$Tor$ over finite rings

To clarify notations, I assume 𝑚 is the unique maximal ideal of the finite local principal ideal ring 𝑅, and 𝑘 = 𝑅/𝑚 is the residue field.

To compute the first torsion group, we consider the tensor product $k \otimes_R D$, which simplifies 𝐷 by considering only its structure over 𝑘, by effectively "modding out" the maximal ideal 𝑚. That is, since 𝑘 = 𝑅/𝑚, any element t ∈ 𝑚 acts trivially on 𝑘, i.e., t ⋅ v = 0  for any v ∈ k.  So $k \otimes_R D$ is a 𝑘-vector space, representing the part of 𝐷 that survives modulo 𝑚.

To compute Tor$_1^𝑅(𝑘,𝐷)$, we look for a projective resolution 𝑃. → 𝐷 → 0 over 𝑅. If $K_1$ is the kernel of the surjection $\varepsilon: P_0 \to D$, then tensoring the exact sequence $0\to K_1 \to P_0 \to D \to 0$ with $k$, we have

0 → Tor$_1^R(k,D) \to k\otimes_R K_1 \to k\otimes_R P_0 \to k\otimes_R D \to 0$.

That is, Tor1$_1^R(k, D)$ is isomorphic to the part of $k \otimes_R K_1$ that is not in the kernel of the map $k \otimes_R K_1 \to k \otimes_R P_0$, i.e., corresponds to the portion of $\ker(\varepsilon)$ that survives in $k$-coefficients.
These are the elements of $D$ that are "annihilated" by multiplication of $m$, which is the submodule $mD$.

Thus, Tor$_1^𝑅$(𝑘,𝐷) ≅ 𝑚𝐷.

I hope this is clear, please feel free to ask if I should elaborate on something.