$\mathrm{Tor}_0^R(A, B) = A \otimes_R B$. This follows from $L_0(A \otimes_R -) = A \otimes_R -$ since the tensor product is right exact.

$\mathrm{Tor}_1^{\mathbb{Z}}(\mathbb{Z}/m, \mathbb{Z}/n) \cong \mathbb{Z}/\gcd(m, n)$. Using the resolution $0 \to \mathbb{Z} \xrightarrow{m} \mathbb{Z} \to \mathbb{Z}/m \to 0$ and tensoring with $\mathbb{Z}/n$: the complex $\mathbb{Z}/n \xrightarrow{m} \mathbb{Z}/n$ has kernel $\mathbb{Z}/\gcd(m, n)$. The name "Tor" comes from this detection of torsion.

Over $\mathbb{Z}$: $\mathrm{Tor}_n^{\mathbb{Z}}(A, B) = 0$ for $n \geq 2$ (since $\mathbb{Z}$ has global dimension 1). $\mathrm{Tor}_1^{\mathbb{Z}}(A, B)$ is the torsion product, detecting common torsion between $A$ and $B$. If $A$ or $B$ is torsion-free, $\mathrm{Tor}_1 = 0$.

Over a field $k$: $\mathrm{Tor}_n^k(V, W) = 0$ for all $n \geq 1$, since every module is free (hence flat). $\mathrm{Tor}_0 = V \otimes_k W$ is the only non-vanishing term.

An $R$-module $M$ is flat iff $\mathrm{Tor}_1^R(M, -) = 0$ iff $\mathrm{Tor}_n^R(M, -) = 0$ for all $n \geq 1$. Flatness means that $M \otimes_R -$ is exact. Free modules and localizations are flat. Over a PID, flat = torsion-free.

Over $R = k[x]/(x^2)$: $\mathrm{Tor}_n^R(k, k) \cong k$ for all $n \geq 0$. The periodic projective resolution $\cdots \xrightarrow{x} R \xrightarrow{x} R \to k \to 0$ gives $k \otimes_R R \cong k$ with differential $x \cdot - = 0$, producing $k$ in every degree.

For $R = k[x_1, \ldots, x_n]$ and the maximal ideal $\mathfrak{m} = (x_1, \ldots, x_n)$: $\mathrm{Tor}_i^R(k, k) \cong k^{\binom{n}{i}}$. The Koszul complex provides the resolution, and $\mathrm{Tor}_i$ is the $i$-th exterior power $\bigwedge^i(k^n)$.

Serre's intersection multiplicity formula: for subvarieties $V, W$ of a smooth variety $X$ meeting properly at a point $p$, the intersection multiplicity is

$i(V, W; p) = \sum_{n \geq 0} (-1)^n \mathrm{length}\, \mathrm{Tor}_n^{\mathcal{O}_{X,p}}(\mathcal{O}_{V,p}, \mathcal{O}_{W,p})$

The higher Tor terms are correction terms to the naive intersection.

For a ring map $R \to S$ and an $R$-module $M$: $\mathrm{Tor}_n^R(S, M)$ measures the failure of $M \otimes_R S$ to be exact in the $S$ variable. If $R \to S$ is flat (e.g., localization), then $\mathrm{Tor}_n^R(S, -) = 0$ for $n \geq 1$, so base change is exact.

For a chain complex $C_\bullet$ of free abelian groups, the universal coefficient theorem gives a short exact sequence:

$0 \to H_n(C) \otimes G \to H_n(C \otimes G) \to \mathrm{Tor}_1^{\mathbb{Z}}(H_{n-1}(C), G) \to 0$

The Tor term is the obstruction to the Kunneth isomorphism.

If $x_1, \ldots, x_r$ is a regular sequence in $R$ and $M = R/(x_1, \ldots, x_r)$, then $\mathrm{Tor}_i^R(M, N)$ for any $R$-module $N$ can be computed using the Koszul complex on $x_1, \ldots, x_r$. In particular, $\mathrm{Tor}_i^R(M, N) = 0$ for $i > r$.

$\mathrm{Tor}_n^R(A, B) \cong \mathrm{Tor}_n^R(B, A)$ for commutative $R$. This symmetry (or "balance") follows from the fact that $\mathrm{Tor}$ can be computed using a projective resolution of either argument. For noncommutative $R$, one must be more careful with left vs right modules.

A short exact sequence $0 \to A' \to A \to A'' \to 0$ of $R$-modules induces:

$\cdots \to \mathrm{Tor}_2(M, A'') \to \mathrm{Tor}_1(M, A') \to \mathrm{Tor}_1(M, A) \to \mathrm{Tor}_1(M, A'') \to M \otimes A' \to M \otimes A \to M \otimes A'' \to 0$

This long exact sequence is the primary computational tool for Tor.