Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $\{Z_i\}$ a directed projective system of quasi-compact topological spaces with projective limit $Z$. Assume we are given open subsets $U_i \subseteq Z_i$ such that:

1) For every $i \leq j$, the preimage $(Z_j \to Z_i)^{-1}(U_i)$ is contained in $U_j$.

2) For every $i$, the preimage $(Z \to Z_i)^{-1}(U_i)$ equals $Z$.

Does it follow that $U_i = Z_i$ for some $i$?

If not, assume that $\{Z_i\}$ is actually a system of affine schemes and scheme morphisms. Then it is true by (EGA IV, Corollaire 8.3.4). But I wonder if there is a direct proof which avoids all these nasty lemmas about ind-constructible sets ...

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.