Localization theorems in topological Hochschild homology and topological cyclic homology

By Andrew Justin Blumberg, Michael A. Mandell. 2008

115 citations recorded by the source corpus.

We construct localization cofibration sequences for the topological Hochschild homology (THH ) and topological cyclic homology (TC ) of small spectral categories. Using a global construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofibration sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of Thomason‐Trobaugh in K ‐theory. We also deduce versions of Thomason’s blow-up formula and the projective bundle formula for THH and TC . 19D55; 14F43

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/34a65311587f0add0e283f7f988174215b1c977d

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Citations in the corpus

Chow motives versus non-commutative motives (2011, background): “Thanks to the work of Blumberg–Mandell, Keller, Schlichting, and Thomason–Trobaugh (see [8], [30], [29], [42], [47]), all the mentioned invariants satisfy localization3 and so give rise to localizing invariants.”
The homotopy theory of cyclotomic spectra (2013, unknown): “In [1], the authors lacked Proposition 2.”
K-theory and topological cyclic homology of henselian pairs (2018, methodology): “By the theorems of Thomason–Trobaugh [74] and Blumberg–Mandell [11] respectively, K(−)/p and TC(−)/p are Nisnevich sheaves on SpecR with values in the ∞-category Sp.”
Differential Graded Categories are k-linear Stable Infinity Categories (2013, unknown): “3. SpectralCategoriesand theEnriched Dold-Kan Correspondence The theory of spectral categories has a similar ﬂavor to the theory of dg categories. The reader is encouraged to read [BGT1], [BGT2], and [BM] for further background in this direction. 3.1. Review of Spectral Categories. We let S denote t”
The Local Structure of Algebraic K-Theory (2012, methodology): “This definition is not the same as what you get by applying topological cyclic homology to (Waldhausen’s S-construction on) the category of vector bundles on X, however a preprint [23] of Blumberg and Mandell show that Geisser and Hesselholt’ definition agrees with the topological cyclic homology of”
A universal characterization of higher algebraic K-theory (2010, background): “y theorem 1.4. Then the following proposition 48 ANDREW J. BLUMBERG, DAVID GEPNER, AND GONC¸ALO TABUADA follows from the analogous properties of THH and TC in the setting of small spectral categories [47, 4]. Proposition 9.1. THH and TC are localizing invariants of small stable ∞- categories. Next, ”
Uniqueness of the multiplicative cyclotomic trace (2011, background): “Since we know that THH(C) is Morita invariant [4], it descends to a functor THH : Cat ∞ −→ S.”