论文标题
动机atiyah-segal完成定理
Motivic Atiyah-Segal completion theorem
论文作者
论文摘要
令t为圆环，x配备了T-Action的平滑准合作分离方案，[x/t]相关的商堆栈。 Given any localizing A1-homotopy invariant of dg categories E, we prove that the derived completion of E([X/T]) at the augmentation ideal I of the representation ring R(T) of T agrees with the Borel construction associated to the T-action on X. Moreover, for certain localizing A1-homotopy invariants, we extend this result to the case of a linearly reductive group scheme G. As a first application, we obtain an Krishna在代数K理论中的克里希纳完成定理的替代证明，Thomason的完成定理在étaleK理论中具有系数，以及Atiyah-Segal的完成定理的拓扑k理论。这些替代证明会导致相应完成定理的光谱丰富以及以下改进：对于托马森（Thomason）的完成定理的情况，基本领域不再需要分离关闭，并且在atiyah-segal的完整空间中，拓扑空间不再需要综合群体，而不再需要综合群体。作为第二个应用程序，我们在L-AdicétaleK理论中获得了新的完整定理，在（实际）半主理论以及定期循环同源性中。作为第三次应用，我们获得了文献中不同eprovariast群体组的纯代数描述（动机，L-ADIC，（真实）形态，Betti，de Rham等）。最后，在两个独立兴趣的附录中，我们扩展了对同质k理论的芯片的结果，从方案的领域到商堆栈的广泛设置，并建立了一些（真实的）半平衡K理论的有用属性。
Let T be a torus, X a smooth quasi-compact separated scheme equipped with a T-action, and [X/T] the associated quotient stack. Given any localizing A1-homotopy invariant of dg categories E, we prove that the derived completion of E([X/T]) at the augmentation ideal I of the representation ring R(T) of T agrees with the Borel construction associated to the T-action on X. Moreover, for certain localizing A1-homotopy invariants, we extend this result to the case of a linearly reductive group scheme G. As a first application, we obtain an alternative proof of Krishna's completion theorem in algebraic K-theory, of Thomason's completion theorem in étale K-theory with coefficients, and also of Atiyah-Segal's completion theorem in topological K-theory. These alternative proofs lead to a spectral enrichment of the corresponding completion theorems and also to the following improvements: in the case of Thomason's completion theorem the base field no longer needs to be separably closed, and in the case of Atiyah-Segal's completion theorem the topological spaces no longer needs to be compact and the equivariant topological K-theory groups no longer need to be finitely generated over the representation ring. As a second application, we obtain new completion theorems in l-adic étale K-theory, in (real) semi-topological K-theory and also in periodic cyclic homology. As a third application, we obtain a purely algebraic description of the different equivariant cohomology groups in the literature (motivic, l-adic, (real) morphic, Betti, de Rham, etc). Finally, in two appendixes of independent interest, we extend a result of Weibel on homotopy K-theory from the realm of schemes to the broad setting of quotient stacks and establish some useful properties of (real) semi-topological K-theory.
