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arXiv 2102.02453math.RT
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扩展Drinfeld double的支撑理论

Support Theory for Extended Drinfeld Doubles

Eric M. Friedlander

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摘要：

继早期的工作与克里斯Negron上同调的Drinfeld双打$D(\mathbb G_{(r)})$，我们开发了一个“几何理论”的支持品种“扩展Drinfeld双打”$\tilde D(\mathbb G_{(r)})$的Frobenius内核$\mathbb G_{(r)}$的光滑线性代数群$\mathbb G$在一个领域$k$的特点$p > 0$。到一个$\tilde D(\mathbb G_{(r)})$ -模块$M$，我们将空间$Π(\tilde D(\mathbb G_{(r)}))_M$的等价类的“对$π$ -点”，并证明了大多数所需的性质的$M \mapsto Π(\tilde D(\mathbb G_{(r)}))_M$。也就是说，这个协会满足“张量积性质”，并承认一个自然的连续映射$Ψ_{\tilde D}$的上同调支持理论。此外，对于$M$有限维和适当的条件下，$\mathbb G_{(r)}$，这种关联提供了一个“投影测试”，$Ψ_{\tilde D}$是同胚，并确定$Π(\tilde D(\mathbb G_{(r)}))_M$与上同调支持各种$M$的各种类型的$\tilde D(\mathbb G_{(r)})$ -模块$M$。

英文摘要：

Following earlier work with Cris Negron on the cohomology of Drinfeld doubles $D(\mathbb G_{(r)})$, we develop a "geometric theory" of support varieties for "extended Drinfeld doubles" $\tilde D(\mathbb G_{(r)})$ of Frobenius kernels $\mathbb G_{(r)}$ of smooth linear algebraic groups $\mathbb G$ over a field $k$ of characteristic $p > 0$. To a $\tilde D(\mathbb G_{(r)})$-module $M$ we associate the space $Π(\tilde D(\mathbb G_{(r)}))_M$ of equivalence classes of "pairs of $π$-points" and prove most of the desired properties of $M \mapsto Π(\tilde D(\mathbb G_{(r)}))_M$. Namely, this association satisfies the "tensor product property" and admits a natural continuous map $Ψ_{\tilde D}$ to cohomological support theory. Moreover, for $M$ finite dimensional and with suitable conditions on $\mathbb G_{(r)}$, this association provides a "projectivity test", $Ψ_{\tilde D}$ is a homeomorphism, and identifies $Π(\tilde D(\mathbb G_{(r)}))_M$ with the cohomological support variety of $M$ for various classes of $\tilde D(\mathbb G_{(r)})$-modules $M$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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