betti f -- display the graded Betti numbers for a Matrix f, regarding it as a complex of length one.
betti G -- display the graded Betti numbers for the matrix of generators of a GroebnerBasis G.
Here is a sample display:
i1 = R = ZZ/101[a..h]
o1 = R
o1 : PolynomialRing
i2 = p = genericMatrix(R,a,2,4)
o2 = | a c e g |
| b d f h |
2 4
o2 : Matrix R <--- R
i3 = q = generators gb p
o3 = | g e c a 0 0 0 0 0 0 |
| h f d b fg-eh dg-ch bg-ah de-cf be-af bc-ad |
2 10
o3 : Matrix R <--- R
i4 = C = resolution cokernel leadTermMatrix q
2 10 14 7 1
o4 = R <-- R <-- R <-- R <-- R
0 1 2 3 4
o4 : ChainComplex
i5 = betti C
total: 2 10 14 7 1
0: 2 4 6 4 1
1: . 6 8 3 .The top row of the display indicates the ranks of the free module C_j
in column j. The entry below in row i column j gives the number of
basis elements of degree i+j.
If these numbers are needed in a program, one way to get them is with tally.
i6 = degrees C_2
o6 = {{2},{2},{2},{2},{2},{2},{3},{3},{3},{3},{3},{3},{3},{3}}
o6 : List
i7 = t2 = tally degrees C_2
o7 = tally {6 : {2}, 8 : {3}}
o7 : Tally
i8 = peek t2
Tally{{2} => 6}
{3} => 8 i9 = t2_{2}
o9 = 6
i10 = t2_{3}
o10 = 8
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