directImageComplex(ChainComplex) -- direct image of a chain complex

Synopsis

Function: directImageComplex
Usage:

piC = directImageComplex C
Inputs:
- C, a chain complex, over a bigraded ring $S = A[y_0..y_n]$, where $A$ is singly graded
Optional inputs:
- Regularity => ..., default value null, direct image complex
Outputs:
- piC, a chain complex, a minimal free complex representing the direct image complex in the derived category

Description

The method is an elaboration of the exterior algebra method for computing cohomology discovered by Eisenbud, Floeystad, Schreyer: Sheaf cohomology and free resolutions over exterior algebras. Trans. Amer. Math. Soc. (2003).

We give an application of this function to create generalized Eagon-Northcott complexes, discovered by Buchsbaum, Eisenbud, and Kirby, and described in Eisenbud, Commutative Algebra, 1995, section A2.6. This method can be generalized to produce pure resolutions of any degree sequence.

These are the complexes associated to a generic 2 by 5 matrix.

i1 : (p,q) = (2,5) -- number of rows and columns

o1 = (2, 5)

o1 : Sequence

i2 : A=ZZ/101[a_(0,0)..a_(p-1,q-1)];

i3 : S = A [x_0..x_(p-1)];

i4 : M = sub(map(A^p, A^{q:-1},transpose genericMatrix(A,a_(0,0),q,p)), S)

o4 = | a_(0,0) a_(0,1) a_(0,2) a_(0,3) a_(0,4) |
     | a_(1,0) a_(1,1) a_(1,2) a_(1,3) a_(1,4) |

             2      5
o4 : Matrix S  <-- S

i5 : Y = map(S^1, S^{q:{-1,-1}}, (vars S)*M)

o5 = | a_(0,0)x_0+a_(1,0)x_1 a_(0,1)x_0+a_(1,1)x_1 a_(0,2)x_0+a_(1,2)x_1
     ------------------------------------------------------------------------
     a_(0,3)x_0+a_(1,3)x_1 a_(0,4)x_0+a_(1,4)x_1 |

             1      5
o5 : Matrix S  <-- S

i6 : F = koszul Y

      1      5      10      10      5      1
o6 = S  <-- S  <-- S   <-- S   <-- S  <-- S
                                           
     0      1      2       3       4      5

o6 : ChainComplex

i7 : L = for i from -1 to q-p+1 list directImageComplex(F**S^{{i,0}});

i8 : L/betti

             0  1  2  3 4         0  1  2  3 4         0 1  2  3 4         0
o8 = {total: 5 20 30 20 5, total: 1 10 20 15 4, total: 2 5 10 10 3, total: 3
          1: 5 20 30 20 5      0: 1  .  .  . .      0: 2 5  .  . .      0: 3
                               1: . 10 20 15 4      1: . . 10 10 3      1: .
     ------------------------------------------------------------------------
      1  2 3 4         0  1  2  3 4         0  1  2  3 4
     10 10 5 2, total: 4 15 20 10 1, total: 5 20 30 20 5}
     10 10 . .      0: 4 15 20 10 .      0: 5 20 30 20 5
      .  . 5 2      1: .  .  .  . 1

o8 : List

Caveat

This function is not yet functorial, i.e., there is no method to take a map of chain complexes and produce the induced map on direct image complexes. Additionally, the input ring must be a tower ring with exactly two gradings, and the variables must have degree $\{0,1\}$ and $\{1,0\}$ A later version will remove these restrictions

Ways to use this method: