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On subdirect factors of a projective module and applications to system theory

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Abstract

We extend a result of Napp Avelli, van der Put, and Rocha with a system-theoretic interpretation to the noncommutative case: Let \(P\) be a f.g. projective module over a two-sided Noetherian domain. If \(P\) admits a subdirect product structure of the form \(P \cong M \times _T N\) over a factor module \(T\) of grade at least \(2\) then the torsion-free factor of \(M\) (resp. \(N\)) is projective.

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Notes

  1. One can take the index set to be the solution space \(I := {{\mathrm{Hom}}}_R(M,{\mathcal {F}})\) and require in the definition that the evaluation map from \(M\) to the direct power \({\mathcal {F}}^I\), sending \(m \in M\) to the map \(I \rightarrow {\mathcal {F}}, {\varphi }\mapsto {\varphi }(m)\), is an embedding.

  2. Cyclic or even simple \(R\)-modules suffice (cf. Ishikawa 1964, Theorem 3.1).

  3. This follows easily from the fact that an exact faithful functor of Abelian categories is conservative, i.e., reflects isomorphisms (see, e.g., Barakat and Lange-Hegermann (2013), Lemma A.1). An exact functor of Abelian categories which also reflects exactness is called “faithfully exact” in Ishikawa (1964, Definition 1). Injective cogenerators are called “faithfully injective” in Ishikawa (1964, Definition 3).

  4. As a referee remarked, this restriction rules out singularities which solutions of ODEs with varying coefficients might generally exhibit.

  5. The “only if”-part follows by setting \(\lambda \) to be the composition of the embedding \(\jmath :M \hookrightarrow R^I\) and the projection \(\pi _\iota : R^I \rightarrow R\) such that \(\pi _\iota (\jmath (m)) \ne 0\). The “if”-part follows by setting \(I=M^*\) and \(\jmath \) to be the evaluation map \({\varepsilon }_M: M \rightarrow M^{**}, m \mapsto (\lambda \mapsto \lambda (m))\) considered as a map to \(R^I \supset M^{**}\).

  6. Recall, \(M\) is called reflexive if \({\varepsilon }_M\) is an isomorphism.

  7. This is a more convenient than defining the grade by an equality.

  8. Proposition 6.3 provides an alternative embedding into a free module of finite rank.

  9. Also called fiber product.

  10. \(R\) two-sided coherent is, as usual, enough but we stick to two-sided Noetherian for lack of references.

  11. Here we need that \({{\mathrm{t}}}(M)\) is torsion and not merely having a zero evaluation map.

  12. Here we need that \(T\) is torsion and not merely having a zero evaluation map.

  13. Finite free resolution ring.

References

  • Auslander, M., & Bridger, M. (1969). Stable module theory. Memoirs of the American Mathematical Society, no. 94. Providence, RI: American Mathematical Society.

  • Auslander, M. (1966). Coherent functors. In Proceedings of the Conference on Categorical Algebra (La Jolla, CA, 1965) (pp. 189–231). New York: Springer.

  • Bruns, W., & Herzog, J. (1993). Cohen–Macaulay rings, Cambridge studies in advanced mathematics (Vol. 39). Cambridge: Cambridge University Press.

    Google Scholar 

  • Barakat, M., & Lange-Hegermann, M. (2011). An axiomatic setup for algorithmic homological algebra and an alternative approach to localization. Journal of Algebra and its Applications, 10(2), 269–293. (arXiv:1003.1943).

    Google Scholar 

  • Barakat, M., & Lange-Hegermann, M. (2013). Characterizing Serre quotients with no section functor and applications to coherent sheaves. Applied Categorical Structures. Published online (arXiv:1210.1425).

  • Chyzak, F., Quadrat, A., & Robertz, D. (2005). Effective algorithms for parametrizing linear control systems over Ore algebras. Applicable Algebra in Engineering, Communication and Computing, 16(5), 319–376. http://www-sop.inria.fr/members/Alban.Quadrat/PubsTemporaire/AAECC.pdf.

  • Chyzak, F., Quadrat, A., & Robertz, D. (2007). OreModules: a symbolic package for the study of multidimensional linear systems. Applications of time delay systems. Lecture notes in control and information science (Vol. 352, pp. 233–264). Berlin: Springer. http://wwwb.math.rwth-aachen.de/OreModules.

  • Eisenbud, D. (1995). Commutative algebra. Graduate texts in mathematics (Vol. 150). New York: Springer. With a view toward algebraic geometry.

  • Fliess, M. (1990). Some basic structural properties of generalized linear systems. Systems & Control Letters, 15(5), 391–396.

    Article  MATH  MathSciNet  Google Scholar 

  • Fröhler, S., & Oberst, U. (1998). Continuous time-varying linear systems. Systems & Control Letters, 35(2), 97–110.

    Article  MATH  MathSciNet  Google Scholar 

  • Hilton, P. J., & Stammbach, U. (1997). A course in homological algebra. Graduate texts in mathematics (2nd ed., Vol. 4). New York: Springer.

