Abstract:
For any $\mathbb{A}^1$-invariant cohomology theory that satisfies Nisnevich excision on the category of smooth schemes over a field $k$ it is proved that the Cousin complex on the complement $U-D$ to the strict normal-crossing divisor $D$ in a local essentially smooth scheme $U$ is acyclic. This claim is also proved for the schemes $(X-D)\times(\mathbb{A}^1_k-Z_0)\times\dots\times(\mathbb{A}^1_k-Z_l)$, where $Z_0,\dots,Z_l$ is a finite family of closed subschemes in the affine line over $k$.
Bibliography: 32 titles.
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