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May 21, 2011 at 6:08 comment added jlk @Yuija Qiu: Sorry for the delayed response. I am slightly confused by your your final conclusion. The vector spaces $F_0$ and $S_0$ could have different dimensions (and similarly for $F_1$ and $S_1$). The fact that the complexes are quasi-isomorphic only tells us that $\operatorname{dim}(F_1) - \operatorname{dim}(F_0) = \operatorname{dim}(S_1) - \operatorname{dim}(S_0)$. When the dimensions are different, what is meant by $\operatorname{det}(F_0)$?
May 6, 2011 at 4:24 comment added jlk @Yujia Qiu: Thanks for the response! I have not thought about this question in a bit. I will try to think about your answer over the weekend.
Apr 29, 2011 at 17:19 history edited Yujia Qiu CC BY-SA 3.0
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Apr 29, 2011 at 16:33 comment added Yujia Qiu i don't know why this looks like a mess, sorry for the inconveniece :(
Apr 29, 2011 at 16:32 history edited Yujia Qiu CC BY-SA 3.0
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Apr 29, 2011 at 16:27 history answered Yujia Qiu CC BY-SA 3.0