minimalPresentation(CoherentSheaf) -- minimal presentation of a coherent sheaf or sheaf map

Given a sheaf $\mathcal F$ represented by some module $M$, there are always isomorphisms $$H^i (X , \mathcal{F}) = H^i_{\mathfrak{m}} (M)_0 \quad \text{for} \ i > 0,$$ where $\mathfrak{m}$ denotes the irrelevant ideal. When $i = 0$ the best one can say in general is that there is an exact sequence $$0 \to H^0_{\mathfrak{m}} (M) \to \bigoplus_{d \geq 0} H^0 (X , \mathfrak{F}(d)) \to M \to H^1_{\mathfrak{m}} (M) \to 0.$$ Thus, the prune command for a coherent sheaf computes a graded module $M'$ representing the same sheaf $\mathca{F}$, but such that there is an honest isomorphism $$M' \cong \bigoplus_{d \geq 0} H^0 (X , \mathcal{F} (d)).$$ In other words, the prune command computes a module with depth at least 2 that represents the same sheaf $\mathcal{F}$.

The pruning isomorphism of sheaves is cached with the key pruningMap. It is an isomorphism of sheaves that is essentially never represented by an isomorphism of the underlying modules (at least in interesting cases).

Pruning also applies in a functorial way to morphisms of sheaves.

The method for computing the pruned representative proceeds with a few steps: first, the representative module $M$ is replaced by $M / H^0_{\mathfrak{m}} (M)$, which kills any torsion. Next, since $M$ is now torsion-free, we have an equality $\operatorname{Hom}_S (N , M) = 0$ for any module $N$ with finite length. Thus, we can choose a large formal Frobenius power $\mathfrak{m}^{[m]} = (x_1^m , \dots , x_n^m)$ of the maximal ideal so that $\operatorname{Hom}_S (\mathfrak{m}^{[m]} , M)$ has depth at least $2$. Applying the functor $\operatorname{Hom}_S ( - , M)$ to the short exact sequence $$0 \to \mathfrak{m}^{[m]} \to S \to S / \mathfrak{m}^{[m]} \to 0$$ yields a short exact sequence $$0 \to \operatorname{Hom}_S (\mathfrak{m}^{[m]} , M) \to M \to \operatorname{Ext}^1_S (S / \mathfrak{m}^{[m]} , M) \to 0.$$ The induced map $\operatorname{Hom}_S (\mathfrak{m}^{[m]} , M) \to M$ is not an isomorphism of modules, but sheafifies to an isomorphism since the cokernel has finite length. This induced sheaf map is the pruning map, and the new representative is the sheaf associated to the module $\operatorname{Hom}_R (\mathfrak{m}^{[m]} , M)$.

For morphisms, the above process can be made compatible with maps without much difficulty, since one only needs to conjugate the map by the pruning maps to get an induced map on the pruned representatives.

Since the pruning operation for sheaves tends to be much more involved, one should be careful about the fact that pruning sheaves at every step of a computation may slow cause significant slowdowns.