Motivic Atiyah-Segal完备定理
Motivic Atiyah-Segal completion theorem
摘要：
设T为环面，X为光滑拟紧分离格式，[X/T]为相应的商栈.给定dg范畴E的任意局部化A1-同伦不变量，证明了E（[X/T]）在T的表示环R（T）的增广理想I上的导出完备化与X上T作用的Borel构造一致.此外，对于某些局部化A1-同伦不变量，我们将这个结果推广到线性约化群概型G的情形。作为第一个应用，我们得到了代数K理论中的Krishna完备定理、带系数的Etale K-理论中的Kesason完备定理以及拓扑K理论中的Atiyah-Segal完备定理的另一种证明。这些替代证明导致相应的完备化定理的谱丰富，并且还导致以下改进：在Mesquason完备定理的情况下，基域不再需要可分离地闭合，在Atiyah-Segal完备定理的情况下，拓扑空间不再需要是紧的，并且等变拓扑K-理论群不再需要在表示环上生成。作为第二个应用，我们在l-adic代数K-理论、（实）半拓扑K理论和周期循环同调中得到了新的完备化定理.作为第三个应用，我们得到了一个纯代数描述的不同的等变上同调群在文献中（motivic，l-adic，（实）morphic，贝蒂，德Rham等）。最后，在两个独立的兴趣附录中，我们推广了韦伯的同伦K理论的结果，从范围内的计划，以广泛的设置商栈和建立（实）半拓扑K理论的一些有用的性质。
英文摘要：
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.
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