Algebraizability of locally free sheaves on a formal completion

The canonical sheaf of \(\PP_k^N\) is \[ \omega_X\cong\mco_X(-N-1). \] By Serre duality, for \(i=0,1\), \[ H^i\bigl(X,\mce(-rd)\bigr)^\vee \cong H^{N-i}\bigl( X,\mce^\vee(rd-N-1) \bigr). \] Because \(N\ge2\), both \[ N>0 \qquad\text{and}\qquad N-1>0. \] The twist on the right tends to \(+\infty\) with \(r\). Hence Serre vanishing makes both right-hand groups zero for \(r\gg0\). Dualizing gives the asserted vanishings.

Since \(\mci^r\cong\mco_X(-rd)\) and \(\mce\) is locally free, tensoring the defining sequence of \(X_r\) with \(\mce\) gives \[ 0 \longrightarrow \mce(-rd) \longrightarrow \mce \longrightarrow \mce_r \longrightarrow 0. \] Its cohomology begins \[ 0 \longrightarrow H^0(X,\mce(-rd)) \longrightarrow H^0(X,\mce) \longrightarrow H^0(X_r,\mce_r) \longrightarrow H^1(X,\mce(-rd)). \] By step 1 the two outer groups vanish for all \(r\gg0\). Therefore \[ H^0(X,\mce)\xrightarrow{\sim}H^0(X_r,\mce_r) \] for every sufficiently large \(r\).

By definition, \[ \hat\mce=\varprojlim_r\mce_r. \] Global sections commute with inverse limits of sheaves, so \[ H^0(\hat X,\hat\mce) = \varprojlim_r H^0(X_r,\mce_r). \] The inverse system on the right is eventually canonically constant with value \(H^0(X,\mce)\) by the preceding paragraph. Its inverse limit is therefore \(H^0(X,\mce)\), and the resulting isomorphism is exactly the natural restriction map.

Let \(\mathfrak F\) be locally free on \(\hat X\). By (i), choose a coherent sheaf \(\mcg\) on \(X\) and an isomorphism \[ \hat\mcg\cong\mathfrak F. \] By Serre's theorem A, there is \(n_0\) such that \[ \mcg(n) \] is generated by finitely many global sections for every \(n\ge n_0\). Completing the corresponding evaluation surjection gives a surjection onto \[ \hat\mcg(n)\cong\mathfrak F(n). \] The completed generating sections are global sections on \(\hat X\), so \(\mathfrak F(n)\) is generated by global sections for all \(n\ge n_0\). Thus (ii) holds.

By (ii), choose \(q_0\) such that \(\mathfrak F(q_0)\) is generated by global sections. Since \(\mathfrak F(q_0)\) is coherent and \(|\hat X|=|Y|\) is quasi-compact, finitely many global sections suffice. Twisting their evaluation map back by \(-q_0\) gives a surjection \[ \hat\mce_0=\mco_{\hat X}(-q_0)^{\oplus a} \twoheadrightarrow \mathfrak F \] for some finite \(a\).

Let \(\mathfrak K\) be its kernel. Locally the target \(\mathfrak F\) is free, so the surjection splits locally. Hence \(\mathfrak K\) is again locally free of finite rank.

Apply (ii) to \(\mathfrak K\). For some \(q_1\) the sheaf \(\mathfrak K(q_1)\) is generated by finitely many global sections. Twisting back gives a surjection \[ \hat\mce_1=\mco_{\hat X}(-q_1)^{\oplus b} \twoheadrightarrow \mathfrak K. \] Composing with \(\mathfrak K\hookrightarrow\hat\mce_0\) gives \(\hat\delta\) and the asserted exact presentation. Taking \[ \mce_0=\mco_X(-q_0)^{\oplus a}, \qquad \mce_1=\mco_X(-q_1)^{\oplus b} \] realizes both formal bundles as completions of the required bundles on \(X\).

The morphism \[ \hat\delta:\hat\mce_1\longrightarrow\hat\mce_0 \] is a global section of \[ \sheafhom_{\mco_{\hat X}}(\hat\mce_1,\hat\mce_0). \] Since \(\mce_1\) and \(\mce_0\) are locally free, there is a canonical identification \[ \sheafhom_{\mco_{\hat X}}(\hat\mce_1,\hat\mce_0) \cong \widehat{\sheafhom_{\mco_X}(\mce_1,\mce_0)}. \] The sheaf \[ \mch=\sheafhom_{\mco_X}(\mce_1,\mce_0) \] is locally free. By part (a), proved in step 2, \[ H^0(X,\mch)\xrightarrow{\sim}H^0(\hat X,\hat\mch). \] Therefore \(\hat\delta\) is the completion of a unique morphism \[ \delta:\mce_1\longrightarrow\mce_0. \]

Set \[ \mcg=\coker(\delta). \] This is coherent on the noetherian scheme \(X\). Adic completion is exact on coherent sheaves; affine-locally this is the exactness of completion on finitely generated modules over a noetherian ring (Stacks Project, Tag 0BNH). Consequently \[ \hat\mcg \cong \coker(\hat\delta) \cong \mathfrak F. \] Thus \(\mathfrak F\) is algebraizable, proving (i). Together with step 3, conditions (i) and (ii) are equivalent.

