On \(\Spec A\), \(M\mapsto\widetilde M\) is an equivalence from \(A\)-modules to quasicoherent sheaves. On \(\Proj S\) it is not: graded \(S\)-modules that agree in all degrees \(\ge n_0\) give isomorphic sheaves, and a graded module annihilated by a power of the irrelevant ideal \(S_+\) gives \(0\).
The sheaf associated to a graded module, and $\Gamma_*$ (Tag D-MODGRMOD)
For \(S\) a graded ring and \(M\) a graded \(S\)-module, \(\tilde M\) is the sheaf on \(\Proj S\) with \(\tilde M(D_+(f)) = (M_f)_0\), the degree-zero part of the localization. Conversely, for \(\mcf \in \mods{\OO_X}\) with \(X = \Proj S\), \[ \Gamma_*(\mcf) \da \bigoplus_{n \in \ZZ} \Gamma(X, \mcf(n)) , \] a graded \(S\)-module, where \(\mcf(n) \da \mcf \tensor_{\OO_X} \OO_X(n)\).
For \(S\) generated in degree \(1\) over a Noetherian ring \(S_0\), every quasicoherent \(\mcf\) on \(\Proj S\) satisfies \(\widetilde{\Gamma_*(\mcf)} \cong \mcf\), and \(\OO_X(n) = \widetilde{S(n)}\).
Under the hypotheses of the theorem, the functor \(M\mapsto\tilde M\) is essentially surjective onto quasicoherent sheaves, with \(\mcf\cong\widetilde{\Gamma_*(\mcf)}\), and it is not an equivalence. A graded homomorphism \(M\to M'\) that is an isomorphism in all sufficiently large degrees induces an isomorphism \(\tilde M\cong\tilde M'\), and a graded module with \(M_n=0\) for all \(n\gg0\) has \(\tilde M=0\); for example, \(\widetilde{S/S_+}=0\) while \(S/S_+\ne0\). The category \(\QCoh(\Proj S)\) is the quotient of the category of graded \(S\)-modules by the modules in which every element is annihilated by a power of \(S_+\).
If \(S\) is generated by finitely many elements of \(S_1\) over \(S_0\), then every quasicoherent \(\mcf\) on \(\Proj S\) satisfies \(\mcf\cong\widetilde{\Gamma_*(\mcf)}\), where \(\Gamma_*(\mcf)=\bigoplus_{n\in\ZZ}\Gamma(\Proj S,\mcf(n))\).
Generation and ampleness
Globally generated sheaves (Tag D-MODGG)
\(\mcf\) is generated by global sections if there are \(\ts{s_i}_{i \in I} \subseteq \Gamma(X, \mcf)\) whose images in \(\mcf_x\) generate that stalk as an \(\OO_{X,x}\)-module for every \(x\). Equivalently, there is a surjection \(\bigoplus_{i \in I} \OO_X \surjects \mcf\).
Every quasicoherent sheaf on an affine scheme is globally generated by any generating set of its corresponding module. For \(n\ge1\), \(\OO_{\PP^n}(d)\) is globally generated if and only if \(d \geq 0\): for \(d\ge0\) the monomials of degree \(d\) have no common zero, and for \(d < 0\) there are no nonzero global sections.
Let \(X\) be a scheme over a ring \(A\) and \(\mcl\) an invertible sheaf generated by global sections \(s_0,\ldots,s_n\). There is a unique \(A\)-morphism \(f\colon X \to \PP^n_A\) with \(f^*\OO(1)\cong\mcl\) and \(f^*x_i=s_i\) [@Har10a, Theorem II.7.1]; \(\mcl\) is very ample over \(\Spec A\) if and only if some such \(f\) is an immersion.
Global sections \(s_0,\ldots,s_n\) generating an invertible sheaf give a morphism to \(\PP^n\). Every quasicoherent sheaf on an affine scheme is globally generated.
