Comparing K-theories for complex varieties
Comparing K-theories for complex varieties
复制标题
比较复杂品种的 K 理论
DOI：
10.1353/ajm.2001.0032
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发表时间：
2001
影响因子：
1.7
通讯作者：
M. Walker
中科院分区：
文献类型：
--
作者：
E. Friedlander;M. Walker
作者在最近的一篇论文中定义了复簇的半拓扑 K 理论，期望它被证明是一种介于簇的代数 K 理论和相关解析空间的拓扑 K 理论之间的理论，因此将与其他理论共享属性。在本文中，我们通过证明以下结果来实现这些期望：（1）具有有限系数的代数 K 理论和具有有限系数的半拓扑 K 理论在所有射影复簇上都一致，（2）半拓扑 K 理论和拓扑 K 理论在某些类型的广义标志簇上一致，以及（3）（假设 Cohen 和 Lima-Filho 断言的结果）任何光滑射影簇的半拓扑 K 理论一旦 Bott 元素反转，它就与基础解析空间的拓扑 K 理论同构。为了说明我们的结果的实用性，我们观察到作为推论获得了平滑、完整曲线的 Quillen-Lichtenbaum 猜想的新证明。
The semi-topological K -theory of a complex variety was defined in a recent paper by the authors, with the expectation that it would prove to be a theory lying "part way" between the algebraic K -theory of the variety and the topological K -theory of the associated analytic space, and thus would share properties with each of these other theories. In this paper, we realize these expectations by proving among other results that (1) the algebraic K -theory with finite coefficients and the semi-topological K -theory with finite coefficients coincide on all projective complex varieties, (2) semi-topological K -theory and topological K -theory agree on certain types of generalized flag varieties, and (3) (assuming a result asserted by Cohen and Lima-Filho) the semi-topological K -theory of any smooth projective variety becomes isomorphic to the topological K -theory of the underlying analytic space once the Bott element is inverted. To illustrate the utility of our results, we observe that a new proof of the Quillen-Lichtenbaum conjecture for smooth, complete curves is obtained as a corollary.