An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization

TL;DR

This paper develops an axiomatic, constructive framework for algorithmic homological algebra in Abelian categories, particularly focusing on finitely presented modules over computable rings and their localizations, enabling new computational methods.

Contribution

It introduces an axiomatic setup for algorithmic homological algebra that explicitly turns existential quantifiers into constructive ones, and applies this to localizations of rings, improving computational approaches.

Findings

Reduced solving systems over localizations to original rings

Established computability of localized module categories

Demonstrated computational advantages over existing algorithms

Abstract

In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring $R$, i.e., a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over $R$. For a finitely generated maximal ideal $\mathfrak{m}$ in a commutative ring $R$ we show how solving (in)homogeneous linear systems over $R_{\mathfrak{m}}$ can be reduced to solving associated systems over $R$. Hence, the computability of $R$ implies that of $R_{\mathfrak{m}}$. As a corollary we obtain the computability of the category of finitely presented $R_{\mathfrak{m}}$-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a by-product, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm.