# Long Exact

### Some articles on long exact, long, exact:

Five Lemma - Applications
... The five lemma is often applied to long exact sequences when computing homology or cohomology of a given object, one typically employs a simpler subobject whose homology/cohomology is known, and arrives at a long ... then a morphism between the respective long exact sequences is induced, and the five lemma can then be used to determine the unknown homology groups ...
Derived Functor - Construction and First Properties
... The right derived functors of the covariant left-exact functor F A → B are then defined as follows ... Because there are enough injectives, we can construct a long exact sequence of the form where the I i are all injective (this is known as an injective resolution of X) ... first term, we obtain the chain complex Note this is in general not an exact sequence anymore ...
Homotopy Groups Of Spheres - General Theory - Hopf Fibrations
... theory of fiber bundles F→E→B shows that there is a long exact sequence of homotopy groups For this specific bundle, each group homomorphism πi(S1)→πi(S3), induced ... Thus the long exact sequence breaks into short exact sequences, Since Sn+1 is a suspension of Sn, these sequences are split by the suspension ... Thus the long exact sequences again break into families of split short exact sequences, implying two families of relations The three fibrations have base space Sn ...

### Famous quotes containing the words exact and/or long:

Better risk loss of truth than chance of error—that is your faith-vetoer’s exact position. He is actively playing his stake as much as the believer is; he is backing the field against the religious hypothesis, just as the believer is backing the religious hypothesis against the field.
William James (1842–1910)

Let me tell you I am better acquainted with you for a long absence, as men are with themselves for a long affliction: absence does but hold off a friend, to make one see him the truer.
Alexander Pope (1688–1744)