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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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Are you offering to write one? –  Andrew Stacey Jun 17 '10 at 8:41
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Heh. Hush, hush. –  Mariano Suárez-Alvarez Jun 17 '10 at 11:29
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Hi Mariano, could you tell me what the "few other" references are? –  Kevin H. Lin Jul 13 '10 at 17:01
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5 Answers

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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What do you mean by "curved"? –  Tom Goodwillie Jun 17 '10 at 14:42
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Mariano's mythical treatise won't be published by Springer if it uses the spelling "Konstant", by the way. –  Jim Humphreys Jun 17 '10 at 14:47
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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. –  Tim Perutz Jun 17 '10 at 14:53
    
@Jim: Oops, sorry!! Thanks for the correction. –  Kevin H. Lin Jun 17 '10 at 17:30
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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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What do you have in mind, specifically, for formality? –  Kevin H. Lin Jun 17 '10 at 18:13
    
Probably you're thinking of "Kontsevich formality"? –  Kevin H. Lin Jun 17 '10 at 18:16
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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) –  David Ben-Zvi Jun 17 '10 at 18:22
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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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Do you know a place where this last formula is proved ? –  Geoffroy Horel Apr 4 '11 at 1:58
    
@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 $M = A$. –  Tyler Lawson Apr 4 '11 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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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.) –  Dev Sinha Jun 17 '10 at 21:14
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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") –  Yemon Choi Jun 18 '10 at 1:06
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