FastNonminimal -- compute a non-minimal graded free resolution

Synopsis

Usage:

C = res(I, FastNonminimal => true, ...)
Inputs:
- I, an ideal, or a matrix or a module, in a polynomial ring or skew commuting polynomial ring, over a finite prime field
Outputs:
- C, a chain complex,

Description

Given an inhomogeneous, singly-graded, or multi-graded ideal or module, this function computes a non-minimal free resolution. If the input is an ideal $I \subset S$, it computes a non-minimal resolution of $S^1/I$.

A key benefit of this function is that it allows a much faster method for computing the betti numbers of the minimal free resolution. If that is your only interest (i.e. you don't need the complex itself), instead use minimalBetti. However, minimalBetti currently only works for single gradings, not multi-gradings.

i1 : I = Grassmannian(1,6, CoefficientRing => ZZ/101);

               ZZ
o1 : Ideal of ---[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   , p   , p   , p   , p   , p   , p   , p   , p   , p   , p   , p   ]
              101  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4   0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6

i2 : S = ring I

o2 = S

o2 : PolynomialRing

i3 : elapsedTime C = res(I, FastNonminimal => true)
 -- 0.850065 seconds elapsed

      1      35      241      841      1781      2464      2294      1432      576      135      14
o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
                                                                                                         
     0      1       2        3        4         5         6         7         8        9        10      11

o3 : ChainComplex

i4 : elapsedTime C1 = res ideal(I_*)
 -- 1.06867 seconds elapsed

      1      35      140      385      819      1080      819      385      140      35      1
o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S    <-- S   <-- S  <-- 0
                                                                                                    
     0      1       2        3        4        5         6        7        8        9       10     11

o4 : ChainComplex

i5 : betti(C, Minimize => true) == betti C1

o5 = true

For a non-minimal resolution, betti gives the actual Betti numbers, and using the betti(...,Minimize=>...) option gives the ranks in a minimal resolution (which is itself not computed).

i6 : betti C

            0  1   2   3    4    5    6    7   8   9 10
o6 = total: 1 35 241 841 1781 2464 2294 1432 576 135 14
         0: 1  .   .   .    .    .    .    .   .   .  .
         1: . 35 140 290  402  402  293  152  53  11  1
         2: .  . 101 514 1174 1577 1365  780 287  62  6
         3: .  .   .  37  204  479  621  480 221  56  6
         4: .  .   .   .    1    6   15   20  15   6  1

o6 : BettiTally

i7 : betti(C, Minimize => true)

            0  1   2   3   4    5   6   7   8  9 10
o7 = total: 1 35 140 385 819 1080 819 385 140 35  1
         0: 1  .   .   .   .    .   .   .   .  .  .
         1: . 35 140 189  84    .   .   .   .  .  .
         2: .  .   . 196 735 1080 735 196   .  .  .
         3: .  .   .   .   .    .  84 189 140 35  .
         4: .  .   .   .   .    .   .   .   .  .  1

o7 : BettiTally

As mentioned above, if you are just interested in the minimal betti numbers of the ideal or module, then use minimalBetti, as it avoids construction of the non-minimal free resolution.

i8 : minimalBetti I

            0  1   2   3   4    5   6   7   8  9 10
o8 = total: 1 35 140 385 819 1080 819 385 140 35  1
         0: 1  .   .   .   .    .   .   .   .  .  .
         1: . 35 140 189  84    .   .   .   .  .  .
         2: .  .   . 196 735 1080 735 196   .  .  .
         3: .  .   .   .   .    .  84 189 140 35  .
         4: .  .   .   .   .    .   .   .   .  .  1

o8 : BettiTally

If the resolution is not large, this function can be slower than the usual function resolution. But for larger examples, if one is only interested in the betti numbers, this function can be hundreds or thousands of times faster.

