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

Understand

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 $R$.