Diagrammatic calculus (contd.)
Recorded at Groupoids & Stacks, IHP, Paris (2007), featuring Jonathan Rosenberg. From the Michael Wright Collection, held by the Archive Trust for Research in Mathematical Sciences & Philosophy.
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mw0000235-cc-b- Format
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- Michael Wright Collection
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- Archive Trust for Research in Mathematical Sciences & Philosophy
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0:00 The theory of A tends to be C. And suppose you have a symmetric fundamental class in both k-theory and syclo-theory, then the modified churn character with a square root of the top class gives you an isometry between those two pairs. So, one pairing comes from the delta, and the other pairing comes from the C. This one's back on k-theory, this one's on, let's say, etymology. Alright, so how do you even know that you have this square root of the Todd class? Well, a Todd class is an invertible element in the endomorphism ring of this. So it's basically just an element of the GLNC, basically. So as such, you can take a square root. I mean, again, a square root's not canonical, but you just choose one. then you can use the in proving this, you use the symmetry condition so you wrote Todd as square root of Todd times square root of Todd and then you use the symmetry condition to move one of those down to the other side and now you can slide it across and now you can commute things and then you've got the proof of it here So it's not particularly difficult, but that now gives you the analog of the formula that we started with. I think that's it. Okay, so on the equations or components, that's, do you know that we're not going to Do you know examples where there is no PCP? That is a good question.
2:30 That's because since you can see it from the left to the right... Actually, no. I don't know of any reason why you couldn't have such examples, but I don't know how to describe it. So, all the examples I know are with nuclear algebras, where you can check the use of the . Not necessarily nuclear, k-nuclear actually. Or k-nuclear, right. Probably the tens of order to take is either minimal or maximum, right? Right. Yeah, I didn't want to talk about that so much. But I guess that we should be k-nuclear. Yeah, the algebra should be k-nuclear or else you run into difficulties. So, I'm mostly thinking here about the spatial tensor product, but for the examples I have in mind, it doesn't make a difference. Other questions? Thank you.
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