Hopf algebra structure on topological Hochschild homology
Hopf algebra structure on topological Hochschild homology
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拓扑 Hochschild 同调上的 Hopf 代数结构
DOI：
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发表时间：
2005
期刊：
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通讯作者：
J. Rognes
中科院分区：
文献类型：
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作者：
V. Angeltveit;J. Rognes
交换S-代数（E_1环谱）R的拓扑Hochschild同调THH（R）自然具有严格意义上的交换R-代数的结构和同伦范畴R上的Hopf代数的结构.我们表明，在平坦性假设下，这使得Bokstedt谱序列收敛到THH（R）的mod p同调到一个Hopf代数谱序列。然后，我们应用这个额外的结构来研究一些有趣的例子，包括交换S-代数ku，ko，tmf，ju和j，并计算同伦群THH（ku）和THH（ko）粉碎后，适当的有限复形。这是利用分圆迹映射到拓扑循环同调来系统计算S-代数的代数K-理论的程序的一部分。AMS分类55 P43、55 S10、55 S12、57 T05; 13 D 03、55 T15
The topological Hochschild homology THH(R) of a commu- tative S-algebra (E1 ring spectrum) R naturally has the structure of a commutative R-algebra in the strict sense, and of a Hopf algebra over R in the homotopy category. We show, under a flatness assumption, that this makes the Bokstedt spectral sequence converging to the mod p homology of THH(R) into a Hopf algebra spectral sequence. We then apply this additional structure to the study of some interesting examples, including the commutative S-algebras ku, ko, tmf, ju and j, and to calculate the homotopy groups of THH(ku) and THH(ko) after smashing with suitable finite complexes. This is part of a program to make systematic computa- tions of the algebraic K-theory of S-algebras, by means of the cyclotomic trace map to topological cyclic homology. AMS Classification 55P43, 55S10, 55S12, 57T05; 13D03, 55T15