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Trace in additive categories

Published online by Cambridge University Press:  24 October 2008

Alan Thomas
Alan Thomas
Affiliation:
University of Warwick
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Extract

1. Let ℳ be an additive category. (We refer to ((1), Ch. IX) for the definition of an additive category and associated terms. In particular a sequence

is exact if i = kerp and p = coker i.) We write End M for Hom(M, M). A trace on ℳ with values in an abelian group G is a collection of (abelian group) homomorphisms

one for each M ∈ ℳ, satisfying the following two conditions:

(i) Exactness. Given a commutative diagram with exact rows,

then tA(f) + tC(h) = tB(g).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 67 , Issue 3 , May 1970 , pp. 541 - 547
Copyright
Copyright © Cambridge Philosophical Society 1970

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References

REFERENCES

(1)MacLane, S.Homology (Springer-Verlag, Berlin, 1963).CrossRefGoogle Scholar
(2)Stallings, J.Centreless Groups—An Algebraic Formulation of Gottlieb's Theorem. Topology 4 (1965).CrossRefGoogle Scholar