Ample and very ample invertible sheaves (Tag D-MODAMPLE)
Let \(f:X\to Y\) be a morphism of schemes and let \(\mcl\) be an invertible sheaf on \(X\). It is very ample over \(Y\) if there is a \(Y\)-immersion \(\iota:X\to\PP_Y^n\), for some finite \(n\), with \(\iota^*\OO(1)\cong\mcl\) [@Har10a, Chapter II, §7].
(Ample)
Let \(X\) be a noetherian scheme and let \(\mcl\) be an invertible sheaf on \(X\). It is ample if for every coherent \(\mcf\) there is an integer \(N_0\) such that \(\mcf\tensor\mcl^{\otimes n}\) is globally generated for every \(n\ge N_0\) [@Har10a, Chapter II, §7].
Let \(X\) be of finite type over a noetherian ring \(A\) and let \(\mcl\) be invertible. Then \(\mcl\) is ample if and only if \(\mcl^{\otimes m}\) is very ample over \(\Spec A\) for some \(m>0\) [@Har10a, Theorem II.7.6].
If \(f:X\to Y\) is proper and \(\mcl\) is very ample over \(Y\), its immersion into \(\PP_Y^n\) is proper, hence a closed immersion. Thus \(f\) is projective in the sense of admitting a closed immersion into some \(\PP_Y^n\). Conversely, such a closed immersion gives a very ample sheaf by pulling back \(\OO(1)\) [@Har10a, Chapter II, §§4 and 7].
(Ampleness on a complete curve)
Let \(X\) be a complete nonsingular curve over an algebraically closed field, and let \(\mcl\) be an invertible sheaf. Then \(\mcl\) is ample if and only if \(\deg\mcl>0\) [@Har10a, Corollary IV.3.3].
A globally generated invertible sheaf gives a morphism to projective space, a very ample one gives an immersion, and \(\mcl\) is ample exactly when \(\mcl^{\otimes m}\) is ample for some \(m>0\).
Serre’s theorems
Serre's theorems A and B (Tag T-MODSERRE)
Let \(X\) be projective over a noetherian ring \(A\), with \(\OO_X(1)\) very ample, and let \(\mcf\) be coherent. Then there is \(N_0\) such that for all \(n \geq N_0\):
\(\mcf(n)\) is generated by finitely many global sections;
\(H^i(X, \mcf(n)) = 0\) for all \(i > 0\).
[@Har10a, Theorems II.5.17 and III.5.2]
(Dimension shifting on projective space)
On \(\PP_A^r\), part (A) gives a surjection from a finite sum of twists to any coherent sheaf \(\mathcal K_0\). Iterating on the kernels gives exact sequences \[ 0\longrightarrow\mathcal K_{j+1}\longrightarrow\mathcal E_j\longrightarrow\mathcal K_j\longrightarrow0, \] where \(\mathcal E_j\) is a finite sum of twists and \(\mathcal K_j\) is coherent. Take the first \(r\) such sequences. For sufficiently large \(n\), the positive-degree cohomology of every \(\mathcal E_j(n)\) in this finite list vanishes by the cohomology calculation for \(\OO(d)\) [@Har10a, Theorem III.5.1]. The long exact sequences then give, for \(i>0\), \[ H^i(\PP_A^r,\mathcal K_0(n))\cong H^{i+r}(\PP_A^r,\mathcal K_r(n))=0. \] The last vanishing follows from the standard cover by \(r+1\) affine opens: its intersections are affine, so its Cech complex computes quasi-coherent cohomology and has no terms in degrees greater than \(r\) [@Har10a, Theorem III.4.5]. For \(r=0\), affineness gives the vanishing directly. Thus a finite initial part of the resolution suffices for this dimension-shifting proof, even when the resolution does not terminate.
