Algebraic and combinatorial properties of ideals and algebras of uniform clutters of TDI systems

Let C be a uniform clutter and let A be the incidence matrix of C. We denote the column vectors of A by v1, . . . , vq . Under certain conditions we prove that C is vertex critical. If C satisfies the max-flow min-cut property, we prove that A diagonalizes over Z to an identity matrix and that v1, . . . , vq form a Hilbert basis. We also prove that if C has a perfect matching such that C has the packing property and its vertex covering number is equal to 2, then A diagonalizes over Z to an identity matrix. If A is a balanced matrix we prove that any regular triangulation of the cone generated by v1, . . . , vq is unimodular. Some examples are presented to show that our results only hold for uniform clutters. These results are closely related to certain algebraic properties, such as the normality or torsion-freeness, of blowup algebras of edge ideals and to finitely generated abelian groups. They are also related to the theory of Gröbner bases of toric ideals and to Ehrhart rings.