Asok--Fasel's exactness conjecture for the \mathbb{A}^1-homotopy sheaf sequence

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Let d≥5d\geq 5. Consider the sequence of strictly invariant sheaves

Kd+1M/24⟶πd−1A1(Ad−1−0)⟶GW~dd−1.K^{\mathrm{M}}_{d+1}/24\longrightarrow \boldsymbol{\pi}^{\mathbb{A}^1}_{d-1}(\mathbb{A}^{d-1}-0)\longrightarrow \widetilde{GW}^{d-1}_d.

Asok--Fasel's exactness conjecture. The sequence

Kd+1M/24⟶πd−1A1(Ad−1−0)⟶GW~dd−1K^{\mathrm{M}}_{d+1}/24\longrightarrow \boldsymbol{\pi}^{\mathbb{A}^1}_{d-1}(\mathbb{A}^{d-1}-0)\longrightarrow \widetilde{GW}^{d-1}_d

is exact and, after (d−4)(d-4)-fold contractions, is surjective on the right. This conjecture would imply the vanishing of the cohomology groups Hd(X,πd−1A1(Ad−1−0))H^d(X,\boldsymbol{\pi}^{\mathbb{A}^1}_{d-1}(\mathbb{A}^{d-1}-0)) and Hd(X,πdA1(Q2d−2))H^d(X,\boldsymbol{\pi}^{\mathbb{A}^1}_d(Q_{2d-2})) considered in the source. Its status is not resolved in the supplied text.

References

Primary source

Rakesh Pawar and Husney Parvez Sarwar, “Cancellation and splitting of Symplectic modules in the critical range and Euler class group”, arXiv:2312.15782 (2026).

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