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The Euler characteristic and Euler defect for comodules over Euler coalgebras

Published online by Cambridge University Press:  13 November 2009

Daniel Simson
Daniel Simson
Affiliation:
Faculty of Mathematics and Computer Sciences, Nicolaus Copernicus University, ul. Chopina 12/18, 87–100 Toruń, Poland, simson@mat.uni.torun.pl

Abstract

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Let K be a field. We study a class of left C-comodules over a basic left Euler coalgebra C by means of the Euler ℤ-bilinear form associated to C, the Euler characteristic χC(M,N) of left C-comodules M,N, and the defect ∂C(M,N) ∊ ℤ associated to any computable Euler pair (M,N) of left C-comodules. We show that (lgthM, lgthN) = χC(M,N) + ∂C(M,N), for any computable Euler pair (M,N) of comodules over a left Euler coalgebra C. One of the main results of the paper asserts that the defect ∂C(M,N) is zero and (lgthM,lgthN) = χC(M,N), if the comodules M,N are finite-dimensional.

Keywords

coalgebracomoduleinjective resolutionEuler characteristicintegral bilinear formGrothendieck groupBetti number

Information

Type
Research Article
Information
Journal of K-Theory , Volume 7 , Issue 1 , February 2011, pp. 91 - 113
Copyright
Copyright © ISOPP 2009

References

1.Assem, I., Simson, D. and Skowroński, A., Elements of the Representation Theory of Associative Algebras, Volume 1. Techniques of Representation Theory, London Math. Soc. Student Texts 65, Cambridge Univ. Press, Cambridge-New York, 2006.CrossRefGoogle Scholar
2.Chin, W., A brief introduction to coalgebra representation theory, Lecture Notes in Pure and Appl. Math., Marcel-Dekker 237(2004), pp. 109–131.Google Scholar
3.Chin, W., Kleiner, M. and Quinn, D., Almost split sequences for comodules, J. Algebra 249(2002), 1–19.CrossRefGoogle Scholar
4.Chin, W. and Simson, D., Coxeter transformation and inverses of Cartan matrices for coalgebras, preprint, 2009.CrossRefGoogle Scholar
5.Kleiner, M. and Reiten, I., Abelian categories, almost split sequences and comodules, >Trans. Amer. Math. Soc. 357(2005), 3201–3214.CrossRefGoogle Scholar
6.Dăscălescu, S., Năstăsescu, C. and Raianu, S., Hopf Algebras. An Introduction, Lecture Notes in Pure and Applied Mathematics 235, Marcel-Dekker, New-York, 2001.Google Scholar
7.Dăscălescu, S., Năstăsescu, C. and Torrecillas, B., Homological dimension of coalgebras and crossed products, K-Theory 23(2001), 53–65.CrossRefGoogle Scholar
8.Lopez-Ramos, J. A., Năstăsescu, C. and Torrecillas, B., Minimal projective resolutions for comodules, K-Theory 32(2004), 357–364.CrossRefGoogle Scholar
9.Montgomery, S., Hopf Algebras and Their Actions on Rings, CMBS No. 82, AMS, 1993.CrossRefGoogle Scholar
10.Simson, D., Coalgebras, comodules, pseudocompact algebras and tame comodule type, >Colloq. Math. 90(2001), 101–150.CrossRefGoogle Scholar
11.Simson, D., Localising embeddings of comodule categories with applications to tame and Euler coalgebras, J. Algebra 312(2007), 455–494.CrossRefGoogle Scholar
12.Simson, D., Hom-computable coalgebras, a composition factors matrix and an Euler bilinear form of an Euler coalgebra, >J. Algebra 315(2007), 42–75.CrossRefGoogle Scholar
13.Simson, D., Representation-directed incidence coalgebras of intervally finite posets and the tame-wild dichotomy, >Comm. Algebra 36(2008), 2764–2784.CrossRefGoogle Scholar
14.Simson, D., Tame-wild dichotomy for coalgebras, >J. LondonMath. Soc. 78(2008), 783–797.Google Scholar
15.Simson, D., Incidence coalgebras of intervally finite posets, their integral quadratic forms and comodule categories, >Colloq. Math. 115(2009), 259–295.CrossRefGoogle Scholar
16.Simson, D. and Skowroński, A., Elements of the Representation Theory of Associative Algebras, Volume 2. Tubes and Concealed Algebras of Euclidean Type, London Math. Soc. Student Texts 71, Cambridge Univ. Press, Cambridge-New York, 2007.Google Scholar