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.

  • $\begingroup$ 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). $\endgroup$ Sep 19, 2013 at 12:40
  • 4
    $\begingroup$ 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
    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?? $\endgroup$ 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? $\endgroup$ 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. $\endgroup$ Sep 19, 2013 at 19:06

1 Answer 1


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.

  • $\begingroup$ Thank you very much for the answer, especially for the edit about KU! What about MU? $\endgroup$ 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? $\endgroup$ Jan 13, 2015 at 4:11
  • 3
    $\begingroup$ @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. $\endgroup$ Jan 13, 2015 at 7:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.