AI 中文总结
本文研究无挠 profinite 环系数导出 Hecke 代数与有限环系数逆向极限及有限支撑子代数的比较，刻画核与像，并计算 $\mathbb{Q}_p$ 及 $\operatorname{GL}_2$ 情形的具体结构。
AI 中文摘要
本文研究了系数在无挠 profinite 环中、并将其视为离散环的导出 Hecke 代数。我们将每个这样的代数与有限环系数的导出 Hecke 代数的逐次逆向极限及其由具有一致有限双陪集支撑的元素构成的子代数进行比较。在同调比较假设下，到该子代数的自然映射的像由其零次部分及其正次挠元组成。我们描述了核，用连续上同调刻画，并证明它是一个可除的平方零理想，被所有正次元素零化。这些比较与 Yoneda 积相容，且对挠 Ext 类的作用通过像分解。我们计算了加法群 $\mathbb{Q}_p$ 的所得代数，并确定了 $p>3$ 时系数在 $\mathbb{Z}_p$ 中的 $(\operatorname{GL}_2(\mathbb{Q}_p),\operatorname{GL}_2(\mathbb{Z}_p))$ 的底层分次阿贝尔群。这些例子展示了系数拓扑与支撑条件的显著不同影响。
英文摘要
In this article, we study derived Hecke algebras with coefficients in torsion-free profinite rings regarded as discrete rings. We compare each such algebra with the degreewise inverse limit of the derived Hecke algebras with coefficients in finite rings and with its subalgebra consisting of elements with uniformly finite double-coset support. Under cohomological comparison hypotheses, the natural map to this subalgebra has image given by its degree-zero part and its positive-degree torsion. We describe the kernel in terms of continuous cohomology and prove that it is a divisible square-zero ideal annihilated by every positive-degree element. These comparisons are compatible with Yoneda products, and the action on torsion Ext classes factors through the image. We compute the resulting algebras for the additive group $\mathbb{Q}_p$ and determine the underlying graded abelian groups for $(\operatorname{GL}_2(\mathbb{Q}_p),\operatorname{GL}_2(\mathbb{Z}_p))$ with coefficients in $\mathbb{Z}_p$, for $p>3$. The examples exhibit distinct effects of the coefficient topology and the support condition.
Comments52 pages
