On the Deformation of Rings and Algebras

1964/01/01 by Murray Gerstenhaber · 1,413 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Mathematics #Deformation (meteorology) #Pure mathematics #Algebra over a field #Geography

paper · doi:10.2307/1970484

published in Annals of Mathematics 79(1), 59 (Princeton University)

openalex publication_date 1964/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/03

Abstract

CHAPTER I. The deformation theory for algebras 1. Infinitesimal deformations of an algebra 2. Obstructions 3. Trivial deformations 4. Obstructions to derivations and the squaring operation 5. Obstructions are cocycles 6. Additivity and integrability of the square 7. Restricted deformation theories and their cohomology theories 8. Rigidity of fields in the commutative theory CHAPTER II. The parameter space 1. The set of structure constants as parameter space for the deformation theory 2. Central algebras and an example justifying the choice of parameter space 3. The automorphism group as a parameter space, and examples of obstructions to derivations 4. A fiber space over the parameter space, and the upper semicontinuity theorem 5. An example of a restricted theory and the corresponding modular group CHAPTER III. The deformation theory for graded and filtered rings 1. Graded, filtered, and developable rings 2. The Hochschild theory for developable rings 3. Developable rings as deformations of their associated graded rings 4. Trivial deformations and a criterion for rigidity 5. Restriction to the commutative theory 6. Deformations of power series rings

Citations

• On the structure and ideal theory of complete local rings
• Galois Theory of Purely Inseparable Fields of Exponent One

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