For any chain complex $C$, a map $f \colon M \to N$ of $R$-modules induces a morphism $C \otimes f$ of chain complexes from $C \otimes M$ to $C \otimes N$. This method returns this map of chain complexes.
i1 : R = ZZ/101[a..d];
|
i2 : I = ideal(c^2-b*d, b*c-a*d, b^2-a*c)
2 2
o2 = ideal (c - b*d, b*c - a*d, b - a*c)
o2 : Ideal of R
|
i3 : J = ideal(I_0, I_1)
2
o3 = ideal (c - b*d, b*c - a*d)
o3 : Ideal of R
|
i4 : C = koszulComplex vars R
1 4 6 4 1
o4 = R <-- R <-- R <-- R <-- R
0 1 2 3 4
o4 : Complex
|
i5 : f = map(R^1/I, R^1/J, 1)
o5 = | 1 |
o5 : Matrix
|
i6 : C ** f
o6 = 0 : cokernel | c2-bd bc-ad b2-ac | <--------- cokernel | c2-bd bc-ad | : 0
| 1 |
1 : cokernel {1} | c2-bd bc-ad b2-ac 0 0 0 0 0 0 0 0 0 | <------------------- cokernel {1} | c2-bd bc-ad 0 0 0 0 0 0 | : 1
{1} | 0 0 0 c2-bd bc-ad b2-ac 0 0 0 0 0 0 | {1} | 1 0 0 0 | {1} | 0 0 c2-bd bc-ad 0 0 0 0 |
{1} | 0 0 0 0 0 0 c2-bd bc-ad b2-ac 0 0 0 | {1} | 0 1 0 0 | {1} | 0 0 0 0 c2-bd bc-ad 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 c2-bd bc-ad b2-ac | {1} | 0 0 1 0 | {1} | 0 0 0 0 0 0 c2-bd bc-ad |
{1} | 0 0 0 1 |
2 : cokernel {2} | c2-bd bc-ad b2-ac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <----------------------- cokernel {2} | c2-bd bc-ad 0 0 0 0 0 0 0 0 0 0 | : 2
{2} | 0 0 0 c2-bd bc-ad b2-ac 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 1 0 0 0 0 0 | {2} | 0 0 c2-bd bc-ad 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 c2-bd bc-ad b2-ac 0 0 0 0 0 0 0 0 0 | {2} | 0 1 0 0 0 0 | {2} | 0 0 0 0 c2-bd bc-ad 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 c2-bd bc-ad b2-ac 0 0 0 0 0 0 | {2} | 0 0 1 0 0 0 | {2} | 0 0 0 0 0 0 c2-bd bc-ad 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 c2-bd bc-ad b2-ac 0 0 0 | {2} | 0 0 0 1 0 0 | {2} | 0 0 0 0 0 0 0 0 c2-bd bc-ad 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c2-bd bc-ad b2-ac | {2} | 0 0 0 0 1 0 | {2} | 0 0 0 0 0 0 0 0 0 0 c2-bd bc-ad |
{2} | 0 0 0 0 0 1 |
3 : cokernel {3} | c2-bd bc-ad b2-ac 0 0 0 0 0 0 0 0 0 | <------------------- cokernel {3} | c2-bd bc-ad 0 0 0 0 0 0 | : 3
{3} | 0 0 0 c2-bd bc-ad b2-ac 0 0 0 0 0 0 | {3} | 1 0 0 0 | {3} | 0 0 c2-bd bc-ad 0 0 0 0 |
{3} | 0 0 0 0 0 0 c2-bd bc-ad b2-ac 0 0 0 | {3} | 0 1 0 0 | {3} | 0 0 0 0 c2-bd bc-ad 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 c2-bd bc-ad b2-ac | {3} | 0 0 1 0 | {3} | 0 0 0 0 0 0 c2-bd bc-ad |
{3} | 0 0 0 1 |
4 : cokernel {4} | c2-bd bc-ad b2-ac | <------------- cokernel {4} | c2-bd bc-ad | : 4
{4} | 1 |
o6 : ComplexMap
|
i7 : f ** C
o7 = 0 : cokernel | c2-bd bc-ad b2-ac | <--------- cokernel | c2-bd bc-ad | : 0
| 1 |
1 : cokernel {1} | c2-bd 0 0 0 bc-ad 0 0 0 b2-ac 0 0 0 | <------------------- cokernel {1} | c2-bd 0 0 0 bc-ad 0 0 0 | : 1
{1} | 0 c2-bd 0 0 0 bc-ad 0 0 0 b2-ac 0 0 | {1} | 1 0 0 0 | {1} | 0 c2-bd 0 0 0 bc-ad 0 0 |
{1} | 0 0 c2-bd 0 0 0 bc-ad 0 0 0 b2-ac 0 | {1} | 0 1 0 0 | {1} | 0 0 c2-bd 0 0 0 bc-ad 0 |
{1} | 0 0 0 c2-bd 0 0 0 bc-ad 0 0 0 b2-ac | {1} | 0 0 1 0 | {1} | 0 0 0 c2-bd 0 0 0 bc-ad |
