Given a list L = {a, b1,…, bn} of positive integers with a= sumi bi, and a field (or ring of integers) kk, the script creates a ring S = kk[x1,…,xn] and a map
f: A →B1⊗…⊗Bn
of LabeledModules over S, where A is a free LabeledModule of rank a and Bi is a free LabeledModule of rank bi. The map f is constructed from symmetric functions, and corresponds to collection of linear forms on Pb1-1×…×ß Pbn-1 as used in the construction of pure resolutions in the paper “Betti numbers of graded modules and cohomology of vector bundles” of Eisenbud and Schreyer.The format of F is the one required by tensorComplex1, namely f: A →B1⊗…⊗Bn, with a = rank A, bi = rank Bi.
There is an OPTION, MonSize => n (where n is 8,16, or 32). This sets the MonomialSize option when the base ring of flattenedESTensor is created.
i1 : kk = ZZ/101 o1 = kk o1 : QuotientRing |
i2 : f = flattenedESTensor({5,2,1,2},kk)
o2 = | x_0 x_1 x_2 0 0 |
| 0 x_0 x_1 x_2 0 |
| 0 x_0 x_1 x_2 0 |
| 0 0 x_0 x_1 x_2 |
4 5
o2 : Matrix (kk[x , x , x ]) <--- (kk[x , x , x ])
0 1 2 0 1 2
|
i3 : numgens ring f o3 = 3 |
i4 : betti matrix f
0 1
o4 = total: 4 5
-1: . 5
0: 4 .
o4 : BettiTally
|
i5 : S = ring f o5 = S o5 : PolynomialRing |
i6 : g = tensorComplex1 f
o6 = | x_0^2 0 x_0x_1 0 x_1^2-x_0x_2 0 x_0x_2 0 x_1x_2 0 x_2^2 0 0 0 0 0 0 0 0 0 |
| 0 x_0^2 x_0^2 x_0x_1 x_0x_1 x_1^2-x_0x_2 x_0x_1 x_0x_2 x_1^2 x_1x_2 x_1x_2 x_2^2 x_0x_2 0 x_1x_2 0 x_2^2 0 0 0 |
| 0 0 0 x_0^2 x_0^2 x_0x_1 0 x_0x_1 x_0x_1 x_1^2 x_1^2-x_0x_2 x_1x_2 0 x_0x_2 x_0x_2 x_1x_2 x_1x_2 x_2^2 x_2^2 0 |
| 0 0 0 0 0 x_0^2 0 0 0 x_0x_1 0 x_1^2-x_0x_2 0 0 0 x_0x_2 0 x_1x_2 0 x_2^2 |
4 20
o6 : Matrix S <--- S
|
i7 : betti res coker g
0 1 2 3
o7 = total: 4 20 20 4
0: 4 . . .
1: . 20 20 .
2: . . . 4
o7 : BettiTally
|