# G Theory Localization Sequence without "quasiseparated"

Let $$U \subseteq X$$ be an open and $$Z := X \setminus U$$ its closed complement. I want a sequence $$G_0(Z) \to G_0(X) \to G_0(U) \to 0.$$ However $$X, U$$ are not quasiseparated and perhaps not even quasicompact. Does this sequence still exist? What should I even mean by $$G_0$$-theory? Would other types of $$K$$ Theory have such a sequence?

I'm interested in the case where $$X, U, Z$$ are Artin Stacks, but I'd be happy to hear words of caution that pertain to schemes as well. What should I be careful of without qcqs? What goes wrong? I assume one takes $$G$$-theory to be coherent sheaves with quasicompact support, but no evident adaptation presents itself without the "qs" part.