If the input module is not graded, or is multi-graded, this function still works. However, minimalBetti does not work in these cases. In the inhomogeneous case, the returned free resolution is often highly non minimal. Of course, there is no notion of minimal resolution in this case, but one can use pruneComplex to clean up the returned complex.

i9 : R = ZZ/101[a..f]

o9 = R

o9 : PolynomialRing

i10 : I = ideal"a3-ab-c2,abc-d2-1, b3-b2-b"

              3          2           2       3    2
o10 = ideal (a  - a*b - c , a*b*c - d  - 1, b  - b  - b)

o10 : Ideal of R

i11 : C = res(I, FastNonminimal => true)

       1      12      28      24      7
o11 = R  <-- R   <-- R   <-- R   <-- R  <-- 0
                                             
      0      1       2       3       4      5

o11 : ChainComplex

i12 : needsPackage "PruneComplex"

o12 = PruneComplex

o12 : Package

i13 : pruneComplex C

       1      3      3      1
o13 = R  <-- R  <-- R  <-- R
                            
      0      1      2      3

o13 : ChainComplex

If one has a specific Groebner basis on which one wants to base the Schreyer resolution, use Strategy=>5. This will not check that the input forms a Groebner basis, but if it does not, then the function will either produce non-sensical answers, or fail.

i14 : R = ZZ/101[a..c,x_1..x_9, MonomialOrder=>{3,9}]

o14 = R

o14 : PolynomialRing

i15 : I = ideal(a^2 - b^2 - x_1 * a*c - x_2 * b*c - x_3 * c^2,
                 a*b - x_4 * a*c - x_5 * b*c - x_6 *c^2,
                 b^2 - x_7 * a*c - x_8 * b*c - x_9 *c^2
                 )

              2    2                      2                             2   
o15 = ideal (a  - b  - a*c*x  - b*c*x  - c x , a*b - a*c*x  - b*c*x  - c x ,
                            1        2      3             4        5      6 
      -----------------------------------------------------------------------
       2                      2
      b  - a*c*x  - b*c*x  - c x )
                7        8      9

o15 : Ideal of R

i16 : C = res(I, FastNonminimal => true, Strategy=>5)

       1      3      2
o16 = R  <-- R  <-- R  <-- 0
                            
      0      1      2      3

o16 : ChainComplex

i17 : C.dd

           1                                                                             3
o17 = 0 : R  <------------------------------------------------------------------------- R  : 1
                | a2-b2-acx_1-bcx_2-c2x_3 ab-acx_4-bcx_5-c2x_6 b2-acx_7-bcx_8-c2x_9 |

           3                                              2
      1 : R  <------------------------------------------ R  : 2
                {2} | -b+cx_4           cx_7         |
                {2} | a-cx_1+cx_5-cx_7  -b-cx_4+cx_8 |
                {2} | -b-cx_2+cx_4-cx_8 a-cx_5+cx_7  |

           2
      2 : R  <----- 0 : 3
                0

o17 : ChainComplexMap

Note that Strategy=>4 or Strategy=>5 implies FastNonminimal.

i18 : C1 = res(ideal I_*, Strategy=>5)

       1      3      2
o18 = R  <-- R  <-- R  <-- 0
                            
      0      1      2      3

o18 : ChainComplex

i19 : C1.dd

           1                                                                             3
o19 = 0 : R  <------------------------------------------------------------------------- R  : 1
                | a2-b2-acx_1-bcx_2-c2x_3 ab-acx_4-bcx_5-c2x_6 b2-acx_7-bcx_8-c2x_9 |

           3                                              2
      1 : R  <------------------------------------------ R  : 2
                {2} | -b+cx_4           cx_7         |
                {2} | a-cx_1+cx_5-cx_7  -b-cx_4+cx_8 |
                {2} | -b-cx_2+cx_4-cx_8 a-cx_5+cx_7  |

           2
      2 : R  <----- 0 : 3
                0

o19 : ChainComplexMap

Caveat

Released in M2 1.9, still experimental. Only works over finite prime fields. Uses quite a lot of memory. For inhomogeneous ideals or modules, the monomial order must be a degree order. For multi-graded ideals or modules, minimalBetti is not yet implemented.

Functions with optional argument named FastNonminimal :

"resolution(...,FastNonminimal=>...)"

For the programmer

The object FastNonminimal is a symbol.