Timeline for Reduced Noetherian ring is the intersection of its localizations at primes associated to a nonzero-divisor

6 events

when toggle format	what		by	license	comment
May 10, 2019 at 7:43	history	edited	Martin Sleziak	CC BY-SA 4.0	Removed the deprecated (abstract-algebra) tag - see the tag info: https://mathoverflow.net/tags/abstract-algebra/info (if there are some other suitable tags, choose them instead.)
May 8, 2019 at 10:11	comment	added	Bin		Ah, I missed that! Thx!
May 8, 2019 at 7:47	comment	added	Luc Guyot		As the last inclusion in the chain above is only an inclusion of $R$-modules, i.e., $K(R)_P = \bigoplus_{\mathfrak{p} \subseteq P} K(R/\mathfrak{p}) \subseteq \bigoplus_{\mathfrak{p}} K(R/\mathfrak{p}) = K(R)$, where $\mathfrak{p}$ ranges in the set of minimal primes, the intersection $\bigcap_P R_P$ is to be understood as an intersection of submodules and not of rings. (This is probably why the proposition 11.3 is not phrased in terms of intersection).
May 7, 2019 at 14:30	comment	added	Luc Guyot		The identification $K(R_P) = K(R)_P$ is needed in the statement and the last step of the proof relies on it; this may not hold if $R$ is not reduced, see Exercise 3.15. If $R$ is reduced, then $R_P \subset K(R_P) = K(R)_P \subset K(R)$, so we might just say that $\bigcap_P R_P = R$ for $P$ in the specified range.
May 7, 2019 at 8:25	review	First posts
May 7, 2019 at 9:31
May 7, 2019 at 8:21	history	asked	Bin	CC BY-SA 4.0