Use (Ex. 9.3) to prove that if \[ 0 \to \mcf' \to \mcf \to \mcf'' \to 0 \] is an exact sequence of \(\OO_\mathfrak{X}\dash\)modules on a noetherian formal scheme \(\mathfrak{X}\), and if \(\mcf', \mcf''\) are coherent, then \(\mcf\) is coherent also.
Solution
Write \(a:\mcf'\hookrightarrow\mcf\) and \(b:\mcf\twoheadrightarrow\mcf''\) for the given maps. Coherence on a noetherian formal scheme is local, so work on an arbitrary affine formal open \(\mathfrak U\subseteq\mathfrak X\) and restrict all sheaves and maps to it. For coherent formal sheaves, finite direct sums, kernels and cokernels of morphisms are coherent [@Har10a, Theorem II.9.7 and Corollary II.9.9]. These closure properties concern morphisms whose source and target are already coherent; no such assumption is made about \(\mcf\).
1. There is a surjection \(q:\OO_{\mathfrak U}^{\oplus s}\twoheadrightarrow\mcf''\) and a lift \(\widetilde q:\OO_{\mathfrak U}^{\oplus s}\to\mcf\) with \(b\widetilde q=q\).
Write \(\mathfrak U=\operatorname{Spf}A\) with \(A\) noetherian and complete for its defining ideal. The affine formal module correspondence identifies \(\mcf''\) with the sheaf associated to the finite \(A\)-module \(\Gamma(\mathfrak U,\mcf'')\) [@Har10a, Theorem II.9.7]. A finite set of module generators therefore gives global sections generating the sheaf and a surjection \(q\) as asserted. By P-AGH293EXACTGLOBAL, applied to the given sequence with coherent kernel \(\mcf'\), the map \[ \Gamma(\mathfrak U,\mcf)\longrightarrow\Gamma(\mathfrak U,\mcf'') \] is surjective. Lift each of the \(s\) chosen sections through this map. Their images define \(\widetilde q\), and \(b\widetilde q=q\) because the maps agree on the standard generators of the free sheaf.
2. Put \(R=\ker q\).There is an exact sequence\[ 0\longrightarrow R\xrightarrow{\gamma} \mcf'\oplus\OO_{\mathfrak U}^{\oplus s} \xrightarrow{\beta}\mcf\longrightarrow0 \]in which the first two sheaves are coherent.
1.The restriction of \(\widetilde q\) to \(R\) has zero composite with \(b\), so it factors uniquely through \(a\) as a map \(\lambda:R\to\mcf'\).Define\[ \gamma(r)=(-\lambda(r),r),\qquad \beta(u,v)=a(u)+\widetilde q(v). \]Here the second component of \(\gamma\) uses \(R\subseteq\OO_{\mathfrak U}^{\oplus s}\).The equality \(a\lambda=\widetilde q|_R\) makes \(\beta\gamma=0\), and the second component makes \(\gamma\) injective.
Exactness of the remaining terms can be checked at a stalk. For an element of \(\mcf\), its image in \(\mcf''\) lifts through \(q\); subtracting the corresponding image under \(\widetilde q\) leaves an element of \(a(\mcf')\). Thus \(\beta\) is surjective. If \(a(u)+\widetilde q(v)=0\), applying \(b\) gives \(q(v)=0\), so \(v\in R\). Then injectivity of \(a\) gives \(u=-\lambda(v)\), which says \((u,v)=\gamma(v)\). This proves exactness. Both \(R\) and \(\mcf'\oplus\OO_{\mathfrak U}^{\oplus s}\) are coherent, as required.
3. Q.E.D.
Step 2 identifies \(\mcf|_{\mathfrak U}\) with the cokernel of a morphism between coherent formal sheaves. It is therefore coherent by [@Har10a, Corollary II.9.9]. The construction applies on every affine formal open, and these opens cover \(\mathfrak X\). Locality of coherence proves that \(\mcf\) is coherent on \(\mathfrak X\).