2_9x Formal Schemes

II.9: Formal Schemes

II.9.1. #to_work

Let \(X\) be a noetherian scheme, \(Y\) a closed subscheme, and \(\widehat{X}\) the completion of \(X\) along \(Y\). We call the ring \(\Gamma\left(\widehat{X}, \mathcal{O}_{\widehat{X}}\right)\) the ring of formal-regular functions on \(X\) along \(Y\). In this exercise we show that if \(Y\) is a connected, nonsingular, positive dimensional subvariety of \(X=\mathbf{P}_k^n\) over an algebraically closed field \(k\), then \(\Gamma\left(\widehat{X}, \mathcal{O}_{\widehat{X}}\right)=k\)

• Let \(\mathcal{I}\) be the ideal sheaf of \(Y\). Use (8.13) and (8.17) to show that there is an inclusion of sheaves on \(Y, \mathcal{I} / \mathcal{I}^2 \hookrightarrow \mathcal{O}_Y(-1)^{n+1}\).

• Show that for any \(r \geqslant 1, \Gamma\left(Y, \mathcal{I}^r / \mathcal{I}^{r+1}\right)=0\).

• Use the exact sequences \begin{align*} 0 \rightarrow \mathcal{I}^r / \mathcal{I}^{r+1} \rightarrow \mathcal{O}_X / \mathcal{I}^{r+1} \rightarrow \mathcal{O}_X / \mathcal{I}^r \rightarrow 0 \end{align*} and induction on \(r\) to show that \(\Gamma\left(Y, \mathcal{O}_X / \mathcal{I}^r\right)=k\) for all \(r \geqslant 1\). ¹

• Conclude that \(\Gamma\left(\widehat{X}, \mathcal{O}_{\widehat{X}}\right)=k\). ²

II.9.2. #to_work

Use the result of (Ex. 9.1) to prove the following geometric result. Let \(Y \subseteq X=\) \(\mathbf{P}_k^n\) be as above, and let \(f: X \rightarrow Z\) be a morphism of \(k\)-varieties. Suppose that \(f(Y)\) is a single closed point \(P \in Z\). Then \(f(X)=P\) also.

II.9.3. #to_work

Prove the analogue of \((5.6)\) for formal schemes, which says, if \(\mathrm{X}\) is an affine formal scheme, and if \begin{align*} 0 \rightarrow \mathrm{F}^{\prime} \rightarrow \mathrm{F} \rightarrow \mathrm{F}^{\prime \prime} \rightarrow 0 \end{align*} is an exact sequence of \(\mathcal{O}_x\)-modules, and if \(\mathrm{F}^{\prime}\) is coherent, then the sequence of global sections \begin{align*} 0 \rightarrow \Gamma\left(\mathrm{X}, \mathrm{F}^{\prime}\right) \rightarrow \Gamma(\mathrm{X}, \mathrm{F}) \rightarrow \Gamma\left(\mathrm{X}, \mathrm{F}^{\prime \prime}\right) \rightarrow 0 \end{align*} is exact. For the proof, proceed in the following steps.

• Let \(\mathrm{I}\) be an ideal of definition for \(\mathrm{X}\), and for each \(n>0\) consider the exact sequence \begin{align*} 0 \rightarrow \mathrm{F}^{\prime} / \mathrm{J}^n \mathrm{F}^{\prime} \rightarrow \mathrm{F} / \mathrm{J}^n \mathrm{F}^{\prime} \rightarrow \mathrm{F}^{\prime \prime} \rightarrow 0 .\end{align*} Use (5.6), slightly modified, to show that for every open affine subset \(\mathrm{U} \subseteq \mathrm{X}\), the sequence \begin{align*} 0 \rightarrow \Gamma\left(\mathrm{U}, \mathrm{F}^{\prime} / \mathrm{I}^n \mathrm{F}^{\prime}\right) \rightarrow \Gamma\left(\mathrm{Y}, \mathrm{F} / \mathrm{J}^n \mathrm{F}^{\prime}\right) \rightarrow \Gamma\left(\mathrm{U}, \mathrm{F}^{\prime \prime}\right) \rightarrow 0 \end{align*} is exact.

