# Difference between Derived Functors $\mathcal{Ext}^i(F,G)$ and $Ext^i_X(F,G)$

Let $$F,G$$ be quasicoherent sheaves of $$\mathcal{O}_X$$modules on a scheme $$X$$.

What is exactly the difference between the derived functors $$\mathcal{Ext}^i(F,G)$$ and $$Ext^i_X(F,G)$$?

By definition $$Ext^i_X(F,G)$$ arises from the the global section functor $$U \to \Gamma(U, F^{\vee} \otimes G)$$ while $$\mathcal{Ext}^i(F,G)$$ from the Hom functor $$U \to \mathcal{Hom}_U(F|_U,G|_U)$$.

Both are calculated using injective resolutions $$I_{\bullet}$$ of $$F$$ or $$G$$.

Futhermore the functors are essentially the same since

(*) $$\Gamma(U, F^{\vee} \otimes G)= \Gamma(U, \mathcal{Hom}(F, \mathcal{O}_X) \otimes G)= \mathcal{Hom}(\mathcal{O}_X, \mathcal{Hom}(F, \mathcal{O}_X)\otimes G)= \mathcal{Hom}_X(\mathcal{F}, \mathcal{G})$$

So why generally $$\mathcal{Ext}^i(F,G)$$ and $$Ext^i_X(F,G)$$ should be distinguished?

Till now - according to line (*) - they arise from the same functor, or am I wrong?

• Just as an example, take $X=\mathbb{P}^1_k$, $k$ a filed. Then $\mathcal{E}xt^1(\mathcal{O},\mathcal{O}(-2))=0$ while $Ext^1(\mathcal{O},\mathcal{O}(-2))=k$. In general there is only a spectral sequence connecting the two. – Mohan Oct 8 '18 at 11:34
• @Mohan: Yes. But where is then the error in my reasonings? Is the line $$$*$$$ in generally wrong? Since if (*) would be true then $\mathcal{Ext}^i(F,G)$ and $Ext^i_X(F,G)$ would be derived from the same (up to natural isomorphism) functor, right? Your conterexample shows indeed that that can't be true. – KarlPeter Oct 8 '18 at 11:54
• The middle equality in (*) is wrong. It is not $\mathcal{H}om(\mathcal{O}_X,-)$, but $\mathrm{Hom}(\mathcal{O}_X,-)$. – Mohan Oct 8 '18 at 12:29
• @Mohan: Ah yes, of course. $\mathcal{H}om(F,G)$ was by consruction a sheaf while $Hom(F,G)$ is just the set of morphism between the two sheaves. I guess that's the cruical point. – KarlPeter Oct 8 '18 at 17:18