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Matroids :: idealChowRing

idealChowRing -- the defining ideal of the Chow ring

Synopsis

Description

The Chow ring of M is the ring R := QQ[xF]/(I1 + I2), where I1 = (∑i1∈F xF - ∑i2∈F xF : i1, i2 ∈ M) and I2 = (xFxF’ : F, F’ incomparable), as F runs over all proper nonempty flats of M. This is the same as the Chow ring of the toric variety associated to the Bergman fan of M. This ring is an Artinian standard graded Gorenstein ring, by a result of Adiprasito, Katz, and Huh: cf. https://arxiv.org/abs/1511.02888, Theorem 6.19.

This method returns the defining ideal of the Chow ring, which lives in a polynomial ring with variable indices equal to the flats of M. To work with these subscripts, use "last baseName v" to get the index of a variable v. For more information, cf. Working with Chow rings of matroids.

i1 : M = matroid completeGraph 4

o1 = a matroid of rank 3 on 6 elements

o1 : Matroid
i2 : I = idealChowRing M

o2 = ideal (x   x   , x   x   , x   x   , x   x   , x   x   , x   x   ,
             {5} {4}   {5} {3}   {4} {3}   {5} {2}   {4} {2}   {3} {2} 
     ------------------------------------------------------------------------
     x   x   , x   x   , x   x   , x   x   , x   x   , x   x   , x   x   ,
      {5} {1}   {4} {1}   {3} {1}   {2} {1}   {5} {0}   {4} {0}   {3} {0} 
     ------------------------------------------------------------------------
     x   x   , x   x   , x   x      , x   x      , x   x      , x   x      ,
      {2} {0}   {1} {0}   {4} {0, 5}   {3} {0, 5}   {2} {0, 5}   {1} {0, 5} 
     ------------------------------------------------------------------------
     x   x      , x   x      , x   x      , x   x      , x      x      ,
      {5} {1, 4}   {3} {1, 4}   {2} {1, 4}   {0} {1, 4}   {0, 5} {1, 4} 
     ------------------------------------------------------------------------
     x   x      , x   x      , x   x      , x   x      , x      x      , x   
      {5} {2, 3}   {4} {2, 3}   {1} {2, 3}   {0} {2, 3}   {0, 5} {2, 3}   {1,
     ------------------------------------------------------------------------
       x      , x   x         , x   x         , x   x         , x      x   
     4} {2, 3}   {2} {3, 4, 5}   {1} {3, 4, 5}   {0} {3, 4, 5}   {0, 5} {3,
     ------------------------------------------------------------------------
          , x      x         , x      x         , x   x         , x   x      
     4, 5}   {1, 4} {3, 4, 5}   {2, 3} {3, 4, 5}   {4} {1, 2, 5}   {3} {1, 2,
     ------------------------------------------------------------------------
       , x   x         , x      x         , x      x         , x      x      
     5}   {0} {1, 2, 5}   {0, 5} {1, 2, 5}   {1, 4} {1, 2, 5}   {2, 3} {1, 2,
     ------------------------------------------------------------------------
       , x         x         , x   x         , x   x         , x   x      
     5}   {3, 4, 5} {1, 2, 5}   {5} {0, 2, 4}   {3} {0, 2, 4}   {1} {0, 2,
     ------------------------------------------------------------------------
       , x      x         , x      x         , x      x         , x      
     4}   {0, 5} {0, 2, 4}   {1, 4} {0, 2, 4}   {2, 3} {0, 2, 4}   {3, 4,
     ------------------------------------------------------------------------
       x         , x         x         , x   x         , x   x         ,
     5} {0, 2, 4}   {1, 2, 5} {0, 2, 4}   {5} {0, 1, 3}   {4} {0, 1, 3} 
     ------------------------------------------------------------------------
     x   x         , x      x         , x      x         , x      x         ,
      {2} {0, 1, 3}   {0, 5} {0, 1, 3}   {1, 4} {0, 1, 3}   {2, 3} {0, 1, 3} 
     ------------------------------------------------------------------------
     x         x         , x         x         , x         x         , x    -
      {3, 4, 5} {0, 1, 3}   {1, 2, 5} {0, 1, 3}   {0, 2, 4} {0, 1, 3}   {1}  
     ------------------------------------------------------------------------
     x    - x       + x       + x          - x         , x    - x    - x   
      {0}    {0, 5}    {1, 4}    {1, 2, 5}    {0, 2, 4}   {2}    {0}    {0,
     ------------------------------------------------------------------------
        + x       + x          - x         , x    - x    - x       + x      
     5}    {2, 3}    {1, 2, 5}    {0, 1, 3}   {3}    {0}    {0, 5}    {2, 3}
     ------------------------------------------------------------------------
     + x          - x         , x    - x    - x       + x       + x         
        {3, 4, 5}    {0, 2, 4}   {4}    {0}    {0, 5}    {1, 4}    {3, 4, 5}
     ------------------------------------------------------------------------
     - x         , x    - x    + x          + x          - x          - x   
        {0, 1, 3}   {5}    {0}    {3, 4, 5}    {1, 2, 5}    {0, 2, 4}    {0,
     ------------------------------------------------------------------------
          )
     1, 3}

o2 : Ideal of QQ[x   , x   , x   , x   , x   , x   , x      , x      , x      , x         , x         , x         , x         ]
                  {5}   {4}   {3}   {2}   {1}   {0}   {0, 5}   {1, 4}   {2, 3}   {3, 4, 5}   {1, 2, 5}   {0, 2, 4}   {0, 1, 3}
i3 : basis comodule I

o3 = | 1 x_{0} x_{0, 5} x_{1, 4} x_{2, 3} x_{3, 4, 5} x_{1, 2, 5} x_{0, 2, 4}
     ------------------------------------------------------------------------
     x_{0, 1, 3} x_{0, 1, 3}^2 |

o3 : Matrix
i4 : (0..<rank M)/(i -> hilbertFunction(i, I))

o4 = (1, 8, 1)

o4 : Sequence
i5 : betti res minimalPresentation I

            0  1   2   3   4   5   6  7 8
o5 = total: 1 35 160 350 448 350 160 35 1
         0: 1  .   .   .   .   .   .  . .
         1: . 35 160 350 448 350 160 35 .
         2: .  .   .   .   .   .   .  . 1

o5 : BettiTally
i6 : apply(gens ring I, v -> last baseName v)

o6 = {{5}, {4}, {3}, {2}, {1}, {0}, {0, 5}, {1, 4}, {2, 3}, {3, 4, 5}, {1, 2,
     ------------------------------------------------------------------------
     5}, {0, 2, 4}, {0, 1, 3}}

o6 : List

See also

Ways to use idealChowRing :