Equivariant Structure on Smash Powers

Since the early 1990's there have been several useful symmetric monoidal model structures on the underlying category of a point set category of spectra. Topological Hochschild homology (THH) is constructed from smash powers of ring-spectra, that is, monoids in the category of spectra. Already in the 1980's such a construction was made. Building on ideas of Goodwillie and Waldhausen, Bökstedt introduced a class of ring-spectra for which he could define THH via a homotopy-invariant ad hoc construction of smash powers [Bö], and he determined the homotopy types of THH applied to Z and to Z/pZ. When symmetric monoidal model categories of spectra appeared (e.g. S-modules in the sense of Elmendorf-Kriz-Mandell-May [EKMM], symmetric spectra [HSS] and orthogonal spectra [MMSS],) it turned out that the Hochschild complex in these model categories provided a construction of THH. For S-modules this was shown already in [EKMM], for symmetric spectra it was shown by Shipley [Sh]. The case of orthogonal spectra was treated thesis [Kr] of Kro. It was discovered by Bökstedt, Hsiang and Madsen that the action of the circle group S 1 on THH contains crucial information about algebraic K-theory [BHM]. However it was surprisingly hard to provide symmetric monoidal model structures of equivariant spectra, and the equivariant homotopy type of the Hochschild complex depends on the chosen point set category of spectra as remarked in [EKMM, IX.3.9]. For a commuta-tive ring-spectrum A there is an alternative description of the Hochschild complex as the categorical tensor A ⊗ S 1 in the category of commutative ring spectra. In order for this to be homotopically meaningful we need a convenient model structure on the category of commutative ring spectra [Sh04]. The action of S 1 on topological Hochschild homol-ogy needed for the construction of Bökstedt, Hsiang and Madsen's Topological Cyclic homology (TC) [BHM] has been addressed by Kro in his thesis [Kr], but he works in the category of ring-spectra as opposed to the category of commutative ring-spectra. Model structures on commutative orthogonal ring-spectra with action of a compact Lie group are also important in the norm construction of Hopkins, Hill and Ravenel in [HHR]. Recently iterated Topological Hochschild homology and its relation to the chromatic filtration has been studied together with its versions of " higher Topological Cyclic ho-mology " (e.g. [BCD], [CDD], [Schl] and [BM]). In the present work we study the categorical tensor A ⊗ X of a commutative orthogonal ring-spectrum …