Motivic Steenrod 代数与无需消解的 Thom 障碍
Motivic Steenrod algebra and Thom obstructions without desingularization
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中文总结 AI 辅助
本文给出无需消解的证明，确认特征零域上双稳定模ℓ motivic上同调运算等同于motivic Steenrod代数，并应用此证明复代数环上所有Thom消解障碍消失。
中文摘要 AI 辅助
我们给出了 Voevodsky 关于双稳定模 $\ell$ motivic 上同调运算与特征零域上 motivic Steenrod 代数等同的无需消解的证明。这提供了 Voevodsky 在研究 motivic Eilenberg--MacLane 空间时所预期的直接论证。作为应用，我们研究了 Thom 关于消解的拓扑障碍，这些障碍源于他在 Steenrod 实现问题中的工作。在不借助消解的情况下，我们证明了初级及所有高阶 Thom 障碍在复代数环上消失。证明利用 Brown--Peterson 塔中连续 $k$-不变量的 motivic 双次数来表明每个提升障碍都消失。
英文摘要
We give a desingularization-free proof of Voevodsky's identification of bistable mod-$\ell$ motivic cohomology operations with the motivic Steenrod algebra over fields of characteristic zero. This supplies the direct argument anticipated by Voevodsky in his study of motivic Eilenberg--MacLane spaces. As an application, we study Thom's topological obstructions to desingularization, which arose in his work on the Steenrod realization problem. Without invoking desingularization, we prove that the primary and all higher Thom obstructions vanish on complex algebraic cycles. The proof uses the motivic bidegrees of the successive $k$-invariants in the Brown--Peterson tower to show that every lifting obstruction vanishes.
