This function truncates the source and target of a module map, and returns the induced map between them.
i1 : R = ZZ/101[a..d, Degrees => {{1,3},{1,0},{1,3},{1,2}}];
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i2 : I = ideal "a,b2,c3,d4"
2 3 4
o2 = ideal (a, b , c , d )
o2 : Ideal of R
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i3 : C = res I
1 4 6 4 1
o3 = R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5
o3 : ChainComplex
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i4 : g1 = truncate({1,1}, C.dd_1)
o4 = {1, 2} | 0 b2 0 0 0 d3 |
{1, 3} | 0 0 b2 0 c2 0 |
{1, 3} | 1 0 0 b2 0 0 |
o4 : Matrix
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i5 : g2 = truncate({1,1}, C.dd_2)
o5 = {1, 3} | -b2 -c3 0 -d4 0 0 |
{3, 2} | 0 0 0 0 -d3 0 |
{3, 3} | 0 0 -c2 0 0 0 |
{3, 3} | 1 0 0 0 0 0 |
{3, 9} | 0 a b2 0 0 -d4 |
{4, 8} | 0 0 0 a b2 c3 |
o5 : Matrix
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i6 : g3 = truncate({1,1}, C.dd_3)
o6 = {3, 3} | c3 d4 0 0 |
{4, 12} | -b2 0 d4 0 |
{5, 9} | a 0 0 d4 |
{5, 11} | 0 -b2 -c3 0 |
{6, 8} | 0 a 0 -c3 |
{7, 17} | 0 0 a b2 |
o6 : Matrix
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i7 : g4 = truncate({1,1}, C.dd_4)
o7 = {6, 12} | -d4 |
{7, 11} | c3 |
{8, 20} | -b2 |
{9, 17} | a |
o7 : Matrix
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i8 : D = chainComplex {g1, g2, g3, g4}
o8 = image | d c a | <-- image {1, 3} | 1 0 0 0 0 0 | <-- image {3, 3} | 1 0 0 0 0 0 | <-- image {6, 12} | 1 0 0 0 | <-- image {10, 20} | 1 |
{2, 0} | 0 d c a 0 0 | {4, 12} | 0 1 0 0 0 0 | {7, 11} | 0 1 0 0 |
0 {3, 9} | 0 0 0 0 1 0 | {5, 9} | 0 0 1 0 0 0 | {8, 20} | 0 0 1 0 | 4
{4, 8} | 0 0 0 0 0 1 | {5, 11} | 0 0 0 1 0 0 | {9, 17} | 0 0 0 1 |
{6, 8} | 0 0 0 0 1 0 |
1 {7, 17} | 0 0 0 0 0 1 | 3
2
o8 : ChainComplex
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This functor is exact.
i9 : prune HH D
o9 = 0 : cokernel {1, 2} | a c 0 b2 0 0 d3 |
{1, 3} | 0 -d a 0 c2 b2 0 |
1 : 0
2 : 0
3 : 0
4 : 0
o9 : GradedModule
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i10 : HH_0 D == truncate({1,1}, comodule ideal"a,b2,c3,d4")
o10 = true
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