Timeline for Does the category of local rings with residue field $F$ have an initial object?
Current License: CC BY-SA 4.0
9 events
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| May 18, 2020 at 15:57 | comment | added | Simon Henry | @MyGrandmother'sCobblestone: By the way, your computation of the product in $C_F$ (or equivalently, fiber product over $\mathbb{F}_9$) is not quite right: you also need to quotient out by the kernel of $\mathbb{Z}[X] \to \mathbb{Z}[\sqrt 2] \times \mathbb{Z}[i]$, that is by $(X^2+1)(X^2-2)$. | |
| May 18, 2020 at 13:25 | history | edited | Simon Henry | CC BY-SA 4.0 |
added 109 characters in body
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| May 15, 2020 at 16:20 | comment | added | Simon Henry | Well, it is about pullback, but pullback over $A_{\ker \phi}$ not pullback over $\mathbb{F}_9$. For a general field $F$ this category has all products (given by fiber product over $F$), but it can be shown by abstract category theory that this category will have an initial object if and only if has all limits, i.e. if and only if it has fiber products. Then I started from an example of a fiber product that do not exists (the fiber product over $A_{\ker \phi}$) and back tracked it to a counter example to the original question. | |
| May 15, 2020 at 15:56 | vote | accept | The Thin Whistler | ||
| May 18, 2020 at 11:29 | |||||
| May 15, 2020 at 15:53 | vote | accept | The Thin Whistler | ||
| May 15, 2020 at 15:53 | |||||
| May 15, 2020 at 15:46 | comment | added | Maxime Ramzi | The argument is not about pullbacks : it looks at the map $R\to A_{\ker\phi}$ (where $R$ is your initial object). Since this map factors through both of the subrings indicated (by uniqueness), its image (as a map of rings) is contained in their intersection; and that is not possible, as the restriction of the quotient map to said intersection is not surjective. | |
| May 15, 2020 at 15:04 | comment | added | The Thin Whistler | The pullback of $\mathbb{Z}[\sqrt{2}]_{(3)}\longrightarrow\mathbb{F}_{9}$ and $\mathbb{Z}[i]_{(3)}\longrightarrow\mathbb{F}_{9}$ is $\mathbb{Z}_{(3)}$ when viewed as morphisms in the category of rings. As morphisms in the category $C_{F}$, their pullback is $\mathbb{Z}[x]_{(3,x^{2}+1)}$ - and that lies in $C_{F}$. | |
| May 15, 2020 at 13:19 | vote | accept | The Thin Whistler | ||
| May 15, 2020 at 14:15 | |||||
| May 15, 2020 at 13:14 | history | answered | Simon Henry | CC BY-SA 4.0 |