Timeline for Primary decomposition for non-affine schemes
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Jul 7, 2010 at 15:58 | history | edited | Charles Staats | CC BY-SA 2.5 |
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| Jul 7, 2010 at 14:53 | comment | added | Charles Staats | Okay, if it's not actually supposed to be primary decomposition in the affine case, then I can believe it. Thanks! | |
| Jul 7, 2010 at 4:13 | comment | added | Boyarsky | One last thing: you should unravel my suggestion (= special case of EGA definition, applied to the structure sheaf) in the affine case to see that it is not primary decomposition, but rather a different kind of expression for $(0)$ as an intersection of ideals with prime radical. It satisfies your desired injectivity, surjectivity, and irreducibility properties, which is ultimately what matters. Your definition of primary scheme has problems because you allow empty or non-affine $U$, but after you read the EGA discussion you'll agree that Grothendieck's way is the right one...as usual. :) | |
| Jul 7, 2010 at 4:06 | comment | added | Boyarsky | Charles, if you follow my reference suggestion you'll quickly come upon EGA IV$_2$, 3.2.5 & 3.2.6, which are precisely a useful notion of primary decomposition for coherent sheaves $\mathcal{F}$ on locally noetherian schemes $X$ (recovering my suggestion for the structure sheaf). Beware of two typos: in 3.2.5 the quotients $\mathcal{F}_ {\alpha}$ of $\mathcal{F}$ should be required to be quasi-coherent (equivalently, coherent), and more importantly in 3.2.6 at the end $\kappa(x)$ should be $\mathcal{O}_ {X,x}$. Hopefully my preceding comment clarifies things, but seriously, just read EGA. | |
| Jul 7, 2010 at 3:34 | comment | added | Charles Staats | Are you sure? What you are describing sounds uniquely defined, which primary decomposition is not, even in the affine case. | |
| Jul 7, 2010 at 2:20 | comment | added | Boyarsky | Charles, you want the concept of "associated point" of a scheme and its good behavior in the (locally) noetherian case, so you then take the $Y_i$ to be the schematic closures of the local rings at the associated points. So by bizarre coincidence, I sort of answered this one yesterday (do search on "prime cycle"): see EGA IV$_2$, 3.1ff. | |
| Jul 7, 2010 at 0:05 | history | asked | Charles Staats | CC BY-SA 2.5 |