Term-ordering free involutive base

In this paper, we consider a monomial ideal JP:=A[x1,..., xn], over a commutative ring A, and we face the problem of the characterization for the family Mf(J) of all homogeneous ideals IP such that the A-module P/I is free with basis given by the set of terms in the Gröbner escalier N(J) of J. This family is in general wider than that of the ideals having J as initial ideal w.r.t. any term-ordering, hence more suited to a computational approach to the study of Hilbert schemes.For this purpose, we exploit and enhance the concepts of multiplicative variables, complete sets and involutive bases introduced by Riquier (1893, 1899, 1910) and in Janet (1920, 1924, 1927) and we generalize the construction of J-marked bases and term-ordering free reduction process introduced and deeply studied in Bertone et al. (2013a), Cioffi and Roggero (2011) for the special case of a strongly stable monomial ideal J.Here, we introduce and characterize for every monomial ideal J a particular complete set of generators F(J), called stably complete, that allows an explicit description of the family Mf(J). We obtain stronger results if J is quasi-stable, proving that F(J) is a Pommaret basis and Mf(J) has a natural structure of affine scheme.The final section presents a detailed analysis of the origin and the historical evolution of the main notions we refer to.