Quasicoherent and coherent sheaves (Tag D-QNTZY)
For an \(A\)-module \(M\), let \(\tilde{M}\) be the sheaf on \(\Spec A\) with \(\tilde{M}(D_f) = M_f\). An \(\OO_X\)-module \(\mcf\) is quasicoherent if \(X\) has a cover by affine opens \(U_i = \Spec A_i\) with \(\mcf|_{U_i} \cong \tilde{M_i}\) for \(A_i\)-modules \(M_i\).
(Coherent module)
Let \(A\) be a ring. An \(A\)-module \(M\) is coherent if it is finitely generated and for every \(n \geq 0\) and every \(A\)-linear map \(\varphi \colon A^n \to M\), not necessarily surjective, the kernel of \(\varphi\) is finitely generated.
(Coherent sheaf)
An \(\OO_X\)-module \(\mcf\) is coherent if \(X\) has a cover by affine opens \(U_i = \Spec A_i\) with \(\mcf|_{U_i} \cong \tilde{M_i}\) for coherent \(A_i\)-modules \(M_i\).
For an \(A\)-module \(M\): coherent \(\implies\) finitely presented \(\implies\) finitely generated. If \(A\) is Noetherian, the three conditions coincide, so on a locally Noetherian scheme \(X\) a quasicoherent \(\mcf\) is coherent exactly when each \(M_i\) in some affine cover is finitely generated.
On \(\Spec A\), the functor \(M \mapsto \tilde{M}\) is an exact equivalence from \(A\)-modules to quasicoherent sheaves, with inverse \(\mcf \mapsto \mcf(X)\).
(An $\OO_X$-module that is not quasicoherent)
Let \(A = k[x]_{(x)}\), so \(X = \Spec A\) has two points: the generic point \(\eta\) and the closed point \((x)\), and the open sets are \(\emptyset\), \(U = \{\eta\} = D(x)\) and \(X\), with \(\OO_X(U) = k(x)\). Define \(\mcf(X) = k(x)\), an \(A\)-module by multiplication in the field \(k(x) \supseteq A\), and \(\mcf(U) = \mcf(\emptyset) = 0\), with zero restriction maps. This is a sheaf of \(\OO_X\)-modules: the only open cover of \(X\) contains \(X\) itself, so the sheaf conditions hold, and the module structures are compatible with restriction because \(\mcf(U) = 0\). It is not quasicoherent: if \(\mcf \cong \tilde{M}\), then \(M \cong \mcf(X) = k(x)\) and \(\mcf(D(x)) \cong M_x = k(x) \neq 0\), while \(\mcf(U) = 0\).
The module structure matters. If instead \(x\) acts by \(0\) on \(k(x)\), through the residue field \(A/(x) = k\), then \(M_x = 0\) and \(\mcf = \tilde{M}\) is quasicoherent: it is the skyscraper sheaf at the closed point. The same holds for the skyscraper sheaf at the origin of \(\AA^1_k = \Spec k[x]\) with value \(k(x)\): with the multiplication action of \(\OO(V) \subseteq k(x)\) on each open \(V \ni 0\) it is an \(\OO\)-module that is not quasicoherent, and with the action through the residue field \(k\) it is quasicoherent.
If \(\mcf\) is quasicoherent, then \(\mcf|_U\cong\widetilde{\mcf(U)}\) for every affine open \(U\subseteq X\), not only for the members of one cover [@Har10a, Proposition II.5.4]. Quasicoherent sheaves on an affine scheme have vanishing higher cohomology.
An \(\OO_X\)-module \(\mcf\) is quasicoherent if and only if every point has an open neighbourhood \(U\) on which \(\mcf\) has a presentation \[ \OO_U^{(I)} \to \OO_U^{(J)} \to \mcf|_U \to 0 . \]
A sheaf of \(\OO_X\)-modules \(\mcf\) is quasicoherent exactly when every point has an open neighbourhood \(U\) with an exact sequence \(\OO_U^{(J)}\to\OO_U^{(I)}\to\mcf|_U\to0\). For a morphism \(X\to S\) and affine opens \(\Spec B\subseteq X\) and \(\Spec A\subseteq S\) with \(\Spec B\) mapping into \(\Spec A\), \(\Omega_{X/S}|_{\Spec B}\cong\widetilde{\Omega_{B/A}}\).
