issues with Hom(CoherentSheaf, CoherentSheaf) for sheaves with zero-dimensional support
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- Macaulay2
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Description
The following Hom calculation is giving `error: matrix not invertible`:
```
S = QQ[x,y,z]
P = sheaf comodule ideal(x, y)
M = sheaf comodule ideal(x^2, y^2)
Hom(M,P)
```
The issue is that `Hom(M,P)` is calculating `prune sheafHom(module M, module P)` and trying to invert the pruning map, which it can't because it calculated the pruning map as multiplication by $z^2$. (Mathematically, `sheafH(M,P)` is supported at the point $[0,0,1]$, so $z^2$ is a unit on this module.)
I think this is really only a problem in the zero-dimensional case, because the point is that $
\sheafHom(module M, module P)$ is giving us $\mathrm{coker}\ (x\,y)$ in degree 2, and prune is giving us the same module but in degree 0. These are isomorphic because the support is zero-dimensional, so shifting the grading doesn't change the sheaf. So here prune is doing something "unnecessary" and which can't be inverted... (The code above works if $S = QQ[x,y,z,w]$.)
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Research direction
Reproduce the issue with the supplied QQ[x,y,z] example, then trace the Hom(M,P) path through sheafHom and prune, focusing on the pruning map inversion. Compare the failing three-variable case with the working QQ[x,y,z,w] case; done means the zero-dimensional-support calculation no longer reports a non-invertible matrix.
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Assessment
- Domain
- tooling
- Issue type
- Bug
- Difficulty
- 5/5
- Estimated time
- Over a week
- Activity status
- Stale
- Clarity
- Mostly clear
- Newbie friendliness
- 25/100