l$-adic常模残差上变定理

arXiv - MATH - Number Theory Pub Date : 2024-09-16 DOI:arxiv-2409.10248

Bruno Kahn

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引用次数: 0

摘要

我们证明了在有限域 $k$ 上光滑 varieties $X$ 的连续同调群$H^n_{\mathrm{cont}}(X,\mathbf{Q}_l(n))$ 作为 $\mathbf{Q}_l$-vector 空间，在所有 $n\ge 0$ 的情况下，都被 Zariski 拓扑的局部 $n$-th Milnor$K$-sheaf 所跨越。这里的 $l$ 是 $k$ 中的可逆prime。这是针对 arXiv:math/9801017 (math.AG)猜想的第一个一般性无条件结果，该猜想将相对于光滑射影$k$变量上代数循环的泰特猜想和贝林森猜想结合在一起。

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An $l$-adic norm residue epimorphism theorem

We show that the continuous \'etale cohomology groups $H^n_{\mathrm{cont}}(X,\mathbf{Q}_l(n))$ of smooth varieties $X$ over a finite field $k$ are spanned as $\mathbf{Q}_l$-vector spaces by the $n$-th Milnor $K$-sheaf locally for the Zariski topology, for all $n\ge 0$. Here $l$ is a prime invertible in $k$. This is the first general unconditional result towards the conjectures of arXiv:math/9801017 (math.AG) which put together the Tate and the Beilinson conjectures relative to algebraic cycles on smooth projective $k$-varieties.

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arXiv - MATH - Number Theory

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