Timeline for Functorial subscheme structure on non-locally closed subsets

7 events

when toggle format	what		by	license	comment
May 20, 2019 at 15:20	comment	added	Jason Starr		* “For every locally ringed space ...” -> “For every such locally ringed space ...”
May 20, 2019 at 15:18	comment	added	Jason Starr		The common feature of all these examples is that the scheme structure in the subset $S$ of the scheme $X$ is a quotient of the inverse image sheaf of rings $i^{-1}\mathcal{O}_X.$. For every locally ringed space $(i,i^):(S,\mathcal{O}_S)\to (X,\mathcal{O}_X),$ this satisfies the universal property that a morphism to $S$ is a morphism to $X$ whose image subset is contained in $S$ and whose pullback map of sheaves of rings factors through $\mathcal{O}_S.$. Thus, the morphism $(i,i^)$ is a universal mono morphism.
May 20, 2019 at 11:11	history	edited	user138661	CC BY-SA 4.0	added 151 characters in body
May 20, 2019 at 10:58	comment	added	Laurent Moret-Bailly		Spec(local ring of a point) is another example.
May 20, 2019 at 10:15	comment	added	Jason Starr		No, locally closed subsets are not the most general subsets. For instance, singleton subsets of non-closed points are (typically) not locally closed subsets, yet the Spec of the residue field is a scheme structure compatible with the subspace topology and the inclusion map.
May 20, 2019 at 9:03	history	edited	user138661	CC BY-SA 4.0	added 154 characters in body
May 20, 2019 at 8:45	history	asked	user138661	CC BY-SA 4.0