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 m in a commutative ring R we
show how solving (in)homogeneous linear systems over Rm​ can be
reduced to solving associated systems over R. Hence, the computability of R
implies that of Rm​. As a corollary we obtain the computability
of the category of finitely presented Rm​-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.Comment: Fixed a typo in the proof of Lemma 4.3 spotted by Sebastian Posu