Bases in Orlik-Solomon Type Algebras

Let M be a matroid on [n] and E be the graded algebra generated over a field k generated by the elements 1, e1, . . . , en . Let =(M) be the ideal of E generated by the squares e2 1, . . . , e 2 n , elements of the form ei e j + ai j e j ei and ‘boundaries of circuits’, i.e., elements of the form ∑ χ j ei1 . . . ei j−1 ei j+1 . . . eim , with χ j ∈ k and ei1 , . . . , eim a circuit of the matroid with some special coefficients. The χ -algebraA(M) is defined as the quotient of E by =(M). Recall that the class of χ -algebras contains several studied algebras and in first place the Orlik–Solomon algebra of a matroid. We will essentially construct the reduced Gröbner basis of =(M) for any term order and give some consequences.