalence after application of ${\bf L}$ . This follows from the fact that for every small stable $\infty$ -category ${\mathcal{A}}$ , we have an equivalence $\gamma(V({\mathcal{A}}))\simeq{\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}})$ : note that we have cofiber sequences in ${\mathcal{M}}_{\mathrm{loc}}^{\kappa}$

{\mathcal{U}}_{\mathrm{loc}}^{\kappa}(\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\longrightarrow{\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\longrightarrow{\mathcal{U}}_{\mathrm{loc}}^{\kappa}(\Sigma_{\kappa}^{(n+1)}({\mathcal{A}}))\,.

∎

0NPU

Proposition 9.26. Let ${\mathcal{A}}$ be small stable $\infty$ -category. We have a natural isomorphism in the stable homotopy category of spectra

\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{S}}_{\infty}^{\omega}),\,{\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}}))\simeq I\mspace{-6.mu}K({\mathcal{A}})\,.

0NPV

Proof. This follows from the following equivalences

(9.27)	$\displaystyle\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{S}}_{\infty}^{\omega}),{\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}}))$	$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\gamma^{\ast}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}})))$
(9.27)		$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\mathrm{Loc}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}})))$
(9.28)		$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),V({\mathcal{A}}))$
(9.29)		$\displaystyle\simeq$	$\displaystyle I\mspace{-6.mu}K({\mathcal{A}})\,.$

Equivalence (9.27) comes from proposition 9.25, equivalence (9.28) comes from corollary 9.24, and equivalence (9.29) is proposition 9.17. ∎

0NPW

Proof of theorem 9.8. Recall from subsection 8.3 that ${\mathcal{M}}_{\mathrm{loc}}$ is obtained by localizing ${\mathcal{M}}_{\mathrm{loc}}^{\kappa}$ with respect to the set ${\mathcal{E}}_{\mathrm{L}}$ . Since ${\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{S}}_{\infty}^{\omega})$ is compact in ${\mathcal{M}}_{\mathrm{loc}}^{\kappa}$ , it is sufficient by proposition 9.26 and the universal property of localization (see section 2.5) to show that the functor

\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{S}}_{\infty}^{\omega}),-)\colon{\mathcal{M}}_{\mathrm{loc}}^{\kappa}\longrightarrow{\mathcal{S}}_{\infty}

sends the elements of ${\mathcal{E}}_{\mathrm{L}}$ to equivalences. This follows from the fact that the non-connective $K$ -theory construction preserves filtered colimits (see [70, §7, Lemma 6]), and so the proof is finished. ∎

9.3. Non-connective $K$ -theory of Waldhausen categories and localization

In particular, theorem 9.8 implies that non-connective $K$ -theory satisfies localization. This is an extremely useful fact in practice; localization sequences provide one of the main computation tools for understanding algebraic $K$ -theory. As such, we state a version of this result in terms of Waldhausen categories. We begin by defining the non-connective $K$ -theory of a Waldhausen category.

0NPX

Definition 9.30. Let ${\mathcal{C}}$ be a DHKS-saturated Waldhausen category with factorization. Then the non-connective $K$ -theory $I\mspace{-6.mu}K({\mathcal{C}})$ of ${\mathcal{C}}$ is defined as the non-connective $K$ -theory $I\mspace{-6.mu}K(\mathrm{N}({\mathcal{C}})[W^{-1}][\Sigma^{-1}])$ of the $\infty$ -category

\mathrm{N}({\mathcal{C}})[W^{-1}][\Sigma^{-1}]\simeq\colim\{\mathrm{N}({\mathcal{C}})[W^{-1}]\overset{\Sigma}{\longrightarrow}\mathrm{N}({\mathcal{C}})[W^{-1}]\overset{\Sigma}{\longrightarrow}\cdots\}

obtained by inverting the suspension on the underlying $\infty$ -category $\mathrm{N}({\mathcal{C}})[W^{-1}]$ in the $\infty$ -category $\Cat_{\infty}^{\mathrm{Rex}}$ of $\infty$ -categories with finite colimits and right-exact functors.

This definition in terms of the stabilization is reasonable because of the following consistency results.

