Exact Sequence - Facts

Facts

The splitting lemma states that if the above short exact sequence admits a morphism t: B → A such that t f is the identity on A or a morphism u: C → B such that g u is the identity on C, then B is a twisted direct sum of A and C. (For groups, a twisted direct sum is a semidirect product; in an abelian category, every twisted direct sum is an ordinary direct sum.) In this case, we say that the short exact sequence splits.

The snake lemma shows how a commutative diagram with two exact rows gives rise to a longer exact sequence. The nine lemma is a special case.

The five lemma gives conditions under which the middle map in a commutative diagram with exact rows of length 5 is an isomorphism; the short five lemma is a special case thereof applying to short exact sequences.

The importance of short exact sequences is underlined by the fact that every exact sequence results from "weaving together" several overlapping short exact sequences. Consider for instance the exact sequence

,

which implies that there exist objects Ck in the category such that

.

Suppose in addition that the cokernel of each morphism exists, and is isomorphic to the image of the next morphism in the sequence:

(This is true for a number of interesting categories, including any abelian category such as the abelian groups; but it is not true for all categories that allow exact sequences, and in particular is not true for the category of groups, in which coker(f): G → H is not H/im(f) but, the quotient of H by the conjugate closure of im(f).) Then we obtain a commutative diagram in which all the diagonals are short exact sequences:

Note that the only portion of this diagram that depends on the cokernel condition is the object C7 and the final pair of morphisms A6 → C7 → 0. If there exists any object and morphism such that is exact, then the exactness of is ensured. Again taking the example of the category of groups, the fact that im(f) is the kernel of some homomorphism on H implies that it is a normal subgroup, which coincides with its conjugate closure; thus coker(f) is isomorphic to the image H/im(f) of the next morphism.

Conversely, given any list of overlapping short exact sequences, their middle terms form an exact sequence in the same manner.

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