Relative determinant of a bilinear module

Przemysław Koprowski

Discussiones Mathematicae - General Algebra and Applications (2014)

  • Volume: 34, Issue: 2, page 203-212
  • ISSN: 1509-9415

Abstract

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The aim of the paper is to generalize the (ultra-classical) notion of the determinant of a bilinear form to the class of bilinear forms on projective modules without assuming that the determinant bundle of the module is free. Successively it is proved that this new definition preserves the basic properties, one expects from the determinant. As an example application, it is shown that the introduced tools can be used to significantly simplify the proof of a recent result by B. Rothkegel.

How to cite

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Przemysław Koprowski. "Relative determinant of a bilinear module." Discussiones Mathematicae - General Algebra and Applications 34.2 (2014): 203-212. <http://eudml.org/doc/270149>.

@article{PrzemysławKoprowski2014,
abstract = {The aim of the paper is to generalize the (ultra-classical) notion of the determinant of a bilinear form to the class of bilinear forms on projective modules without assuming that the determinant bundle of the module is free. Successively it is proved that this new definition preserves the basic properties, one expects from the determinant. As an example application, it is shown that the introduced tools can be used to significantly simplify the proof of a recent result by B. Rothkegel.},
author = {Przemysław Koprowski},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {determinant; bilinear forms; projective modules},
language = {eng},
number = {2},
pages = {203-212},
title = {Relative determinant of a bilinear module},
url = {http://eudml.org/doc/270149},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Przemysław Koprowski
TI - Relative determinant of a bilinear module
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2014
VL - 34
IS - 2
SP - 203
EP - 212
AB - The aim of the paper is to generalize the (ultra-classical) notion of the determinant of a bilinear form to the class of bilinear forms on projective modules without assuming that the determinant bundle of the module is free. Successively it is proved that this new definition preserves the basic properties, one expects from the determinant. As an example application, it is shown that the introduced tools can be used to significantly simplify the proof of a recent result by B. Rothkegel.
LA - eng
KW - determinant; bilinear forms; projective modules
UR - http://eudml.org/doc/270149
ER -

References

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  1. [1] M. Ciemała and K. Szymiczek, On the existence of nonsingular bilinear forms on projective modules, Tatra Mt. Math. Publ. 32 (2005) 1-13. Zbl1150.11419
  2. [2] O. Goldman, Determinants in projective modules, Nagoya Math. J. 18 (1961) 27-36. Zbl0103.27001
  3. [3] M.A. Marshall, Bilinear forms and orderings on commutative rings, volume~71 of Queen's Papers in Pure and Applied Mathematics (Queen's University, Kingston, ON, 1985). 
  4. [4] J. Milnor and D. Husemoller, Symmetric bilinear forms (Springer-Verlag, New York, 1973). Zbl0292.10016
  5. [5] H.P. Petersson, Polar decompositions of quaternion algebras over arbitrary rings, preprint, 2008. http://www.fernuni-hagen.de/petersson/download/polar-quat-l.pdf 
  6. [6] B. Rothkegel, Nonsingular bilinear forms on direct sums of ideals, Math. Slovaca 63(4) (2013) 707-724. doi: 10.2478/s12175-013-0130-5. Zbl06258761
  7. [7] C.A. Weibel, The K-book. An introduction to algebraic K-theory. volume 145 of Graduate Studies in Mathematics (American Mathematical Society, Providence, 2013). 

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