0NPY

Proposition 9.31. Let $\mathcal{C}$ be a presentable $\infty$ -category with a zero object, and let $\mathcal{C}^{\omega}[\Sigma^{-1}]$ denote the colimit

\mathcal{C}^{\omega}[\Sigma^{-1}]\simeq\colim\{\mathcal{C}^{\omega}\overset{\Sigma}{\longrightarrow}\mathcal{C}^{\omega}\overset{\Sigma}{\longrightarrow}\cdots\}

in $\Cat_{\infty}^{\mathrm{Rex}}$ . Then $\mathcal{C}^{\omega}$ is stable, and the induced functor

\mathcal{C}^{\omega}[\Sigma^{-1}]\longrightarrow\Stab(\mathcal{C})

identifies the idempotent-completion of $\mathcal{C}^{\omega}[\Sigma^{-1}]$ with $\Stab(\mathcal{C})^{\omega}$ .

0NPZ

Proof. Let ${\mathcal{D}}$ be an idempotent-complete stable $\infty$ -category. Then

	$\displaystyle\mathrm{Fun}^{\ex}(\mathcal{C}^{\omega}[\Sigma^{-1}],{\mathcal{D}})$	$\displaystyle\simeq\lim\mathrm{Fun}^{\ex}(\mathcal{C}^{\omega},{\mathcal{D}})\simeq\lim\mathrm{Fun}^{\mathrm{L}}_{\omega}(\mathcal{C},\Ind({\mathcal{D}}))$
		$\displaystyle\simeq\mathrm{Fun}^{\mathrm{L}}_{\omega}(\Stab(\mathcal{C}),\Ind({\mathcal{D}}))\simeq\mathrm{Fun}^{\mathrm{\ex}}(\Stab(\mathcal{C})^{\omega},{\mathcal{D}}).$

Since $\Stab(\mathcal{C})^{\omega}$ is necessarily idempotent-complete, we conclude that it is equivalent to the idempotent-completion of $\mathcal{C}^{\omega}[\Sigma^{-1}]$ . ∎

0NQ0

Proposition 9.32. Let ${\mathcal{C}}$ be a DHKS-saturated Waldhausen category with factorization. Then the natural map $\mathrm{N}({\mathcal{C}})[W^{-1}]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\\ N({\mathcal{C}})[W^{-1}][\Sigma^{-1}]$ induces a natural equivalence

K({\mathcal{C}})\longrightarrow K(\mathrm{N}({\mathcal{C}})[W^{-1}][\Sigma^{-1}]).

0NQ1

Proof. The additivity theorem implies that, for Waldhausen categories with factorization, the suspension endomorphism $\Sigma:{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ induces $-\id:K({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}K({\mathcal{C}})$ . By naturality, we conclude that $\Sigma:\mathrm{N}({\mathcal{C}})[W^{-1}])\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}({\mathcal{C}})[W^{-1}]$ acts invertibly on $K$ -theory. Finally, since $K$ -theory (viewed as a functor of small $\infty$ -categories with finite colimits and a zero object and right-exact functors) preserves filtered colimits, we see that

K(\mathrm{N}({\mathcal{C}})[W^{-1}][\Sigma^{-1}])\simeq\colim K(\mathrm{N}({\mathcal{C}})[W^{-1}])\simeq K(\mathrm{N}({\mathcal{C}})[W^{-1}])\simeq K({\mathcal{C}}),

where the last equivalence follows from Corollary 7.12. ∎

0NQ2

Remark 9.33. On $0$ -connective covers there is an equivalence $K({\mathcal{C}})_{>0}\simeq I\mspace{-6.mu}K({\mathcal{C}})_{>0}$ between this notion of non-connective $K$ -theory and the usual connective $K$ -theory of ${\mathcal{C}}$ . In degree $0$ , there an isomorphism $\pi_{0}K({\mathcal{C}})\cong\pi_{0}I\mspace{-6.mu}K({\mathcal{C}})$ if the underlying $\infty$ -category of ${\mathcal{C}}$ is idempotent complete.

