AI 中文总结
本文通过引入反射函子，证明了拓扑空间上C*-代数的双变量K-理论相关范畴的等价性，还将其应用于滤过K-理论的通用系数定理证明。
AI 中文摘要
本文针对各类拓扑空间对$X$和$Y$，在以$X$和$Y$为基的可分C*-代数的基尔希伯格理想相关KK-理论范畴$\boldsymbol{\frak{KK}}(X)$与$\boldsymbol{\frak{KK}}(Y)$之间引入反射函子。我们证明这些函子诱导出由Meyer和Nest引入的局部化子范畴之间的等价关系$\boldsymbol{\frak{KK}}(X)_{\text{loc}} \backsimeq \boldsymbol{\frak{KK}}(Y)_{\text{loc}}$，且该等价关系进一步限制为bootstrap范畴之间的等价关系$\boldsymbol{\frak{B}}(X) \backsimeq \boldsymbol{\frak{B}}(Y)$。将这些等价关系与Bernstein、Gelfand和Ponomarev在箭图表示论中的组合论证相结合，我们证明对于哈塞图为树定向的有限$T_0$-空间$X$，范畴$\boldsymbol{\frak{KK}}(X)_{\text{loc}}$和$\boldsymbol{\frak{B}}(X)$仅依赖于基础树，与其定向无关。我们还为Dadarlat和Meyer的理想相关E-理论证明了类似结果。此外，我们证明反射函子的重排性质，由此得到考克斯特函子的类似物。最后，我们应用反射函子证明，对于哈塞图为A型Dynkin图定向的有限$T_0$-空间上的C*-代数，滤过K-理论满足通用系数定理。
英文摘要
In this paper, for various pairs of topological spaces $X$ and $Y$, we introduce reflection functors between the categories $\mathfrak{KK}(X)$ and $\mathfrak{KK}(Y)$ of Kirchberg's ideal-related KK-theory for separable C*-algebras over $X$ and $Y$. We prove that these functors induce an equivalence $\mathfrak{KK}(X)_{\mathrm{loc}} \simeq \mathfrak{KK}(Y)_{\mathrm{loc}}$ between the localizing subcategories introduced by Meyer and Nest, and that this equivalence restricts to an equivalence $\mathcal{B}(X) \simeq \mathcal{B}(Y)$ between the bootstrap categories. Combining these equivalences with a combinatorial argument due to Bernstein, Gelfand, and Ponomarev from the representation theory of quivers, we show that, for a finite $T_0$-space $X$ whose Hasse diagram is an orientation of a tree, the categories $\mathfrak{KK}(X)_{\mathrm{loc}}$ and $\mathcal{B}(X)$ depend only on the underlying tree and not on its orientation. We also prove the analogous results for Dadarlat and Meyer's ideal-related E-theory. Moreover, we prove a rearrangement property of reflection functors, which yields an analogue of Coxeter functors. Finally, we apply reflection functors to prove that filtrated K-theory satisfies the universal coefficient theorem for C*-algebras over finite $T_0$-spaces whose Hasse diagrams are orientations of Dynkin diagrams of type A.
Comments108 pages
