Timeline for Does Peetre's theorem hold in complex analysis?
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| Mar 2, 2021 at 2:04 | history | edited | Carlos Esparza | CC BY-SA 4.0 |
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| May 14, 2020 at 15:51 | vote | accept | Carlos Esparza | ||
| Nov 12, 2019 at 22:36 | answer | added | Igor Khavkine | timeline score: 7 | |
| Nov 11, 2019 at 23:31 | comment | added | Carlos Esparza | @Qfwfq Well, Peetre's theorem says a lot more - namely that every differential operator locally is a polynomial with $C^\infty$ coefficients in the partial derivatives. Q1 is whether that is true in the holomorphic category. For example on $\mathbb{C}$ this would mean that locally $D = f_0 + f_1 \partial + f_2 \partial^2 + \cdots + f_n \partial^n$. | |
| Nov 11, 2019 at 16:09 | comment | added | Qfwfq | Ok, I see. - Assume we're on a space $(X,O_X)$ locally ringed in algebras over a char zero field $K$, and $E$ and $F$ are two (finite rank) locally free sheaves on $X$. We have: 1) the set $Hom_K (E,F)$ of $K$-linear maps of sheaves of $K$-vector spaces. 2) the set $Diff_K(E,F)$ of the $D\in Hom_K(E,F)$ locally satisfying the recursive $[D,M_f]\in Diff^{n-1}$ type relation. So, if I've understood well, smooth Peetre's theorem says the sets 1) and 2) are equal in this case? and your Q.1 was whether that was also true in the holomorphic category? | |
| Nov 11, 2019 at 15:36 | comment | added | Carlos Esparza | @Qfwfq on a (paracompact) manifold that's equivalent to saying that $D$ is a morphism if sheaves. But that's just a particularity if the real case, in general you should define differential operators as morphisms of sheaves (e.g. the support of an nonzero holomorphic function on a connected domain is the entire domain) | |
| Nov 11, 2019 at 15:32 | comment | added | Qfwfq | According to Wikipedia, Peetre's theorem needs a support-decreasing hypothesis (in the real differentiable case). | |
| Nov 11, 2019 at 13:50 | history | edited | Carlos Esparza | CC BY-SA 4.0 |
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| Nov 11, 2019 at 9:57 | answer | added | Michael Bächtold | timeline score: 1 | |
| S Nov 9, 2019 at 20:03 | history | suggested | Amir Sagiv | CC BY-SA 4.0 |
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| Nov 9, 2019 at 19:34 | comment | added | Carlos Esparza | Éléments de Géometrie Algébrique by Grothendieck | |
| Nov 9, 2019 at 19:32 | comment | added | Amir Sagiv | Welcome to Math Overflow. What is EGA? | |
| Nov 9, 2019 at 19:32 | review | Suggested edits | |||
| S Nov 9, 2019 at 20:03 | |||||
| Nov 9, 2019 at 18:39 | history | edited | Carlos Esparza | CC BY-SA 4.0 |
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| Nov 9, 2019 at 18:33 | history | edited | Carlos Esparza | CC BY-SA 4.0 |
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| Nov 9, 2019 at 18:30 | review | First posts | |||
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| Nov 9, 2019 at 18:25 | history | asked | Carlos Esparza | CC BY-SA 4.0 |