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An axiomatic setup for algorithmic homological algebra and an alternative approach to localization. (English) Zbl 1227.18009

Homological algebra of module categories can be viewed as linear algebra over general rings. In analogy to linear algebra over a field, one would expect that solving linear systems would play an important role. In this paper the authors show that solving linear systems is, as expected, the key to the complete computability of the category of finitely presented modules, merely viewed as an abelian category. They list all the existential quantifiers entering the definition of an abelian category.
Then the authors consider the computability of the abelian categories of finitely presented modules over so-called computable rings, i.e., the ones that one can effectively solve (in)homogeneous linear systems. The authors also show that if \(R_{\mathcal{M}}\) is the localization of the commutative ring \(R\) at a finitely generated maximal ideal \(\mathcal{M}\), then the computation in the abelian category of finitely presented modules over the local ring \(R_{\mathcal{M}}\) can be reduced to computations over \(R.\) Finally in this paper the authors describe some implementation and some examples that illustrate the computational advantage of their approach.

MSC:

18E10 Abelian categories, Grothendieck categories
18E25 Derived functors and satellites (MSC2010)
18G05 Projectives and injectives (category-theoretic aspects)
18G10 Resolutions; derived functors (category-theoretic aspects)
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
13H99 Local rings and semilocal rings
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13P20 Computational homological algebra
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References:

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