We illustrate the image of a complex under a ring map.
i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing |
i2 : S = QQ[s,t] o2 = S o2 : PolynomialRing |
i3 : phi = map(S, R, {s, s+t, t})
o3 = map (S, R, {s, s + t, t})
o3 : RingMap S <--- R
|
i4 : I = ideal(x^3, x^2*y, x*y^4, y*z^5)
3 2 4 5
o4 = ideal (x , x y, x*y , y*z )
o4 : Ideal of R
|
i5 : C = freeResolution I
1 4 4 1
o5 = R <-- R <-- R <-- R
0 1 2 3
o5 : Complex
|
i6 : D = phi C
1 4 4 1
o6 = S <-- S <-- S <-- S
0 1 2 3
o6 : Complex
|
i7 : isWellDefined D o7 = true |
i8 : dd^D
1 4
o8 = 0 : S <------------------------------------------------ S : 1
| s3 s3+s2t s5+4s4t+6s3t2+4s2t3+st4 st5+t6 |
4 4
1 : S <------------------------------------------------------- S : 2
{3} | -s-t 0 0 0 |
{3} | s -s3-3s2t-3st2-t3 -t5 0 |
{5} | 0 s 0 -t5 |
{6} | 0 0 s2 s4+3s3t+3s2t2+st3 |
4 1
2 : S <----------------------------- S : 3
{4} | 0 |
{6} | t5 |
{8} | -s3-3s2t-3st2-t3 |
{10} | s |
o8 : ComplexMap
|
i9 : prune HH D
o9 = cokernel | s2t s3 st4 t6 | <-- cokernel {7} | s t3 |
0 1
o9 : Complex
|
When the ring map doesn't preserve homogeneity, the DegreeMap option is needed to determine the degrees of the image free modules in the complex.
i10 : R = ZZ/101[a..d] o10 = R o10 : PolynomialRing |
i11 : S = ZZ/101[s,t] o11 = S o11 : PolynomialRing |
i12 : phi = map(S, R, {s^4, s^3*t, s*t^3, t^4}, DegreeMap => i -> 4*i)
4 3 3 4
o12 = map (S, R, {s , s t, s*t , t })
o12 : RingMap S <--- R
|
i13 : C = freeResolution coker vars R
1 4 6 4 1
o13 = R <-- R <-- R <-- R <-- R
0 1 2 3 4
o13 : Complex
|
i14 : D = phi C
1 4 6 4 1
o14 = S <-- S <-- S <-- S <-- S
0 1 2 3 4
o14 : Complex
|
i15 : assert isWellDefined D |
i16 : assert isHomogeneous D |
i17 : prune HH D
o17 = cokernel | t4 st3 s3t s4 | <-- cokernel {5} | s3 0 t3 0 0 st2 | <-- cokernel {10} | s2 0 0 t2 |
{5} | 0 t3 s3 s2t 0 0 | {11} | t s 0 0 |
0 {6} | 0 0 0 t2 st s2 | {11} | 0 0 t s |
1 2
o17 : Complex
|
Every term in the complex must be free or a submodule of a free module. Otherwise, use tensor(RingMap,Complex).