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:
“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)
“I think, therefore I am is the statement of an intellectual who underrates toothaches.”
—Milan Kundera (b. 1929)
“He has the common feeling of his profession. He enjoys a statement twice as much if it appears in fine print, and anything that turns up in a footnote ... takes on the character of divine revelation.”
—Margaret Halsey (b. 1910)