Timeline for Which ring spectra have some kind of exponential map turning addition into multiplication?
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| Jul 12, 2017 at 21:09 | comment | added | skd | For $KO$ (completed at $p$), the Rezk logarithm is a map $gl_1 KO \to KO$. By Theorem 4.11 of math.uiuc.edu/~mando/papers/koandtmf.pdf, this map is a weak equivalence on 3-connected covers. This means that you get an "exponential" map $KSO(X) \to (1+KSO(X))^\times$, which seems like something you might be interested in. (This is probably in Rezk's logarithm paper.) | |
| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| Jun 5, 2014 at 14:56 | comment | added | მამუკა ჯიბლაძე | @Fernando Yes yes thanks, something like this, and in this case $x+y$ does indeed go to $(1+x)(1+y)$, but this is quite rare of course... | |
| Jun 5, 2014 at 14:36 | comment | added | Fernando Muro | Mamuka, do you mean something like this toy commutative algebra example? Let $(R,\frak m)$ be a local ring with $\frak m^2=0$. Then there is an injective homomorphism ${\frak m}\rightarrow R^\times\colon x\mapsto 1+x$. In this case, you would like to study the `connected component of $1$ in the spectrum of unit of $R$'. For such a thing, I believe you have different versions in the literature, and it's a matter of current research to dilucidate which is the most appropriate one (I may be saying something silly here, since I'm not an expert, but this is what I've understood from experts). | |
| Jun 5, 2014 at 14:26 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 3.0 |
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| Jun 3, 2014 at 19:37 | comment | added | Qiaochu Yuan | It seems that there is a kind of logarithm, though: ncatlab.org/nlab/show/logarithmic+cohomology+operation | |
| Jun 3, 2014 at 14:18 | comment | added | Charles Rezk | I don't understand your examples. In neither case is there an equivalence between $R_0$ (with additive structure) and $R_1$ (with multiplicative structure) which is compatible with the given $H$-space structures. | |
| Jun 3, 2014 at 10:39 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 3.0 |
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| Jun 3, 2014 at 10:20 | history | asked | მამუკა ჯიბლაძე | CC BY-SA 3.0 |