If \(\mcf\) is a coherent sheaf on a noetherian formal scheme \(\mathfrak{X}\) which can be generated by global sections, show in fact that it can be generated by a finite number of its global sections.
(Quasi-compactness cannot simply be omitted)
Let \(\mathfrak X=\coprod_{j\ge1}\Spec k\), viewed as a formal scheme with the zero ideal of definition, and let \(\mcf\) restrict to \(\OO^{\oplus j}\) on its \(j\)th component. This formal scheme is locally noetherian but not quasi-compact, and \(\mcf\) is coherent because coherence is local and each component has a finite free module. The sections given by one coordinate vector on one component and zero on all other components generate \(\mcf\) globally. However, any \(N\) global sections give at most \(N\) vectors on the \((N+1)\)st component, and cannot span its \((N+1)\)-dimensional fibre. Thus no finite global generating family exists in this locally noetherian example.
Solution
Choose a family \((s_\lambda)_{\lambda\in\Lambda}\) of global sections generating \(\mcf\). We will find a finite subfamily which still generates it. The noetherian hypothesis includes quasi-compactness of the underlying space.
1. For each \(x\in\mathfrak X\), finitely many of the \(s_\lambda\) generate \(\mcf\) on an open neighborhood of \(x\).
The coherent sheaf \(\mcf\) is locally of finite type, so \(\mcf_x\) is a finite \(\OO_{\mathfrak X,x}\)-module [@Har10a, Chapter II, §9]. Express a finite generating set of this module as finite linear combinations of the germs \((s_\lambda)_x\). Only finitely many indices occur; let \(\Lambda_x\subseteq\Lambda\) be their union. The corresponding map \[ \OO_{\mathfrak X}^{\oplus\Lambda_x}\longrightarrow\mcf \] is surjective at \(x\).
Its cokernel \(Q_x\) is locally of finite type because it is a quotient of the locally finite-type sheaf \(\mcf\). Choose an open neighborhood \(V\) of \(x\) and finitely many sections generating \(Q_x|_V\). Their germs at \(x\) are all zero, since \((Q_x)_x=0\). Each section therefore vanishes on a neighborhood of \(x\); intersect these finitely many neighborhoods to obtain an open \(U_x\subseteq V\) on which all the generators vanish. It follows that \(Q_x|_{U_x}=0\). The sections indexed by \(\Lambda_x\) thus generate \(\mcf\) throughout \(U_x\).
2. A finite subfamily generates \(\mcf\) on all of \(\mathfrak X\).
The open sets \(U_x\) of step 1 cover \(\mathfrak X\). By quasi-compactness, choose a finite subcover \(U_{x_1},\ldots,U_{x_t}\). The set \[ \Lambda_0=\Lambda_{x_1}\cup\cdots\cup\Lambda_{x_t} \] is finite. At every point, one of these finite subsets already generates the stalk, so the sections indexed by \(\Lambda_0\) generate every stalk. Equivalently, with \(N=\#\Lambda_0\), their evaluation map is a surjection \[ \boxed{\OO_{\mathfrak X}^{\oplus N}\twoheadrightarrow\mcf.} \] For the empty formal scheme or the zero sheaf, an empty generating family suffices.
3. Q.E.D.
Step 1 gives finite generating subfamilies on neighborhoods, and step 2 selects finitely many of those neighborhoods to give the required global finite family.