The twists
The twisting sheaves $\OO(n)$, and $\Pic(\PP^n) = \ZZ$ (Tag D-CB9XS)
\(\OO_{\PP^n}(-1)\) is the tautological bundle: its fibre over \(p \in \PP^n\) is the line in \(\AA^{n+1}\) that \(p\) is. Its dual is \(\OO(1)\), and \(\OO(n) = \OO(1)^{\tensor n}\); on \(\Proj S\) these are the sheaves \(\widetilde{S(n)}\) associated to the shifted graded modules.
\(\Pic(\PP^n_k) = \ZZ\), generated by \(\OO(1)\), and \[ H^0(\PP^n, \OO(d)) = S_d = \ts{\text{homogeneous polynomials of degree } d} , \] of dimension \(\binom{n+d}{n}\) for \(d \geq 0\) and zero for \(d < 0\).
For \(n \geq 1\), \(\OO(-1)\) has no nonzero global sections, whereas the global sections of \(\OO(1)\) are the linear forms.
By Serre’s theorem, every coherent sheaf on \(\PP^n_k\) is a quotient of a finite sum of twists \(\OO(-d_i)\). On a smooth projective curve, line bundles correspond to divisors modulo linear equivalence. On a curve of positive genus, two line bundles of the same degree can be non-isomorphic: on an elliptic curve \(E\) with origin \(O\) and a point \(P\ne O\), \(\OO_E(P-O)\) has degree \(0\) and is not isomorphic to \(\OO_E\).
Twisting sheaves and a skyscraper sequence on the projective line (Tag FE-SHFPONE)
Let \(X = \PP^1_k = \Proj k[x_0, x_1]\) and \(p = [1:0]\), the zero of \(x_1\). Multiplication by \(x_1\) gives \(\OO(-1) \to \OO\), with image the ideal sheaf \(\mathcal{I}_p\) of the reduced point \(p\).
On \(D_+(x_1)\), \(x_1\) is a unit, so the map is an isomorphism and both \(\mathcal{I}_p\) and the quotient are trivial away from \(p\).
On \(D_+(x_0) = \Spec k[t]\) with \(t = x_1/x_0\), the map is \(k[t] \xrightarrow{t} k[t]\), which is injective with cokernel \(k[t]/(t) = k\) supported at \(t = 0\), that is, at \(p\).
So \(0 \to \OO(-1) \xrightarrow{x_1} \OO \to k_p \to 0\) is exact on stalks, hence exact, where \(k_p\) is the skyscraper sheaf at \(p\) with value \(k\).
(Tensor products and duals of twists)
On each chart \(D_+(x_i)\) the sheaf \(\OO(d)\) is free with generator \(x_i^d\), and on overlaps the generators differ by the transition functions \((x_i/x_j)^d\). Multiplication \(\OO(d_1) \otimes \OO(d_2) \to \OO(d_1 + d_2)\), \(s \otimes s' \mapsto s s'\), sends the local generator \(x_i^{d_1} \otimes x_i^{d_2}\) to the local generator \(x_i^{d_1 + d_2}\), so it is an isomorphism. Taking \(d_2 = -d_1\) gives \(\OO(d) \otimes \OO(-d) \cong \OO\), and a perfect pairing of invertible sheaves identifies \(\OO(-d) \cong \sheafhom(\OO(d), \OO) = \OO(d)^\vee\).
For \(S=k[x_0,\ldots,x_n]\), \(H^0(\PP^n_k,\OO(d))=S_d\) for \(d\ge0\). For a closed subscheme \(X=V(I)\subseteq\PP^n_k\), the Hilbert polynomial of \(S/I\) from dimension and degree equals \(h^0(X,\OO_X(d))\) for \(d\gg0\).
Every coherent sheaf on \(\PP^n_A\) is a quotient of a finite sum of twists \(\OO(-d_i)\). By descending induction on \(i\), starting above \(n\) where \(H^i\) vanishes, finiteness of \(H^i(\PP^n_A,\mcf)\) for coherent \(\mcf\) and noetherian \(A\) follows from finiteness of \(H^i(\PP^n_A,\OO(d))\).