@tb I'm a bit lost, but I think I get you. If I show that the choice of lifting is irrelevant to applying $Hom(-,M)$ then it will magically imply that the map between $Ef$ and $Ep$ is an ismorphism.
@AsafKaragila It will not quite be irrelevant to applying $\operatorname{Hom}{({-},M)}$ (the homotopies will still be there, but it will be irrelevant after passing to $Ef$ and $Ep$).
@AsafKaragila Yes, and $f_1$ is a map onto the image of $f_1$. So you can apply freeness of $P_0$ to find a lift $h : P_0 \to F_1$ such that $f_1 h = d_0 - d_0'$.
Given $e,e' \in P_0$ such that $e - e' = p(x)$ (so they give the same class in $Ep$), we have that $d_0(e-e') = d_0p(x) = fd_1(x)$, so it makes sense to define $d^\ast[e] = [d_0 e]$, right?
@AsafKaragila we'll be right there, a few more minutes :)
@AsafKaragila Wait, we apply $\operatorname{Hom}{(-,M)}$ to $p: P_1 \to P_0$ to get a complex $p^\ast: \operatorname{Hom}(P_0, M) \to \operatorname{Hom}(P_1,M)$ and $Ep = \operatorname{Hom}{(P_1,M)}/\operatorname{im}{p^\ast}$, right?
And the maps $d_0: P_0 \to F_0$ and $d_1: P_1 \to F_1$ give us maps $d_0^\ast: \operatorname{Hom}{(F_0,M)} \to \operatorname{Hom}{(P_0,M)}$, and similarly for $d_1$ and the $'$ versions.
@robjohn Is this Gradient correct? I may be doing the substitution in the wrong spot (look gradient for $x$ has only scalars...looks worrying, I think I did substitution too early)...
@robjohn for this kind of writing with mistakes, original writing is much more revealing than the LaTex...I will do LaTex on the final document or when things are more clear... until then I need help to find the error spots because I cannot find them...
@tb Still NARA. At first I thought this one involved Hawaiian earrings but it doesn't fit. You seem to have a lot of the thing starting with f in your life. Maybe you need to be more pacifist. Have you considered trying these? I'm taking these and it works like a charm.
@robjohn basically the problem is to calculate the $\nabla$ in some point $\bar{x}_0$, I am not sure whether I should do the substitution when doing $$\partial_{x} f(\bar{x},t)_{\bar{x}=(0,0,1)},$$ or only $$\partial_{x} f(\bar{x},t)_{x=1},$$ ideas?
@AsafKaragila The lemma we just discussed is the fact that given two resolutions $P_\bullet$ and $F_\bullet$ you get a map $Ef \to Ep$ which doesn't depend on anything but $A$, the two resolutions and $M$. This gives you a unique abelian group associated to $A$ and $M$ and now you have to think about what happens if you change $M$ to $M'$, but this is pretty much the same as what we just discussed. I think you can take it from here.
@Srivatsan Thanks :-) I had actually written out complete answer, but then saw that it was a homework, so I cut the answer back to a hint. So I never saw ADF's comment.
@KannappanSampath Spamming? Really? I find it a bit harsh. (But I have not been around for a while, so I don't know if iyengar posted too many such questions in a short interval.)
BTW, Kannappan, I don't find the question that wrong. I simply interpret the question as: "Tell me some easy-to-follow book that motivates cohomology well." As for "DOWN TO EARTH" (I don't know why this ought to be in caps ;)), I am just ignoring it...
@Srivatsan Well, I don't say that his questions are wrong. He says "even the basic concept of exact sequence is used widely in advanced mathematics". Is this not a reason enough to master this tool then? This is what I would say is implicit troll or some such thing!
@KannappanSampath Too many people have tried to reason with him. The only real change in his questions is better language, and less of the "Prof. Dr. Riemann Sir"'s. :)
@KannappanSampath Yes. What I wanted to say was: If I take this question in isolation (not very justified, given the history behind the user), I feel it is an acceptable question. However, it is true that the user is not very willing to do the mathematics the way others do.
One of these days, I was solving some exercise in Artin. I came across a rather curious question. Can you look at it and tell me if it's a well-known problem?
@Jonas the issue is about what the author means by a sentence.....the OP is not getting into such technical issue at that detail...the book he is referring to is written for non mathematicians
Have a look at page 54 on the book he linked, if possible