Combinatorial free chain complexes over quotient polynomial rings

TL;DR

This paper introduces a combinatorial method to construct free resolutions over quotient polynomial rings with quadratic monomial ideals, demonstrating exactness and infinite injective dimension in specific cases, and conjecturing general applicability.

Contribution

The paper presents a novel combinatorial procedure for constructing free resolutions over quotient polynomial rings, with proofs of exactness and properties in particular cases, and proposes it as a general method.

Findings

Constructed explicit free resolutions for specific quotient rings.

Proved the resulting chain complexes are exact and thus form free resolutions.

Showed that modules over these rings have infinite injective dimension.

Abstract

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.