I am starting to learn the $K$-theory of triangulated categories and is stuck with the following.
Let $\mathcal{T}$ be a triangulated category having a semi-orthogonal decomposition $\langle \mathcal{A\_1},\mathcal{A\_2},…,\mathcal{A\_n}\rangle$.
I saw in the paper of Orlov titled Smooth and Proper Noncommutative Schemes and Gluing of DG Categories mentioning an isomorphism $K\_i(\mathcal{T})\cong K\_i(\mathcal{A\_1})\oplus K\_i(\mathcal{A\_2})\oplus…\oplus K\_i(\mathcal{A\_n})$ for all $i$.How do we prove this result?
Are there any detailed references for this result?
I am thinking of applying this result to the case where $\mathcal{T}$ is the bounded derived category of coherent sheaves on an algebraic variety.



How to compute the higher $K$-theory of a triangulated category having a semi-orthogonal decomposition?
Assume $\mathcal{T} = D^b(X)$ and $X$ is smooth. Then $K\_0(X \times X)$ has an algebra structure (with the convolution product) and $K\_\bullet(X)$ is a graded module over this algebra. So, the idea is to use a decomposition of $1 \in K\_0(X \times X)$ (the class of the diagonal $\Delta\_X$) into a sum of idempotents, then the action of these idempotents will give the required direct sum decomposition.


To construct the required decomposition of the diagonal, note that there are objects
$$
R\_i \in D^b(X \times X)
$$
such that the corresponding Fourrier--Mukai functors are the projection functors for the semiorthogonal decomposition, see Theorem 7.1 in [Kuznetsov, Alexander. Base change for semiorthogonal decompositions. Compos. Math. 147 (2011), no. 3, 852--876]. Moreover,
$$
[\Delta\_X] = \sum [R\_i]
$$
and $[R\_i] \in K\_0(X \times X)$ are the required idempotents.


If $X$ is singular, the argument is essentially the same, but you need to replace $K\_0(X \times X)$ by the subgroup of $G\_0(X \times X)$ formed by classes that are perfect over each factor.


If $\mathcal{T}$ does not have a geometric origin, but is enhanced, say, by a DG-algebra $A$, the argument is still the same, but you need to replace $K\_0(X \times X)$ by the Grothendieck group of appropriate category of $A$-bimodules.


