Prove the analogue of (5.6) for formal schemes, which says: if \(\mathfrak{X}\) is an affine noetherian formal scheme, and if \[ 0 \to \mcf' \to \mcf \to \mcf'' \to 0 \] is an exact sequence of \(\OO_\mathfrak{X}\dash\)modules, and if \(\mcf'\) is coherent, then the sequence of global sections \[ 0 \to \Gamma(\mathfrak{X}, \mcf') \to \Gamma(\mathfrak{X}, \mcf) \to \Gamma(\mathfrak{X}, \mcf'') \to 0 \] is exact. Proceed in the following steps.
Let \(\mci\) be an ideal of definition for \(\mathfrak{X}\), and for each \(n > 0\) consider the exact sequence \[ 0 \to \mcf'/\mci^n \mcf' \to \mcf/\mci^n \mcf' \to \mcf'' \to 0 . \] Use (5.6), slightly modified, to show that for every open affine subset \(\mcu \subseteq \mathfrak{X}\) the sequence \[ 0 \to \Gamma(\mcu, \mcf'/\mci^n \mcf') \to \Gamma(\mcu, \mcf/\mci^n \mcf') \to \Gamma(\mcu, \mcf'') \to 0 \] is exact.
Now pass to the limit, using (9.1), (9.2), and (9.6). Conclude that \(\mcf \cong \inverselim_n \mcf/\mci^n \mcf'\) and that the sequence of global sections above is exact.
Solution
Identify \(\mcf'\) with its image in \(\mcf\) and put \[ K_n=\mcf'/\mci^n\mcf',\qquad G_n=\mcf/\mci^n\mcf', \qquad \mathfrak X_n=(|\mathfrak X|,\OO_{\mathfrak X}/\mci^n). \] The schemes \(\mathfrak X_n\) are the ordinary infinitesimal layers on the common underlying space of the formal scheme. On an affine formal open \(\mathfrak U\), their restrictions \(\mathfrak U_n\) are affine schemes [@Har10a, Propositions II.9.4 and II.9.5]. The sheaf \(K_n\) is coherent on \(\mathfrak X_n\), and \[ \mcf'\xrightarrow{\cong}\varprojlim_n K_n \] by [@Har10a, Proposition II.9.6]. Neither \(\mcf\) nor \(\mcf''\) is assumed coherent or annihilated by a power of \(\mci\).
1. For every affine formal open \(\mathfrak U\) and \(n\ge1\), the sequence\[ 0\to K_n(\mathfrak U)\to G_n(\mathfrak U) \to\mcf''(\mathfrak U)\to0 \]is exact.
The sheaf sequence \(0\to K_n\to G_n\to\mcf''\to0\) is exact by taking the indicated quotient of the original sequence. Left exactness on sections gives exactness except possibly at the last term.
Take \(s\in\mcf''(\mathfrak U)\). Surjectivity as sheaves gives local lifts to \(G_n\). Refine their domains to a finite cover by opens which are distinguished affine opens of \(\mathfrak U_n\); the topology is the same, and the affine scheme \(\mathfrak U_n\) is quasi-compact. Write the lifts as \(s_i\). On overlaps their differences \(c_{ij}=s_i-s_j\) are sections of \(K_n\) and satisfy \(c_{ij}+c_{jk}=c_{ik}\). The usual affine gluing argument for a quasi-coherent sheaf gives sections \(b_i\) of \(K_n\) with \(c_{ij}=b_i-b_j\) [@Har10a, Proposition II.5.6], as in the affine Čech calculation in T-COHAFF. Thus \(s_i-b_i\) agree on overlaps and glue to a section of \(G_n(\mathfrak U)\) lifting \(s\).
This argument only uses the quasi-coherence of \(K_n\) on the affine scheme \(\mathfrak U_n\). The middle and last sheaves need only be sheaves of abelian groups for the correction and gluing, so they need not be modules over \(\OO_{\mathfrak U_n}\). This proves the required modification of Proposition II.5.6 and all of (a).
2. The transition maps \(K_{n+1}(\mathfrak U)\to K_n(\mathfrak U)\) are surjective for every affine formal open \(\mathfrak U\).
On the affine scheme \(\mathfrak U_{n+1}\) there is an exact sequence of coherent sheaves \[ 0\to\mci^n\mcf'/\mci^{n+1}\mcf' \to K_{n+1}\to K_n\to0. \] The last sheaf is regarded as a sheaf on that scheme through its quotient structure sheaf. Exactness of global sections for quasi-coherent sheaves on an affine scheme gives the asserted surjectivity [@Har10a, Proposition II.5.6]. Consequently the inverse system of groups \(K_n(\mathfrak U)\) satisfies the Mittag–Leffler condition.
3. For \(H=\varprojlim_n G_n\), there is an exact sequence of sheaves\[ 0\longrightarrow\mcf'\longrightarrow H\longrightarrow\mcf''\longrightarrow0, \]and it is exact on sections over every affine formal open.
1.Its discrepancy at level \(n\) lies in \(K_n(\mathfrak U)\); lift that discrepancy through the surjection of step 2 and subtract it at level \(n+1\).This produces compatible lifts at every level.
Limits of sheaves are computed on sections [@Har10a, Proposition II.9.2]. Together with \(\mcf'\cong\varprojlim_n K_n\), this identifies the displayed sequence with \[ 0\to\mcf'(\mathfrak U)\to H(\mathfrak U) \to\mcf''(\mathfrak U)\to0. \] These maps commute with restriction. Since affine formal opens form a basis, the resulting sequence of sheaves is exact. In particular, exactness has been proved on sections before passing to sheaves; it is not an application of a generally false exactness assertion for inverse limits of sheaves.
4. The natural map \(\mcf\to H\) is an isomorphism, and the desired global-section sequence is exact.
The quotient maps \(\mcf\to G_n\) give a natural map \(\eta:\mcf\to H\). It takes the original short exact sequence to the short exact sequence in step 3, with the identity on \(\mcf'\) and on \(\mcf''\). At a stalk, an element in the kernel of \(\eta\) maps to zero in \(\mcf''\) and hence comes from \(\mcf'\); the identity on that term forces it to be zero. Conversely, an element of \(H\) at a stalk has an image in \(\mcf''\) which lifts to \(\mcf\) at that stalk. Subtracting the image of this lift leaves an element of \(\mcf'\), also in the image of \(\eta\). Thus \(\eta\) is an isomorphism at every stalk and therefore an isomorphism of sheaves.
We have proved the specific completion assertion \[ \boxed{\mcf\xrightarrow{\cong}\varprojlim_n\mcf/\mci^n\mcf'.} \] This uses the filtration by \(\mci^n\mcf'\), not the possibly different filtration by \(\mci^n\mcf\). Finally, take \(\mathfrak U=\mathfrak X\) in step 3 and replace \(H\) by \(\mcf\) through \(\eta\). The result is the full short exact sequence of global sections requested in (b).
5. Q.E.D.
Step 1 proves (a), and steps <1>2–<1>4 prove both the inverse-limit identification and global exactness in (b).