Why does K-theory need schemes to be Noetherian? The definition of K-theory of a scheme $X$ is defined as
$G_i(X):=K_i(\mathrm{Coh}(X))$ or $K_i(X):=K_i(\mathrm{Vec}(X))$.
But usually the schemes are required to be (at least locally) Noetherian, and usually it is said that if it is not then the $G_i$'s are pretty bad.
But for what reasons that we really need the condition of being noetherian? (If it is not, then $\mathrm{Coh}(X)$ is not abelian, but we only need it to be exact, which might be satisfied.)
 A: You don't need the Noetherianness hypothesis to talk about K-theory. But the definition you propose in your question is not suited for the most general case. From a notion of K-theory we want at least the following properties


*

*K-theory of an affine scheme $\mathrm{Spec}\,R$ is given by the algebraic K-theory of projective $R$-modules in the sense of Quillen

*K-theory satisfies Zariski descent (as a spectrum)
Since every scheme has a Zariski cover by affine schemes it is clear that there's at most one definition that satisfies the above two properties. At least when $X$ is quasi compact quasi separated this can be expressed more geometrically as the algebraic K-theory of perfect complexes. An introduction that does not require much background can be found in the classical paper by Thomason and Trobaugh, Higher Algebraic K-theory of schemes and of Derived categories (although if you want to invest some time learning some modern homotopy theory can only be beneficial -- many of the proofs in Thomason-Trobaugh can be simplified if you have more modern technology at your disposal).
Similarly for G-theory the "correct" analogue for qcqs schemes is the algebraic K-theory spectrum of bounded pseudocoherent complexes (also known as "almost perfect" complexes in parts of the literature).
Both notions of K-theory and G-theory recover Quillen's definition when working on Noetherian schemes (and for K-theory in a much greater generality -- whenever the scheme has an ample family of line bundles).
