Timeline for Which endomorphism algebras are not Morita-trivial?
Current License: CC BY-SA 3.0
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| Mar 4, 2014 at 17:36 | comment | added | Theo Johnson-Freyd | @მამუკაჯიბლაძე I don't know much about ring spectra, but I'd love to learn. Maybe you can describe some such examples in an answer? | |
| Mar 4, 2014 at 17:36 | comment | added | Theo Johnson-Freyd | @WillSawin Yes, I believe that will work. So the reason I tend to believe (mistakenly) that all endomorphism algebras are Morita trivial is because Vect does not split as a product of categories (so the only non-Morita-trivial endomorphism algebra is 0). | |
| Mar 4, 2014 at 2:21 | comment | added | Will Sawin | $ (X \otimes A X^*) \otimes_1 (X \otimes_A X^*) = X \otimes_A ( X^* \otimes_1 X) \otimes_A X^*= X \otimes_A A \otimes_A X^*= X \otimes X_*$. So this object is an idempotent, and I think it is self-dual, which I think means one can without loss of generality take $A$. Can one argue from there to split the category into a product category? I am thinking taking the map $e \to 1$ and the dual $1 \to e$ and using them to define the complement in each object $Y$ of $e \otimes Y$. I haven't checked that all the diagrams and such commute, though. | |
| Mar 3, 2014 at 8:05 | comment | added | მამუკა ჯიბლაძე | There must be lots of examples in the stable homotopy category - many nontrivial ring spectra arise as $X\wedge DX$ | |
| Mar 3, 2014 at 3:58 | history | asked | Theo Johnson-Freyd | CC BY-SA 3.0 |