Stably free modules without free summands

We study the question of when a generic stably free module admits a free summand of a given rank. By work of M. Raynaud, this question can be reinterpreted in terms of the existence (or nonexistence) of a section of a natural projection map between Stiefel varieties given by "forgetting frames." Using tools from motivic homotopy theory, we show that a section does not exist in many cases, thereby constructing examples of stably free modules that do not admit free summands of given ranks.

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