Suppose $X$ is a scheme and $f_1,...,f_n\in \Gamma(X,\mathcal O)$ are global sections.
One often reads about the ideal sheaf $\mathcal I=\mathcal (f_1,...,f_n)\subset \mathcal O$, but I have never seen it defined.
The definition should be that $\mathcal I$ is the sheaf associated to the presheaf $\mathcal I^$ whose value on an open subset $U\subset X$ is $\mathcal I^(U)=({f_1}{\vert_U},...,{f_n}{\vert_U})\cdot\mathcal O(U)$. Right?
My question is simply: Is there a nontautological sufficient condition on $U$ and the $f_i$'s guaranteeing that $\mathcal I(U)=\mathcal I^(U)$?
The only sufficient condition I can think of is that $n=1$ and $U$ is an arbitrary open subset of an integral scheme $X$.
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5$\begingroup$ A sufficient condition is that $U$ is affine. Indeed $\mathcal{I}$ fits in a s.e.s. $0\to \mathcal{K}\to \mathcal{O}^n\to \mathcal{I}\to 0$ and $H^1(U,\mathcal{K}) = 0$ for $U$ affine. $\endgroup$– ChrisJul 13, 2020 at 16:08

1$\begingroup$ @Christ: Thank you very much for this fine answer: +1. Why not transform this comment in a genuine answer? $\endgroup$– lefunesteJul 14, 2020 at 11:26
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