Calculate the first cohomoly group of a sheaf of quotients in Macaulay2

Question

I want to calculate the cohomology of the sheaf Q in the exact sequence on $\mathbb P^3$:

0\rightarrow\mathcal O(-1)\rightarrow \mathcal O^{\oplus 4}\rightarrow Q\rightarrow 0

To do it, I want to calculate the cohomology of Q, using Macaulay2.

I write the follow (using the tutorial in CoherentSheaf)

X=Proj(QQ[x,y,z,w])
A=OO_X^4
B=OO_X^1 (-1)
C=A/B

But, in the four line the output say:

C=A/B stdio:15:3:(3): error: expected modules with the same ambient module.

I'm having trouble writing the correct code using Macaulay2. I'm following the tutorial (linked above), but I keep getting an error message. Can anyone help me troubleshoot this?

Answer 1

The link you posted gives a 404 error. The error you encountered in Macaulay2 is due to the mismatch in the ambient module of the two modules A and B that you are trying to compute the quotient of. To fix this error, you need to ensure that the ambient modules of A and B are the same.

i1 : X = Proj(QQ[x, y, z, w])

o1 = X

o1 : ProjectiveVariety

i2 : A = OO_X^4

        4
o2 = OO
       X

o2 : coherent sheaf on X, free

i3 : B = OO_X^4

        4
o3 = OO
       X

o3 : coherent sheaf on X, free

i4 : C = A/B

o4 = cokernel | 1 0 0 0 |
              | 0 1 0 0 |
              | 0 0 1 0 |
              | 0 0 0 1 |

                                         4
o4 : coherent sheaf on X, quotient of OO  
                                        X

p.s: For Macaulay2 questions, I recommend you to ask in its own Google group.