Let $X$ be a scheme and $mathscr{I}$ a sheaf of beliefs. Let $U$ be an open set of $X$ and $s_1,dots,s_n in Gamma(U,mathscr{I})$. I’m in search of a clarification of what we imply by “the best sheaf $mathscr{I}$‘ generated by the $s_i$‘s”. It appears to me that that is the extension to $X$ by zero (within the sense of Train II.1.19 of Hartshorne) of the subsheaf $mathscr{J}$ of $mathscr{I}|_U$ generated by the $s_i$‘s. That is the sheaf of $X$ related to the presheaf $V mapsto mathscr{J}(V)$ for $V subseteq U$ and $mathscr{J}(V)=0$ in any other case. Utilizing the truth that a piece of that sheaf is a group of suitable native sections of the presheaf and the next algebraic reality we lastly get that $mathscr{I}’$ is a perfect sheaf. Algebraic reality: let $A$ be ring and $f_j in A$ such that $(f_1,…,f_m)=A$. Let $s_j in A_{f_j}$ such that $s_j = s_k$ in $A_{f_j f_k}$. Then there’s an $a in A$ such that $s_j = a$ in $A_{f_j}$ for each $j$. Do i’ve it proper?