Lifting Gröbner Bases from the Exterior Algebra

In [3], the authors proved a number of results regarding Gröbner bases and initial ideals of those ideals J , in the free associative algebra K 〈X1, . . . , Xn〉, which contain the commutator ideal. We prove similar results for ideals which contains the anti-commutator ideal (the defining ideal of the exterior algebra). We define one notion of generic initial ideals in K 〈X1, . . . , Xn〉, and show that gin’s of ideals containing the commutator ideal, or the anti-commutator ideal, are finitely generated. 1. Notations Let K be an infinite field, and let V be a vector space of dimension n over K. Let X = {X1, . . . ,Xn} be an ordered basis of V . Then the tensor algebra T (V ) may be identified with the free associate (non-commutative) polynomial ring K 〈X1, . . . ,Xn〉, which is the monoid ring on the free monoid 〈X〉 = 〈X1, . . . ,Xn〉. The symmetric algebra S(V ) is then the quotient of K 〈X1, . . . ,Xn〉 by the commutator ideal, the ideal generated by all XiXj−XjXi for i < j. Alternatively, it is the monoid algebra on the free abelian monoid [X]. The exterior algebra over V , E(V ), can be identified with the quotient of K 〈X1, . . . ,Xn〉 by the anti-commutator ideal C generated by XiXj +XjXi, 1 ≤ i ≤ j ≤ n. We denote the quotient epimorphism from K 〈X1, . . . ,Xn〉 to E(V ) by π, and put π(Xi) = xi. If we denote the set of square-free products of elements in π(X) by Y , then Y is a K-basis of E(V ). For any subset I = {i1, . . . , ir} ⊂ {1, . . . , n}, i1 < · · · < ir, we denote by xI the element xi1 · · · xir ∈ Y . If σ is any permutation of {1, . . . , r} then xσ(i1) . . . xσ(ir) = sgn(σ)xI . For j ∈ I, we define xI xj as xI\{j}. Thus xj xI xj = + − xI . We let ≺s be (the strict part of) a monoid well-order on [X], and we let ≺e be its restriction to Y . Then m,n, t ∈ Y, mt 6= 0, nt 6= 0, m ≺e n =⇒ mt ≺e nt. (1) Definition 1.1. We denote by <lex the lexicographic order on 〈X〉; it is supposed to order the variables X1 <lex X2 <lex · · · <lex Xn. We define a monoid well order on 〈X〉 by