# Algebraic K-theory of a category containing all perfect complexes

Let $$R$$ be a ring and let $$\mathcal{C}$$ be the category of perfect $$R$$-complexes. Suppose that $$S=\bigoplus_{i=1}^{\infty}R$$

Let us define $$\mathcal{D}$$ the smallest thick category generated by $$S$$.

Clearly $$\mathcal{C}$$ is a full subcategory of $$\mathcal{D}$$

The natural embedding $$i:\mathcal{C}\rightarrow \mathcal{D}$$ induces a morphism in algebraic $$K$$-theory given by $$K(i):K(\mathcal{C})\rightarrow K(\mathcal{D})$$

My question is the following: What can be said about $$K_{n}(\mathcal{C})\rightarrow K_{n}(\mathcal{D})$$ for each $$n\in \mathbf{N}$$, is it an injective/surjective homomorphism?

• K-theory of $\mathcal{D}$ is just zero due to the Eilenberg swindle, or am I missing something? Jun 30 at 16:05

The point is as follows : let $$E$$ be the subcategory of $$D$$ on those objects $$X$$ such that $$\bigoplus_{i\in\mathbb N} X \in D$$. Then $$E$$ is a thick subcategory of $$D$$, and it contains $$S$$ (because $$\mathbb N\times \mathbb N \sim \mathbb N$$), thus it contains $$D$$. In particular $$\bigoplus_\mathbb N : D\to D$$ is well-defined and allows the Eilenberg swindle argument to go through.
For the sake of self-containment, let me recall how this argument goes : $$\bigoplus_\mathbb N \simeq \mathrm{id}_D \oplus \bigoplus_\mathbb N$$.
Therefore by additivity, $$\mathrm{id}_{K(D)} + K(F) \simeq K(F)$$ (where $$F = \bigoplus_\mathbb N$$), and so $$\mathrm{id}_{K(D)} \simeq 0$$, in particular $$K(D)=0$$.