Provides explicit triples (r, n, ℓ) with r ≥ 2 for which the natural projection V_{r+ℓ}(A^n) → V_r(A^n) admits no section, yielding stably free modules without free summands, plus a motivic stable splitting of V_2(A^n).
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Symplectic modules satisfy cancellation and splitting in the critical range via vanishing of top A1-cohomology, partially answering a question on isomorphism between (d-1)th Euler class group and Chow group for smooth affine varieties of dimension d.
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The nonexistence of sections of Stiefel varieties and stably free modules
Provides explicit triples (r, n, ℓ) with r ≥ 2 for which the natural projection V_{r+ℓ}(A^n) → V_r(A^n) admits no section, yielding stably free modules without free summands, plus a motivic stable splitting of V_2(A^n).
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Cancellation and splitting of Symplectic modules in the critical range and Euler class group
Symplectic modules satisfy cancellation and splitting in the critical range via vanishing of top A1-cohomology, partially answering a question on isomorphism between (d-1)th Euler class group and Chow group for smooth affine varieties of dimension d.