Is there some textbook for the details of the computation of the homology groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T08:27:00Z http://mathoverflow.net/feeds/question/70163 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70163/is-there-some-textbook-for-the-details-of-the-computation-of-the-homology-groups Is there some textbook for the details of the computation of the homology groups ren l 2011-07-12T18:55:30Z 2011-07-12T20:28:20Z <p>Is there some results for the cyclic homology group $HC_1(A)$, for example, when it is zero, or which case we can compute out it explictly, here $A$ is a commutative algebra over the complex field.</p> http://mathoverflow.net/questions/70163/is-there-some-textbook-for-the-details-of-the-computation-of-the-homology-groups/70170#70170 Answer by Mariano Suárez-Alvarez for Is there some textbook for the details of the computation of the homology groups Mariano Suárez-Alvarez 2011-07-12T19:56:39Z 2011-07-12T19:56:39Z <p>You should provide more information about your algebra $A$ if you intend a useful, non-generic answer. I doubt you will find a textbook exposition---but for appropriate classes fo algebras, one can surely direct you to papers where computations are carried out.</p> <p>The one general approach to computing $HC_\bullet$ inthe commutative case is to mimick rational homotopy theorty and construct a <em>model</em> of your algebra $A$, that is, a differential graded algebra $\mathcal A$, and then use the fact that $HC(A)$ and $HC(\mathcal A)$ (this last homology is the homology of <em>graded differential algebras</em>) are isomorphic.</p> http://mathoverflow.net/questions/70163/is-there-some-textbook-for-the-details-of-the-computation-of-the-homology-groups/70172#70172 Answer by Daniel Pomerleano for Is there some textbook for the details of the computation of the homology groups Daniel Pomerleano 2011-07-12T20:00:42Z 2011-07-12T20:28:20Z <p>There is a formula in the commutative case, $HC_1(A) \cong \Omega^1(A)/(dA)$. Namely there is a Connes exact sequence $HH_0(A) \to HH_1(A) \to HC_1(A) \to 0$. In the case of a commutative algebra over a field $HH_0(A) \cong A$ and $HH_1(A) \cong \Omega^1(A)$. The left hand map is d, giving the above formula. For smooth algebras, you have a similar formula for all of the cyclic homology groups. You can surely find all of this in Loday. </p>