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Truncations :: truncate(List,Matrix)

truncate(List,Matrix) -- truncation of a matrix

Synopsis

Description

This function truncates the source and target of f, and returns the induced map between them.

i1 : R = ZZ/101[a..d, Degrees=>{{1,3},{1,0},{1,3},{1,2}}]

o1 = R

o1 : PolynomialRing
i2 : C = res coker vars R

      1      4      6      4      1
o2 = R  <-- R  <-- R  <-- R  <-- R  <-- 0
                                         
     0      1      2      3      4      5

o2 : ChainComplex
i3 : g1 = truncate({1,1},C.dd_1)

o3 = {1, 2} | 0 b 0 0 0 1 |
     {1, 3} | 0 0 b 0 1 0 |
     {1, 3} | 1 0 0 b 0 0 |

o3 : Matrix
i4 : g2 = truncate({1,1},C.dd_2)

o4 = {1, 3} | -b -c 0  -d 0  0  |
     {2, 2} | 0  0  0  0  -1 0  |
     {2, 3} | 0  0  -1 0  0  0  |
     {2, 3} | 1  0  0  0  0  0  |
     {1, 3} | 0  a  b  0  0  -d |
     {1, 2} | 0  0  0  a  b  c  |

o4 : Matrix
i5 : g3 = truncate({1,1},C.dd_3)

o5 = {2, 3} | c  d  0  0  |
     {2, 6} | -b 0  d  0  |
     {2, 3} | a  0  0  d  |
     {2, 5} | 0  -b -c 0  |
     {2, 2} | 0  a  0  -c |
     {2, 5} | 0  0  a  b  |

o5 : Matrix
i6 : g4 = truncate({1,1},C.dd_4)

o6 = {3, 6} | -d |
     {3, 5} | c  |
     {3, 8} | -b |
     {3, 5} | a  |

o6 : Matrix
i7 : assert(g1 * g2 == 0)
i8 : assert(g2 * g3 == 0)
i9 : assert(g3 * g4 == 0)

This functor is exact.

i10 : assert(ker g1 == image g2)
i11 : assert(ker g2 == image g3)
i12 : assert(ker g3 == image g4)

See also