×

homalg: a meta-package for homological algebra. (English) Zbl 1151.13306

Summary: The central notion of this work is that of a functor between categories of finitely presented modules over so-called computable rings, i.e. rings \(R\) where one can algorithmically solve inhomogeneous linear equations with coefficients in \(R\). The paper describes a way allowing one to realize such functors, e.g. \(\operatorname{Hom}_R\), \(\otimes_R\), \(\text{Ext}_R^i\), \(\text{Tor}_i^R\), as a mathematical object in a computer algebra system. Once this is achieved, one can compose and derive functors and even iterate this process without the need of any specific knowledge of these functors. These ideas are realized in the ring independent package homalg. It is designed to extend any computer algebra software implementing the arithmetics of a computable ring R, as soon as the latter contains algorithms to solve inhomogeneous linear equations with coefficients in R. Beside explaining how this suffices, the paper describes the nature of the extensions provided by homalg.

MSC:

13D02 Syzygies, resolutions, complexes and commutative rings
13D05 Homological dimension and commutative rings
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
16E05 Syzygies, resolutions, complexes in associative algebras
16E10 Homological dimension in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] DOI: 10.1007/978-94-017-0285-0
[2] Björk J.-E., North-Holland Mathematical Library 21, in: Rings of Differential Operators (1979)
[3] DOI: 10.1016/j.jsc.2005.09.007 · Zbl 1158.01307
[4] Cartan H., Princeton Landmarks in Mathematics, in: Homological Algebra (1999)
[5] DOI: 10.1007/978-1-4757-2181-2
[6] DOI: 10.1007/978-3-540-49556-7_15 · Zbl 1248.93006
[7] Decker W., Algorithms and Computation in Mathematics 16, in: Computing in Algebraic Geometry (2006) · Zbl 1095.14001
[8] DOI: 10.1007/978-1-4612-5350-1
[9] A. Fabianska and A. Quadrat, Gröbner Bases in Control Theory and Signal Processing, Radon Series on Computational and Applied Mathematics 3, eds. H. Park and G. Regensburger (de Gruyter, 2007) pp. 23–106.
[10] V. P. Gerdt, Computational Commutative and Non-Commutative Algebraic Geometry, NATO Sci. Ser. III Comput. Syst. Sci. 196 (IOS, Amsterdam, 2005) pp. 199–225.
[11] DOI: 10.1007/978-3-662-12492-5
[12] DOI: 10.1007/978-3-662-04963-1
[13] DOI: 10.1007/978-1-4419-8566-8
[14] DOI: 10.1007/978-3-540-70628-1
[15] Kreuzer M., Computational Commutative Algebra. 2 (2005) · Zbl 1090.13021
[16] DOI: 10.1090/gsm/030
[17] DOI: 10.1007/s00013-004-1282-x · Zbl 1091.13018
[18] DOI: 10.1016/j.jsc.2007.06.005 · Zbl 1137.16002
[19] Roos J.-E., C. R. Acad. Sci. Paris 254 pp 1556–
[20] Rotman J. J., Pure and Applied Mathematics 85, in: An Introduction to Homological Algebra (1979)
[21] DOI: 10.1017/CBO9780511756320
[22] DOI: 10.1017/CBO9781139644136
[23] DOI: 10.1137/S0363012900374749 · Zbl 1030.93012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.