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:
“One is apt to be discouraged by the frequency with which Mr. Hardy has persuaded himself that a macabre subject is a poem in itself; that, if there be enough of death and the tomb in one’s theme, it needs no translation into art, the bold statement of it being sufficient.”
—Rebecca West (1892–1983)
“I think, therefore I am is the statement of an intellectual who underrates toothaches.”
—Milan Kundera (b. 1929)
“A sentence is made up of words, a statement is made in words.... Statements are made, words or sentences are used.”
—J.L. (John Langshaw)