Let \(X\) be a noetherian scheme, and suppose that every coherent sheaf on \(X\) is a quotient of a locally free sheaf of finite rank. In this case we say \(\Coh(X)\) has enough locally frees. Then for any \(\mcg \in \Mod(X)\), show that the \(\delta\)-functor \((\mathcal{E}xt^i(\wait, \mcg))\), from \(\Coh(X)\) to \(\Mod(X)\) is a contravariant universal \(\delta\)-functor.
Hint
Show \(\mathcal{E}xt^i(\wait, \mcg)\) is coeffaceable for \(i>0\).
Solution
Fix \(\mcg\) and put \(T^i(F)=\mathcal{E}xt_X^i(F,\mcg)\) for \(F\in\Coh(X)\). Regard this contravariant functor as a covariant functor on the abelian category \(\Coh(X)^{\mathrm{op}}\).
1. The functors \(T^i\) form a cohomological \(\delta\)-functor on \(\Coh(X)^{\mathrm{op}}\), with \(T^0(F)=\sheafhom_X(F,\mcg)\).
The category of coherent sheaves on a noetherian scheme is abelian, and its inclusion into \(\Mod(X)\) is exact [@Har10a, Proposition II.5.7]. The long exact sheaf-Ext sequence in the first variable therefore applies to every short exact sequence in \(\Coh(X)\) [@Har10a, Proposition III.6.4]. Reversing that short exact sequence gives a short exact sequence in the opposite category, with the variance required for a covariant cohomological \(\delta\)-functor. The degree-zero identification is the definition of sheaf Ext as the right derived functors of sheaf Hom in the second variable. The boundary maps are natural by that same long exact sequence construction.
2. If \(E\) is locally free of finite rank, then \(\mathcal{E}xt_X^i(E,\mcg)=0\) for every \(i>0\).
The functor \(\sheafhom_X(E,-)\) is isomorphic to \(E^\vee\otimes_{\OO_X}-\). In a local frame it is a finite direct sum of copies of the identity functor, and hence is exact. Applying it to an injective resolution of \(\mcg\) consequently leaves an exact augmented complex. Its positive-degree cohomology sheaves, which define \(\mathcal{E}xt_X^i(E,\mcg)\), vanish. This is also the locally free special case of [@Har10a, Proposition III.6.5].
3. Each \(T^i\) for \(i>0\) is coeffaceable on \(\Coh(X)\), and hence effaceable on its opposite category.
2.Thus the induced map is zero, which is exactly coeffaceability.In \(\Coh(X)^{\mathrm{op}}\), the same arrow is a monomorphism \(F\to E\), and it effaces \(T^i(F)\) in the covariant sense.The same epimorphism works for every positive degree.
4. Q.E.D.
3.Consequently every natural transformation from \(T^0\) to the degree-zero part of another contravariant \(\delta\)-functor extends uniquely to a morphism of \(\delta\)-functors.This is the requested contravariant universality.