The comparison map from a graded module to its sections

For \(m\in M_d\), the element \(m/1\) lies in the degree-zero part of \(M(d)_f\) for every homogeneous \(f\in S\) of positive degree. These elements agree after further localization, so they define a global section of \(\widetilde{M(d)}\). Since \(S\) is generated in degree one, the opens \(D_+(f)\) with \(f\in S_1\) cover \(X\). On each of them, powers of \(f\) trivialize \(\OO_X(d)\) and give the canonical isomorphism \[ \widetilde{M(d)}\cong\widetilde M\otimes_{\OO_X}\OO_X(d)=\widetilde M(d) \] [@Har10a, Proposition II.5.12]. Composing with this isomorphism defines \(\alpha_d\). Multiplication of the local fractions by a homogeneous \(s\in S_e\) sends the section defined by \(m\) to that defined by \(sm\). Thus the direct sum \(\alpha_M=\bigoplus_d\alpha_d\) is graded \(S\)-linear. The construction also commutes with every graded homomorphism \(M\to N\), proving naturality.

The rings \(A\) and \(S\) are noetherian by the Hilbert basis theorem. Choose a finite set of degree-one generators of \(S\) over \(A\), giving a graded surjection \[ P=A[x_0,\ldots,x_r]\longrightarrow S. \] A zero generator may be included if \(S_1=0\). This gives a closed immersion \(i:X\hookrightarrow\PP_A^r\). Regard \(M\) as a finite graded \(P\)-module. The sheaf associated to this \(P\)-module is \(i_*\widetilde M\), as follows by comparing the degree-zero localizations on the standard affine charts. The same identification holds after twisting.

Choose finite graded free \(P\)-modules \(F_0,F_1\) and exact sequences \[ 0\longrightarrow K\longrightarrow F_0\longrightarrow M\longrightarrow0, \qquad 0\longrightarrow L\longrightarrow F_1\longrightarrow K\longrightarrow0. \] The kernels are finite because \(P\) is noetherian. Sheafification on \(\PP_A^r\) is exact, and Serre vanishing gives \[ H^1(\PP_A^r,\widetilde K(d))=H^1(\PP_A^r,\widetilde L(d))=0 \] for all sufficiently large \(d\) [@Har10a, Theorem III.5.2]. The associated long exact sequences then give a right-exact sequence \[ \Gamma(\PP_A^r,\widetilde F_1(d)) \longrightarrow\Gamma(\PP_A^r,\widetilde F_0(d)) \longrightarrow\Gamma(X,\widetilde M(d))\longrightarrow0. \] For \(j=0,1\), the comparison map from \((F_j)_d\) to the corresponding section module is an isomorphism for large \(d\): each \(F_j\) is a finite sum of shifts of \(P\), and the canonical map \[ P_q\longrightarrow\Gamma(\PP_A^r,\OO(q)) \] is an isomorphism for \(q\ge0\) [@Har10a, Theorem III.5.1]. For \(r=0\) this is the identification \(Ax_0^q\cong A\) on \(\Spec A\). Consequently \(M_d\) and \(\Gamma(X,\widetilde M(d))\) are cokernels of the same map \((F_1)_d\to(F_0)_d\). Naturality from step 1 identifies the induced isomorphism with \(\alpha_d\).

Let \(\mathcal Q\) have quasi-finitely generated graded \(S\)-modules as objects and set \[ \Hom_{\mathcal Q}(M,N) \coloneqq\varinjlim_{d\to+\infty} \Hom_{\mathrm{gr},S}(M_{\ge d},N_{\ge d}). \] The transition maps restrict a degree-zero homomorphism to a higher truncation. Two representatives are equal when their restrictions agree in all sufficiently large degrees. Composition is defined by restricting both representatives to a common truncation and composing there; further restriction preserves this composite. Identities are represented by identity homomorphisms.

A graded isomorphism between truncations gives an isomorphism in \(\mathcal Q\). Conversely, representatives of inverse morphisms in \(\mathcal Q\) become mutually inverse after one sufficiently large common truncation. Thus the isomorphism relation on its objects is exactly \(\approx\). This also proves transitivity of \(\approx\); reflexivity and symmetry follow from identity and inverse graded isomorphisms.

The inclusion \(M_{\ge d}\to M\) induces an isomorphism of associated sheaves. Indeed, on \(D_+(f)\) with \(f\in S_1\), every homogeneous fraction of degree zero has the form \(m/f^n\) with \(m\in M_n\). Multiplying its numerator and denominator by a sufficiently large power of \(f\) makes the numerator lie in \(M_{\ge d}\). Injectivity follows from exactness of localization. Hence sheafification defines a functor \(T:\mathcal Q\to\operatorname{Coh}(X)\). It takes values in coherent sheaves because an object of \(\mathcal Q\) has a truncation isomorphic to a truncation of a finite module, and \(S\) is noetherian.

By Serre generation, choose a surjection \[ \mathcal E=\bigoplus_{j=1}^t\OO_X(-a_j)\longrightarrow\mcf \] with coherent kernel \(\mathcal K\) [@Har10a, Theorem II.5.17]. By Serre vanishing, \(H^1(X,\mathcal K(d))=0\) for all sufficiently large \(d\). Thus \(\Gamma_*(\mathcal E)_{\ge d_0}\to\Gamma_*(\mcf)_{\ge d_0}\) is surjective for a suitable \(d_0\). Set \(F=\bigoplus_j S(-a_j)\). Step 2 identifies \(\Gamma_*(\mathcal E)_{\ge d_0}\) with \(F_{\ge d_0}\) after increasing \(d_0\). The latter is a finite \(S\)-module because it is a submodule of the finite module \(F\) over the noetherian ring \(S\). Its quotient \(\Gamma_*(\mcf)_{\ge d_0}\) is finite as well. This is precisely quasi-finite generation.

On a standard affine open \(D_+(f)\) with \(f\in S_1\), define the evaluation map by \[ s/f^n\longmapsto s|_{D_+(f)}\otimes f^{-n}, \qquad s\in\Gamma(X,\mcf(n)). \] The factor \(f^{-n}\) trivializes \(\OO_X(-n)\) on this open set. The rule respects equality of fractions, commutes with restriction to intersections, and is linear. It therefore gives \(\beta_{\mcf}\), naturally in \(\mcf\).

Step 1 constructs the natural morphism; steps 2 and 3 prove that it is an isomorphism.

Step 4 makes \(G\) a functor from \(\operatorname{Coh}(X)\) to \(\mathcal Q\), and step 5 gives a natural isomorphism \(TG\cong\operatorname{id}\). For \(M\in\mathcal Q\), choose \(e\) with \(M_{\ge e}\) finite. Step 2 applied to \(M_{\ge e}\), together with the naturality of \(\alpha\) for \(M_{\ge e}\to M\), shows that \(\alpha_M\) is an isomorphism in all sufficiently large degrees. Thus \(\alpha\) gives a natural isomorphism \(\operatorname{id}_{\mathcal Q}\cong GT\).

Step 1 proves part (a), step 2 proves part (b), and steps <1>3–<1>6 construct and prove the equivalence in part (c).