Universal Coefficient Theorem - Statement

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:

    The most distinct and beautiful statement of any truth must take at last the mathematical form.
    Henry David Thoreau (1817–1862)

    Children should know there are limits to family finances or they will confuse “we can’t afford that” with “they don’t want me to have it.” The first statement is a realistic and objective assessment of a situation, while the other carries an emotional message.
    Jean Ross Peterson (20th century)

    Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.
    Mario Vargas Llosa (b. 1936)