I'm reading Gathmanns's lecture notes on Algebraic Geometry where he defines a Prevariety as ringed space $V$ which has a finite open cover of affine varieties $U_i$. I assume this not just a set-theoretic covering, and that the topologies/structure sheaves of the $U_i$ has to play nice with the topology/structure sheaf of $V$, but I can't find a definition which specifies this anywhere. For example, it seems sensible to require that the subspace topologies $V \cap U_i$ agree with the original topologies of the $U_i$ and that $\mathcal{O}_V(U_i) = \mathcal{O}_{U_i}(U_i)$. Is there such compatibility condition, and in that case, what is it?
2026-02-23 06:59:33.1771829973
Definition of a Prevariety - Specifically a covering by ringed spaces
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