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.
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 = | x_0a_(0,0)+x_1a_(1,0) x_0a_(0,1)+x_1a_(1,1) x_0a_(0,2)+x_1a_(1,2) ------------------------------------------------------------------------ x_0a_(0,3)+x_1a_(1,3) x_0a_(0,4)+x_1a_(1,4) | 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 |