标题： 关于非Hausdorff群胚的Hausdorff覆盖 显示英文标题

标题： On Hausdorff covers for non-Hausdorff groupoids

摘要： 我们开发了一种新的方法来研究非Hausdorff的埃塔勒群胚及其代数，该方法基于Timmermann对Hausdorff覆盖的构造。 作为应用，我们完全表征了在任意环上的Steinberg代数中奇异理想何时消失。 我们还完全表征了$C^*$-代数奇异理想与连续紧支撑函数子代数的非Hausdorff类比之间的交集何时为平凡。 这导致了对满足有限性条件的群胚上$C^*$-代数奇异理想何时消失的表征。 此外，我们的方法进一步给出了奇异理想的消失充分条件，并将关于非Hausdorff埃塔勒群胚的简单性、理想交性质、可换性和核性的疑问简化为Hausdorff的情况。 显示英文摘要

摘要： We develop a new approach to non-Hausdorff \'etale groupoids and their algebras based on Timmermann's construction of Hausdorff covers. As an application, we completely characterise when singular ideals vanish in Steinberg algebras over arbitrary rings. We also completely characterise when $C^*$-algebraic singular ideals have trivial intersection with the non-Hausdorff analogue of subalgebras of continuous, compactly supported functions. This leads to a characterisation when $C^*$-algebraic singular ideals vanish for groupoids satisfying a finiteness condition. Moreover, our approach leads to further sufficient vanishing criteria for singular ideals and reduces questions about simplicity, the ideal intersection property, amenability and nuclearity for non-Hausdorff \'etale groupoids to the Hausdorff case.