nonminimalMaps -- find the degree zero maps in the Schreyer resolution of an ideal

Synopsis

Usage:

(C, H) = nonminimalMaps I
Inputs:
- I, an ideal, in a polynomial ring $S$ over a base field or coefficient ring $A$. The lead terms of the generators of $I$ should be the initial ideal of $I$, and should be monic.
Outputs:
- C, a chain complex, A complex over a polynomial ring where any parameters in the base ring are set to have degree 0, and the variables of the ring of $I$ are set to have degree one.
- H, a hash table, Whose keys describe which submatrix in the resolution this is, and whose values are those submatrices (placed into the original coefficient ring $A$)

Description

The Schreyer resolution of $I$ (which is generally non-minimal) is computed. The nonminimal parts are the submatrices in this resolution which do not involve the variables in $S$. They are elements in the base ring $A$. For instance, H#(\ell, d) is the submatrix of the matrix from $C_{\ell+1} \to C_{\ell}$ sending degree $d$ to degree $d$.

The ranks of these matrices for a specific parameter value determine exactly the minimal Betti table for the ideal $I$, evaluated at that parameter point.

Now for our example.

i1 : kk = ZZ/101;

i2 : S = kk[a..d];

i3 : F = groebnerFamily ideal"a2,ab,ac,b2,bc2,c3"

             2                      2                      2               
o3 = ideal (a  + t b*c + t a*d + t c  + t b*d + t c*d + t d , a*b + t b*c +
                  1       3       2      4       5       6           7     
     ------------------------------------------------------------------------
                2                         2                              2  
     t a*d + t c  + t  b*d + t  c*d + t  d , a*c + t  b*c + t  a*d + t  c  +
      9       8      10       11       12           13       15       14    
     ------------------------------------------------------------------------
                           2   2                         2                  
     t  b*d + t  c*d + t  d , b  + t  b*c + t  a*d + t  c  + t  b*d + t  c*d
      16       17       18          19       21       20      22       23   
     ------------------------------------------------------------------------
           2     2                    2       2          2         2       3 
     + t  d , b*c  + t  b*c*d + t  a*d  + t  c d + t  b*d  + t  c*d  + t  d ,
        24            25         27        26       28        29        30   
     ------------------------------------------------------------------------
      3                    2       2          2         2       3
     c  + t  b*c*d + t  a*d  + t  c d + t  b*d  + t  c*d  + t  d )
           31         33        32       34        35        36

o3 : Ideal of kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ][a..d]
                  6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31

i4 : (C, H) = nonminimalMaps F;

i5 : betti(C, Weights => {1,1,1,1})

            0 1  2 3 4
o5 = total: 1 6 10 6 1
         0: 1 .  . . .
         1: . 4  4 2 .
         2: . 2  5 3 1
         3: . .  1 1 .

o5 : BettiTally

We see that there are 4 maps that are nonminimal (of sizes $2 \times 4$, $5 \times 2$, $1 \times 3$, and $1 \times 1$).

i6 : keys H

o6 = {(3, 4), (3, 5), (4, 6), (2, 3)}

o6 : List

i7 : H#(2,3)

o7 = {3} | -t_8-t_20t_13      t_7t_20-t_14t_20+t_20t_13t_19            
     {3} | -t_7+t_14-t_13t_19 -t_8-t_20t_13+t_7t_19-t_14t_19+t_13t_19^2
     ------------------------------------------------------------------------
     -t_2-t_14^2+t_20t_13^2    -t_8t_14+t_1t_20+t_7t_20t_13             |
     -t_1-2t_14t_13+t_13^2t_19 -t_2-t_7t_14-t_8t_13+t_1t_19+t_7t_13t_19 |

                                                                                                                                                                                           2                                                                                                                                                                                    4
o7 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])
                 6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31             6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31

i8 : H#(3,4)

o8 = {4} | -t_20                                   
     {4} | -1                                      
     {4} | t_8+t_20t_13-t_7t_19+t_14t_19-t_13t_19^2
     {4} | -t_7+t_14-t_13t_19                      
     {4} | 0                                       
     ------------------------------------------------------------------------
     -t_8                                    |
     t_13                                    |
     t_2+t_7t_14+t_8t_13-t_1t_19-t_7t_13t_19 |
     -t_1-2t_14t_13+t_13^2t_19               |
     t_7-t_14+t_13t_19                       |

                                                                                                                                                                                           5                                                                                                                                                                                    2
o8 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])
                 6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31             6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31

i9 : H#(3,5)

o9 = {5} | -1 t_13 -t_14 |

                                                                                                                                                                                           1                                                                                                                                                                                    3
o9 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])
                 6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31             6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31

i10 : H#(4,6)

o10 = {6} | -1 |

                                                                                                                                                                                            1                                                                                                                                                                                    1
o10 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])
                  6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31             6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31

