Localization theorems in topological Hochschild homology and topological cyclic homology

Abstract

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 .

Multiplicative properties of Quinn spectra

“…sic idea of this machine by considering the example of topological bordism. 1 Let T .Top k / denote the Thom space of the universal R k -bundle with structure group Top k . The usual simplicial model for this space, denoted by S T .Top k /, has as n-simplices the continuous maps f W n !…”

“…Section 7 below). Other important examples are Ranicki's quadratic and symmetric algebraic 1 The smooth case is technically more difficult because it requires careful attention to manifolds with corners; the second author plans to pursue this in a future paper. 2 In the literature these are often called -sets, but that terminology seems infelicitous since the category that governs simplicial sets is called .…”

Transfer maps and projection formulas

“…Connective algebraic K-theory (K), non-connective algebraic K-theory (IK), Hochschild homology (HH), cyclic homology (HC), negative cyclic homology (HN ), periodic cyclic homology (HP ), topological Hochschild homology (T HH), and topological cyclic homology (T C) are all examples of derived Morita invariant functors; see [1,10,17,21,23]. Therefore, by Theorem 1.3(i) we obtain transfer maps :…”

Universal suspension via noncommutative motives

“…It is well known that the category of perfect complexes in the (unbounded) derived category of quasi-coherent sheaves on X admits a dg-enhancement perf dg .X/; see for instance [2], [20] or [6], Example 4.5. Due to [1], Theorem 1.3, [13], §5.2, and [22], §8, Theorem 5, the algebraic K-theory and the (topological) cyclic homology 3 of the scheme X can be obtained from the dg category perf dg .X/ by applying the corresponding invariant. Therefore, when A D perf dg .X/, the above isomorphisms (1.3)-(1.9) suggest that the dg category †.perf dg .X// should be considered as the "noncommutative suspension" or "noncommutative delooping" of the scheme X.…”

“…to distinguished triangles in the base category D.e/ of D. Due to the work of Keller [15], [13], , Schlichting [22], and Blumberg-Mandell [1] (see also [29]) all the mentioned invariants satisfy localization 1 , and so give rise to localizing invariants. In [28], the author constructed the universal localizing invariant [5], [6], [28] for several applications of this theory of noncommutative motives.…”