Vol. 26, No. 2, 1968

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Euler characteristics

J. L. Kelley and Edwin Spanier

Vol. 26 (1968), No. 2, 317–339

Given a suitable category of R-modules, a generalized Euler characteristic is defined for each finite sequence of modules in the category, and is characterized by simple properties. For many categories, including the category of all finitely generated R-modules, this generalized characteristic has the following two properties. First, it assigns the same value to isomorphic sequences. Second, for any chain complex of R-modules in the category, the characteristic of the sequence of chain modules equals the characteristic of the sequence of homology modules. For such categories our results imply that any function having these two properties is itself a function of the characteristic so that the generalized Euler characteristic is essentially the only such function. For the special case of the category of all finitely generated modules over a principal ideal domain the generalized Euler characteristic can be identified with the integer valued function which is the classical Euler characteristic. By considering the special case of the category of all finitely generated torsion modules over the polynomial ring F[x] over a field F we obtain a generalized Euler characteristic for the case of a linear endomorphism of a finite sequence of finite dimensional vector spaces over F. In this case we establish the relations between the characteristic and the sequence of Lefschez numbers of the endomorphism and its iterates.

Mathematical Subject Classification
Primary: 18.20
Secondary: 16.00
Received: 6 February 1967
Published: 1 August 1968
J. L. Kelley
Edwin Spanier