Periodicity of ideals of minors in free resolutions

TL;DR

This paper investigates the long-term behavior of ideals of minors in minimal free resolutions over local rings, revealing their eventual periodicity in specific classes and exploring their stable properties and connections to other algebraic concepts.

Contribution

It proves that ideals of minors become 2-periodic in complete intersections and Golod rings, and provides general results on their stable behavior in infinite resolutions.

Findings

Ideals of minors are eventually 2-periodic in complete intersections and Golod rings.

Stable behavior of ideals of minors can be explicitly characterized and computed.

Connections between ideals of minors, trace ideals, and cohomology annihilators are established.

Abstract

We study the asymptotic behavior of the ideals of minors in minimal free resolutions over local rings. In particular, we prove that such ideals are eventually 2-periodic over complete intersections and Golod rings. We also establish general results on the stable behavior of ideals of minors in any infinite minimal free resolution. These ideals have intimate connections to trace ideals and cohomology annihilators. Constraints on the stable values attained by the ideals of minors in many situations are obtained, and they can be explicitly computed in certain cases.