{1} | 0 0 0 1 |
2 : cokernel {2} | c2-bd 0 0 0 0 0 bc-ad 0 0 0 0 0 b2-ac 0 0 0 0 0 | <----------------------- cokernel {2} | c2-bd 0 0 0 0 0 bc-ad 0 0 0 0 0 | : 2
{2} | 0 c2-bd 0 0 0 0 0 bc-ad 0 0 0 0 0 b2-ac 0 0 0 0 | {2} | 1 0 0 0 0 0 | {2} | 0 c2-bd 0 0 0 0 0 bc-ad 0 0 0 0 |
{2} | 0 0 c2-bd 0 0 0 0 0 bc-ad 0 0 0 0 0 b2-ac 0 0 0 | {2} | 0 1 0 0 0 0 | {2} | 0 0 c2-bd 0 0 0 0 0 bc-ad 0 0 0 |
{2} | 0 0 0 c2-bd 0 0 0 0 0 bc-ad 0 0 0 0 0 b2-ac 0 0 | {2} | 0 0 1 0 0 0 | {2} | 0 0 0 c2-bd 0 0 0 0 0 bc-ad 0 0 |
{2} | 0 0 0 0 c2-bd 0 0 0 0 0 bc-ad 0 0 0 0 0 b2-ac 0 | {2} | 0 0 0 1 0 0 | {2} | 0 0 0 0 c2-bd 0 0 0 0 0 bc-ad 0 |
{2} | 0 0 0 0 0 c2-bd 0 0 0 0 0 bc-ad 0 0 0 0 0 b2-ac | {2} | 0 0 0 0 1 0 | {2} | 0 0 0 0 0 c2-bd 0 0 0 0 0 bc-ad |
{2} | 0 0 0 0 0 1 |
3 : cokernel {3} | c2-bd 0 0 0 bc-ad 0 0 0 b2-ac 0 0 0 | <------------------- cokernel {3} | c2-bd 0 0 0 bc-ad 0 0 0 | : 3
{3} | 0 c2-bd 0 0 0 bc-ad 0 0 0 b2-ac 0 0 | {3} | 1 0 0 0 | {3} | 0 c2-bd 0 0 0 bc-ad 0 0 |
{3} | 0 0 c2-bd 0 0 0 bc-ad 0 0 0 b2-ac 0 | {3} | 0 1 0 0 | {3} | 0 0 c2-bd 0 0 0 bc-ad 0 |
{3} | 0 0 0 c2-bd 0 0 0 bc-ad 0 0 0 b2-ac | {3} | 0 0 1 0 | {3} | 0 0 0 c2-bd 0 0 0 bc-ad |
{3} | 0 0 0 1 |
4 : cokernel {4} | c2-bd bc-ad b2-ac | <------------- cokernel {4} | c2-bd bc-ad | : 4
{4} | 1 |
o7 : ComplexMap
|
i8 : f' = random(R^2, R^{-1, -1, -1})
o8 = | 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d |
| 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |
2 3
o8 : Matrix R <-- R
|
i9 : C ** f'
2 3
o9 = 0 : R <--------------------------------------------------------- R : 0
| 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d |
| 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |
8 12
1 : R <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R : 1
{1} | 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d 0 0 0 0 0 0 0 0 0 |
{1} | 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d 0 0 0 0 0 0 |
{1} | 0 0 0 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d 0 0 0 |
{1} | 0 0 0 0 0 0 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d |
{1} | 0 0 0 0 0 0 0 0 0 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |
12 18
2 : R <----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R : 2
{2} | 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d 0 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |
8 12
3 : R <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R : 3
{3} | 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d 0 0 0 0 0 0 0 0 0 |
{3} | 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d 0 0 0 0 0 0 |
{3} | 0 0 0 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d 0 0 0 |
{3} | 0 0 0 0 0 0 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d |
{3} | 0 0 0 0 0 0 0 0 0 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |
2 3
4 : R <------------------------------------------------------------- R : 4
{4} | 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d |
{4} | 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |
