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.
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1$\begingroup$ ncatlab.org/nlab/show/Hochschild-Kostant-Rosenberg+theorem $\endgroup$– Simon WadsleyCommented Mar 20, 2013 at 14:49
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$\begingroup$ can HKR be globalised to schemes? $\endgroup$– Jacob BellCommented Mar 20, 2013 at 15:14
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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}, }
