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Abstract: In this paper we develop an axiomatic setup for algorithmic homologicalalgebra of Abelian categories. This is done by exhibiting all existentialquantifiers entering the definition of an Abelian category, which for the sakeof computability need to be turned into constructive ones. We do thisexplicitly for the often-studied example Abelian category of finitely presentedmodules over a so-called computable ring $R$, i.e., a ring with an explicitalgorithm to solve one-sided inhomogeneous linear systems over $R$. For afinitely generated maximal ideal $\mathfrak{m}$ in a commutative ring $R$ weshow how solving inhomogeneous linear systems over $R {\mathfrak{m}}$ can bereduced to solving associated systems over $R$. Hence, the computability of $R$implies that of $R {\mathfrak{m}}$. As a corollary we obtain the computabilityof the category of finitely presented $R {\mathfrak{m}}$-modules as an Abeliancategory, without the need of a Mora-like algorithm. The reduction also yields,as a by-product, a complexity estimation for the ideal membership problem overlocal polynomial rings. Finally, in the case of localized polynomial rings wedemonstrate the computational advantage of our homologically motivatedalternative approach in comparison to an existing implementation of Mora-salgorithm.



Autor: Mohamed Barakat, Markus Lange-Hegermann

Fuente: https://arxiv.org/







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