# Differential Homological Algebra and General Relativity

## TL;DR

This paper explores the application of differential homological algebra to general relativity, revealing new parametrization techniques for stress equations and providing mathematical insights into gauge theories.

## Contribution

It extends the theory of pure differential modules to arbitrary modules and connects differential homological algebra with the mathematical foundations of general relativity.

## Key findings

- General relativity can be parametrized using the formal adjoint of the Ricci operator.
- Introduction of a 'relative parametrization' with differential constraints.
- Mathematical framework clarifies the structure of gauge theories.

## Abstract

In 1916, F.S. Macaulay developed specific localization techniques for dealing with "unmixed polynomial ideals" in commutative algebra, transforming them into what he called "inverse systems" of partial differential equations. In 1970, D.C. Spencer and coworkers studied the formal theory of such systems, using methods of homological algebra that were giving rise to "differential homological algebra", replacing unmixed polynomial ideals by "pure differential modules". The use of "extension modules" and "differential double duality" is essential for such a purpose. In particular, 0-pure differential modules are torsion-free and admit an "absolute parametrization" by means of arbitrary potential like functions. In 2012, we have been able to extend this result to arbitrary pure modules, introducing a "relative parametrization" where the potentials should satisfy compatible "differential constraints". We recently discovered that General Relativity is just a way to parametrize the Cauchy stress equations by means of the formal adjoint of the Ricci operator in order to obtain a "minimum parametrization" by adding sufficiently many compatible differential constraints, exactly like the Lorenz condition in electromagnetism. These unusual purely mathematical results are illustrated by many explicit examples and even strengthen the comments we recently provided on the mathematical foundations of General Relativity and Gauge Theory.