(A coherent sheaf with no finite resolution by twists)
Let \(k\) be a field, \(A=k[\varepsilon]/(\varepsilon^2)\), and \(X=\PP_A^0\cong\Spec A\). Every twist of \(\OO_X\) is trivial. The finite \(A\)-module \(k=A/(\varepsilon)\) has the free resolution \[ \cdots\xrightarrow{\varepsilon}A\xrightarrow{\varepsilon}A\xrightarrow{\varepsilon}A\longrightarrow k\longrightarrow0. \] It is exact because the kernel and image of multiplication by \(\varepsilon\) are both \((\varepsilon)\). Tensoring with \(k\) gives zero differentials, so \(\operatorname{Tor}_i^A(k,k)\cong k\) for every \(i\ge0\). A finite free resolution would make these groups zero in sufficiently high degrees. Therefore the associated coherent sheaf has no finite resolution by finite sums of twists.
(An explicit twist bound)
Let \(k\) be a field, let \(m\ge0\) be an integer, and take \(\mcf=\OO_{\PP_k^1}(-m)\). For every \(n\ge m\), the sheaf \(\mcf(n)=\OO(n-m)\) is generated by its degree-\((n-m)\) monomials, and its positive-degree cohomology vanishes. Indeed, \(H^1(\PP_k^1,\OO(q))=0\) for \(q\ge-1\), and all higher groups vanish [@Har10a, Theorem III.5.1]. Thus \(N_0=m\) is a numerical bound for both (A) and (B). There is no common vanishing bound for all these sheaves on \(\PP_k^1\): for any fixed \(n\), choosing \(m\ge\max(0,n+2)\) gives \(\dim_k H^1(\PP_k^1,\mcf(n))=m-n-1>0\).
(Finiteness over the base)
By finiteness of coherent cohomology, each \(H^i(X,\mcf)\) is a finite \(A\)-module. When \(A=k\) is a field, the dimensions \(h^i(X,\mcf)=\dim_k H^i(X,\mcf)\) are finite and the Euler characteristic \(\chi(X,\mcf)=\sum_i(-1)^i h^i(X,\mcf)\) is defined. For all sufficiently large \(n\), part (B) gives \(\chi(X,\mcf(n))=h^0(X,\mcf(n))\).
On an affine scheme every quasicoherent sheaf is globally generated and has vanishing higher cohomology; Serre’s theorems give both properties for \(\mcf(n)\) with \(n\ge N_0(\mcf)\) on a projective scheme.
Coherent sheaves on projective space have finite resolutions by twists (Tag T-MODRES)
Every coherent \(\mcf\) on \(\PP^n_k\) admits a finite resolution \[ 0 \to \mce_m \to \cdots \to \mce_1 \to \mce_0 \to \mcf \to 0 \] with each \(\mce_j\) a finite direct sum of twists \(\OO(-d)\), and one may take \(m \leq n\).
Serre’s theorem A supplies the surjection \(\mce_0 \surjects \mcf\); iterating it on the kernel gives a possibly infinite resolution, and the Hilbert syzygy theorem for \(k[x_0,\dots,x_n]\) is what truncates it. So the statement is a graded-module theorem read through \(\tilde{\wait\,}\), and its length bound is the global dimension \(n+1\) of the polynomial ring, dropping by one on passage to \(\Proj\).
This is the standard device for reducing any additive invariant of coherent sheaves to the twists. Cohomology, the Euler characteristic, and the Hilbert polynomial are all computed this way: break the resolution into short exact sequences, apply the known values on \(\OO(d)\), and alternate signs. It is also why \(K_0(\PP^n) \cong \ZZ^{n+1}\), generated by the classes of the twists.
For an additive function \(\lambda\) on coherent sheaves on \(\PP^n_k\), such as \(\chi\), the resolution gives \(\lambda(\mcf)=\sum_j(-1)^j\lambda(\mce_j)\), a combination of the values \(\lambda(\OO(-d))\); for \(\chi\) these come from \(H^*(\PP^n,\OO(d))\) in computing cohomology.