Ext^ZZ(CoherentSheaf,SheafMap) -- maps on global Ext induced by a sheaf map

As explained in Ext^ZZ(CoherentSheaf,CoherentSheaf), the global Ext module $\operatorname{Ext}^i_X (\mathcal F, \mathcal G)$ is a vector space, defined as the $i$th right derived functor of the global Hom functor $\operatorname{Hom}_X (\mathcal{F}, -)$. In particular, this implies that for any morphism of sheaves $g : \mathcal{G} \to \mathcal{G}'$ there is an induced map $$\operatorname{Ext}^i_X (\mathcal F , g) : \operatorname{Ext}^i_X (\mathcal F , \mathcal G) \to \operatorname{Ext}^i_X (\mathcal F , \mathcal{G}').$$

In Macaulay2, these vector spaces are not computed using injective resolutions of sheaves. Instead, a result of Greg Smith is used that shows that if $\mathca{F}$ and $\mathcal{G}$ are sheaves represented by modules $M$ and $N$, respectively, then there exists an integer $d$ (depending on $M$, $N$, and $i$) such that $$\operatorname{Ext}^i_X (\mathcal F, \mathcal G) = \operatorname{Ext}^i_S (M_{\geq d} , N)_0,$$ where in the above $S$ is some polynomial ring over a field, $M_{\geq d}$ denotes truncation, and $(-)_0$ denotes the degree $0$ part of a graded module. Moreover, the modules $M$ and $N$ are being viewed as modules over the polynomial ring $S$ via restriction of scalars along the canonical surjection $S \to R$, where $X = \operatorname{Proj} (R)$. In particular, if $\mathcal{G}'$ is represented by some module $N'$, then after taking $d$ to be the maximum of the integers required to satisfy the assumptions of G. Smith's result, the induced map on global Ext may be computed as $$\operatorname{Ext}^i_X (\mathcal F , g) = \operatorname{Ext}_S^i (M_{\geq d} , \widehat{g} )_0,$$ where $\widetilde{g}$ is some map of modules whose associated sheaf is equal to $g$.

If F or G is a sheaf of rings, it is regarded as a free $\mathcal{O}_X$-module of rank $1$. In particular, if $\mathcal{F} = \mathcal{O}_X$ then the map being computed is just the induced map on cohomology $$H^i (g) : H^i (\mathcal{G}) \to H^i (\mathcal{G}').$$

Both F and G must be coherent sheaves on the same projective variety or scheme $X$.

As an example, we compute $\mathrm{Hom}_X(\mathcal I_X,\mathcal O_X)$ and $\mathrm{Ext}^1_X(\mathcal I_X,\mathcal O_X)$, for the rational quartic curve in $\PP^3$.

The $\mathrm{Ext}^1$ being zero says that the point corresponding to $I$ on the Hilbert scheme is smooth (unobstructed), and vector space dimension of $\mathrm{Hom}$ tells us that the dimension of the component at the point $I$ is 16.

The algorithm used may be found in:

If the module $\bigoplus_{a\geq 0} \mathrm{Ext}^i(M, N(a))$ is desired, see Ext^ZZ(CoherentSheaf,SumOfTwists).