Suppose $A$ is an augmented commutative algebra over a field $k$. What is the relation between Hochschild homology $H\_n(A,k)$ and Kahler differential $\Omega\_{A/k}$? The same question is also asked about $H^n(A,k)$ and $\Omega\_{A/k}$. Here $k$ is considered as the trivial $A$-bimodule via the augmentation.



Hochschild (co)homology and Kahler differentials
Xingting, I do not know the answer for your question. Nevertheless, the concepts involved in such questions are nicely developed in the following book (expressed in BibTex format):


@book {MR2640631,
 AUTHOR = {Majadas, Javier and Rodicio, Antonio G.},
 TITLE = {Smoothness, regularity and complete intersection},
 SERIES = {London Mathematical Society Lecture Note Series},
 VOLUME = {373},
 PUBLISHER = {Cambridge University Press},
 ADDRESS = {Cambridge},
 YEAR = {2010},
 PAGES = {vi+134},
 ISBN = {978-0-521-12572-7},
 MRCLASS = {13D03 (13B10)},
 MRNUMBER = {2640631 (2011m:13028)},
MRREVIEWER = {Srikanth B. Iyengar},
 DOI = {10.1017/CBO9781139107181},
 URL = {<http://dx.doi.org/10.1017/CBO9781139107181>},
}


