HH^ZZ SumOfTwists -- coherent sheaf cohomology

HH^ZZ SumOfTwists -- coherent sheaf cohomology

Synopsis:

Usage: M = HH^i(F(>=d))

Function: cohomology -- general cohomology functor

Input:

i, an instance of class ZZ.

F(>=d), an instance of class SumOfTwists: notation representing the sum of the twists F(n) for all n greater than or equal to d, where F is a coherent sheaf on a variety X.

Output:

M, an instance of class Module: a module over the homogeneous coordinate ring of the variety X which agrees, at least in degrees n greater than or equal to d, with the graded module which in degree n is thei-th cohomology group of F(n).

Optional arguments :

cohomology(..., Degree)

To discard the part of the module M of degree less than d, use truncate(d,M).

Use HH^i(F(>d)) to request the twists strictly greater than n.

Note: use HH^i(F(*)) to try to compute the whole graded module. The computation will fail if the module is not finitely generated.

See also:

HH -- general homology and cohomology functor

HH^ZZ CoherentSheaf -- coherent sheaf cohomology

Code:

     -- ../../../Macaulay2/m2/varieties.m2:192-197
     cohomology(ZZ,SumOfTwists) :=  Module => opts -> (i,S) -> (
          F := S#"object";
          R := ring F;
          if not isAffineRing R then error "expected coherent sheaf over a variety over a field";
          b := first S#"bound";
          if i == 0 then globalSectionsModule(F,b) else HH^(i+1)(module F,Degree => b))