通过截断布朗-佩特松上同调反驳卡帕内克猜想
Counter-examples to a conjecture of Karpenko via truncated Brown-Peterson cohomology
AI总结：
通过截断布朗-佩特松上同调与连接K-理论的联系，构造反例反驳卡帕内克猜想，特别针对Spin_{15}群。
摘要：
设G为一个分裂半简单线性代数群，令X表示G中Borel子群的普遍扭曲变元。Nikita Karpenko猜测从X的Chow环到关联格里索特滤波在X的Grothendieck环上的关联格里索特滤波的映射是同构。在被验证许多G后，Yagita通过某些旋量群反驳了该猜想。后来，Baek-Karpenko和Baek-Devyatov分别构造了其他反例。我们提出一种新的方法来构造反例，该方法基于截断布朗-佩特松上同调与连接K-理论之间的联系。使用这种方法，我们反驳了新的群的猜想，包括Spin_{15}，这现在是已知最小的旋量群，其中该猜想不成立。
英文摘要：
Let $G$ be a split semisimple linear algebraic group and let $X$ denote the generically twisted variety of Borel subgroups in $G$. Nikita Karpenko conjectured that the map from the Chow ring of $X$ to the associated graded ring of the topological filtration on the Grothendieck ring of $X$ is an isomorphism. After having been verified for many $G$, the conjecture was disproved by Nobuaki Yagita for some spinor groups. Later, other counter-examples were constructed by Baek-Karpenko and Baek-Devyatov. We present a new method for constructing counter-examples that is based on the connection of the truncated Brown-Peterson cohomology with the connective K-theory. Using this method, we disprove the conjecture for new groups, including $\mathrm{Spin}_{15}$, which is now the smallest known spinor group for which the conjecture fails.
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