Ext can be computed using either a projective resolution of the first variable or an injective resolution of the second: $\text{Ext}^n_R(M, N) \cong H^n(\text{Hom}_R(P_\bullet, N)) \cong H^n(\text{Hom}_R(M, I^\bullet))$

where $P_\bullet \to M$ is projective and $0 \to N \to I^\bullet$ is injective.

Tor is symmetric in its arguments: $\text{Tor}_n^R(M, N) \cong \text{Tor}_n^R(N, M)$

This follows from the symmetry of tensor product and the fact that we can resolve either variable.

Given a short exact sequence $0 \to A \to B \to C \to 0$, there is a long exact sequence: $0 \to \text{Hom}(C, N) \to \text{Hom}(B, N) \to \text{Hom}(A, N) \to \text{Ext}^1(C, N) \to \text{Ext}^1(B, N) \to \cdots$

Given a short exact sequence $0 \to A \to B \to C \to 0$, there is a long exact sequence: $0 \to \text{Hom}(M, A) \to \text{Hom}(M, B) \to \text{Hom}(M, C) \to \text{Ext}^1(M, A) \to \text{Ext}^1(M, B) \to \cdots$

Given a short exact sequence $0 \to A \to B \to C \to 0$ where $A, B, C$ are flat or the sequence is split, there is a long exact sequence: $\cdots \to \text{Tor}_1(M, B) \to \text{Tor}_1(M, C) \to M \otimes A \to M \otimes B \to M \otimes C \to 0$