Let \(\mathfrak{X}\) be a noetherian formal scheme, let \(\mci\) be an ideal of definition, and for each \(n\ge1\) let \(Y_n\) be the scheme \((\mathfrak{X}, \OO_\mathfrak{X}/\mci^n)\). Assume that the inverse system of groups \((\Gamma(Y_n, \OO_{Y_n}))\) satisfies the Mittag-Leffler condition. Then prove that \(\Pic \mathfrak{X} = \inverselim_n \Pic Y_n\). As in the case of a scheme, we define \(\Pic \mathfrak{X}\) to be the group of locally free \(\OO_\mathfrak{X}\dash\)modules of rank \(1\) under \(\tensor\). Proceed in the following steps.
Use the fact that \(\ker(\Gamma(Y_{n+1}, \OO_{Y_{n+1}}) \to \Gamma(Y_n, \OO_{Y_n}))\) is a nilpotent ideal to show that the inverse system \((\Gamma(Y_n, \OO_{Y_n}^*))\) of units in the respective rings also satisfies (ML).
Let \(\mcf\) be a coherent sheaf of \(\OO_\mathfrak{X}\dash\)modules, and assume that for each \(n\) there is some isomorphism \(\varphi_n: \mcf/\mci^n \mcf \cong \OO_{Y_n}\). Then show that there is an isomorphism \(\mcf \cong \OO_\mathfrak{X}\). Be careful, because the \(\varphi_n\) may not be compatible with the maps in the two inverse systems \((\mcf/\mci^n \mcf)\) and \((\OO_{Y_n})\). Conclude that the natural map \(\Pic \mathfrak{X} \to \inverselim_n \Pic Y_n\) is injective.
Given an invertible sheaf \(\mcl_n\) on \(Y_n\) for each \(n\), and given isomorphisms \(\mcl_{n+1} \tensor \OO_{Y_n} \cong \mcl_n\), construct maps \(\mcl_{n'} \to \mcl_n\) for each \(n' \geq n\) so as to make an inverse system, and show that \(\mcl = \inverselim_n \mcl_n\) is a coherent sheaf on \(\mathfrak{X}\). Then show that \(\mcl\) is locally free of rank \(1\), and thus conclude that the map \(\Pic \mathfrak{X} \to \inverselim_n \Pic Y_n\) is surjective. Be careful here, because even though each \(\mcl_n\) is locally free of rank \(1\), the open sets needed to make them free might get smaller and smaller with \(n\).
Show that the hypothesis that \((\Gamma(Y_n, \OO_{Y_n}))\) satisfies (ML) holds if either \(\mathfrak{X}\) is affine, or the \(Y_n\) are projective over a common field \(k\) with the transition morphisms taken over \(k\). See (III, Ex. 11.5–11.7) for further examples and applications.
Solution
Put \(A_n=\Gamma(Y_n,\OO_{Y_n})\) and \(G_n=A_n^\times\). The latter is also \(\Gamma(Y_n,\OO_{Y_n}^*)\): the local inverses of an everywhere invertible section agree and glue. For an inverse system, the Mittag–Leffler condition means that the images in each fixed term stabilize [@Har10a, Chapter II, §9]. All the sheaves on the thickenings are regarded as sheaves on their common underlying space \(|\mathfrak X|\).
1. For \(m\ge n\), the kernel of \(A_m\to A_n\) is nilpotent, and\[ \im(G_m\to G_n)=\bigl(\im(A_m\to A_n)\bigr)^\times. \]
A section in the kernel belongs to the ideal sheaf \(\mci^n/\mci^m\) on \(Y_m\). A product of \(q\) such sections vanishes whenever \(nq\ge m\). Thus the ring kernel is nilpotent.
For any ring \(R\) and nilpotent ideal \(J\), the map \(R^\times\to(R/J)^\times\) is surjective. Indeed, lift a unit and its inverse to \(a,b\in R\). Then \(ab=1+c\) with \(c\in J\), and \(1+c\) has inverse \(1-c+c^2-\cdots\), a finite sum because \(c\) is nilpotent. Hence \(a\) is a unit with inverse \(b(1+c)^{-1}\). Apply this to the surjection from \(A_m\) onto its image in \(A_n\). It gives the displayed equality, with the units on the right taken in that image subring. It does not assume that \(A_m\to A_n\) is surjective or that an arbitrary subring contains inverses of all its elements which are units in the ambient ring.
2. The inverse system \((G_n)\) satisfies the Mittag–Leffler condition, proving (a).
Fix \(n\). The assumed Mittag–Leffler condition on \((A_n)\) gives \(N\ge n\) such that \[ \im(A_m\to A_n)=\im(A_N\to A_n)\qquad(m\ge N). \] Equality of these additive-group images is equality of their image subrings, with the same identity and multiplication. Their unit groups are consequently equal. Step 1 identifies these groups with the images of \(G_m\) in \(G_n\), so those images stabilize as well.
3. In (b), the isomorphisms \(\mcf/\mci^n\mcf\cong\OO_{Y_n}\) can be chosen compatibly for all \(n\).
Write \(F_n=\mcf/\mci^n\mcf\) and let \[ T_n=\operatorname{Isom}_{\OO_{Y_n}}(F_n,\OO_{Y_n}). \] Each \(T_n\) is nonempty by hypothesis. Postmultiplication by a unit gives a free transitive action of \(G_n\) on \(T_n\): two trivializations differ by a unique automorphism of \(\OO_{Y_n}\), hence by a unique global unit. Reduction gives maps \(T_m\to T_n\) compatible with \(G_m\to G_n\) because \(F_m\otimes\OO_{Y_n}=F_n\).
