Lower bounds on projective levels of complexes

TL;DR

This paper establishes lower bounds on the projective level of complexes over rings based on homology vanishing, leading to a new version of the Intersection Theorem for commutative Noetherian local rings.

Contribution

It introduces novel lower bounds for the projective level of complexes and applies them to derive an improved Intersection Theorem in commutative algebra.

Findings

Lower bounds relate projective level to homology vanishing.

Derived a new version of the Intersection Theorem.

Provides tools for analyzing complexes over rings.

Abstract

For an associative ring $R$, the projective level of a complex $F$ is the smallest number of mapping cones needed to build $F$ from projective $R$-modules. We establish lower bounds for the projective level of $F$ in terms of the vanishing of homology of $F$. We then use these bounds to derive a new version of The New Intersection Theorem for level when $R$ is a commutative Noetherian local ring.