Timeline for A simple proof of the Weyl algebra's rigidity.
Current License: CC BY-SA 3.0
13 events
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| Apr 13, 2014 at 21:25 | comment | added | Mariano Suárez-Álvarez | It is not true that HH^1(A)=0 implies A is a projective bimodule. It is very easyy to construct counterexamples for that claim. (It is true that if H^1(A,M)=0 for all A-bimodules M, then A is projective, but that is rather irrelevant here) In any case, the Weyl álgebras are definitely not projective as bimodules over themselves. | |
| Apr 13, 2014 at 21:14 | comment | added | ABIM | oh, and the part about the Wodzicki theorem is quite cool, but its alot to juse use my remark above and then note that $HH^0(A)\cong Z(A)$ and it is easy to see that $Z(A)\cong k$. I do admit my approach is less mathematically elegant but its a tiny bit quicker and requires much less background. | |
| Apr 13, 2014 at 21:09 | comment | added | ABIM | Actually you don't need to compute $ A1/[A1,A1]$ since it can be shown that: if $HH^1(A)\cong 0$ then $A$ is projective as an $A^e$-module. Moreover in the case the algebra $A$ is unital, such as the case of the $A_n$ this is the same as $HH^i(A)\cong 0$ for positive $i$ (since, $HH^i(A)\cong Ext_i^{A^e}(A,A)\cong$ because $A$ is projective). Btw, that comment at the beginning is amazing emperor :P | |
| Jul 25, 2011 at 19:43 | comment | added | Mariano Suárez-Álvarez | Don't hesitate to contact me off-MO on the siubject :) | |
| Jul 25, 2011 at 18:43 | comment | added | B. Bischof | @Mariano, presumably you'll need some condition on the underlying ring similar to your condition that it is polys over one var, and that the central elements are all just series in this variable. Blah blah blah. There is some irony about this entire situation, in particular, I asked the question(on GWAs) above to my advisor, and then asked this question on MO because I wondered about rigidity of hyperbolic algebras(which are GWAs). I guess what I am trying to say is "thanks", or something like that... | |
| Jul 25, 2011 at 18:19 | comment | added | Mariano Suárez-Álvarez | @Bischof: regarding GWAs: the method should work, but I don't know what the end result will be. In particular, you surely can get a resolution by the same means. | |
| Jul 25, 2011 at 18:02 | comment | added | B. Bischof | Additionally, after finding this paper: I reiterate my previous comment. | |
| Jul 25, 2011 at 17:56 | comment | added | B. Bischof | @Mariano, in your paper on cohomology of GWAs, does your result hold for higher "dimensional" GWAs(I mean multiple central elements and automorphisms)? I may be being dumb. | |
| Jun 28, 2011 at 23:47 | comment | added | Mariano Suárez-Álvarez | I am pretty sure I had an argument avoind all computation starting from Block's theorem that tells us that the inclusion $k\to A_n$ induces an isomorphism $HC_\bullet^{\mathrm{per}}(k)\to HC_\bullet^{\mathrm{per}}(A_n)$ and various commutative diagrams, but I cannot see what it was now :/ | |
| Jun 28, 2011 at 23:01 | history | edited | Mariano Suárez-Álvarez | CC BY-SA 3.0 |
deleted 7 characters in body
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| Jun 28, 2011 at 23:01 | comment | added | B. Bischof | You're a cool dude. And to be honest, I also prefer HH. Thanks. :) | |
| Jun 28, 2011 at 23:00 | vote | accept | B. Bischof | ||
| Jun 28, 2011 at 22:55 | history | answered | Mariano Suárez-Álvarez | CC BY-SA 3.0 |