o9 : ComplexMap
|
i10 : f' ** C
2 3
o10 = 0 : R <--------------------------------------------------------- R : 0
| 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d |
| 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |
8 12
1 : R <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R : 1
{1} | 24a-36b-30c-29d 0 0 0 -8a-22b-29c-24d 0 0 0 34a+19b-47c-39d 0 0 0 |
{1} | 0 24a-36b-30c-29d 0 0 0 -8a-22b-29c-24d 0 0 0 34a+19b-47c-39d 0 0 |
{1} | 0 0 24a-36b-30c-29d 0 0 0 -8a-22b-29c-24d 0 0 0 34a+19b-47c-39d 0 |
{1} | 0 0 0 24a-36b-30c-29d 0 0 0 -8a-22b-29c-24d 0 0 0 34a+19b-47c-39d |
{1} | 19a+19b-10c-29d 0 0 0 -38a-16b+39c+21d 0 0 0 -18a-13b-43c-15d 0 0 0 |
{1} | 0 19a+19b-10c-29d 0 0 0 -38a-16b+39c+21d 0 0 0 -18a-13b-43c-15d 0 0 |
{1} | 0 0 19a+19b-10c-29d 0 0 0 -38a-16b+39c+21d 0 0 0 -18a-13b-43c-15d 0 |
{1} | 0 0 0 19a+19b-10c-29d 0 0 0 -38a-16b+39c+21d 0 0 0 -18a-13b-43c-15d |
12 18
2 : R <----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R : 2
{2} | 24a-36b-30c-29d 0 0 0 0 0 -8a-22b-29c-24d 0 0 0 0 0 34a+19b-47c-39d 0 0 0 0 0 |
{2} | 0 24a-36b-30c-29d 0 0 0 0 0 -8a-22b-29c-24d 0 0 0 0 0 34a+19b-47c-39d 0 0 0 0 |
{2} | 0 0 24a-36b-30c-29d 0 0 0 0 0 -8a-22b-29c-24d 0 0 0 0 0 34a+19b-47c-39d 0 0 0 |
{2} | 0 0 0 24a-36b-30c-29d 0 0 0 0 0 -8a-22b-29c-24d 0 0 0 0 0 34a+19b-47c-39d 0 0 |
{2} | 0 0 0 0 24a-36b-30c-29d 0 0 0 0 0 -8a-22b-29c-24d 0 0 0 0 0 34a+19b-47c-39d 0 |
{2} | 0 0 0 0 0 24a-36b-30c-29d 0 0 0 0 0 -8a-22b-29c-24d 0 0 0 0 0 34a+19b-47c-39d |
{2} | 19a+19b-10c-29d 0 0 0 0 0 -38a-16b+39c+21d 0 0 0 0 0 -18a-13b-43c-15d 0 0 0 0 0 |
{2} | 0 19a+19b-10c-29d 0 0 0 0 0 -38a-16b+39c+21d 0 0 0 0 0 -18a-13b-43c-15d 0 0 0 0 |
{2} | 0 0 19a+19b-10c-29d 0 0 0 0 0 -38a-16b+39c+21d 0 0 0 0 0 -18a-13b-43c-15d 0 0 0 |
{2} | 0 0 0 19a+19b-10c-29d 0 0 0 0 0 -38a-16b+39c+21d 0 0 0 0 0 -18a-13b-43c-15d 0 0 |
{2} | 0 0 0 0 19a+19b-10c-29d 0 0 0 0 0 -38a-16b+39c+21d 0 0 0 0 0 -18a-13b-43c-15d 0 |
{2} | 0 0 0 0 0 19a+19b-10c-29d 0 0 0 0 0 -38a-16b+39c+21d 0 0 0 0 0 -18a-13b-43c-15d |
8 12
3 : R <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R : 3
{3} | 24a-36b-30c-29d 0 0 0 -8a-22b-29c-24d 0 0 0 34a+19b-47c-39d 0 0 0 |
{3} | 0 24a-36b-30c-29d 0 0 0 -8a-22b-29c-24d 0 0 0 34a+19b-47c-39d 0 0 |
{3} | 0 0 24a-36b-30c-29d 0 0 0 -8a-22b-29c-24d 0 0 0 34a+19b-47c-39d 0 |
{3} | 0 0 0 24a-36b-30c-29d 0 0 0 -8a-22b-29c-24d 0 0 0 34a+19b-47c-39d |
{3} | 19a+19b-10c-29d 0 0 0 -38a-16b+39c+21d 0 0 0 -18a-13b-43c-15d 0 0 0 |
{3} | 0 19a+19b-10c-29d 0 0 0 -38a-16b+39c+21d 0 0 0 -18a-13b-43c-15d 0 0 |
{3} | 0 0 19a+19b-10c-29d 0 0 0 -38a-16b+39c+21d 0 0 0 -18a-13b-43c-15d 0 |
{3} | 0 0 0 19a+19b-10c-29d 0 0 0 -38a-16b+39c+21d 0 0 0 -18a-13b-43c-15d |
2 3
4 : R <------------------------------------------------------------- R : 4
{4} | 24a-36b-30c-29d -8a-22b-29c-24d 34a+19b-47c-39d |
{4} | 19a+19b-10c-29d -38a-16b+39c+21d -18a-13b-43c-15d |
o10 : ComplexMap
|
i11 : assert isWellDefined(C ** f')
|
i12 : assert isWellDefined(f' ** C)
|