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idealChowRing -- the defining ideal of the Chow ring

Synopsis

Description

The Chow ring of M is the ring R := QQ[x_F]/(I1 + I2), where $I1 = (\sum_{i_1\in F} x_F - \sum_{i_2\in F} x_F : i_1, i_2 \in M)$ and $I2 = (x_Fx_{F'} : F, F' incomparable)$, 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         ,
      {2} {0}   {1} {0}   {2} {3, 4, 5}   {1} {3, 4, 5}   {0} {3, 4, 5} 
     ------------------------------------------------------------------------
     x   x         , x   x         , x   x         , x         x         ,
      {4} {1, 2, 5}   {3} {1, 2, 5}   {0} {1, 2, 5}   {3, 4, 5} {1, 2, 5} 
     ------------------------------------------------------------------------
     x   x      , x   x      , x   x      , x   x      , x         x      ,
      {4} {0, 5}   {3} {0, 5}   {2} {0, 5}   {1} {0, 5}   {3, 4, 5} {0, 5} 
     ------------------------------------------------------------------------
     x         x      , x   x         , x   x         , x   x         , x   
      {1, 2, 5} {0, 5}   {5} {0, 2, 4}   {3} {0, 2, 4}   {1} {0, 2, 4}   {3,
     ------------------------------------------------------------------------
          x         , x         x         , x      x         , x   x      ,
     4, 5} {0, 2, 4}   {1, 2, 5} {0, 2, 4}   {0, 5} {0, 2, 4}   {5} {1, 4} 
     ------------------------------------------------------------------------
     x   x      , x   x      , x   x      , x         x      , x         x   
      {3} {1, 4}   {2} {1, 4}   {0} {1, 4}   {3, 4, 5} {1, 4}   {1, 2, 5} {1,
     ------------------------------------------------------------------------
       , x      x      , x         x      , x   x      , x   x      ,
     4}   {0, 5} {1, 4}   {0, 2, 4} {1, 4}   {5} {2, 3}   {4} {2, 3} 
     ------------------------------------------------------------------------
     x   x      , x   x      , x         x      , x         x      , x   
      {1} {2, 3}   {0} {2, 3}   {3, 4, 5} {2, 3}   {1, 2, 5} {2, 3}   {0,
     ------------------------------------------------------------------------
       x      , x         x      , x      x      , x   x         , x   x   
     5} {2, 3}   {0, 2, 4} {2, 3}   {1, 4} {2, 3}   {5} {0, 1, 3}   {4} {0,
     ------------------------------------------------------------------------
          , x   x         , x         x         , x         x         , x   
     1, 3}   {2} {0, 1, 3}   {3, 4, 5} {0, 1, 3}   {1, 2, 5} {0, 1, 3}   {0,
     ------------------------------------------------------------------------
       x         , x         x         , x      x         , x      x      
     5} {0, 1, 3}   {0, 2, 4} {0, 1, 3}   {1, 4} {0, 1, 3}   {2, 3} {0, 1,
     ------------------------------------------------------------------------
       , x    - x    + x          - x       - x          + x      , x    -
     3}   {1}    {0}    {1, 2, 5}    {0, 5}    {0, 2, 4}    {1, 4}   {2}  
     ------------------------------------------------------------------------
     x    + x          - x       + x       - x         , x    - x    + x   
      {0}    {1, 2, 5}    {0, 5}    {2, 3}    {0, 1, 3}   {3}    {0}    {3,
     ------------------------------------------------------------------------
           - x       - x          + x      , x    - x    + x          - x   
     4, 5}    {0, 5}    {0, 2, 4}    {2, 3}   {4}    {0}    {3, 4, 5}    {0,
     ------------------------------------------------------------------------
        + x       - x         , x    - x    + x          + x          - x   
     5}    {1, 4}    {0, 1, 3}   {5}    {0}    {3, 4, 5}    {1, 2, 5}    {0,
     ------------------------------------------------------------------------
           - x         )
     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}   {3, 4, 5}   {1, 2, 5}   {0, 5}   {0, 2, 4}   {1, 4}   {2, 3}   {0, 1, 3}
 
i3 : basis comodule I

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

o6 : List

See also

Ways to use idealChowRing:

For the programmer

The object idealChowRing is a method function with options.