The following is Hartshorne’s definition of a coherent module:
Definition
Let \((X,\mathcal O_X)\) be a scheme. We say that an \(\mathcal O_X\)-module is quasi-coherent if there exists an affine cover \(\{U_i\}\) of \(X\) and \(\mathcal O_X(U_i)\)-modules \(M_i\) such that \(\mathcal F|_{U_i}\cong \widetilde M_i\) for all \(i\), where \(\widetilde M_{i}\) is the sheaf associated to the module \(M_i\) (defined on distinguished opens via localization of \(M_i\)). We say that \(\mathcal F\) is coherent if in addition each \(M_i\) can be taken to be a finitely generated \(\mathcal O_X(U_i)\)-module.
Compare this to the definition in the Stack’s project:
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.
Using Hartshorne’s definition it is quite clear that \(\mathcal O_X\) is a coherent \(\mathcal O_X\)-module for every scheme \(X\), but this is not true if we take the Stacks project definition. The two definitions are equivalent, however, if we assume \(X\) to be locally Noetherian.
Proposition
If \(X\) is a locally Noetherian scheme, then the two notions of coherent agree.
Proof
It suffices to assume \(X = \Spec R\) is affine since in both cases coherence is a local property. Let \(M\) be a coherent \(R\) module in the sense of Stacks. Then \(M\) is finite-type which is equivalent to being finitely-generated for affines. Thus Stacks coherent implies Hartshorne coherent. (Notice we did not need locally Noetherian for this! This means Stacks coherence is a strictly stronger notion.)
Now assume that \(M\) is coherent in the sense of Hartshorne. Then \(M\) is finitely generated and in particular finite type, so we need only prove condition (2) in the Stacks definition holds. Since \(X\) is affine and locally Noetherian, \(R\) is Noetherian and hence \(R^{\oplus n}\) is Noetherian for every \(n\in \mathbb N\). The kernel \(\ker \varphi\) of any morphism \(\varphi:R^{\oplus n}\to M\) is a submodule of \(R^{\oplus n}\), hence finitely generated as an \(R\)-module, hence finite-type.