Stable triviality and stable bundle equivalence

PDF p. 2: Mohan Kumar fourfold example

Let $X$ be a scheme. A vector bundle on $X$ means a locally free ${\mathcal{O}}_X$-module of finite rank. Write ${\mathcal{O}}_X^N$ for the direct sum of $N$ copies of the structure sheaf, with ${\mathcal{O}}_X^0=0$; these are the trivial bundles. Direct sums and isomorphisms below are taken as ${\mathcal{O}}_X$-modules.

Bundles $E$ and $F$ are stably isomorphic if, for some integer $N\ge 0$, there is an isomorphism $E\oplus {\mathcal{O}}_X^N\cong F\oplus {\mathcal{O}}_X^N$. A bundle $E$ of constant rank $r\ge 0$ is stably trivial if, for some integer $N\ge 0$, $E\oplus {\mathcal{O}}_X^N\cong {\mathcal{O}}_X^{r+N}$. It is trivial if $E\cong {\mathcal{O}}_X^r$. Stable triviality is stable isomorphism to ${\mathcal{O}}_X^r$; the definition itself does not assert cancellation of the trivial summand.

Locally free sheaves of finite rank

Given: A scheme $X$ and finite-rank vector bundles on $X$.

Goal: The displayed conditions are well-defined and stable isomorphism is an equivalence relation.

1.1 On common local trivializing neighbourhoods, the direct sum of bundles with local ranks $r$ and $s$ is isomorphic to ${\mathcal{O}}_X^{r+s}$. Thus the direct sums in the definition are bundles; their ranks and the stated constant rank $r$ are independent of the chosen charts by [F1]. [F1]

2.1 The identity isomorphism with $N=0$ gives reflexivity, and an isomorphism can be inverted, giving symmetry. If $E\oplus {\mathcal{O}}_X^N\cong F\oplus {\mathcal{O}}_X^N$ and $F\oplus {\mathcal{O}}_X^M\cong G\oplus {\mathcal{O}}_X^M$, add ${\mathcal{O}}_X^M$ to the first isomorphism and ${\mathcal{O}}_X^N$ to the second, then compose them using the canonical reassociation and permutation of finite direct sums. This gives $E\oplus {\mathcal{O}}_X^{N+M}\cong G\oplus {\mathcal{O}}_X^{N+M}$, proving transitivity. Finally, ${\mathcal{O}}_X^r\oplus {\mathcal{O}}_X^N\cong {\mathcal{O}}_X^{r+N}$, so the stated form of stable triviality is exactly stable isomorphism to ${\mathcal{O}}_X^r$. [step 1.1, algebra] ∎

The article reports a nontrivial stably trivial rank $2$ bundle on a smooth complex affine fourfold, attributed to Mohan Kumar. This definition does not establish that example or a cancellation theorem.