Nickolas Rollick, Department of Pure Mathematics, University of Waterloo
"The ideal tool"
Last week, we observed that closed embeddings of schemes are modeled after the affine case. In particular, in each affine open set Spec B of the target, the image of the map cuts out a closed subset Spec B/I for some ideal I, thus associating an ideal to each affine open set. This week, we see to what extent this process can be reversed. Given a scheme Y, under what conditions can we attach an ideal I(B) to each affine open subset Spec B, in such a way that the closed subsets Spec B/I(B) glue together into a well-defined closed subscheme?
Once we know how to construct closed subschemes in this way, it will give us an astounding amount of power. For starters, we will very quickly be able to define "the vanishing scheme of a global section", as well as the union and intersection of closed subschemes. A few talks down the road, it will provide a quick route to constructing the "induced reduced subscheme structure" on a closed subset as well.
MC 5417