Unwinding the relative Tate diagonal

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

[1]  Robert Lipshitz,et al.  Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers , 2012, 1203.2963.

[2]  Ib Madsen,et al.  On the K-theory of finite algebras over witt vectors of perfect fields , 1997 .

[3]  Peter Scholze,et al.  Topological cyclic homology , 2019, Handbook of Homotopy Theory.

[4]  R. M. Vogt,et al.  Topological Hochschild Homology , 2000 .

[5]  B. Shipley,et al.  Stable model categories are categories of modules , 2003 .

[6]  HKR Theorem for Smooth S-algebras , 2003, math/0306243.

[7]  M. Hopkins,et al.  On the nonexistence of elements of Kervaire invariant one , 2009, 0908.3724.

[8]  Ib Madsen,et al.  The cyclotomic trace and algebraic K-theory of spaces , 1993 .

[9]  R. Thomason.  Algebraic $K$-theory and etale cohomology , 1985 .

[10]  C. Schlichtkrull,et al.  Higher topological Hochschild homology of Thom spectra , 2008, 0811.0597.

[11]  A. Khoroshkin,et al.  Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie , 2006, math/0611623.

[12]  Alina Iacob,et al.  Generalized Tate cohomology , 2005 .

[13]  A. Lindenstrauss,et al.  On the Taylor Tower of Relative K-theory , 2009, 0903.2248.

[14]  Michael A. Mandell,et al.  Topological Cyclic Homology Via the Norm , 2014, Documenta Mathematica.

[15]  Michael Cole,et al.  Rings, Modules, and Algebras in Stable Homotopy Theory , 2007 .

[16]  Achim Krause,et al.  Bökstedt periodicity and quotients of DVRs , 2019, Compositio Mathematica.

[17]  J. Lurie.  Higher Topos Theory , 2006, math/0608040.

[18]  HZ -algebra spectra are differential graded algebras , 2002, math/0209215.

[19]  A. Mathew.  Kaledin’s degeneration theorem and topological Hochschild homology , 2017, Geometry & Topology.

[20]  J. Francis.  The tangent complex and Hochschild cohomology of $\mathcal {E}_n$-rings , 2011, Compositio Mathematica.

[21]  K. Ponto,et al.  Periodic Points and Topological Restriction Homology , 2018, International Mathematics Research Notices.

[22]  D. Kaledin,et al.  Spectral Algebras and Non-commutative Hodge-to-de Rham Degeneration , 2019, Proceedings of the Steklov Institute of Mathematics.

[23]  Jacob Lurie,et al.  Higher Topos Theory (AM-170) , 2009 .