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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?

Protágoras
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1 Answers1

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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.

tarashi
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  • The link to the Macaulay2 page for the CoherentSheaf package is CoherentSheaf/CoherentSheaf. I have visited the page and found it to be informative without any difficulties. However, in my specific case, I need to calculate the quotient using the module O_X(-1), whereas in your example, you used the module O_X^4. Thank you for providing the answer. I will visit the link you shared. – Protágoras Jun 29 '23 at 16:41
  • @Protágoras Yes, I saw it now, but at that time it was giving 404, maybe it was a temporary problem and it has been solved. And yes, now that I saw the page, I realized that what I wrote was not what you meant. I don't remember much about cohomology but asked in the Google group I linked before, and you will get an answer. Whatever I saw, Dr. Grayson or others always answered to questions there. – tarashi Jul 08 '23 at 17:33