Timeline for Total ring of fractions vs. Localization

11 events

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Oct 27, 2023 at 10:44	comment	added	Elías Guisado Villalgordo		I don't know which among the examples in Kleiman's article is the one where $H$ is not onto (I'm not reading it in detail). Anyway, here's an explicit example: let $k$ be a field and let $R=k[X_1,\dots,X_n,\dots]/(X_iX_j\mid i\neq j)$ be the coordinate ring of the union of the coordinate axes of $\mathbb{A}_k^\mathbb{N}$. The set of zerodivisors of $R$ is $\mathfrak{m}=(X_1,X_2,\dots)$, whence $Q(R)=R_\mathfrak{m}=R$. On the other hand, $R_{X_1}=k[X_1,X_1^{-1}]$, which is a domain. Thus, setting $S=(X_1)$, the map $H$ identifies with $R_{X_1}\to\operatorname{Frac}(R_{X_1})=k(X_1)$.
S Oct 26, 2023 at 18:24	history	suggested	Elías Guisado Villalgordo	CC BY-SA 4.0	Substituted broken link
Oct 26, 2023 at 17:53	review	Suggested edits
S Oct 26, 2023 at 18:24
Apr 13, 2017 at 12:58	history	edited	CommunityBot		replaced http://mathoverflow.net/ with https://mathoverflow.net/
Jun 19, 2012 at 13:15	vote	accept	Martin Brandenburg
Jun 19, 2012 at 10:20	answer	added	David E Speyer		timeline score: 16
Jun 19, 2012 at 9:30	answer	added	user24527		timeline score: 0
Jun 18, 2012 at 18:53	history	edited	Martin Brandenburg	CC BY-SA 3.0	added 12 characters in body
Jun 17, 2012 at 18:52	answer	added	Will Sawin		timeline score: 9
Jun 17, 2012 at 10:25	history	edited	Martin Brandenburg	CC BY-SA 3.0	added 85 characters in body
Jun 17, 2012 at 9:58	history	asked	Martin Brandenburg	CC BY-SA 3.0