It is not too hard, in the theory of vector bundles over manifolds (or nice topological spaces, say locally contractible with finite covering dimension), to arrive at a splitting theorem. This essentially says that for a vector bundle $E$ of large rank (bigger than some explicit bound given by the dimension of the base), one can split off a trivial line bundle. More generally one should probably write this as the existence of an exact sequence $$ L \to E \to F, $$ for $L$ a trivialisable line bundle and $F$ some other vector bundle. One can also show that stably isomorphic vector bundles are actually isomorphic, once their rank is large enough. Similar results are true by results of Bass/Serre over Noetherian affine schemes and projective modules. I may be missing some niceness adjectives here, but the affine bit is the most important bit.

I would like to know some non-affine situations where these results hold, in particular, I've got a separated Noetherian scheme which is a finite union of affines. Possibly I also have an ample line bundle. But, alas, the K-book says

The strict analogue of the Cancellation Theorem 2.3 does not hold for projective schemes.

However, I am emboldened by the fact exercise 5.7 a) asks to prove the analogue of the Serre's cancellation theorem for a projective variety over an algebraically closed field.

One might also ask for analogous results for locally free finitely generated sheaves; I'm not familiar with how different these are from vector bundles.

So what can one say?

In the end, I'm interested in knowing whether Gabber's 'Lemma K' from part 2 of the published version of his thesis is true for more general schemes than affine ones.