Cancellation and splitting theorems for vector bundles etc over schemes

Question

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.

Answer 1

Weibel seems to have inspired his exercise from Hartshorne's exercise 8.2 in Chapter II of Algebraic geometry - a little quickly, as he says that the existence of an everywhere nonvanishing section follows from Bertini's theorem. Actually, Hartshorne hints to use the same method as in the proof of Bertini's theorem, which is easy (seesaw argument).

Answer 2

Part (i) of Gabber's "Lemma K" was generalized to quasi-compact quasi-separated schemes by Tabuada, van den Bergh in Theorem 2.3 of Noncommutative motives of Azumaya algebras:

Let $X$ be a quasi-compact quasi-separated scheme. Then the kernel of the morphism $\text{rank} : K_{0}(X) \to \mathbb{Z}$ is nilpotent.

See also Lemma 3.1.4.3 of Lieblich Twisted sheaves and the period-index problem for a writeup of the affine case.