On the Structure of the Homotopy Lie Algebra of Local Rings

Author

• Dawson M. Strong, The University of Texas Rio Grande Valley

Date of Award

5-1-2026

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics

First Advisor

Luigi Ferraro

Second Advisor

Sergey Grigorian

Third Advisor

Brandt Kronholm

Abstract

This thesis investigates the construction and homological properties of the homotopy Lie algebra π(R) of a commutative local ring (R,m,k). Drawing upon the theoretical framework of differential graded (DG) algebras, we first establish the theory of minimal free resolutions and other standard topics in homological algebra. The core of this work details the iterative construction of the acyclic closure R⟨Y⟩ of k over R, which is achieved by the systematic adjunction of exterior and divided power variables to eliminate cycles in homology. We demonstrate that this acyclic closure serves as a minimal free resolution and provides the means to define the homotopy Lie algebra π(R) as the homology of the complex of Γ-derivations on the resolution.

Comments

Copyright 2026 Dawson M. Strong. All Rights Reserved. https://proquest.com/docview/3371129023

Recommended Citation

Strong, D. (2026). On the Structure of the Homotopy Lie Algebra of Local Rings [Master's thesis, The University of Texas Rio Grande Valley]. ScholarWorks @ UTRGV. https://scholarworks.utrgv.edu/etd/1949