The topological cyclic homology of the dual circle

We give a new proof of a result of Lazarev, that the dual of the circle $S^1_+$ in the category of spectra is equivalent to a strictly square-zero extension as an associative ring spectrum. As an application, we calculate the topological cyclic homology of $DS^1$ and rule out a Koszul-dual reformulation of the Novikov conjecture.
Comment: 18 pages, 2 figures, 2 tables. Replaces the second half of the earlier preprint "On the topological Hochschild homology of $DX$."