Structure of derived Hecke algebras with coefficients in torsion-free profinite rings

Abstract

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.