Subsection 9.3. (0NSI) · ScalingStacks

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. ∎

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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9.3. Non-connective KK-theory of Waldhausen categories and localization

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