Foundations of Algebraic Geometry Class 27
0.1. Here is a notion I should have introduced earlier: induced reduced subscheme structure. Suppose X is a scheme, and S is a closed subset of X. Then there is a unique reduced closed subscheme Z of X “supported on S”. More precisely, it can be defined by the following universal property: for any morphism from a reduced scheme Y to X, whose image lies in S (as a set), this morphism factors through Z uniquely. Over an affine X = Spec R, we get Spec R/I(S). (Exercise: verify this.) For example, if S is the entire underlying set of X, we get X.