Statement
Consider the tensor product of modules Hi(X, Z) ⊗ A.The theorem states that there is an injective group homomorphism ι from this group to Hi(X, A), which has cokernel Tor(Hi-1(X, Z), A). In other words, there is a natural short exact sequence
Furthermore, this is a split sequence (but the splitting is not natural).
The Tor group on the right can be thought of as the obstruction to ι being an isomorphism.
Read more about this topic: Universal Coefficient Theorem
Famous quotes containing the word statement:
“No statement about God is simply, literally true. God is far more than can be measured, described, defined in ordinary language, or pinned down to any particular happening.”
—David Jenkins (b. 1925)
“The force of truth that a statement imparts, then, its prominence among the hordes of recorded observations that I may optionally apply to my own life, depends, in addition to the sense that it is argumentatively defensible, on the sense that someone like me, and someone I like, whose voice is audible and who is at least notionally in the same room with me, does or can possibly hold it to be compellingly true.”
—Nicholson Baker (b. 1957)
“It is commonplace that a problem stated is well on its way to solution, for statement of the nature of a problem signifies that the underlying quality is being transformed into determinate distinctions of terms and relations or has become an object of articulate thought.”
—John Dewey (1859–1952)