Let \(X = \Spec A\) be an affine scheme. Show that the functors \(\sim\) and \(\Gamma\) are adjoint in the following sense: for any \(A\)-module \(M\), and for any sheaf of \(\OO_X\)-modules \(\mcf\), there is a natural isomorphism \[ \Hom_A(M,\Gamma(X,\mcf))\cong\Hom_{\OO_X}(\widetilde M,\mcf). \]
Solution
Use the canonical identifications \(\widetilde M(D(f))\cong M_f\) and \(\Gamma(X,\widetilde M)\cong M\) [@Har10a, Proposition II.5.1]. For \(f\in A\), write \(r_f:\mcf(X)\to\mcf(D(f))\) for restriction.
1. Every \(A\)-linear map \(u:M\to\Gamma(X,\mcf)\) determines an \(\OO_X\)-linear morphism \(\Phi(u):\widetilde M\to\mcf\).
The module \(\mcf(D(f))\) is an \(A_f\)-module, so multiplication by \(f\) is invertible on it. The universal property of localization therefore extends \(r_f\circ u\) uniquely to an \(A_f\)-linear map \[ u_f:M_f\longrightarrow\mcf(D(f)), \qquad u_f(m/f^n)=f^{-n}r_f(u(m)). \] In particular, the formula is independent of the chosen fraction.
For an inclusion \(D(g)\subseteq D(f)\), the restriction of \(f\) is a unit on \(D(g)\). Both composites from \(M_f\) to \(\mcf(D(g))\) obtained using \(u_f\), \(u_g\), and restriction send \(m/1\) to \(u(m)|_{D(g)}\) and respect multiplication by \(f^{-1}\). The uniqueness in localization makes these composites equal. Thus the maps \(u_f\) commute with all restrictions between distinguished opens. Distinguished opens form a basis, so the sheaf gluing axiom extends them uniquely to a morphism \(\Phi(u):\widetilde M\to\mcf\). Its linearity follows on this basis, hence on every open set.
2. Taking global sections defines a map \(\Psi:\Hom_{\OO_X}(\widetilde M,\mcf)\to\Hom_A(M,\Gamma(X,\mcf))\) inverse to \(\Phi\).
For \(v:\widetilde M\to\mcf\), let \(\Psi(v)=v_X\) under \(\Gamma(X,\widetilde M)\cong M\). Taking \(f=1\) in step 1 gives \(\Psi(\Phi(u))=u\).
Conversely, compatibility of \(v\) with restriction and its \(A_f\)-linearity give \[ v_{D(f)}(m/f^n)=f^{-n}r_f(v_X(m)). \] This is precisely the formula for \(\Phi(\Psi(v))\) on \(D(f)\). The morphisms agree on a basis, so \(\Phi(\Psi(v))=v\). Both constructions preserve addition and multiplication by elements of \(A\), giving an isomorphism of \(A\)-modules.
3. The isomorphism is natural in \(M\) and \(\mcf\).
Let \(a:M'\to M\) be \(A\)-linear and let \(b:\mcf\to\mcg\) be an \(\OO_X\)-linear morphism. The formula of step 1 gives \[ \Phi(u\circ a)=\Phi(u)\circ\widetilde a, \qquad \Phi(\Gamma(X,b)\circ u)=b\circ\Phi(u). \] Indeed, on each \(D(f)\) both sides of the first equality send \(m'/f^n\) to \(f^{-n}u(a(m'))|_{D(f)}\). Both sides of the second send \(m/f^n\) to \(f^{-n}b_{D(f)}(u(m)|_{D(f)})\). These are the naturality identities in the two variables.
4. Q.E.D.
Steps 1 and 2 construct the inverse isomorphisms, and step 3 proves naturality. Thus \(M\mapsto\widetilde M\) is left adjoint to \(\mcf\mapsto\Gamma(X,\mcf)\).