0NQ3

Theorem 9.34. Let ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ be a sequence of DHKS-saturated Waldhausen categories with factorization such that

\Ho(\mathrm{N}({\mathcal{A}})[W^{-1}][\Sigma^{-1}])\longrightarrow\Ho(\mathrm{N}({\mathcal{B}})[W^{-1}][\Sigma^{-1}])\longrightarrow\Ho(\mathrm{N}({\mathcal{C}})[W^{-1}][\Sigma^{-1}])

is a localization sequence of triangulated categories. Then the induced map

I\mspace{-6.mu}K({\mathcal{A}})\longrightarrow I\mspace{-6.mu}K({\mathcal{B}})\longrightarrow I\mspace{-6.mu}K({\mathcal{C}})

is a cofiber sequence of spectra.

0NQ4

Proof. This follows from the natural equivalence $I\mspace{-6.mu}K(-)\simeq I\mspace{-6.mu}K(\mathrm{N}(-)[W^{-1}][\Sigma^{-1}])$ and the fact that cofiber sequence

I\mspace{-6.mu}K(\mathrm{N}({\mathcal{A}})[W^{-1}][\Sigma^{-1}])\longrightarrow I\mspace{-6.mu}K(\mathrm{N}({\mathcal{B}})[W^{-1}][\Sigma^{-1}])\longrightarrow I\mspace{-6.mu}K(\mathrm{N}({\mathcal{C}})[W^{-1}][\Sigma^{-1}])

is a cofiber sequence because $I\mspace{-6.mu}K(-)$ is a localizing invariant. ∎

9.4. Extending co-representability

In this section, we show how to extend the co-representability of negative $K$ -theory obtained in theorem 9.8 to maps out of any dualizable object, using the theory developed in section 3. We begin with the following technical lemma:

0NQ5

Lemma 9.35. Let ${\mathcal{B}}$ be a small stable idempotent-complete $\infty$ -category. Then the functor given by $(-)\widehat{\otimes}{\mathcal{B}}$ preserves equivalences, filtered colimits, the point, and exact sequences.

0NQ6

Proof. It follows from the definition that $(-)\widehat{\otimes}{\mathcal{B}}$ preserves equivalences, filtered colimits, and the point. The characterization of [53, 6.3.1.16] implies that it preserves exact sequences. ∎

We can now prove the main theorem of this section:

0NQ7

Theorem 9.36. Let ${\mathcal{B}}$ be a smooth and proper small stable $\infty$ -category in the sense of definitions 3.4 and 3.5. Then ${\mathcal{U}}_{\mathrm{loc}}({\mathcal{B}})$ is compact in ${\mathcal{M}}_{\mathrm{loc}}$ and for every small stable $\infty$ -category ${\mathcal{A}}$ , we have a natural equivalence of spectra

\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}({\mathcal{B}}),{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}))\simeq I\mspace{-6.mu}K({\mathcal{B}}^{\op}\widehat{\otimes}{\mathcal{A}})

0NQ8

Proof. For any small stable idempotent-complete $\infty$ -category ${\mathcal{B}}$ , we can consider the functor

(-)\widehat{\otimes}{\mathcal{B}}\colon\Cat_{\infty}^{\perf}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\perf}.

By lemma 9.35, the composed morphism

\Cat_{\infty}^{\perf}\stackrel{{\scriptstyle(-)\widehat{\otimes}{\mathcal{B}}}}{{\longrightarrow}}\Cat_{\infty}^{\perf}\stackrel{{\scriptstyle{\mathcal{U}}_{\mathrm{loc}}}}{{\longrightarrow}}{\mathcal{M}}_{\mathrm{loc}}

is a localizing invariant. Thus, we obtain a commutative diagram

with $\overline{(-)\widehat{\otimes}{\mathcal{B}}}$ a colimit-preserving functor such that

\overline{{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}})\widehat{\otimes}{\mathcal{B}}}\simeq{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}\widehat{\otimes}{\mathcal{B}})\,.

Now, recall from theorem 3.7 that since ${\mathcal{B}}$ is smooth and proper, it is also dualizable (in the symmetric monoidal $\infty$ -category $\Cat_{\infty}^{\perf}$ of idempotent-complete small stable $\infty$ -categories). Therefore, we have an adjunction (on the left) [53, 4.2.5.6], which induces an adjunction (on the right)

with $\overline{(-)\widehat{\otimes}{\mathcal{B}}^{\op}}$ a colimit preserving morphism, such that

\overline{{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}})\widehat{\otimes}{\mathcal{B}}^{\op}}\simeq{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}\otimes{\mathcal{B}}^{\op})\,.

