Stable bundles over rig categories
Stable bundles over rig categories
复制标题
钻机类别上的稳定捆绑
DOI：
10.1112/jtopol/jtr016
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发表时间：
2009
影响因子：
1.1
通讯作者：
J. Rognes
中科院分区：
文献类型：
--
作者：
N. Baas;B. Dundas;Birgit Richter;J. Rognes
本文的目的是证明虚拟 2-向量束可以由拓扑 K 理论的代数 K 理论 K(ku) 分类的猜想。因此，通过 Ausoni 和第四作者的工作，虚拟 2 向量丛为我们提供了与椭圆上同调具有相同伸缩复杂性的几何上同调理论。主要技术步骤是表明，对于表现良好的小型钻机类别 ℛ（也称为双幺半群类别），与 ℛ 相关的环谱 Hℛ 的代数 K 理论空间 K(Hℛ) 等价于 κ(ℛ) ≃ ℤ ×|BGL(ℛ)|+，其中 GL(ℛ) 是弱可逆的幺半群类别ℛ 上的矩阵。标题指的是BGL(ℛ)等价于BGL(Hℛ)的更清晰的结果。如果 π0ℛ 是一个环，这几乎是正式的，我们的方法是将 ℛ 替换为由 Baas、Dundas、Richter 和 Rognes 提供的环完整版本 ℛ¯ [J.雷内·安格乌。数学., 出现] 与 Hℛ ≃ H ℛ́ 和 π0 ℛ́ π0ℛ 的环完成。剩下的步骤是证明“stable ℛ ‐bundles”和“stable ℛ́ ‐bundles”是相同的，这是通过针对 BGL(ℛ) 和 BGL(ℛ́ ) 之间的差异手动收缩定制模型来完成的。
The point of this paper is to prove the conjecture that virtual 2‐vector bundles are classified by K(ku), the algebraic K‐theory of topological K‐theory. Hence, by the work of Ausoni and the fourth author, virtual 2‐vector bundles give us a geometric cohomology theory of the same telescopic complexity as elliptic cohomology. The main technical step is showing that for well‐behaved small rig categories ℛ (also known as bimonoidal categories), the algebraic K‐theory space, K(Hℛ), of the ring spectrum Hℛ associated to ℛ is equivalent to κ(ℛ) ≃ ℤ ×|BGL(ℛ)|+, where GL(ℛ) is the monoidal category of weakly invertible matrices over ℛ. The title refers to the sharper result that BGL(ℛ) is equivalent to BGL(Hℛ). If π0ℛ is a ring, this is almost formal, and our approach is to replace ℛ by a ring completed version, ℛ¯ , provided by Baas, Dundas, Richter, and Rognes [J. reine angew. Math., to appear] with Hℛ ≃ H ℛ¯ and π0 ℛ¯ the ring completion of π0ℛ. The remaining step is then to show that ‘stable ℛ ‐bundles’ and ‘stable ℛ¯ ‐bundles’ are the same, which is done by a hands‐on contraction of a custom‐built model for the difference between BGL(ℛ) and BGL( ℛ¯ ).