• Now pass to the limit, using (9.1), (9.2), and (9.6). Conclude that \(\mathrm{F} \cong \lim \mathrm{F} / \mathrm{J}^n \mathrm{F}^{\prime}\) and that the sequence of global sections above is exact.

II.9.4. #to_work

Use (Ex. 9.3) to prove that if \begin{align*} 0 \rightarrow \mathrm{F}^{\prime} \rightarrow \mathrm{F} \rightarrow \mathrm{F}^{\prime \prime} \rightarrow 0 \end{align*} is an exact sequence of \(\mathcal{O}_x\)-modules on a noetherian formal scheme \(\mathrm{X}\), and if \(\mathrm{F}^{\prime}, \mathrm{F}^{\prime \prime}\) are coherent, then \(\mathrm{F}\) is coherent also.

II.9.5. #to_work

If \(\mathrm{F}\) is a coherent sheaf on a noetherian formal scheme \(\mathrm{X}\), which can be generated by global sections, show in fact that it can be generated by a finite number of its global sections.

II.9.6. #to_work

Let \(\mathrm{X}\) be a noetherian formal scheme, let \(\mathrm{I}\) be an ideal of definition, and for each \(n\), let \(Y_n\) be the scheme \(\left(\mathrm{X}, \mathcal{O}_x / \mathrm{J}^n\right)\). Assume that the inverse system of groups \(\left(\Gamma\left(Y_n, \mathcal{O}_{Y_n}\right)\right)\) satisfies the Mittag-Leffler condition. Then prove that \(\operatorname{Pic}\mathrm{X}=\lim \operatorname{Pic}Y_n\).

As in the case of a scheme, we define \(\operatorname{Pic}\mathrm{X}\) to be the group of locally free \(\mathcal{O}_x\)-modules of rank 1 under the operation \(\otimes\). Proceed in the following steps.

• Use the fact that \(\operatorname{ker}\left(\Gamma\left(Y_{n+1}, \mathcal{O}_{Y_{n+1}}\right) \rightarrow \Gamma\left(Y_n, \mathcal{O}_{Y_n}\right)\right)\) is a nilpotent ideal to show that the inverse system \(\left(\Gamma\left(Y_n, \mathcal{O}_{Y_n}^*\right)\right)\) of units in the respective rings also satisfies (ML).

• Let \(\mathrm{F}\) be a coherent sheaf of \(\mathcal{O}_x\)-modules, and assume that for each \(n\), there is some isomorphism \(\varphi_n: \tilde{F} / \mathrm{J}^n \mathrm{F} \cong \mathcal{O}_{Y_n}\). Then show that there is an isomorphism \(\tilde{F} \cong \mathcal{O}_x\). ³

  Conclude that the natural map Pic \(\mathrm{X} \rightarrow \cocolim \operatorname{Pic}Y_n\) is injective.

• Given an invertible sheaf \(\mathcal{L}_n\) on \(Y_n\) for each \(n\), and given isomorphisms \(\mathcal{L}_{n+1} \otimes\) \(\mathcal{O}_{Y_n} \cong \mathcal{L}_n\), construct maps \(\mathcal{L}_{n^{\prime}} \rightarrow \mathcal{L}_n\) for each \(n^{\prime} \geqslant n\) so as to make an inverse system, and show that \(\mathrm{L}=\lim \mathcal{L}_n\) is a coherent sheaf on \(\mathrm{X}\).

  Then show that \(\mathrm{L}\) is locally free of rank 1, and thus conclude that the map \(\operatorname{Pic}\mathrm{X} \rightarrow \lim \operatorname{Pic}Y_n\) is surjective. ⁴

• Show that the hypothesis “\(\left(\Gamma\left(Y_n, \mathcal{O}_{Y_n}\right)\right)\) satisfies (ML)” is satisfied if either \(\mathrm{X}\) is affine, or each \(Y_n\) is projective over a field \(k\). ⁵