There is currently no treatise treating Hochschild (co)homology systematically. There is a chapter in Weibel's book, there's parts of Loday's and a few others...
What should be covered by such a mythical treatise?
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Sign up to join this communityThere is currently no treatise treating Hochschild (co)homology systematically. There is a chapter in Weibel's book, there's parts of Loday's and a few others...
What should be covered by such a mythical treatise?
In addition to Kevin's excellent list:
Formality
The relation of Hochschild and cyclic homology with loop spaces (eg Jones' theorem) and the circle action on Hochschild homology
operadic structure of $(HH^\ast,HH_*)$ (ie "calculus" a la Tsygan-Tamarkin), in particular the BV structure in the Calabi-Yau case
Relation to the cotangent complex/ Andre-Quillen homology in the commutative case
The role of Hochschild homology as recipient of characters (eg Chern characters and characters of representations) -- more generally the relation with algebraic K-theory
topological Hochschild and cyclic homology, the cyclotomic trace, $K^S=THH$
HH for E_n algebras and the Deligne-Kontsevich conjecture
Lie theoretic perspective ($HH^\ast$ as universal enveloping algebra of the Atiyah bracket on the shifted tangent complex, HKR theorem as PBW, $HH^*$ as the Lie algebra of autoequivalences of the derived category...)
My wishlist:
Hochschild (co)homology of (curved) A-infinity algebras/categories
Relation to deformation theory
Hochschild-Kostant-Rosenberg
Hochschild-cyclic spectral sequence and relation to Hodge-de Rham spectral sequence
Deligne conjecture
Relation to Drinfeld center
2D TQFTs (Costello/Kontsevich/Hopkins-Lurie)
A personal list.
I am working on producing an account from a modern perspective at Hochschild cohomology on the $n$Lab.
Some of the wishlist items expressed here are already being covered to some extent. But clearly more needs to be done.
Sarah Witherspoon's book on Hochschild cohomology for Algebras is now published. See https://www.ams.org/publications/authors/books/postpub/gsm-204
I really want to know more about Hochschild cohomology of commutative algebras, and its relation to the representations of $S_n$ and to free Lie algebras (beyond "there are these strange idempotents in $\mathbb Q\left[S_n\right]$ which happen to occur in both fields).
$k$-algebras for $k\supseteq{\mathbb Q}$
, which notes how it respects the usual gadgets like localization or base change. In particular, I'd quite like an explanation of why it behaves well with the Künneth formula, without needing to use simplicial resolutions by polynomial algebras (that tactic runs aground very quickly in "Banach-world")
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– Yemon Choi
Jun 18 '10 at 1:06
There is so much on this subject coming from different perspectives. I would add to all of the above a complete, up to date, exposition regarding the relationship of Hochschild homology with loop spaces (for example including the non-simply connected case) and string topology (the structure of the hochschild chain complex of an algebra with a homotopy version of Poincaré duality, etc...)