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Let $A$ be a $E_\infty$-ring spectrum. By EKMM, it may be treated as a commutative algebra in the appropriate category. In particular, one may define topological Hochschild homology as $A\wedge_{A\wedge A} A$, like for usual commutative algebra.

My question is: can one say something about $F_{A\wedge A}(A,A)$ (function spectum), that is about topological Hochschild cohomology? Does the Gerstenhaber bracket make sense in this context? If $A$ is, say, the K-theory, does homotopy groups of its Hochschild cohomology contain some interesting elements?

Of course, it is enough to have $A_\infty$-ring structure on $A$ for this questions, but I am interested only in $E_\infty$. Besides, I am interested in spectra $F_{A\otimes S^n}(A, A)$, where $S^n$ is the $n$-sphere.

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  • $\begingroup$ A definition of topological Hochschild cohomology can be found, for example, in this paper by V. Angeltveit math.uchicago.edu/~vigleik/THHAinfty.pdf (Definition 2.1). $\endgroup$
    – Martin Brandenburg
    Commented Sep 19, 2013 at 12:40
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    $\begingroup$ I'm confused by the question. The relevant chapter of EKMM is entitled ``Topological Hochschild homology and cohomology'' and the definition in terms of function spectra is part of Defn IX.1.1, p 168. It is true that we focused most on homology, but it was meant to be entirely clear that there is a parallel development of cohomological spectral sequences $\endgroup$
    – Peter May
    Commented Sep 19, 2013 at 18:11
  • $\begingroup$ Aren't $A \wedge A$ and $A \otimes S^n$ are fairly different? Is it clear that $A$ is a module over $A \otimes S^n$? have you looked at any papers about iterated THH?? $\endgroup$
    – Sean Tilson
    Commented Sep 19, 2013 at 18:26
  • $\begingroup$ @PeterMay Dear Peter, thank you for the comment and the reference! I thought that $THH_R$ is the object dual to $THH^R$... But anyway, let me repeat the question. Does Hochschild cohomology have any geometric meaning (say, Hochschild homology is connected with K-theory, what about cohomology)? Is there some structure on it that corresponds to the Gerstenhaber bracket on usual Hochschild cohomology? $\endgroup$
    – nikitamarkarian
    Commented Sep 19, 2013 at 19:03
  • $\begingroup$ @MartinBrandenburg Thank you,Martin! I've seen this paper, but did not find answers on my questions there. $\endgroup$
    – nikitamarkarian
    Commented Sep 19, 2013 at 19:06

1 Answer 1

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The topological Hochschild cohomology (that I'll denote now THC) makes sense whenever $A$ is at least an $E_1$-algebra. In particular, you can construct THC of an $E_\infty$-algebra. There is a result called Deligne's conjecture but which is now a theorem stating that THC of an $E_1$-algebra is an $E_2$-algebra. In particular, if you take the homology of THC of something, the resulting graded abelian group has a Gerstenhaber algebra structure. If you take homotopy groups, you get a commutative algebra with a degree 1 bracket but I don't think it's going to satisfy the axioms of a Gerstenhaber algebra in general.

Taking the endomorphisms over $A\otimes S^{n-1}$ is a perfectly fine construction called higher THC. It can be defined as soon as $A$ is an $E_{n}$-algebra although the definition is slightly more involved (a good reference is http://www.math.northwestern.edu/~jnkf/writ/cotangentcomplex.pdf). Higher Deligne's conjecture tells you that this higer Hoschild cohomology is an $E_{n+1}$-algebra. In particular taking homology, you get a Gerstenhaber algebra with a bracket of degree $n$.

Note that in the case where $A$ is $E_\infty$, there is a nice construction of higher THC in the following paper of Ginot Tradler and Zeinalian (they restrict to $E_\infty$-algebras in chain complexes but the case of spectra is similar) http://arxiv.org/abs/1205.7056

Edit: I just noticed that you were asking more specifically what THC of $KU$ is. It turns out that the unit map $KU\to F_{KU\wedge KU}(KU,KU)$ is an equivalence. The same is true if you replace $KU$ by $E_n$ (the height $n$ Lubin-Tate spectrum). This remains true for the higher dimensional versions of THC. The unit map $E_n\to F_{S^d\otimes E_n}(E_n,E_n)$ is an equivalence. The reason for this is essentially the fact that $E_n$ is étale aver the $K(n)$-local sphere. You can look at http://geoffroy.horel.org/HHC%20of%20the%20LT%20ring%20spectrum.pdf for more details.

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  • $\begingroup$ Thank you very much for the answer, especially for the edit about KU! What about MU? $\endgroup$
    – nikitamarkarian
    Commented Oct 1, 2013 at 8:36
  • $\begingroup$ Can you explain why what you call THC takes $E_n$ ring spectra to $E_{n+1}$ ring spectra? In Brun, Fiedorowicz, Vogt arxiv.org/abs/math/0410367, they show THH takes $E_n$ ring spectra to $E_{n-1}$ ring spectra. Can you see the difference in the function spectra construction somehow? $\endgroup$
    – Gabriel Angelini-Knoll
    Commented Jan 13, 2015 at 4:11
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    $\begingroup$ @GabrielAngelini-Knoll This is a non-trivial fact that is called the higher Deligne conjecture. Roughly the idea is that $THC(A)$ has an $E_n$-structure coming from the multiplication on $A$ and an $E_1$-structure coming from the fact that it is given as the endomorphisms of something. These two structure commute with one another inducing an $E_{n+1}$-structure by Dunn's additivity theorem. $\endgroup$
    – Geoffroy Horel
    Commented Jan 13, 2015 at 7:18

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