Unwinding the relative Tate diagonal

We show that a spectral sequence developed by Lipshitz and Treumann, for application to Heegaard Floer theory, converges localized form of topological Hochschild homology with coefficients. This allows us the target this can be identified when is torsion-free as module over $\mathrm{THH}_*(\mathbb{F}_2)$, parallel results Mathew on degeneration Hodge-to-de Rham sequence. To carry out, we apply work Nikolaus-Scholze develop general Tate diagonal Hochschild-like diagrams spectra respect decomposition into tensor products. discuss extent which there relative homology.