Definition
Let \((X,\mathcal O_X)\) be a ringed space. We say that a sheaf of \(\mathcal O_X\)-modules \(\mathcal F\) is coherent if
- \(\mathcal F\) is of finite-type
- For every choice of sections \(s_1,...,s_n \in \mathcal F(U)\) the kernel \(\ker \varphi\) of the induced map \(\varphi: \mathcal O_X^{\oplus n} \to \mathcal F\) is finite type.