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D.5.7.7 dimH

Procedure from library sheafcoh.lib (see sheafcoh_lib).

Usage:
dimH(i,M,d); M module, i,d int

Assume:
M is graded, and it comes assigned with an admissible degree vector as an attribute, h>=l, and the basering S has n+1 variables.

Return:
int, vector space dimension of $H^i(F(d))$ for F the coherent sheaf on P^n associated to coker(M).

Note:
The procedure is based on local duality as described in [Eisenbud: Computing cohomology. In Vasconcelos: Computational methods in commutative algebra and algebraic geometry. Springer (1998)].

Example:
 
LIB "sheafcoh.lib";
ring R=0,(x,y,z,u),dp;
resolution T1=mres(maxideal(1),0);
module M=T1[3];
intvec v=2,2,2,2,2,2;
attrib(M,"isHomog",v);
dimH(0,M,2);
==> 6
dimH(1,M,0);
==> 1
dimH(2,M,1);
==> 0
dimH(3,M,-5);
==> 36
See also: sheafCoh; sheafCohBGG.