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On higher topological Hochschild homology of rings of integers. (English) Zbl 1405.55007

This paper describes the higher topological Hochschild homology of rings of integers in number fields with coefficients in suitable residue fields. The answers are formulated in terms of iterated Tor groups, starting with a polynomial algebra on one generator.
Topological Hochschild homology of a ring spectrum with coefficients in a module over the ring is defined via a simplicial object built using the standard simplicial model of the circle \(S^1\). Higher topological Hochschild homology for commutative ring spectra is obtained by replacing the simplicial circle in this construction with a simplicial model of the \(n\)-sphere. Higher \(THH\) provides important invariants, with potential applications to the red-shift conjecture of John Rognes.
The results of M. Bökstedt [“The topological Hochschild homology of \(\mathbb Z\) and of \(\mathbb Z/p\mathbb Z\)”, Preprint] and A. Lindenstrauss and I. Madsen [Trans. Am. Math. Soc. 352, No. 5, 2179–2204 (2000; Zbl 0949.19003)] on ordinary topological Hochschild homology provide the base cases and an iterative description of \(THH\) is used. Consideration of the Bökstedt spectral sequence with rational coefficients shows that in positive degrees \(THH\) is all torsion. The torsion is analyzed by calculating the answer for the completion at each prime ideal.

MSC:

55P42 Stable homotopy theory, spectra
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)

Citations:

Zbl 0949.19003
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