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smallerBettiTables -- Find all (potentially) smaller Betti tables that could degenerate to given table

Synopsis

Description

Given a complex over a graded ring, with Betti table $B$, whenever there is an entry of degree zero, if that entry is nonzero, then one can use that as a pivot, and cancel that row and column creating a smaller complex. This function returns the Betti tables of all possible such cancellations that may be able to occur. Some of these might not be valid for actual complexes, as one might obtain a complex with no non-zero scalar entries. But, the list of every smaller Betti table that could possibly be the minimal Betti diagram of such a module is returned.

i1 : S = ZZ/101[a..d]

o1 = S

o1 : PolynomialRing
 
i2 : I = ideal(a*c, a*b, a^2, c^3, b*c^2, b^2*c, b^3)

                       2   3     2   2    3
o2 = ideal (a*c, a*b, a , c , b*c , b c, b )

o2 : Ideal of S
 
i3 : B = betti res I

            0 1  2 3
o3 = total: 1 7 10 4
         0: 1 .  . .
         1: . 3  3 1
         2: . 4  7 3

o3 : BettiTally
 
i4 : smallerBettiTables B

             0 1  2 3         0 1 2 3         0 1 2 3         0 1 2 3        
o4 = {total: 1 7 10 4, total: 1 7 9 3, total: 1 6 9 4, total: 1 6 8 3, total:
          0: 1 .  . .      0: 1 . . .      0: 1 . . .      0: 1 . . .      0:
          1: . 3  3 1      1: . 3 3 .      1: . 3 2 1      1: . 3 2 .      1:
          2: . 4  7 3      2: . 4 6 3      2: . 3 7 3      2: . 3 6 3      2:
     ------------------------------------------------------------------------
     0 1 2 3         0 1 2 3         0 1 2 3         0 1 2 3
     1 5 8 4, total: 1 5 7 3, total: 1 4 7 4, total: 1 4 6 3}
     1 . . .      0: 1 . . .      0: 1 . . .      0: 1 . . .
     . 3 1 1      1: . 3 1 .      1: . 3 . 1      1: . 3 . .
     . 2 7 3      2: . 2 6 3      2: . 1 7 3      2: . 1 6 3

o4 : List
 
i5 : netList pack(4, oo)

     +---------------+--------------+--------------+--------------+
     |       0 1  2 3|       0 1 2 3|       0 1 2 3|       0 1 2 3|
o5 = |total: 1 7 10 4|total: 1 7 9 3|total: 1 6 9 4|total: 1 6 8 3|
     |    0: 1 .  . .|    0: 1 . . .|    0: 1 . . .|    0: 1 . . .|
     |    1: . 3  3 1|    1: . 3 3 .|    1: . 3 2 1|    1: . 3 2 .|
     |    2: . 4  7 3|    2: . 4 6 3|    2: . 3 7 3|    2: . 3 6 3|
     +---------------+--------------+--------------+--------------+
     |       0 1 2 3 |       0 1 2 3|       0 1 2 3|       0 1 2 3|
     |total: 1 5 8 4 |total: 1 5 7 3|total: 1 4 7 4|total: 1 4 6 3|
     |    0: 1 . . . |    0: 1 . . .|    0: 1 . . .|    0: 1 . . .|
     |    1: . 3 1 1 |    1: . 3 1 .|    1: . 3 . 1|    1: . 3 . .|
     |    2: . 2 7 3 |    2: . 2 6 3|    2: . 1 7 3|    2: . 1 6 3|
     +---------------+--------------+--------------+--------------+

Note that from the Betti table there are 2 maps of degree 0. The first is a $4 \times 3$ matrix, and the second is a $7 \times 1$ matrix. There are 4 possible ranks for the first matrix, and 2 for the second, giving 8 Betti tables in the result. No further knowledge is used to remove possible tables from the output list.

All actual Betti diagrams of ideals with $I$ as its initial ideal will be among this list. Clearly, some of these cannot occur. The ones indexed 2, 4 and 6 cannot occur. One can use the package GroebnerStrata to help determine which can possibly occur.

See also

Ways to use smallerBettiTables :

For the programmer

The object smallerBettiTables is a method function.