The proof now follows from the following equivalences of spectra

$\displaystyle\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}({\mathcal{B}}),{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}))$	$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\overline{{\mathcal{U}}_{\mathrm{loc}}({\mathcal{S}}_{\infty}^{\omega})\widehat{\otimes}{\mathcal{B}}},{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}))$
	$\displaystyle\simeq$	$\displaystyle\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}({\mathcal{S}}_{\infty}^{\omega}),\overline{{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}})\widehat{\otimes}{\mathcal{B}}^{\op}})$
	$\displaystyle\simeq$	$\displaystyle\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}({\mathcal{S}}_{\infty}^{\omega}),{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}\widehat{\otimes}{\mathcal{B}}^{\op}))$
	$\displaystyle\simeq$	$\displaystyle I\mspace{-6.mu}K({\mathcal{A}}\widehat{\otimes}{\mathcal{B}}^{\op})$
	$\displaystyle\simeq$	$\displaystyle I\mspace{-6.mu}K(\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}})))\,.$

Finally, since in the adjunction

the morphism $\overline{(-)\widehat{\otimes}{\mathcal{B}}^{\op}}$ preserves colimits, the object ${\mathcal{U}}_{\mathrm{loc}}({\mathcal{S}}_{\infty}^{\omega})$ is compact, and $\overline{{\mathcal{U}}_{\mathrm{loc}}({\mathcal{S}}_{\infty}^{\omega})\otimes{\mathcal{B}}}\simeq{\mathcal{U}}_{\mathrm{loc}}({\mathcal{B}})$ , we conclude that ${\mathcal{U}}_{\mathrm{loc}}({\mathcal{B}})$ is compact. ∎

9.5. Non-connective $K$ -theory of connective ring spectra

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

K(R)_{n}=K_{0}(\mu^{n}R)\times BGL^{+}(\mu^{n}R).

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

(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].

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

BGL(R)\cong\hocolim_{n}BGL_{n}(R).

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:

0NQ9

Lemma 9.39. Let $R$ be a connective $A_{\infty}$ ring spectrum. There is an equivalence of infinite loop spaces

\Omega^{\infty}K(R)\simeq K_{0}(\pi_{0}R)\times BGL(R)^{+}.

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:

0NQA

Proposition 9.40. For a connective ring spectrum $R$ , the connective algebraic $K$ -theory space $BGL(R)^{+}$ is equivalent to the algebraic $K$ -theory space $BGL^{+}(\Omega^{\infty}R)$ .

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.

0NQB

Definition 9.41. Let $R$ be a connective $A_{\infty}$ ring spectrum. We set

MR=\colim_{n}\mathrm{Map}_{R}(R^{\lor n},R^{\lor n}),

the nonunital $A_{\infty}$ ring spectrum of finite $R$ -valued matrices. We write $LR$ for the $A_{\infty}$ ring spectrum of locally finite matrices, i.e. the connective $A_{\infty}$ ring spectrum obtained as the homotopy pullback

where here $H$ denotes the Eilenberg-Mac Lane spectrum functor.

0NQC

Remark 9.42. These definitions are consistent with the those of $m$ and $\ell$ of [33] in the case of a ringlike $A_{\infty}$ space of [33] — one can construct equivalences $\Omega^{\infty}MR\simeq m(\Omega^{\infty}R)$ and $\Omega^{\infty}LR\simeq\ell(\Omega^{\infty}R)$ , although we leave the details to the interested reader, in order avoid a detailed discussion of the technology for $A_{\infty}$ ring spaces.

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.

0NQD

Proposition 9.43. The unit natural transformation $\Id\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}i^{*}i_{!}$ is an equivalence.

0NQE

Proof. This is follows from the fact that $i_{!}\colon\Psi(F_{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi(F^{\infty}_{R})$ is fully faithful, which in turn follows from the fact that $i$ is a fully faithful functor of spectral categories. ∎

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

\Psi(F_{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi(F^{\infty}_{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}},

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 $R^{\lor n}$ .

