Deformations associated to rigid algebras

TL;DR

This paper explores how the deformation theory of infinite dimensional algebras can be understood through associated diagrams of algebras, revealing parameters and rigidity not apparent in classical approaches.

Contribution

It demonstrates that cohomology of diagrams of algebras can reveal deformation parameters and rigidity, and establishes a theorem linking diagram cohomology to that of a single algebra.

Findings

Cohomology of diagrams can detect deformation parameters.

Certain algebras are rigid in classical theory but deform via diagrams.

Cohomology Comparison Theorem equates diagram and single algebra cohomology.

Abstract

The deformations of an infinite dimensional algebra may be controlled not just by its own cohomology but by that of an associated diagram of algebras, since an infinite dimensional algebra may be absolutely rigid in the classical deformation theory for single algebras while depending essentially on some parameters. Two examples studied here, the function field of a sphere with four marked points and the first Weyl algebra, show, however, that the existence of these parameters may be made evident by the cohomology of a diagram (presheaf) of algebras constructed from the original. The Cohomology Comparison Theorem asserts, on the other hand, that the cohomology and deformation theory of a diagram of algebras is always the same as that of a single, but generally rather large, algebra constructed from the diagram.