Timeline for Points on algebraic stacks
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Dec 11, 2009 at 20:50 | comment | added | Alicia Garcia-Raboso | Glad to be of assistance ;) | |
| Dec 11, 2009 at 20:49 | comment | added | Philipp Hartwig | Yes. You could equivalently describe it as the residue field k(x) of the point x of a scheme X, i.e. the quotient of the local ring O_{X,x} by its maximal ideal. | |
| Dec 11, 2009 at 20:43 | comment | added | Daniel Larsson | Ah, ok, maybe I see where I'm thinking incorrectly. So in the case of a point in an affine scheme $Spec(A)$ corresponding to a non-maximal ideal $p$, the natural field appearing is the fraction field of the residue ring, i.e., $Frac(A/p)$. I mean if take the schematic analog of your explaination. | |
| Dec 11, 2009 at 20:31 | comment | added | Alicia Garcia-Raboso | Images of spectra of field are not necessarily closed points: Spec Q -> Spec Z is the inclusion of the generic point of Spec Z, and Q is a field. | |
| Dec 11, 2009 at 20:28 | comment | added | Philipp Hartwig | Maybe this helps: If X is a scheme, the set of points of X is in canonical bijection to the "set" of morphisms Spec K -> X, where K runs through all fields, modulo the equivalence relation that we identify two morphisms Spec K -> X and Spec K' -> X if there is a field K'' with morphisms Spec K'' -> Spec K' and Spec K'' -> Spec K such that the two compositions Spec K'' -> X coincide. This has nothing to do with closed points and the definition in LMB is the obvious generalization of this fact. | |
| Dec 11, 2009 at 20:17 | comment | added | Daniel Larsson | No it says so in my copy also. But given the way LMB define points (5.2) it seems to me they define closed points: images of spectra of fields (modulo some equivalence relation coming from the stackyness). In my world that is a closed point. | |
| Dec 11, 2009 at 20:01 | history | edited | Philipp Hartwig | CC BY-SA 2.5 |
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| Dec 11, 2009 at 19:40 | history | answered | Philipp Hartwig | CC BY-SA 2.5 |