Galois 丛的中心代数上同调
Central algebraic cohomology of Galois gerbes
- Tel Aviv University（特拉维夫大学）
- Ben-Gurion University of the Negev（内盖夫本-古里安大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究 Galois 丛的中心代数上同调，给出交换化上同调的纤维积公式，并在数域上通过笛卡尔方块恢复完整上同调集，且应用于 Kottwitz 丛。
AI中文摘要：
我们研究了特征零局部域和数域上 Galois 丛的中心代数上同调，系数取为连通约化群和代数中心子群，该中心子群不一定是有限群或连通群。对于存在从 Kaletha 刚性丛到其上的比较态射的丛，我们获得了关于有限中心商代数基本群以及从带（band）到中心子群的同态的交换化上同调的纤维积公式。当带为有限型时，这样的比较态射总是存在。在数域上，独立于比较假设，完整的上同调集通过一个笛卡尔方块从其交换化和实局部化恢复。在非阿基米德局部域和全虚数域上，交换化是双射的。我们还将这些公式专门应用于 Kottwitz 的局部和整体丛。
英文摘要：
We study central algebraic cohomology of Galois gerbes over local fields of characteristic zero and number fields, with coefficients in a connected reductive group and an algebraic central subgroup, not necessarily finite or connected. For gerbes admitting a comparison morphism from Kaletha's rigid gerbe, we obtain fiber product formulas for abelianized cohomology in terms of algebraic fundamental groups of finite central quotients and homomorphisms from the band to the central subgroup. Such comparisons exist whenever the band is of finite type. Over number fields, independently of the comparison hypothesis, the full cohomology set is recovered from its abelianization and its real localizations through a cartesian square. Over non-Archimedean local fields and totally imaginary number fields, abelianization is bijective. We also specialize the formulas to Kottwitz's local and global gerbes.
