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

Let d5d\geq 5. Consider the sequence of strictly invariant sheaves

Kd+1M/24πd1A1(Ad10)GW~dd1.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/24πd1A1(Ad10)GW~dd1K^{\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 (d4)(d-4)-fold contractions, is surjective on the right. This conjecture would imply the vanishing of the cohomology groups Hd(X,πd1A1(Ad10))H^d(X,\boldsymbol{\pi}^{\mathbb{A}^1}_{d-1}(\mathbb{A}^{d-1}-0)) and Hd(X,πdA1(Q2d2))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.

Sources & referencesView supporting material

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).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.