For fixed \(n\), the image of \(T_m\) in \(T_n\) is one orbit of the subgroup \(\im(G_m\to G_n)\). These images are nonempty and form a decreasing family. By step 2 their acting subgroups stabilize. Once the subgroups agree, two nested nonempty orbits must coincide: if they share a point, both are the orbit of that point under the same subgroup. Thus the images of \(T_m\) in \(T_n\) stabilize. Denote the stable image by \(T_n^\infty\).
The maps \(T_{n+1}^\infty\to T_n^\infty\) are surjective. To see this, choose \(m\) so large that the images from \(T_m\) have stabilized at both \(n\) and \(n+1\). Any element of \(T_n^\infty\) lifts from \(T_m\), and the image of that lift at \(n+1\) lies in \(T_{n+1}^\infty\). Choose an element in \(T_1^\infty\) and recursively lift it through these surjections. The resulting sequence \((\psi_n)\) is a compatible family of trivializations, rather than the initially unrelated family \((\varphi_n)\).
4. The sheaf in (b) is trivial, and the restriction homomorphism\[ \Phi:\Pic(\mathfrak X)\longrightarrow\varprojlim_n\Pic(Y_n) \]is injective.
Coherent formal sheaves are recovered from their compatible reductions [@Har10a, Proposition II.9.6], so \[ \mcf\cong\varprojlim_n F_n. \] The compatible isomorphisms \(\psi_n\) from step 3, together with their compatible inverses, give \[ \mcf\cong\varprojlim_n\OO_{Y_n}=\OO_{\mathfrak X}. \] This proves the first assertion of (b).
For an invertible sheaf \(L\) on \(\mathfrak X\), set \(L_n=L\otimes\OO_{Y_n}\). These restrictions preserve tensor products, so the assignment \([L]\mapsto([L_n])\) defines the homomorphism \(\Phi\). Invertible sheaves on a noetherian formal scheme are coherent. If \([L]\) belongs to its kernel, every \(L_n\) is trivial, so the preceding argument with \(\mcf=L\) gives \(L\cong\OO_{\mathfrak X}\). Thus \(\Phi\) has zero kernel.
5. The data in (c) give a coherent sheaf \(L=\varprojlim_n L_n\) with \(L/\mci^nL\cong L_n\).
Let \(\alpha_n:L_{n+1}\otimes\OO_{Y_n}\xrightarrow{\cong}L_n\) be the given isomorphism. Define \(t_{n+1,n}\) as the quotient map \(L_{n+1}\to L_{n+1}\otimes\OO_{Y_n}\) followed by \(\alpha_n\). Define every longer transition as the composite of these adjacent transitions and \(t_{n,n}=\id\). The composition identities then hold by construction.
The adjacent maps are surjective with kernel \(\mci^n L_{n+1}\). More generally, repeated reduction gives \(L_m\otimes\OO_{Y_n}\cong L_n\) for every \(m\ge n\), with the composite just defined as its quotient map. Each \(L_n\) is coherent on the noetherian scheme \(Y_n\). These are precisely the compatibility conditions of [@Har10a, Proposition II.9.6]. That proposition gives coherence of \(L\) and the asserted identifications of all its reductions.
6. The sheaf \(L\) in step 5 is locally free of rank one, and \(\Phi\) is surjective.
Fix a point of \(\mathfrak X\) and choose an affine formal neighborhood \(\mathfrak U\) on which \(L_1\) is trivial. This is possible because the affine formal opens form a basis and \(L_1\) is invertible. Write \(U_n\) for the corresponding affine open of \(Y_n\), and choose a frame \(s_1\in\Gamma(U_1,L_1)\).
1.Hence \(s_{n+1}\) is a frame on all of \(U_{n+1}\).
Induction gives compatible frames on the fixed neighborhood \(\mathfrak U\), without shrinking it as \(n\) increases. They define compatible isomorphisms \(\OO_{U_n}\xrightarrow{\cong}L_n|_{U_n}\). Taking their inverse limits and using the sectionwise construction of limits [@Har10a, Proposition II.9.2] gives \[ \OO_{\mathfrak U}\xrightarrow{\cong}L|_{\mathfrak U}. \] Thus \(L\) is invertible.
5.The invertible \(L\) just constructed restricts to every prescribed class, so it maps to that element under \(\Phi\).This proves surjectivity; the uniqueness of its class follows from step
4.
7. Both sufficient hypotheses in (d) imply the required Mittag–Leffler condition.
If \(\mathfrak X\) is affine, every \(Y_n\) is affine. The sequence \[ 0\to\mci^n/\mci^{n+1}\to\OO_{Y_{n+1}} \to\OO_{Y_n}\to0 \] is exact as a sequence of quasi-coherent sheaves on the affine scheme \(Y_{n+1}\). Taking sections gives a surjection \(A_{n+1}\to A_n\). All longer transition maps are then surjective, so their images in a fixed \(A_n\) are always \(A_n\).
In the projective case, each \(A_n\) is finite-dimensional over \(k\) by finiteness of coherent cohomology [@Har10a, Theorem III.5.2]. The transition maps are \(k\)-linear. For fixed \(n\), the images of \(A_m\) in \(A_n\) form a descending sequence of vector subspaces of a finite-dimensional space. Their dimensions eventually stop decreasing, after which nested subspaces of equal dimension are equal. Thus these images stabilize, proving the Mittag–Leffler condition.
8. Q.E.D.
Steps <1>1–<1>2 prove (a), steps <1>3–<1>4 prove (b), steps <1>5–<1>6 prove (c), and step 7 proves (d). Since restriction is a homomorphism and is both injective and surjective, the requested canonical isomorphism is \[ \boxed{\Pic(\mathfrak X)\xrightarrow{\cong}\varprojlim_n\Pic(Y_n), \qquad [L]\longmapsto([L\otimes\OO_{Y_n}])_{n\ge1}.} \]