    Book  Google Scholar 

  • Ishikawa, T. (1964). Faithfully exact functors and their applications to projective modules and injective modules. Nagoya Mathematical Journal, 24, 29–42.

    MATH  MathSciNet  Google Scholar 

  • Lam, T. Y. (1999). Lectures on modules and rings. Graduate texts in mathematics (Vol. 189). New York: Springer.

    Book  Google Scholar 

  • Lam, T. Y. (2006). Serre’s problem on projective modules. Springer monographs in mathematics. Berlin: Springer.

    Book  Google Scholar 

  • Napp Avelli, D., & Rocha, P. (2010). Autonomous multidimensional systems and their implementation by behavioral control. Systems & Control Letters, 59(3–4), 203–208.

    Article  MATH  MathSciNet  Google Scholar 

  • Oberst, U. (1990). Multidimensional constant linear systems. Acta Applicandae Mathematica, 20(1–2), 1–175.

    Article  MATH  MathSciNet  Google Scholar 

  • Pommaret, J. F., & Quadrat, A. (1999). Algebraic analysis of linear multidimensional control systems. IMA Journal of Mathematical Control and Information, 16(3), 275–297.

    Article  MATH  MathSciNet  Google Scholar 

  • Robertz, D. (2006). Formal computational methods for control theory. Ph.D. thesis, RWTH Aachen, Germany, June 2006. This thesis is available at http://darwin.bth.rwth-aachen.de/opus/volltexte/2006/1586.

  • Stenström, B. (1975). Rings of quotients. New York: Springer. Die Grundlehren der Mathematischen Wissenschaften, Band 217. An introduction to methods of ring theory.

  • The homalg project authors. (2003–2013). The  homalg  project—algorithmic homological algebra. http://homalg.math.rwth-aachen.de/.

  • Zerz, E. (2006). An algebraic analysis approach to linear time-varying systems. IMA Journal of Mathematical Control and Information, 23(1), 113–126.

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

I would like to thank Diego Napp Avelli for communicating the problem to me and the referees for various improvement suggestions. I am very grateful to Alban Quadrat who showed us how to do homological algebra in a constructive way. His influence is everywhere in this work.

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Authors and Affiliations

  1. Department of Mathematics, University of Kaiserslautern, 67653 , Kaiserslautern, Germany

    Mohamed Barakat

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  1. Mohamed Barakat
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Correspondence to Mohamed Barakat.

Appendix: A converse of Lemma 5.2

Appendix: A converse of Lemma 5.2

Let \({\mathcal {A}}\) be an Abelian category and \(P \cong M \times _T N \in {\mathcal {A}}\) a fiber product of two objects \(M\) and \(N\) over a common factor object \(T\). Again we set \(A := \ker (P \twoheadrightarrow M)\), \(B := \ker (P \twoheadrightarrow N)\), and \(S := A + B\).

figure c

The four factors

$$\begin{aligned} \underbrace{P/S}_{\cong T} \cong (\underbrace{P/B}_{\cong N})/(\underbrace{S/B}_{\cong A}) \end{aligned}$$

in the first isomorphism theorem applied to \(B \le S \le P\) can be expressed by four commuting short exact sequences yielding the above diagram.

We now formulate the converse of Lemma 5.2 under the assumption that \({{\mathrm{Ext}}}^1(T,P)=0\).

Proposition 8.1

Under the assumption that \({{\mathrm{Ext}}}^1(T,P)=0\) the following two conditions become equivalent:

  1. (1)

    The extension \(0 \rightarrow A \rightarrow N \rightarrow T \rightarrow 0\) is trivial.

  2. (2)

    \({{\mathrm{Ext}}}^1(T,A)=0\).

Proof

For the nontrivial implication (1) \(\implies \) (2) consider the braid diagram below. Condition (1) implies that the connecting homomorphism \({{\mathrm{Hom}}}(T,T) \rightarrow {{\mathrm{Ext}}}^1(T,A)\) is zero, i.e., that \({{\mathrm{Ext}}}^1(T,A)\) embeds into \({{\mathrm{Ext}}}^1(T,N)\). The homomorphism \({\varphi }: {{\mathrm{Ext}}}^1(T,S) \rightarrow {{\mathrm{Ext}}}^1(T,N)\) can be written as the composition \({{\mathrm{Ext}}}^1(T,S) = {{\mathrm{Ext}}}^1(T,A+B) \cong {{\mathrm{Ext}}}^1(T,A) + {{\mathrm{Ext}}}^1(T,B) \twoheadrightarrow {{\mathrm{Ext}}}^1(T,A) \hookrightarrow {{\mathrm{Ext}}}^1(T,N)\), showing that the image of \({\varphi }\) is isomorphic to \({{\mathrm{Ext}}}^1(T,A)\). But \({\varphi }\) factors through \({{\mathrm{Ext}}}^1(T,P) = 0\) and is hence zero, together with its image \({{\mathrm{Ext}}}^1(T,A)\). \(\square \)

figure d

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Barakat, M. On subdirect factors of a projective module and applications to system theory. Multidim Syst Sign Process 26, 339–348 (2015). https://doi.org/10.1007/s11045-013-0258-z

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