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This is false even for $\mathrm H^0$: take $X$ to be $\mathbb A^2 \smallsetminus \{0\}$, and as $F$ the structure sheaf of $L \smallsetminus \{0\}$, where $L$ is a line through $0$.

About your first question: the relation is transitive. If $a$ is equivalent to $b$ using a covering, and $b$ is equivalent to $c$ using another covering, it is easy to see that $a$ is equivalent to $c …

Of course, the correct definition of coherence is that in your Question 2. It just so happens that for a sheaf of modules on a scheme it is equivalent to the easier one. As far as a I know, the notio …

Ok, so what will the spectral sequence give you? This is a very easy exercise, but since Altgr is not experienced, here is the solution. The term $E_2^{p,q}$ is the $p^{\rm th}$ cohomology group of th …

This is not true; for example, take $X = \mathbb R^2$, $U = \mathbb R^2 \smallsetminus \{(0,0)\}$. Then your $\mathbb Z_U$ coincides with $\mathbb Z_X$, and $Hom(\mathbb Z_U, F)$ is $F(X)$, not $F(U)$ …

Taking equalizers, or limits, is also the standard algebraic geometry way. So, for example, say that $X \to S$ and $Y \to S$ are finitely presented and proper, with $X\to S$ flat, and have sections $S …