Homogeneous part of degree d

[Andre Saint Eudes MIALEBAMA BOUESSO]

Hi,

I am interested in computing the dimension of the kernel of the map 

H^5(P^5,O(-8))^3------>H^5(P^5,O(-7))^12. 

Since this map is nothing but the map 

(K[x_1,....,x_6]_2^*)^3-------->(K[x_1,....,x_6]_1^*)^12

then what I need is the dimension of the cokernel of the map 

T : (K[x_1,....,x_6]_1)^12 -------> (K[x_1,....,x_6]_2)^3

i.e    63 - rank T. 

Problem: I am struggling to find a command defining K[x_1,....,x_6]_d. Is there a command in Macaulay 2 for this ? Thanks in advance. 

Best,

Andre

  

[Mahrud Sayrafi]

Hi Andre,

For any graded module M, basis(d, M) gives a matrix whose columns are the generators of the degree d component of M. Importantly, given a map f : N--> M, basis(d, f) is the induced map on the degree d parts.

Regarding your main question though, the most recent version of Macaulay2 can actually compute induced maps on sheaf cohomology given a map of sheaves. So if you define your map f : O(-8)^3 --> O(-7)^12 first, HH^5(X, f) will give you the induced map, and you can compute it's kernel directly.

Best,

Mahrud

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[Andre Saint Eudes MIALEBAMA BOUESSO]

Dear Mahrud,

Thank you very much. My problems are now solved and I am happy. 

Best,

Andre

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