#### Definition 9.1.1.

Let \(\calA'\) be a second abelian category. A covariant functor \(F\colon \calA \to \calA'\) is left exact if every exact sequence

\begin{equation*}
0 \to M_1 \to M \to M_2
\end{equation*}

yields an exact sequence

\begin{equation*}
0 \to F(M_1) \to F(M) \to F(M_2).
\end{equation*}

Under suitable conditions (namely, that \(\calA\) has enough injectives), we can “fill in the gap” on the right: if the original sequence extends to an exact sequence

\begin{equation*}
0 \to M_1 \to M \to M_2 \to 0,
\end{equation*}

then we get a long exact sequence

\begin{equation*}
0 \to R^0 F(M_1) \to R^0F (M) \to R^0F(M_2) \to R^1 F(M_1) \to R^1 F(M) \to R^1 F(M_2) \to R^1 F(M_1) \to \cdots
\end{equation*}

where \(R^i F\) are the right derived functors of \(F\) (with \(R^0 F = F\)). These functors can be evaluated at \(M\) by forming an injective resolution of \(M\text{,}\) i.e., a complex

\begin{equation}
0 \to I^0 \to I^1 \to \dots\tag{9.1}
\end{equation}

in which each object \(I^j \in \calA\) is injective (that is, \(\Hom(N, I^j) \to \Hom(N', I^j)\) is surjective whenever \(N' \to N\) is a monomorphism) and the augmented sequence

\begin{equation*}
0 \to M \to I^0 \to I^1 \to \cdots
\end{equation*}

is exact; then \(RF^i\) is the cohomology at \(i\) of the complex

\begin{equation*}
0 \to F(I^0) \to F(I^1) \to \cdots.
\end{equation*}

However, there is some work to be done to confirm that these are well-defined functors.