Inverse Limit


The inverse limit of a family of R-modules is the dual notion of a direct limit and is characterized by the following mapping property. For a directed set I and a family of R-modules {M_i}_(i in I), let (M_i,sigma_(ji)) be an inverse system. lim_(<--)M_i is some R-module with some homomorphisms sigma_i, where for each i in I, i<=j

 sigma_i:lim_(<--)M_i->M_i with the property sigma_i=sigma_(ji) degreessigma_j

such that if there exists some R-module N with homomorphisms alpha_i, where for each i in I, i<=j

 alpha_i:N->M_i with the property alpha_i=sigma_(ji) degreesalpha_j,

then a unique homomorphism alpha:N->lim_(<--)M_i is induced and the above diagram commutes.

The inverse limit can be constructed as follows. For a given inverse system, (M_i,sigma_(ji)), write

 lim_(<--)M_i={(m_i)_(i in I): if i<=j, then m_i=sigma_(ji)(m_j)} subset product_(i in I)M_i.

See also

Commutative Diagram, Direct Limit, Direct Sum, Direct System, Directed Set, Module, Module Homomorphism, Quotient Module

This entry contributed by Bart Snapp

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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.Rotman, J. J. Advanced Modern Algebra. Upper Saddle River, NJ: Prentice Hall, 2002.

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Inverse Limit

Cite this as:

Snapp, Bart. "Inverse Limit." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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