Can you help in proving the isomorphism going between the torsion of $H_p$ and $\left(\bigoplus_{i} R / d_{p i} R\right)$?
Where $H_p$ is the p-th homolgy module, R is a commutative PID, and $d_{i}$'s are the diagonal elements in the Smith normal form of the boundary matrix.
Any hints would be highly appreciated!
Let $Z_p$ be the kernel of $\partial_p: C_p \to C_{p-1}$; then $Z_p$ will be a free module, since submodules of free modules are free over a PID. Then we have a module map $\partial_{p+1}: C_{p+1} \to Z_p$. Choose bases for both of these free modules and then find the matrix representation $m$ for $\partial_{p+1}$; it will look like $m: R^{n} \to R^k$ where $m$ is a $k \times n$ matrix. By construction, the Smith normal form $m'$ for $m$ is a composite $$ R^n \xrightarrow{\alpha} R^n \xrightarrow{m} R^k \xrightarrow{\beta} R^k $$ where $\alpha$ and $\beta$ are invertible matrices, i.e. isomorphisms of $R$-modules. I hope it is clear that the cokernel of the Smith normal form matrix $m'$ has the form $\bigoplus_i R/d_i R$, where the $d_i$ are the diagonal entries of $m'$. Since $\alpha$ and $\beta$ are isomorphisms, they induce an isomorphism between $\text{coker } m$ and $\text{coker } m'$. In particular, the nonzero diagonal entries in $m'$ give the "torsion" part of the cokernel.