Timeline for Morita equivalence and isomorphisms in cohomology theories
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 1, 2017 at 15:26 | comment | added | hänsel | The map $K(A) =KK(\mathbb{C},A)\rightarrow K(B) = KK(\mathbb{C},B)$ could be realized by multiplication with the Kasparov element $[P,0] \in KK(A,B)$, where $P$ is the Morita $A,B$-bimodule. | |
| Sep 30, 2017 at 16:37 | vote | accept | truebaran | ||
| Sep 28, 2017 at 20:57 | answer | added | Qiaochu Yuan | timeline score: 6 | |
| Sep 26, 2017 at 22:33 | comment | added | Dylan Wilson | One quick way to do the Hochschild case is to use the fact that $HH_*(A/k) \cong \mathrm{Tor}_*^{A \otimes A^{op}}(A,A)$, in other words Tor in the category of A-bimodules. Now, the Morita equivalence yields an equivalence between categories of bimodules by $M \mapsto Q \otimes_AM\otimes_AP$ and similarly the other way. This sends $A$ to $B$, so the two Tor groups are the same. Same argument for Ext and $HH^*$. | |
| Sep 26, 2017 at 17:02 | comment | added | მამუკა ჯიბლაძე | Did you consult the literature? The Hochschild and cyclic case is in several books, e. g. "Cyclic homology" by Loday. The K-theoretic case is straightforward: K-theory of a ring can be defined solely in terms of the category of modules over the ring, and Morita equivalence is more or less the same as equivalence of the categories of modules. | |
| Sep 26, 2017 at 14:20 | history | asked | truebaran | CC BY-SA 3.0 |