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Inverse System

For {M_i}_(i in I) a family of R-modules indexed by a directed set I, let

sigma_(ji):M_j->M_i i<=j

be an R-module homomorphism. Call (M_i,sigma_(ji)) an inverse system over I when

1. sigma_(ii)(m)=m for all m in M_i.

2. When i<=j<=k, sigma_(ki)=sigma_(ji) degreessigma_(kj).

See also

Directed Set, Direct System, Module, Module Homomorphism

This entry contributed by Bart Snapp

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References

Atiyah, M. F. and Macdonald, I. G. Introduction to Commutative Algebra. Menlo Park, CA: Addison-Wesley, 1969.Matsumura, H. Commutative Ring Theory. New York: Cambridge University Press, 1986.

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Inverse System

Cite this as:

Snapp, Bart. "Inverse System." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/InverseSystem.html

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