2010/03/31 by Mohamed Barakat, MOHAMED BARAKAT, Markus Lange-Hegermann +1 · 19 citations
Computer Science · Mathematics · #Abelian category #Abelian group #Algebra over a field #Axiom #Commutative property #Computability #Computable analysis #Homotopy and Cohomology in Algebraic Topology #Ideal (ethics) #Polynomial and algebraic computation #Polynomial ring #Topological and Geometric Data Analysis #math.AC #msc:13H99 #msc:13P10 #msc:13P20 #msc:18E10 #msc:18E25 #msc:18G05 #msc:18G10 #msc:18G15
paper · pdf · doi:10.1142/s0219498811004562
published in Journal of Algebra and Its Applications 10(02), 269-293 (World Scientific) · Fixed a typo in the proof of Lemma 4.3 spotted by Sebastian Posur
openalex publication_date 2011/04/01 · openalex created_date 2016/06/24 · arxiv created 2017/10/26 · arxiv updated 2017/10/27 · openalex updated_date 2026/08/05
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 R 𝔪 can be reduced to solving associated systems over R. Hence, the computability of R implies that of R 𝔪 . As a corollary, we obtain the computability of the category of finitely presented R 𝔪 -modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a byproduct, 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.