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We have: $$M \cong Z/2 \oplus( Z/4 \oplus Z/9) \oplus (Z/8 \oplus Z/27) \cong Z/2 \oplus Z/36 \oplus Z/216$$


Very simple: take $A$ a noetherian domain. $\operatorname{Ass}A=\{0\}$ and $\operatorname{Supp}A=\operatorname{Spec}A$. They're different unless $A$ is a field. Note: If $A$ is reduced, $\operatorname{Ass}A=\operatorname{Min}A$, which is different from $\operatorname{Spec}A$ if the Krull dimension of $A$ is positive.


Hint : try to identify $C_0/\mathrm{Im} f = C_0/\langle v_1-v_0, v_2-v_1, v_0-v_2\rangle$... Edit : quotienting by $\mathrm{Im} f$ means making $v_1-v_0 = v_2-v_1= v_0-v_2=0$, i.e. $v_0=v_1=v_2$.


They are distinguished by their $\pi_1X$ action. Your definition of coefficients in a module, is the so-called untwisted homology by: $$ C_n(X;M) := M \otimes_R C_n(X;R) = M \otimes_R R^{|n|} = M^{|n|}, $$ where $|n|$ denotes the number of $n$-cells of $X$ with no $\pi_1$ action on it. (3) in contrast is the twisted homology. As suggested by the name, (2) ...

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