Let \(X=\PP_k^1\), with \(k\) an infinite field.
Show that there does not exist a projective object \(\mcp \in \Mod(X)\), together with a surjective map \(\mcp \to \mco_X \to 0\).
Show that there does not exist a projective object \(\mcp\) in either \(\QCoh(X)\) or \(\Coh(X)\) together with a surjection \(\mcp \to \mco_X \to 0\).
Hint
For (a), consider surjections \(\mco_V\to k(x)\to0\), where \(x\in X\) is a closed point, \(V\) is an open neighborhood of \(x\), and \(\mco_V=j_!(\mco_X|_V)\) for \(j:V\hookrightarrow X\). For (b), consider surjections \(\mcl\to\mcl\otimes k(x)\to0\) with \(\mcl\) invertible.
Solution
For a closed point \(x\), write \(k(x)\) for the skyscraper \(\OO_X\)-module with fibre \(\kappa(x)\). An object is projective when every morphism from it to the target of an epimorphism lifts through that epimorphism. Suppose, in each of the indicated categories, that \(p:\mcp\twoheadrightarrow\OO_X\) is an epimorphism with \(\mcp\) projective.
1. In \(\Mod(X)\), there is a nonempty open \(W\) and a section \(s\in\mcp(W)\) with \(p(s)=1\), and \(W\) contains two distinct \(k\)-rational points \(x,y\).
Surjectivity of \(p\) as a sheaf morphism gives a local lift of the section \(1\) near any chosen point. After shrinking that neighborhood, the lift is a section \(s\) on \(W\) with \(p(s)=1\). Every nonempty open in \(\PP_k^1\) omits only finitely many closed points: a proper closed subset has dimension zero and is finite. There are infinitely many \(k\)-rational points because \(k\) is infinite. Thus two distinct such points lie in \(W\).
2. No projective object of \(\Mod(X)\) can admit the epimorphism \(p\).
Choose \(W,s,x,y\) as in step 1, put \(V=X\setminus\{y\}\), and let \(j:V\hookrightarrow X\). The residue map on \(V\) extends by zero to an epimorphism \[ q:j_!(\OO_X|_V)\twoheadrightarrow k(x). \] Its stalk at \(x\) is the residue surjection, and the target has zero stalk at every other point. Projectivity would lift the composite \(\mcp\xrightarrow{p}\OO_X\to k(x)\) to \(h:\mcp\to j_!(\OO_X|_V)\).
There is a natural injection from the extension by zero \(j_!(\OO_X|_V)\) into \(\OO_X\): its stalk is the identity on \(V\) and is the map from zero outside \(V\) [@Har10a, Exercise II.1.19]. The image of \(h(s)\) under this injection is a regular function on \(W\) with zero germ at \(y\). It therefore vanishes on a nonempty open neighborhood of \(y\) in \(W\). Since \(W\) is integral, a regular function with zero generic value is zero everywhere. Injectivity then gives \(h(s)=0\). However, its image under \(q\) must be the residue of \(p(s)=1\) at \(x\), which is nonzero. This contradiction proves (a).
3. For every coherent sheaf \(F\) on \(X\), there is an integer \(m\) such that \(\Hom_X(F,\OO_X(-m))=0\).
For sufficiently large \(a\), the sheaf \(F(a)\) is generated by finitely many global sections [@Har10a, Theorem II.5.17]. Thus there is an epimorphism \(\OO_X(-a)^{\oplus N}\twoheadrightarrow F\). Precomposition embeds \(\Hom_X(F,\OO_X(-m))\) into \[ \Hom_X(\OO_X(-a)^{\oplus N},\OO_X(-m)) \cong H^0(X,\OO_X(a-m))^{\oplus N}. \] For \(m>a\) this is zero, since negative twists on \(\PP^1\) have no global sections [@Har10a, Proposition II.5.13].
4. No projective object of \(\QCoh(X)\) or \(\Coh(X)\) can admit the epimorphism \(p\).
First work in \(\QCoh(X)\) and fix a \(k\)-rational point \(x\). The composite \(\mcp\to k(x)\) is nonzero and surjective on its stalk at \(x\). A quasi-coherent sheaf on a noetherian scheme is the union of its coherent subsheaves, as proved in P-AGH2515EXTCOH, part (e). Choose a germ mapping to \(1\in\kappa(x)\) and a coherent subsheaf \(F\subseteq\mcp\) containing a representative of that germ. The restricted map \(F\to k(x)\) is nonzero.
Choose \(m\) as in step 3 and put \(L=\OO_X(-m)\). Trivializing its one-dimensional fibre at \(x\) gives an epimorphism \[ L\twoheadrightarrow L\otimes k(x)\xrightarrow{\cong}k(x) \] in \(\QCoh(X)\). Projectivity of \(\mcp\) lifts its fixed map to \(k(x)\) through this epimorphism, giving \(h:\mcp\to L\). Its restriction to \(F\) is zero by step 3, whereas the composite \(F\to\mcp\xrightarrow{h}L\to k(x)\) was chosen nonzero. This is a contradiction.
In \(\Coh(X)\) the same proof uses \(F=\mcp\) directly. The sheaves \(L\) and \(k(x)\) and the epimorphism between them are coherent, so the required lifting problem lies in that category. It yields the same contradiction and proves (b).
5. Q.E.D.
Steps <1>1–<1>2 prove (a), and steps <1>3–<1>4 prove (b). In particular none of the three categories has enough projectives, since enough projectives would supply an epimorphism from a projective object onto \(\OO_X\).