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recursiveDividedCohomology -- Computes dimension or character of sheaf cohomology of twists of divided powers of the cotangent sheaf on projective space

Description

recursiveDividedCohomology({i,p,d,e,n}) computes the dimension of $H^i(\mathbb{P}^{n-1}, D^d R(e))$, where $D^d \mathcal{R}$ is the d-th divided power of the universal rank (n-1) subsheaf $\mathcal{R}$.

The cohomology index i is 0 or 1 and the underlying characteristic of the field is p (prime or 0).

i1 : i = 1; p = 3; d = 7; e = 6; n = 3;
i6 : recursiveDividedCohomology({i,p,d,e,n})

o6 = 9
i7 : i = 1; p = 0; d = 7; e = 6; n = 3;
i12 : recursiveDividedCohomology({i,p,d,e,n})

o12 = 0

We could instead ask for the character instead of just the dimension, setting the option FindCharacter to be true.

i13 : i = 1; p = 3; d = 7; e = 6; n = 3;
i18 : recursiveDividedCohomology({i,p,d,e,n}, FindCharacter => true)

       6 5 2    5 6 2    5 5 3    6 2 5    5 3 5    3 5 5    2 6 5    5 2 6  
o18 = x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  +
       1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3  
      -----------------------------------------------------------------------
       2 5 6
      x x x
       1 2 3

o18 : ZZ[x ..x ]
          1   3
i19 : i = 1; p = 0; d = 5; e = 3; n = 3;
i24 : recursiveDividedCohomology({i,p,d,e,n}, FindCharacter => true)

       4 4    4 3      3 4      4 2 2    3 3 2    2 4 2    4   3    3 2 3  
o24 = x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  +
       1 2    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3  
      -----------------------------------------------------------------------
       2 3 3      4 3    4 4    3   4    2 2 4      3 4    4 4
      x x x  + x x x  + x x  + x x x  + x x x  + x x x  + x x
       1 2 3    1 2 3    1 3    1 2 3    1 2 3    1 2 3    2 3

o24 : ZZ[x ..x ]
          1   3

Additional input of a polynomial ring allows the user to control the ambient ring of the character.

i25 : i = 1; p = 3; d = 7; e = 6; R = ZZ[x_1..x_3];
i30 : recursiveDividedCohomology({i,p,d,e},R)

       6 5 2    5 6 2    5 5 3    6 2 5    5 3 5    3 5 5    2 6 5    5 2 6  
o30 = x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  +
       1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3  
      -----------------------------------------------------------------------
       2 5 6
      x x x
       1 2 3

o30 : R

Ways to use recursiveDividedCohomology:

  • recursiveDividedCohomology(List)
  • recursiveDividedCohomology(List,PolynomialRing)

For the programmer

The object recursiveDividedCohomology is a method function with options.


The source of this document is in IncidenceCorrespondenceCohomology.m2:1648:0.