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I am trying to write up an elementary exposition for how the Hochschild chain complex for a commutative associate algebra is the normalized chains/Moore complex of the simplicial algebra that one gets by tensoring the algebra $A$ with the simplicial set $\Delta[1]/\partial \Delta[1]$:
$C_\bullet(A,A) = N_\bullet( (\Delta[1]/\partial \Delta[1]) \cdot A ) \,.$I would like to get feedback on whether or not my exposition is in fact understandable in an elementary way.
The section that contains this material is the section
at the entry Hochschild cohomology. Just this one section. It’s not long.
It describes first the simplicial set $\Delta[1]/\partial \Delta[1]$, then discusses how the coproduct in $CAlg_k$ is given by the tensor product over $k$, and deduces from that what the simplicial algebra $(\Delta[1]/\partial \Delta[1])$ is like.
After taking the normalized chains of that, the result is Pirashvili’s construction of a chain complex from a simplicial set and a commutative algebra. I just think it is important to amplify that this construction of Pirashvili’s is a categorical tensoring=copower operation. Because that connects the construction to general abstract constructions. That’s what the beginning of the above entry is about. But for the moment I would just like to make the elementary exposition of the tensoring operation itself pretty and understandable.
I have further expanded the section
Identification with Kähler differential forms and have now some discussion in The simplicial circle action. This gives so far at least a hint of how to prove that under the identification $(Spec A)^{S^1} \simeq Spec(C_\bullet(A,A))$ and the HKR-theorem, the canonical circle action induced the de Rham differential.
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