0NQF

Proposition 9.44. The $F_{R}$ -module represented by any finite wedge $R^{\lor n}$ is a compact generator of $\Psi(F_{R})$ and the $F^{\infty}_{R}$ -module represented by the countably infinite wedge $R^{\lor\infty}$ is a compact generator of $\Psi(F^{\infty}R)$ . In particular, we have equivalences $\Psi(R)\simeq\Psi(\End_{R}(R^{\lor n}))\simeq\Psi(F_{R})$ and $\Psi(\End_{R}(R^{\lor\infty}))\simeq\Psi(F^{\infty}_{R})$ .

0NQG

Proof. The statement about compact generators essentially follows by construction. Then the $\infty$ -categorical version of Schwede-Shipley’s Morita theorem [69], [53, §7.1.2], allows us to characterize these categories in terms of endomorphisms of the compact generator. ∎

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.

0NQH

Proposition 9.45. An $F^{\infty}_{R}$ -module $M$ is in the full subcategory ${\mathcal{C}}\subseteq\Psi(F^{\infty}_{R})$ spanned by the local objects if and only if $i^{*}M\simeq 0$ in $\Psi(F_{R})$ . Similarly, a map of $F^{\infty}_{R}$ -modules $f\colon M\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M^{\prime}$ is a local equivalence if and only if the cofiber of $f$ lies in the essential image of $i_{!}$ .

0NQI

Proof. The first claim follows from the fact that $i^{*}M\simeq 0$ if and only if for all $F_{R}$ -modules $N$ , $\mathrm{Map}_{F^{\infty}_{R}}(i_{!}N,M)\simeq 0$ . In turn, this holds if and only if for any map of $F^{\infty}_{R}$ -modules $Q\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}P$ with cofiber of the form $i_{!}N$ ,

\mathrm{Map}_{F^{\infty}_{R}}(P,M)\simeq\mathrm{Map}_{F^{\infty}_{R}}(Q,M).

The second claim follows from the fact that, if the cofiber of $f$ lies in the essential image of $i_{!}$ , then for any local object $L$ , $\mathrm{Map}_{F^{\infty}_{R}}(M^{\prime},L)\simeq\mathrm{Map}_{F^{\infty}_{R}}(M,L)$ . ∎

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

0NQJ

Corollary 9.46. Let $G$ denote the cofiber of the counit $i_{!}i^{*}R^{\lor\infty}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}R^{\lor\infty}$ in $\Psi(F^{\infty}_{R})$ . Then $G$ lies in the full subcategory ${\mathcal{C}}\subseteq\Psi(F^{\infty}_{R})$ , i.e. $G$ is a local object, and the map $R^{\lor\infty}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}G$ is a local equivalence. Furthermore, $G$ is a compact generator of ${\mathcal{C}}$ .

0NQK

Proof. By the previous proposition, $G$ is a local object, and the cofiber

\Sigma i_{!}i^{*}R^{\lor\infty}\simeq i_{!}\Sigma i^{*}R^{\lor\infty}

of $R^{\lor\infty}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}G$ is in the image of $i_{!}$ . $G$ is compact because $R^{\lor\infty}$ is a compact generator of $F^{\infty}_{R}$ and the functor $\Psi(F^{\infty}_{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ preserves compact objects [70, 2.9]. ∎

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

\mathrm{Map}_{F^{\infty}_{R}}(R^{\lor\infty},i_{!}i^{*}R^{\lor\infty})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Map}_{F^{\infty}_{R}}(R^{\lor\infty},R^{\lor\infty}).

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

\pi_{0}\tilde{F}_{R}^{\infty}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\pi_{0}F_{R}^{\infty}

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

f\in\pi_{0}\mathrm{Map}(R^{\lor m},R^{\lor n})\cong\Hom_{\pi_{0}R}(\pi_{0}R^{\lor m},\pi_{0}R^{\lor n}),

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].

0NQL

Lemma 9.47. The symmetric monoidal functor $\pi_{0}\colon\mathrm{Sp}_{\geq 0}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Ab}$ is right adjoint to the Eilenberg-MacLane spectrum functor $H$ , which is lax symmetric monoidal. It induces a functor

\pi_{0}\colon\Cat_{\mathrm{Sp}_{\geq 0}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\mathrm{Ab}},

from categories enriched in connective symmetric spectra to categories enriched in abelian groups with right adjoint $H$ .

