Convert CoherentSheaf to Complex concentrated in deg 0

[Chris Dare]

Hello All —

I appreciate all the help given in terms of my previous questions on Fano threefolds. I'm now attempting to work a bit on the derived geometry side, and my problem is pretty straightforward: given a (prime) Fano threefold (of index 1), functorality of Serre duality says there should be a morphism f : OO_X ---> OO_X(K_X)[3] corresponding to the identity map id : OO_X ---> OO_X, and I want to find the cone of this morphism so that I can look at some of its homological data.

Based on the fact that I want the functionality of cones and shifts, it seems like I should be using the Complexes package — the issue is I'm not entirely sure how to convert an object of type CoherentSheaf to a Complex. There's functionality for this in terms of modules, but I'm not sure I want to reduce to the level of global sections as I may get different results.

For an easy example, if I am working with a (2, 3) complete-intersection:

i1 : loadPackage "Divisor"

i2 : Q5 = QQ[x_0..x_5]

i3 : f = substitute(random(2, ZZ[x_0..x_5]), Q5)

i4 : g = substitute(random(3, ZZ[x_0..x_5]), Q5)

i5 : I = ideal(f, g)

i6 : R = minimalPresentation(Q5/I)

i7 : X = Proj R

i8 : KX = canonicalDivisor(R, IsGraded=>true)

what I am looking to do is something like :

C = Complex(sheaf OO(KX))[3]

Any help would be appreciated.


— Best
Chris

[Greg]

Dear Chris,

Currently, the Complexes package only supports chain complexes formed by modules over a ring.  As far as I know, one cannot construct a complex of sheaves directly in Macaulay2. 

In principle, one can represent each sheaf (on a projective subscheme) by an appropriate graded module (over the homogeneous coordinate ring for the subscheme or the polynomial ring corresponding to the ambient projective space).   However, one needs to deal with the ambiguity in the choice of representative.  

Cheers,

Greg.