I met so many problems when I study homological algebra by myself. Thus, I really would like to see the answers. Hopefully, everyone can help me (my big thanks).
1) When we create torsion functor from a resolution of M illustrated by:
$...\overset{d_{3}}{\rightarrow} X_{2} \overset{d_{2}}{\rightarrow} X_{1} \overset{d_{1}}{\rightarrow} X_{0} \overset{\epsilon}{\rightarrow}M \rightarrow 0$
We eliminate M in above resolution then we have:
$...\overset{d*_{3}}{\rightarrow} X_{2} \overset{d*_{2}}{\rightarrow} X_{1} \overset{d*_{1}}{\rightarrow} X_{0} \rightarrow 0$
Now the above sequence is semi-exact. After taking homology from the product tensor of it and module B, we have torsion functor. I would like to ask that $d_{i}$ and $d*_{i}$ is the same with $i\ge 1$ or not? If it is the same (I mean that when we take the homology), $Tor_{n}(M,B)$ will be equal zero with $n\ge 1$ because $Imd*_{i+1}=Imd_{i+1}=Kerd_{i}=Kerd*_{i}$ then $Im(d*_{i+1}\otimes 1) = Ker(d*_{i}\otimes 1)$
2) We have cohomology by effecting functor $Hom(-,B)$ to semi-exact sequence and then take homology. I would like to ask that it will be also called cohomology when we use functor $Hom(B,-)$ or not?
