Realization of graded monomial ideal rings modulo torsion

TL;DR

This paper constructs a topological space whose cohomology ring, modulo torsion, is isomorphic to a class of graded monomial ideal rings, linking algebraic and topological structures.

Contribution

It introduces a method to realize graded monomial ideal rings as cohomology rings of specific topological spaces, extending the understanding of their algebraic-topological correspondence.

Findings

Constructed a space $X_A$ for a given monomial ideal ring $A$.

Proved that $A$ is isomorphic to $H^*(X_A)$ modulo torsion.

Bridged algebraic monomial rings with topological space realizations.

Abstract

Let $A$ be the quotient of a graded polynomial ring $\mathbb{Z}[x_1,\cdots,x_m]\otimes\Lambda[y_1,\cdots,y_n]$ by an ideal generated by monomials with leading coefficients 1. Then we constructed a space~$X_A$ such that $A$ is isomorphic to $H^*(X_A)$ modulo torsion elements.