# idealChowRing -- the defining ideal of the Chow ring

## Synopsis

• Usage:
idealChowRing M
• Inputs:
• M, ,
• Outputs:
• an ideal, the defining ideal of the Chow ring of M

## Description

The Chow ring of M is the ring R := QQ[x_F]/(I1 + I2), where $I1 = (\sum_{i_1\in F} x_F - \sum_{i_2\in F} x_F : i_1, i_2 \in&nbsp;M)$ 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 , 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.. 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