Topological cyclic homology via the norm

Abstract

We describe a construction of the cyclotomic structure on topological Hochschild homology (T HH) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes place entirely in the category of equivariant orthogonal spectra, avoiding use of the Bökstedt coherence machinery. We are also able to define two relative versions of topological cyclic homology (T C) and T R-theory: one starting with a ring Cn-spectrum and one starting with an algebra over a cyclotomic commutative ring spectrum A. We describe spectral sequences computing the relative theory A T R in terms of T R over the sphere spectrum and vice versa. Furthermore, our construction permits a straightforward definition of the Adams operations on T R and T C. ContentsProposition 2.2 ([29, II.4.3]). The category S G of orthogonal G-spectra is equivalent to the category of J G -spaces, i.e., the continuous equivariant functors from J G to T G . The symmetric monoidal structure is given by the Day convolution.This description provides simple formulas for suspension spectra and desuspension spectra in orthogonal G-spectra.Definition 2.3 ([29, II.4.6]). For any finite-dimensional G-inner product space V we have the shift desuspension spectrum functorThis is the left adjoint to the evaluation functor which evaluates an orthogonal G-spectrum at V . Remark 2.4. In [20], the desuspension spectrum F V S 0 is denoted as S −V and F 0 A is denoted as Σ ∞ A in a nod to the classical notation. (They write S −V ∧ A for F V A ∼ = F V S 0 ∧ A.

On curves in K-theory and TR

“…. with y = y [1] and the relations that y [i] y [j] = i+j i y [i+j] for every pair of positive integers i and j. We have the following result:…”

“…[19] and Hesselholt-Madsen [41] prove that THH(R) admits the structure of a genuine cyclotomic spectrum by using the Bökstedt construction. Angeltveit-Blumberg-Gerhardt-Hill-Lawson-Mandell [1] construct the genuine cyclotomic structure on THH(R) using the Hill-Hopkins-Ravenel norm [45], and these constructions are equivalent by the work of Dotto-Malkiewich-Patchkoria-Sagave-Woo [27]. Nikolaus-Scholze [59] construct a cyclotomic structure on THH(R) using the Tatevalued diagonal.…”

“…This computation was previously obtained by Guccione-Guccione-Redondo-Solotar-Villamayor [37]. We let Z⟨y⟩ denote the free divided power algebra which has generators y [1] , y [2] , . .…”

The strong Künneth theorem for topological periodic cyclic homology

“…In S G [L(1)], the fibrations are also determined by the forgetful functor to S G . Note that the lax symmetric monoidal fibrant replacement functor R G on S G induces a lax symmetric monoidal fibrant replacement functor on S G [L(1)]: as the fibrations in S G and S G [L] are the same, R G X is clearly fibrant, and a straightforward verification shows that R G is lax symmetric monoidal on S G [L (1)].…”

“…We now turn to the case of the circle group T. We write C(1) for C with the standard action of T as the group of unit complex numbers, S(C(1) n ) for the unit sphere in C(1) n , and S C(1) n for the one-point compactification of C(1) n . When working in the T-equivariant stable category, we write S nC (1) for the suspension spectrum of S C(1) n for n ∈ N, and we extend this notation to representation spheres S nC (1) for all n ∈ Z. We have the standard bar construction model for ET, which comes with a filtration from geometric realization, but to better match the numbering in the finite group case, we define ET 2n+1 = ET 2n and let ET 2n be the geometric realization n-skeleton; we use the corresponding numbering for the filtration on ET.…”