Let \(\mcl\) be an invertible sheaf on \(X\) whose completion is trivial, and choose an isomorphism \[ \hat\varphi:\mco_{\hat X}\xrightarrow{\sim}\hat\mcl. \] Its inverse is \[ \hat\psi:\hat\mcl\xrightarrow{\sim}\mco_{\hat X}. \]

Apply part (a) to the locally free sheaves \[ \sheafhom(\mco_X,\mcl)\cong\mcl \] and \[ \sheafhom(\mcl,\mco_X)\cong\mcl^\vee. \] It lifts \(\hat\varphi\) and \(\hat\psi\) uniquely to morphisms \[ \varphi:\mco_X\longrightarrow\mcl, \qquad \psi:\mcl\longrightarrow\mco_X. \] The two composites are endomorphisms of line bundles, hence global sections of \(\mco_X\): \[ \psi\circ\varphi\in H^0(X,\mco_X), \qquad \varphi\circ\psi\in H^0(X,\sheafhom(\mcl,\mcl)) \cong H^0(X,\mco_X). \] Their completions are the identity because \(\hat\varphi\) and \(\hat\psi\) are inverse. Part (a), now applied to \(\mco_X\), makes the map on these global sections injective. Hence \[ \psi\circ\varphi=\id_{\mco_X}, \qquad \varphi\circ\psi=\id_{\mcl}. \] Thus \(\mcl\cong\mco_X\), so the kernel of \(\Pic X\to\Pic\hat X\) is zero.

Fix \(y\in Y\) and put \[ A=\mco_{X,y}, \qquad I=\mci_y, \qquad M=\mcg_y. \] The coherent completion theorem identifies the completed module with \[ \hat M\cong M\otimes_A\hat A, \] where \(\hat A\) is the \(I\)-adic completion of \(A\). The isomorphism \(\hat\mcg\cong\mathfrak L\) makes \(\hat M\) a free \(\hat A\)-module of rank one.

Since \(A\) is noetherian local and \(I\subseteq\mathfrak m_A\), the map \[ A\longrightarrow\hat A \] is faithfully flat (Stacks Project, Tag 00MC). Finite projectivity and rank descend under faithfully flat maps (Stacks Project, Tag 058S). Hence \(M\) is a projective \(A\)-module of rank one. Because \(A\) is local, \[ M\cong A. \] Thus \(\mcg\) is locally free of rank one at every point of \(Y\).

The locally-free-of-rank-one locus of a coherent sheaf of finite presentation is open. Therefore there is an open neighbourhood \[ V\supseteq Y \] on which \(\mcg|_V\) is invertible.

The open set \(V\) of step 7 is nonempty. Since \(X=\PP_k^N\) is integral, its generic point lies in \(V\). Hence \(\mcg\) has generic rank one, so its reflexive hull \(\mcl=\mcg^{\vee\vee}\) is a coherent reflexive sheaf of rank one; see Stacks Project, Tag 0AVT.

Projective space is regular, hence normal and locally factorial. On an integral locally noetherian normal scheme, rank-one coherent reflexive sheaves represent Weil divisor classes (Stacks Project, Tag 0EBM). On a locally factorial scheme every Weil divisor is Cartier, so the Cartier--Weil comparison shows that every such rank-one reflexive sheaf is invertible. Thus \(\mcl\) is an invertible sheaf on \(X\).

On \(V\), the sheaf \(\mcg\) is already locally free, so the natural map \[ \mcg\longrightarrow\mcg^{\vee\vee}=\mcl \] is an isomorphism. Since \(V\) is an actual open neighbourhood of \(Y\), two coherent sheaves which agree on \(V\) have the same formal completion along \(Y\). Therefore \[ \hat\mcl \cong \hat\mcg \cong \mathfrak L. \] Thus every class in \(\Pic\hat X\) lies in the image of \(\Pic X\).

Steps <1>1–<1>2 prove (a). Steps <1>3–<1>5 prove the equivalence in (b). Step 6 proves injectivity of \[ \Pic X\longrightarrow\Pic\hat X, \] and steps <1>7–<1>8 prove surjectivity under those equivalent conditions. Hence \[ \boxed{\Pic X\xrightarrow{\sim}\Pic\hat X}. \] This proves (c).