Using this we obtain a morphism of spectral categories

H\pi_{0}\tilde{F}_{R}^{\infty}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H\pi_{0}F_{R}^{\infty}.

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

\mathrm{Map}_{H\pi_{0}\tilde{F}_{R}^{\infty}}(R^{\lor m},R^{\lor n})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Map}_{H\pi_{0}F_{R}^{\infty}}(R^{\lor m},R^{\lor n})

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

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

0NQM

Proposition 9.48. The spectral functor $\tilde{F}_{R}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F_{R}$ is a weak equivalence of spectral categories, and there is an equivalence of $A_{\infty}$ ring spectra $\End_{\tilde{F}^{\infty}_{R}}(R^{\lor\infty})\simeq LR$ .

0NQN

Proof. As the functor is actually surjective on objects, it is enough to show that it is fully faithful. This follows from the fact that mapping spectra in the homotopy pullback spectral category are computed as the homotopy pullbacks of the mapping spectra. Applying the long exact sequence to the homotopy pullback

implies the desired equivalence. A similar computation with $m=n=\infty$ implies the second statement. ∎

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

\Psi(\tilde{F}_{R}^{\infty})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi(F_{R}^{\infty}),

on $\infty$ -categories of modules.

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

{\mathcal{C}}^{\prime}=\Psi(\tilde{F}_{R}^{\infty})/\Psi(F_{R})

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

i_{!}i^{*}R^{\lor\infty}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}R^{\lor\infty}

(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

(9.49)

\mathrm{Map}_{\Psi(\tilde{F}_{R}^{\infty})}(R^{\lor\infty},i_{!}i^{*}R^{\lor\infty})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\End_{\Psi(\tilde{F}_{R}^{\infty})}(R^{\lor\infty}).

0NQP

Lemma 9.50. There is an equivalence of rings $\pi_{0}(\End_{{\mathcal{C}}^{\prime}}(G^{\prime}))\simeq\pi_{0}(\mu\Omega^{\infty}R)$ .

0NQQ

Proof. Regarding $\tilde{F}_{R}^{\infty}$ as a simplicial category, $\End_{\tilde{F}_{R}^{\infty}}(R^{\infty})$ is (by construction) the $A_{\infty}$ ring space $\ell R$ . Furthermore, we have that

\pi_{0}(\mathrm{Map}_{\Psi(\tilde{F}_{R}^{\infty})}(R^{\lor\infty},i_{!}i^{*}R^{\lor\infty}))\cong\pi_{0}(i_{!}i^{*}R^{\lor\infty})\cong m\pi_{0}R

and by construction

\pi_{0}(\End_{\Psi(\tilde{F}_{R}^{\infty})}(R^{\lor\infty}))\cong\ell\pi_{0}R.

Therefore, equation 9.49 implies that as groups there is an isomorphism

	$\displaystyle\pi_{0}(\End_{{\mathcal{C}}^{\prime}}(G^{\prime}))\cong\ell(\pi_{0}(R))/m(\pi_{0}(R))\cong\ell(\pi_{0}(\Omega^{\infty}R))/m(\pi_{0}(\Omega^{\infty}R))$
	$\displaystyle\cong\mu\pi_{0}(\Omega^{\infty}R)\cong\pi_{0}(\mu\Omega^{\infty}R),$

where the last isomorphism follows from [33, 5.1]. Finally, the universal property of the cofiber in spectra implies that there is a ring structure induced on $\pi_{0}(\End_{{\mathcal{C}}^{\prime}}(G^{\prime}))$ induced by the ring structure on $m\pi_{0}(R)$ quotiented by the two-sided ideal $\ell\pi_{0}(R)$ . Inspection of $\pi_{0}$ shows that this multiplication coincides with the ring structure on $\pi_{0}(\End_{{\mathcal{C}}^{\prime}}(G^{\prime}))$ induced by composition. ∎

Based on this, we define

\hat{\mu}R\cong\End_{{\mathcal{C}}^{\prime}}(G^{\prime}),

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 $\hat{\mu}R$ .

