Timeline for What is the theory of local rings and local ring homomorphisms?
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6 events
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| May 26, 2023 at 14:08 | comment | added | Morgan Rogers | Ha I agree, but the point remains that your construction does satisfy your proposed conditions 1 and 2 (and I think also 3 where relevant). You found a theory of local rings and local ring homomorphisms, which answers the question in your title, but you also identified why Grothendieck and co didn't use this theory. For the record, I don't think you can do any better: your example 3 comments up is as far from being a local ring homomorphism as one can get! | |
| May 9, 2023 at 3:51 | comment | added | Zhen Lin | @MorganRogers Yes, but actually for this question we also know what $\mathbb{T} ([\mathbf{2}, \textbf{Set}])$ is, or at least what some of its objects are. This is because $[\mathbf{2}, \textbf{Set}]$ is equivalent to sheaves on the Sierpiński space, which is homeomorphic to the prime spectrum of any discrete valuation ring. We want the structure sheaf to be a $\mathbb{T}$-model – otherwise this $\mathbb{T}$ would not be useful in algebraic geometry! | |
| Feb 19, 2021 at 13:10 | comment | added | Morgan Rogers | This answer is very misleading. It relies on $\mathbb{T}$ being fixed, and the whole point is that we are looking for a distinct $\mathbb{T}$. OP asked for a topos whose $\mathbf{Set}$-models are local rings and local ring homs. Your answer doesn't exclude this. The theory of decidable objects has a distinct classifying topos from the theory of objects; the models of both of these theories in $\mathbf{Set}$ are precisely sets, but the maps of the former are monomorphisms, whereas the maps of the latter are arbitrary functions. | |
| Jun 11, 2013 at 17:19 | vote | accept | Zhen Lin | ||
| Jun 11, 2013 at 17:19 | comment | added | Zhen Lin | More explicitly, the local ring corresponding to $\operatorname{Spec} A$ for a discrete valuation ring $A$ corresponds to the ring homomorphism $A \to \operatorname{Frac} A$, which is not a local ring homomorphism. Hence not every local ring in $[\mathbb{2}, \mathbf{Set}]$ comes from a local ring homomorphism in $\mathbf{Set}$. | |
| Jun 11, 2013 at 16:06 | history | answered | Achilleas K | CC BY-SA 3.0 |