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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
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Abstract

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

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

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