Let $F$ be a left exact functor and $0 \to A \to B \to C \to 0$ a short exact sequence. Then there exists a long exact sequence:

$0 \to F(A) \to F(B) \to F(C) \xrightarrow{\delta} R^1F(A) \to R^1F(B) \to R^1F(C) \xrightarrow{\delta} R^2F(A) \to \cdots$

The connecting homomorphisms $\delta$ are natural in the short exact sequence.

For any short exact sequence $0 \to A \to E \to B \to 0$ where $E$ is $F$-acyclic (i.e., $R^iF(E) = 0$ for $i > 0$):

$R^{n+1}F(B) \cong R^nF(A)$

for all $n \geq 1$. This allows computation of higher derived functors from lower ones.

Let $F: \mathcal{A} \to \mathcal{B}$ and $G: \mathcal{B} \to \mathcal{C}$ be left exact functors. If $F$ sends injectives to $G$-acyclic objects, there is a spectral sequence:

$E_2^{p,q} = R^pG(R^qF(M)) \Rightarrow R^{p+q}(G \circ F)(M)$