Combinatorial free chain complexes over quotient polynomial rings

We present a procedure that constructs, in a combinatorial manner, a chain complex of free modules over a polynomial ring in finitely many variables, modulo an ideal generated by quadratic monomials. Applying this procedure to two specific rings and one family of rings, we demonstrate that the resulting chain complex is indeed an exact chain complex and thus a free resolution. Utilizing this free resolution, we show that, for these rings, the injective dimension is infinite, as modules over itself. Finally, we propose the conjecture that this procedure always yields a free resolution.