Periodicity of ideals of minors in free resolutions

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.

• The paper establishes that ideals of minors in minimal free resolutions exhibit eventual 2-periodicity over complete intersection and Golod rings.
• Specifically, the study proves that over complete intersection rings, these ideals become 2-periodic, leveraging concepts like Shamash resolutions.
• Over Golod rings, the paper confirms similar 2-periodic behavior for ideals of minors using structure theorems involving A-infinity algebra operations or related techniques from the field of abstract algebra dealing with algebraic structures and their properties related to resolutions of modules.

Periodicity of Ideals of Minors in Free Resolutions

The paper "Periodicity of Ideals of Minors in Free Resolutions" investigates the asymptotic behavior of ideals of minors within minimal free resolutions over local rings. Authored by Michael K. Brown, Hailong Dao, and Prashanth Sridhar, the study establishes that these ideals exhibit eventual periodicity in specific algebraic settings, namely complete intersections and Golod rings, and develops results concerning the stable behavior of such ideals.

Key Contributions

1. Complete Intersections: The authors prove that over complete intersection rings, the ideals of minors in minimal free resolutions become 2-periodic. The pivotal result here is structured as Theorem 1, which provides conditions for the periodic occurrence of ideals of minors. This is distinguished further for Shamash resolutions, where a universal bound on homological index for the onset of periodicity is established (Theorem 2).
2. Golod Rings: On the side of Golod rings, which exhibit maximal growth of Betti numbers, the paper confirms that similar 2-periodic behavior is present for ideals of minors. This result hinges upon structure theorems concerning module resolutions that incorporate $A_\infty$-algebra operations (Theorem 3).

The paper builds upon several pivotal algebraic concepts:

• Shamash Resolutions: The authors leverage an interpretation involving matrix factorizations to draw connections between complete intersections and the structure of free resolutions.
• Stable Ideals of Minors: Definitions and propositions outline how these ideals can be characterized in both finite and infinite resolutions, with a focus on eventual periodic behavior.
• Cohomological Annihilators: The extension of Eisenbud-Green's and Wang's results on ideals of minors facilitates the understanding of implications regarding the singularities of underlying rings.

Implications and Future Work

The implications of these findings touch upon both theoretic and computational aspects of commutative algebra and algebraic geometry. Practically, these results offer new tools for handling complexity in computations involving free resolutions, which are foundational for researchers working with syzygies and module theory.

The work also suggests intriguing avenues for future research, such as:

• Identifying broader classes of rings where 2-periodicity of ideals of minors can be ensured.
• Exploring deeper algebraic properties tied to 2-periodicity, such as impacts on the geometric characteristics of associated schemes.

The paper stands as a comprehensive examination of periodic patterns in algebraic structures that were previously understood primarily in terms of exceptional cases. It not only extends known results but also opens up new questions about the behavior of resolutions across varied algebraic terrains.