简单环的控制代数与代数k理论

Mark Ullmann
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引用次数: 1

摘要

我们发展了一个简单环的控制代数。这推广了对一大类群的代数K-理论同构猜想(Farrell-Jones猜想)的成功证明方法。这是证明简单环的代数k理论同构猜想的第一步。构造了一个控制简单模的范畴,证明了它具有Waldhausen范畴的结构,并讨论了它的代数k理论。我们强调详细的证明。重点包括简单柱面函子的讨论,胶合引理,分裂相干同伦幂等的简单映射望远镜,以及简单环的弱等价在其代数k理论上推导出等价的直接证明。因为代数k理论需要一定的共通性定理,我们提供了一个证明,并表明某些假设,有时在文献中省略,是必要的。最后,我们注意到我们的设置与环光谱的关系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Controlled Algebra for Simplicial Rings and Algebraic K-theory
We develop a version of controlled algebra for simplicial rings. This generalizes the methods which lead to successful proofs of the algebraic K- theory isomorphism conjecture (Farrell-Jones Conjecture) for a large class of groups. This is the first step to prove the algebraic K-theory isomorphism conjecture for simplicial rings. We construct a category of controlled simplicial modules, show that it has the structure of a Waldhausen category and discuss its algebraic K-theory. We lay emphasis on detailed proofs. Highlights include the discussion of a simplicial cylinder functor, the gluing lemma, a simplicial mapping telescope to split coherent homotopy idempotents, and a direct proof that a weak equivalence of simplicial rings induces an equivalence on their algebraic K-theory. Because we need a certain cofinality theorem for algebraic K-theory, we provide a proof and show that a certain assumption, sometimes omitted in the literature, is necessary. Last, we remark how our setup relates to ring spectra.
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