Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I´m reading the Zig Zag lemma in Cohomology and i want to prove the exactness of cohomology sequence at $ H^k(A)$ and $H^k(B)$ :

A short exact sequence of cochain complexes $ 0 \to A \ \xrightarrow{i} \ B \ \xrightarrow{j} \ C \to 0$ gives rise to a long exact sequence in cohomology:

$ ... \ \xrightarrow{j^*} \ H^{k-1}(C) \ \xrightarrow{d^*} \ H^k(A) \ \xrightarrow{i^*} H^k(B) \ \xrightarrow{j^*} H^k(C) \ \xrightarrow{d^*} H^{k+1}(A) \ \xrightarrow{i^*} ...$

where $i^∗$ and $j^∗$ are the maps in cohomology induced from the cochain maps i and j,and $d^∗$ is the connecting homomorphism.

I think first i need to prove that $im(d^∗) = ker(i^∗)$ for exactness in $H^k(A)$ . Help please…..

I prove the exactness of $H^k(C)$:

First I prove that $im( j^*)\subseteq ker (d^*)$. Let $[b]\in H^k(B) $ then $d^* j^* [b] = d^*[j(b)]$. In the recipe above for $d^*$ , we can choose the element in $B^k$ that maps to $j(b)$ to be b. Then $db \in B^{k+1}$. Because b is a cocycle, $db=0$. Following the Zig-Zag diagram we see that since $i(0) = 0 = db$, we must have $d^*[j(b)] = [0]$, so $j^*[b]\in ker(d^*)$.

The other way, i.e., $ker(d^*) \subseteq im(j^*)$: suppose $d^*[c] = [a]=0$, where $[c] \in H^k(C) $, this means that $a=da´$ for some $ a´ \in A^k$.i calculate the $d^*$ again by the diagram and take an element $ b \in B^k$with $j(b) = c$ and $i(a) = db$. Then $b - i(a´)$ is a cocycle in $B^k$ that maps to c under j:

$d(b - i(a´)) = db-di(a´) = db - id(a´) = db - ia = 0$, $j(b - i(a´)) = db-ji(a´) = j(b) = c$ Therefore, $ j^*[b - i(a´)]= [c]$.

share|improve this question
    
The interested reader can find this proof of exactness at $H^k(C)$ in An introduction to manifolds by Loring W. Tu. –  Vladhagen Nov 11 at 23:55

1 Answer 1

up vote 1 down vote accepted

First of all the proofs aren't very difficult and go through without any complications as well as your proof the exactness at $H^k(C)$ given. For a proof of this I would recommend any standard-literature such as Neukirch, p. 24 (in German) or NSW p.26 using the Snake lemma.

share|improve this answer
    
Yes thanks I could prove what I needed just taking elements in the sequence , thanks :) –  Knight Oct 17 '13 at 21:09

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.