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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23$\begingroup$ Are you offering to write one? $\endgroup$– Andrew StaceyCommented Jun 17, 2010 at 8:41
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3$\begingroup$ Heh. Hush, hush. $\endgroup$– Mariano Suárez-ÁlvarezCommented Jun 17, 2010 at 11:29
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4$\begingroup$ Hi Mariano, could you tell me what the "few other" references are? $\endgroup$– Kevin H. LinCommented Jul 13, 2010 at 17:01
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8$\begingroup$ Sarah Witherspoon now has a draft for a book on Hochschild cohomology on her publications page. $\endgroup$– pbelmansCommented Sep 13, 2017 at 17:03
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7 Answers
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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...)
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$\begingroup$ What do you have in mind, specifically, for formality? $\endgroup$ Commented Jun 17, 2010 at 18:13
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$\begingroup$ Probably you're thinking of "Kontsevich formality"? $\endgroup$ Commented Jun 17, 2010 at 18:16
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1$\begingroup$ Well that's a whole story in itself, but the Kontsevich formality theorem for the Hochschild dgla (which is crucial for applications of the deformation-theoretic story, in particular for deformation quantization), and its various refinements (Tamarkin, Tsygan, Dolgushev, etc etc) for the entire "calculus" structure on Hochschild chains and cochains. (this is closely related to your Deligne conjecture suggestion, but the latter I think is a more "formal"/structural statement, no pun intended, which is at least morally clear once the right language is set up) $\endgroup$ Commented Jun 17, 2010 at 18:22
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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)
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$\begingroup$ What do you mean by "curved"? $\endgroup$ Commented Jun 17, 2010 at 14:42
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5$\begingroup$ Mariano's mythical treatise won't be published by Springer if it uses the spelling "Konstant", by the way. $\endgroup$ Commented Jun 17, 2010 at 14:47
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5$\begingroup$ It's high time for such a book! Tom: Curved (a.k.a. obstructed, a.k.a. weak) means that the $A_\infty$ operations $m^k$ on $A$ begin not at the differential $m^1$ but at $m^0: k \to A$. The "curvature" is $m_1\circ m_1$ which now involves $m_2$ and $m_0$. Notorious example: (endomorphisms in) Fukaya categories. $\endgroup$ Commented Jun 17, 2010 at 14:53
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$\begingroup$ @Jim: Oops, sorry!! Thanks for the correction. $\endgroup$ Commented Jun 17, 2010 at 17:30
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A personal list.
- Hope the hypothetical author would not restrict the base to be a field, even perhaps allowing another DGA as base. (Most of the following comments probably assume this.)
- The flatness hypotheses over the base and Shukla homology as a repair for this. Invariance of Shukla homology under weak equivalences of DGAs.
- Relationship to TorR⊗Rop(R,M).
- For R→S a map of commutative DGAs with S flat over R, the Hochschild homology spectral sequence starting with HHR(S, TorR(M,N)) and converging to TorS(M,N).
- Base-change formulas, ranging from simple ones to the following: $$ S \otimes_{HH^R(S)}^{\mathbb L} HH^R(A,M) \simeq HH^S(A,M) $$
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$\begingroup$ Do you know a place where this last formula is proved ? $\endgroup$ Commented Apr 4, 2011 at 1:58
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1$\begingroup$ @Geoffroy: There is a paper of McCarthy and Minasian ("HKR theorem for smooth S-algebras") that covers this as formula (8) on page 250; it is stated as a theorem about topological Hochschild homology but it recovers the statement for ordinary Hochschild homology as a special case. I believe that in that paper it was the case that $M = A$. $\endgroup$ Commented Apr 4, 2011 at 2:50
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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.
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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).
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4$\begingroup$ Note: when you consider Hochschild homology of commutative algebras, you should instead pass to Harrison/Andre-Quillen homology, and there the connection to free Lie algebras (or really, cofree Lie coalgebras) is more apparent. (I don't know the analogous passage you want to make for Hochschild cohomology.) $\endgroup$ Commented Jun 17, 2010 at 21:14
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1$\begingroup$ From the (rather niche) perspective of someone doing Hochschild cohomology of Banach algebras -- it would be nice to see a hands-on approach to Gerstenhaber-Schack's Hodge decomposition of not-necessarily-unital commutative
$k$-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") $\endgroup$ Commented Jun 18, 2010 at 1:06
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Sarah Witherspoon's book on Hochschild cohomology for Algebras is now published. See https://www.ams.org/publications/authors/books/postpub/gsm-204
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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...)