(1) Suppose that $Z\subset X$ is a closed embedding, $U = X\setminus Z$ is the complement. If relevant, suppose that both $X, Z$ are smooth and even (if relevant) that the normal bundle of $Z\subset X$ is trivial. Then is it true that there is an exact triangle of complexes $K^\* (Z)\to K^\* (X)\to K^\* (U) \to$? Here by $K$ theory I mean (connective) K theory of the category of perfect complexes. For what I need (this came up in the context of trace formulae), rational coefficients are enough. I found many places where very similar results are written down, but not this one explicitly.


(2) Suppose that, further, in the example above we have a (suitably flat) sheaf of smooth, compact (DG) algebras, $\mathcal{A}$, over $X$. Then we can study $K$ theory of categories of bundles of $\mathcal{A}$-modules over the three spaces. Again, is it true that these fall into an exact triangle as above?


It seems like both of these statements should follow from some colimit-compatibility of the $K$ theory functor, but I can't find a reference.



K theory long exact sequence
Regarding the first question:


If $X$ is quasi-compact quasi-separated and $U \to X$ is a quasi-compact open immersion, then Thomason-Trobaugh showed that there is a "proto-localization sequence", i.e. a fibre sequence of spaces
 $$ K\_{Z}(X) \to K(X) \to K(U) $$
where $K\_{Z}(X)$ is the K-theory of perfect complexes on $X$ that vanish on $U$.
This follows from the fact that you have a corresponding exact sequence at the level of stable $(\infty,1)$-categories of perfect complexes, and that these categories are compactly generated, so that the sufficient conditions provided by Thomason's localization theorem, for the induced sequence on K-theory to be a fibre sequence, are verified.


On the other hand, for G-theory (= K-theory of coherent sheaves), one has an "honest" localization sequence.
That is, for $X$ is noetherian, there is a fibre sequence
 $$ G(Z) \to G(X) \to G(U). $$
This follows from Quillen's devissage theorem, which implies that $G(Z) \simeq G\_{Z}(X)$, where $G\_{Z}(X)$ is the K-theory of coherent sheaves on $X$ which vanish on $U$.


When $X$ is regular, K-theory is identified with G-theory (because any complex of quasi-coherent sheaves with coherent cohomology is quasi-isomorphic to a perfect complex).


If by "smooth" you meant regular, e.g. smooth over a field, then putting these together you get the fibre sequence
 $$ K(Z) \to K(X) \to K(U) $$
desired.


For the second question, I am not sure.
By the same type of argument as above you should get a proto-localization sequence, but I don't know if you can say more.


