Realization of graded monomial ideal rings modulo torsion (Q6109251)

A classical problem in algebraic topology asks: which commutative graded \(R\)-algebras \(A\) are isomorphic to \(H^\ast(X_A,R)\) for some space \(X_A\)? The space \(X_A\), if it exists, is called a realization of \(A\). According to \textit{J. Aguadé} [Publ., Secc. Mat., Univ. Autòn. Barc. 26, No. 2, 25--68 (1982; Zbl 0595.55001)] the problem goes back to at least Hopf, and was later explicitly stated by \textit{N. E. Steenrod} [Enseign. Math. (2) 7, 153--178 (1962; Zbl 0104.39604)]. To solve the problem in general is probably too ambitious, but many special cases have been proven. One of \textit{D. Quillen}'s motivations for his seminal work on rational homotopy theory [Ann. Math. (2) 90, 205--295 (1969; Zbl 0191.53702)] was to solve this problem over the field of rationals \(\mathbb{Q}\). \par Let \(A\) be the quotient of a graded polynomial ring \(\mathbb{Z}[Ex_1,\ldots,x_m]\otimes\Lambda[y_1,\ldots,y_n]\) by an ideal generated by monomials with leading coefficients \(1\). The authors construct a space \(X_A\) such that \(A\) is isomorphic to \(H^\ast(X_A)\) modulo torsion elements.