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Do rationally contractible presheaves have rationally contractible injective resolution

Given a presheaf $\mathcal{F}: Sm/k\rightarrow Ab$ we define a new presheaf $C\mathcal{F}= \varinjlim\limits_{X\times \{0,1\}\subset U \subset X\times \mathbb{A}^1}\mathcal{F}(U)$. The presheaf $\...