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On topological cyclic homology
Acta Mathematica ( IF 3.7 ) Pub Date : 2018-01-01 , DOI: 10.4310/acta.2018.v221.n2.a1 Thomas Nikolaus 1 , Peter Scholze 2
Acta Mathematica ( IF 3.7 ) Pub Date : 2018-01-01 , DOI: 10.4310/acta.2018.v221.n2.a1 Thomas Nikolaus 1 , Peter Scholze 2
Affiliation
Topological cyclic homology is a refinement of Connes--Tsygan's cyclic homology which was introduced by Bokstedt--Hsiang--Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace map from algebraic $K$-theory to topological cyclic homology, and a theorem of Dundas--Goodwillie--McCarthy asserts that this induces an equivalence of relative theories for nilpotent immersions, which gives a way for computing $K$-theory in various situations. The construction of topological cyclic homology is based on genuine equivariant homotopy theory, the use of explicit point-set models, and the elaborate notion of a cyclotomic spectrum. The goal of this paper is to revisit this theory using only homotopy-invariant notions. In particular, we give a new construction of topological cyclic homology. This is based on a new definition of the $\infty$-category of cyclotomic spectra: We define a cyclotomic spectrum to be a spectrum $X$ with $S^1$-action (in the most naive sense) together with $S^1$-equivariant maps $\varphi_p: X\to X^{tC_p}$ for all primes $p$. Here $X^{tC_p}=\mathrm{cofib}(\mathrm{Nm}: X_{hC_p}\to X^{hC_p})$ is the Tate construction. On bounded below spectra, we prove that this agrees with previous definitions. As a consequence, we obtain a new and simple formula for topological cyclic homology. In order to construct the maps $\varphi_p: X\to X^{tC_p}$ in the example of topological Hochschild homology we introduce and study Tate diagonals for spectra and Frobenius homomorphisms of commutative ring spectra. In particular we prove a version of the Segal conjecture for the Tate diagonals and relate these Frobenius homomorphisms to power operations.
中文翻译:
关于拓扑循环同源性
拓扑循环同调是对 Connes--Tsygan 循环同调的改进,它由 Bokstedt--Hsiang--Madsen 在 1993 年引入,作为代数 $K$-理论的近似。有一个从代数 $K$-理论到拓扑循环同调的迹图,以及 Dundas--Goodwillie--McCarthy 的一个定理,这导致了幂零浸入的相关理论的等价,这给出了计算 $K$ 的方法- 各种情况下的理论。拓扑循环同调的构建基于真正的等变同伦理论、显式点集模型的使用以及循环谱的详细概念。本文的目标是仅使用同伦不变的概念重新审视这一理论。特别地,我们给出了拓扑循环同调的新构造。这是基于对分圆光谱的 $\infty$ 类别的新定义:我们将分圆光谱定义为具有 $S^1$-action(在最幼稚的意义上)以及 $S 的光谱 $X$ ^1$-equivariant 将 $\varphi_p: X\to X^{tC_p}$ 映射到所有素数 $p$。这里 $X^{tC_p}=\mathrm{cofib}(\mathrm{Nm}: X_{hC_p}\to X^{hC_p})$ 是 Tate 结构。在下界谱上,我们证明这与先前的定义一致。因此,我们获得了一个新的简单的拓扑循环同调公式。为了在拓扑Hochschild 同调的例子中构造$\varphi_p: X\to X^{tC_p}$ 的映射,我们引入并研究了谱的Tate 对角线和交换环谱的Frobenius 同态。
更新日期:2018-01-01
中文翻译:
关于拓扑循环同源性
拓扑循环同调是对 Connes--Tsygan 循环同调的改进,它由 Bokstedt--Hsiang--Madsen 在 1993 年引入,作为代数 $K$-理论的近似。有一个从代数 $K$-理论到拓扑循环同调的迹图,以及 Dundas--Goodwillie--McCarthy 的一个定理,这导致了幂零浸入的相关理论的等价,这给出了计算 $K$ 的方法- 各种情况下的理论。拓扑循环同调的构建基于真正的等变同伦理论、显式点集模型的使用以及循环谱的详细概念。本文的目标是仅使用同伦不变的概念重新审视这一理论。特别地,我们给出了拓扑循环同调的新构造。这是基于对分圆光谱的 $\infty$ 类别的新定义:我们将分圆光谱定义为具有 $S^1$-action(在最幼稚的意义上)以及 $S 的光谱 $X$ ^1$-equivariant 将 $\varphi_p: X\to X^{tC_p}$ 映射到所有素数 $p$。这里 $X^{tC_p}=\mathrm{cofib}(\mathrm{Nm}: X_{hC_p}\to X^{hC_p})$ 是 Tate 结构。在下界谱上,我们证明这与先前的定义一致。因此,我们获得了一个新的简单的拓扑循环同调公式。为了在拓扑Hochschild 同调的例子中构造$\varphi_p: X\to X^{tC_p}$ 的映射,我们引入并研究了谱的Tate 对角线和交换环谱的Frobenius 同态。