0NQR

Proposition 9.51. Let $R$ be a connective $A_{\infty}$ ring spectrum. We have a natural equivalence of spectra

for any infinite cardinal $\kappa>\omega$ .

0NQS

Proof. For any infinite cardinal $\kappa>\omega$ , there is a natural inclusion map

\Psi_{\perf}(F_{R}^{\infty})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\Ind(\Psi_{\perf}(R)))^{\kappa}

induced by the fact that any countable wedge of copies of $R$ is in $(\Ind(\Psi(R)_{\perf}))^{\kappa}$ , and the latter is closed under retracts and stable under finite colimits. Since the inclusion $\Psi_{\perf}(F_{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi_{\perf}(F_{R}^{\infty})$ is compatible with the (Yoneda) inclusion $\Psi_{\perf}(R)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\Ind(\Psi_{\perf}(R)))^{\kappa}$ , we have a commutative diagram

Combining this with $\Psi_{\perf}(\tilde{F}_{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi_{\perf}(\tilde{F}^{\infty}_{R})$ we obtain the commutative diagram

(9.52)

and hence an induced composite map of quotients

	$\displaystyle\alpha\colon\Psi_{\perf}(\tilde{F}_{R}^{\infty})/\Psi_{\perf}(F_{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi_{\perf}(F_{R}^{\infty})/\Psi_{\perf}(F_{R})$
	$\displaystyle\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\Ind(\Psi_{\perf}(R)))^{\kappa}/\Psi_{\perf}(R).$

By the work above, $\alpha$ can be described as a map

\Psi_{\tri}(\hat{\mu}R)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\Ind(\Psi_{\perf}(R)))^{\kappa}/\Psi_{\perf}(R).

Finally, since $F_{R}^{\infty}$ has countable coproducts, the usual Eilenberg swindle implies that $K(F_{R}^{\infty})$ is contractible. We also know that $K(\Psi_{\perf}(\tilde{F}_{R}^{\infty})$ is contractible [33, 6.1,6.3]. Therefore, applying $\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\,\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(-))$ to the commutative diagram, the fact that all of the horizontal sequences are strict-exact allows us to apply theorem 9.10 to conclude that $\alpha$ induces an equivalence on $K$ -theory spectra. ∎

Proposition 9.51 allows us finally to establish the desired result.

0NQT

Theorem 9.53. Let $R$ be a connective $A_{\infty}$ ring spectrum. Then for $i\leq 0$ , the natural map $R\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H\pi_{0}R$ induces isomorphisms $\pi_{i}I\mspace{-6.mu}K(R)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\pi_{i}I\mspace{-6.mu}K(H\pi_{0}R)\cong\pi_{i}I\mspace{-6.mu}K(\pi_{0}R)$ .

0NQU

Proof. Using the proof of proposition 9.51 and mimicking definition 9.6, we can define a spectrum

{I\mspace{-6.mu}K}^{{}^{\prime}}(R):=\colim_{n}\Omega^{n}K((\Psi_{\tri}(\hat{\mu}^{n}R)).

The conclusion of proposition 9.51 along with diagram 9.52 (which implies compatibility of the structure maps) yields an equivalence ${I\mspace{-6.mu}K}^{{}^{\prime}}(R)\simeq I\mspace{-6.mu}K(R)$ . By the argument for [70, 11.7], we see that we can compute the homotopy groups of ${I\mspace{-6.mu}K}^{{}^{\prime}}(R)$ using a fibrant model that is a spectrum with $n$ th space given by the space

\Omega^{\infty}K(\Psi_{\perf}(\hat{\mu}^{n}R)).

Lastly, lemma 9.39 and lemma 9.50 implies that there is an equivalence

\Omega^{\infty}K(\Psi_{\perf}(\hat{\mu}^{n}R))\simeq K_{0}(\mu^{n}\pi_{0}R)\times BGL^{+}(\hat{\mu}^{n}R).

Therefore, for $n>1$ , $\pi_{0}\Omega^{\infty}K(\Psi_{\perf}(\hat{\mu}^{n}R))$ is $K_{0}(\mu^{n}\pi_{0}R)=K_{-n}(\pi_{0}R)$ . ∎

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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9. Non-connective KK-theory

9.1. Non-connective KK-theory of ∞\infty-categories