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I suppose it's a common exam question to show that a certain sequence actually is a chain complex. What is it that has be shown, minimally? A chain complex is a sequence of modules and module maps, and any two maps succeeding each other must compose to the zero map. So, do I have to show that the sets in the sequence are modules, and that the maps are module maps, and the final condition on composition?

Erik Vesterlund
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  • I guess it all depends on the flavor of exam they give you. Many times, the interesting part of showing that something is a chain complex is showing that the composition gives 0. You may want to ask your professor in the exam what he's looking for you to prove. – rfauffar Sep 06 '13 at 17:02
  • @RobertAuffarth Well I have a homework/exam assignment with specific modules and maps, but I don't want to post it here in case it counts as cheating :) I guess I'll give all the information in my answer just to be safe. – Erik Vesterlund Sep 06 '13 at 17:20
  • Sounds good! In the worst case scenario your teacher will tell you that you proved everything in too much detail! That's always better :) – rfauffar Sep 06 '13 at 18:44

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A $\mathbb Z$-graded chain complex $(C_{\bullet},d_{\bullet})$ of $R$-modules is a collection

$$C_{\bullet}=\{C_i\}_{i\in\mathbb Z} $$

of $R$-modules and $R$-module morphisms

$$d_i: C_i\rightarrow C_{i-1}$$

s.t. $d_i\circ d_{i-1}=0_{C_{i+1}}$, for all $i\in\mathbb Z$.

You can similalry introduce chain complexes of vector spaces etc... Dually (in some sense) one can define the cochain complexes: one inverts arrows considering "+1-degree" morphisms. A small $caveat$: in some references you can find as chain complex the direct sum

$$C_{\bullet}=\bigoplus_i C_i $$

with the aforementioned morphisms $d_i$; this is a bit "dangerous" definition due to completion issues that can arise in applications (I mean "direct sum vs. direct product"). Using the word "collection" is sufficient to bypass most of these technicalities. $$d_i: C_i\rightarrow C_{i+1}$$

Avitus
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    This is a nice post, but it seems to be an answer to the question "what is a chain complex", which was not my question :) – Erik Vesterlund Sep 07 '13 at 19:55
  • well, to show that something is equal to "A", you need to know what "A" is, right? :) In any case, what you need to "show" are just the pieces of information in my answer: the collection of modules, the morphisms between them and the $d^2=0$ condition. – Avitus Sep 07 '13 at 20:12
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    Of course, but it was clear from my post that I know what a CC. Your post did not answer the actual question. – Erik Vesterlund Sep 08 '13 at 18:26
  • I disagree: you cannot know what a CC of modules is if you ask, in particular: "do I have to show that the sets in the sequence are modules, and that the maps are module maps?" This shows a not clear vision of the setting imho. This is my last word on the topic. – Avitus Sep 09 '13 at 13:14
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    What are you talking about? The question was whether of the three definitions it is sufficient to show to the last one. @RobertAuffarth indicated that you only need to show the last one, and you, well you decided to answer something other than the question.. – Erik Vesterlund Sep 10 '13 at 16:19