\( \usepackage{mathrsfs} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\cHom}{\underline{Hom}} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\inj}{inj} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\coim}{coim} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\eq}{eq} \DeclareMathOperator{\op}{op} \DeclareMathOperator{\ob}{ob} \DeclareMathOperator{\coeq}{coeq} \DeclareMathOperator{\cone}{cone} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\Ch}{Ch} \DeclareMathOperator{\Ab}{\text{Ab}} \DeclareMathOperator{\Top}{\text{Top}} \DeclareMathOperator{\Rmod}{R\text{mod}} \DeclareMathOperator{\SSet}{\text{SSet}} \DeclareMathOperator{\sAb}{\text{sAb}} \DeclareMathOperator{\cA}{\mathcal{A}} \DeclareMathOperator{\cB}{\mathcal{B}} \DeclareMathOperator{\cC}{\mathcal{C}} \DeclareMathOperator{\cO}{\mathcal{O}} \DeclareMathOperator{\cM}{\mathcal{M}} \DeclareMathOperator{\sC}{\mathscr{C}} \DeclareMathOperator{\sF}{\mathscr{F}} \DeclareMathOperator{\sW}{\mathscr{W}} \DeclareMathOperator{\dd}{\partial} \DeclareMathOperator{\rd}{\mathrm{d}} % Rings \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\CC}{\mathbf{C}} \DeclareMathOperator{\PP}{\mathbf{P}} \DeclareMathOperator{\QQ}{\mathbf{Q}} \DeclareMathOperator{\RR}{\mathbf{R}} \DeclareMathOperator{\ZZ}{\mathbf{Z}} \)

Section1.5Exact Functors

Definition1.5.1Exact functors
An additive functor \(F\) is exact if it preserves short exact sequences. It is left exact if it sends a short exact sequence of the form \[ 0\to A \to B \to C \to 0 \] to an exact sequence \[ 0 \to FA \to FB \to FC. \] We have a similar definition for right exact.
Example1.5.2
The functor \(\Hom_{\cA}(M, -)\) is left exact from \(\cA\) to \(\Ab = \ZZ\text{mod}\). The functor \(\Hom_{\cA}( -, M)\colon \cA^{\op} \to\Ab\) is left exact.

Note that left adjoint functors are right exact as they preserve colimits.

Example1.5.3
Let \(M\) be an \(R,S\)-bimodule (i.e. a left \(R\)-module and a right \(S\)-module). Then for \(A \in S\text{mod}\), \(B\in \Rmod\)\[ \Hom_{R}(M \otimes_S A, B) \cong Hom_{S}(M, \Hom_R(A,B)) \]

Clearly not all functors are exact. However they all preserve split exact sequences, i.e. those of the form \[0 \to A \to A\oplus C \to C.\] Because they preserve finite direct sums \[ \xymatrix{ A \ar@<-0.5ex>[r]_f \ar@{=}[d]& B\ar@<-0.5ex>[l]_r \ar@<-0.5ex>[d]_{(f,g)} \ar@<-0.5ex>[r]_g & C\ar@<-0.5ex>[l]_s \ar@{=}[d]\\ A \ar[r]& A \oplus C \ar@<-0.5ex>[u]_{(r,s)}\ar[r] &C } \] \((r,g),(f,s)\) are inverse isomorphisms if and only if \(_B = fr + sg\).