Failure of cancellation for direct sums of line bundles

Swan, Richard G.

doi:10.1090/S0002-9947-1993-1091709-9

Failure of cancellation for direct sums of line bundles
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Trans. Amer. Math. Soc. 336 (1993), 581-605 Request permission

Abstract:

In answer to a question of Murthy and Wiegand, examples are given of finitely generated projective modules $L$ of rank $1$ over a commutative ring $R$ such that $L \oplus {L^{ - 1}}$ is stably free but not free. Examples are also given of projective modules for which the determinant map det: $\operatorname {Aut}(P) \to {R^{\ast } }$ is not onto. Some related topological questions are also discussed.

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