Short exact sequence

We split this sequence into the following short exact sequences. 

By Proposition 1.1 we only have to show that a short exact sequence of C(n)s0... 

If Cv is a point, or more generally Hodd Cv) = 0, the cohomology long exact sequence (5.2) splits into short exact sequences and we have R(t) = 0, i.e. the Morse function is perfect. 

As in (2.1.4), short exact sequences in either the A or F variable give rise to long exact h3rpertor sequences. 

Note that i,) OY. = OxJT. From the short exact sequence... 

Some Structure Theory of Long and Short Exact Sequences... 

To start with, even the prehomology picture is not at all too clear. We certainly do not in general have B = A(B B/A when A and B are 7 .-modules, not even when TZ = Z, i.e., when A and B are abelian groups. As an example, take B = Z, and take A = 2Z. We have B/A = Z2, but Z Z Z2, since Z does not have nonzero elements of finite order, while Z Z2 does. The exact relationship between B, A, and B/A is best phrased in the language of short exact sequences, which we now proceed to introduce. 

Definition 5.5. An exact sequence (of TZ-modules or of chain complexes) is called a short exact sequence if all the terms are 0 except possibly three... 

Exact sequences form a subcategory of the category of all chain complexes and chain complex homomorphisms. The same is true for short exact sequences, or in fact for any set of exact sequences with a fixed set of indices where the terms are allowed to be nontrivial. 

This means that A is isomorphic to (j4), while C is isomorphic to B/Ker p) = B/i A). In other words, all short exact sequences can essentially be written in the form (5.3). To underline this fact, one says that the term 1 is an extension of A by C. Being able to write the middle term as the direct smn of the two others is an additional property that one sometimes has. It is formally defined as follows. 

As we have already noted, passing to homology is bound to make things more complex, since, for example, injective maps may cease to be injective. It turns out that instead of one short exact sequence, the relationship between Hn A), Hn B), and Hn B/A) is best described as a long exact sequence, which involves the groups H A), H, B), and H B/A) in all dimensions. 

We also note that by Proposition 5.3, any map between short exact sequences... 

Remark 5.12. In full analogy with Definition 5.9 one can define the relative cohomology of a pair (X,A) as the cohomology of the cochain complex Cging(X, A), where C sing( j ) consists of all functions on n-cells that have value 0 on the singular simplices inside A. Theorem 5.7 can be applied to the short exact sequence... 

However, if TZ is an arbitrary ring (for example TZ = 7,), then one may need to solve a number of extension problems before obtaining the final answer. This has to do with the fact that, as mentioned above, in a short exact sequence of 72.-modules... 

SO that there is defined a short exact sequence of... 

A [x),x) in the range n 0 is made up of naturally split short exact sequences... 

The short exact sequence of finite-dimensional E (x)-"lodule chain complexes... 

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