Five Lemma - Statements

Statements

Consider the following commutative diagram in any abelian category (such as the category of abelian groups or the category of vector spaces over a given field) or in the category of groups.

The five lemma states that, if the rows are exact, m and p are isomorphisms, l is an epimorphism, and q is a monomorphism, then n is also an isomorphism.

The two four-lemmas state:
(1) If the rows in the commutative diagram

are exact and m and p are epimorphisms and q is a monomorphism, then n is an epimorphism.

(2) If the rows in the commutative diagram

are exact and m and p are monomorphisms and l is an epimorphism, then n is a monomorphism.

Read more about this topic:  Five Lemma

Famous quotes containing the word statements:

    The statements of science are hearsay, reports from a world outside the world we know. What the poet tells us has long been known to us all, and forgotten. His knowledge is of our world, the world we are both doomed and privileged to live in, and it is a knowledge of ourselves, of the human condition, the human predicament.
    John Hall Wheelock (1886–1978)

    The true critic is a scrupulous avoider of formulae; he refrains from statements which pretend to be literally true; he finds fact nowhere and approximation always.
    —T.S. (Thomas Stearns)

    Science is a system of statements based on direct experience, and controlled by experimental verification. Verification in science is not, however, of single statements but of the entire system or a sub-system of such statements.
    Rudolf Carnap (1891–1970)