Let's impose the condition that the map H#(2,3) vanishes (so has rank 0). The Betti diagram of such ideals is not the one for a set of 6 generic points in $\PP^3$.

i11 : J = trim(minors(1, H#(2,3)) + groebnerStratum F);

o11 : Ideal of kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ]
                   6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31

i12 : compsJ = decompose J;

i13 : #compsJ

o13 = 2

i14 : pt1 = randomPointOnRationalVariety compsJ_0

o14 = | 42 9 39 9 34 7 -12 -17 -29 -35 50 2 13 19 -44 50 2 -29 15 2 -27 21
      -----------------------------------------------------------------------
      -36 -29 -39 -10 24 -16 19 -29 39 -38 -22 -8 -30 -24 |

               1       36
o14 : Matrix kk  <-- kk

i15 : pt2 = randomPointOnRationalVariety compsJ_1

o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16
      -----------------------------------------------------------------------
      -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 |

               1       36
o15 : Matrix kk  <-- kk

i16 : F1 = sub(F, (vars S)|pt1)

              2              2                            2               
o16 = ideal (a  - 44b*c - 35c  + 2a*d + 7b*d + 39c*d + 42d , a*b - 39b*c +
      -----------------------------------------------------------------------
         2                             2                   2                 
      15c  - 27a*d + 13b*d - 29c*d + 9d , a*c - 38b*c - 10c  - 16a*d + 2b*d +
      -----------------------------------------------------------------------
                 2   2              2                              2     2  
      19c*d + 34d , b  - 30b*c + 19c  - 22a*d + 21b*d + 50c*d - 12d , b*c  -
      -----------------------------------------------------------------------
                   2         2       2        2     3   3                2   
      29b*c*d - 36c d + 24a*d  + 2b*d  + 50c*d  + 9d , c  - 24b*c*d + 39c d -
      -----------------------------------------------------------------------
          2        2        2      3
      8a*d  - 29b*d  - 29c*d  - 17d )

o16 : Ideal of S

i17 : betti res F1

             0 1 2 3
o17 = total: 1 6 8 3
          0: 1 . . .
          1: . 4 4 1
          2: . 2 4 2

o17 : BettiTally

i18 : F2 = sub(F, (vars S)|pt2)

              2              2                              2               
o18 = ideal (a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c -
      -----------------------------------------------------------------------
         2                              2                  2                
      45c  - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c  + 33a*d + 16b*d
      -----------------------------------------------------------------------
                   2   2              2                   2     2            
      - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , b*c  - 43b*c*d +
      -----------------------------------------------------------------------
         2         2        2        2      3   3                2         2
      19c d + 34a*d  + 21b*d  + 46c*d  + 20d , c  + 38b*c*d + 22c d - 15a*d 
      -----------------------------------------------------------------------
             2        2      3
      - 47b*d  - 39c*d  + 40d )

o18 : Ideal of S

i19 : betti res F2

             0 1 2 3
o19 = total: 1 6 8 3
          0: 1 . . .
          1: . 4 4 1
          2: . 2 4 2

o19 : BettiTally

What are the ideals F1 and F2?

i20 : netList decompose F1

      +------------------------------------------------------+
o20 = |ideal (c - 13d, b + 32d, a + 36d)                     |
      +------------------------------------------------------+
      |ideal (c - 16d, b + d, a + 16d)                       |
      +------------------------------------------------------+
      |                                    2            2    |
      |ideal (b - 6c + 33d, a - 36c + 2d, c  + 43c*d - d )   |
      +------------------------------------------------------+
      |                                     2              2 |
      |ideal (b + 29c + 7d, a - 19c + 24d, c  - 20c*d - 30d )|
      +------------------------------------------------------+

i21 : netList decompose F2

      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                       2                              2   2              2                   2                   2                              2   2              2                              2   3                2         2        2        2      3     2                2         2        2        2      3 |
o21 = |ideal (a*c + 2b*c - 13c  + 33a*d + 16b*d - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , a*b - 20b*c - 45c  - 35a*d + 21b*d - 34c*d + 10d , a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , c  + 38b*c*d + 22c d - 15a*d  - 47b*d  - 39c*d  + 40d , b*c  - 43b*c*d + 19c d + 34a*d  + 21b*d  + 46c*d  + 20d )|
      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+

We can determine what these represent. One should be a set of 6 points, where 5 lie on a plane. The other should be 6 points with 3 points on one line, and the other 3 points on a skew line.

Ways to use nonminimalMaps :

nonminimalMaps(Ideal)

For the programmer

The object nonminimalMaps is a method function.

nonminimalMaps -- find the degree zero maps in the Schreyer resolution of an ideal

Synopsis

Description

See also

Ways to use nonminimalMaps :

For the programmer