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

Section1.8The Derived Category

Idea: We want to talk about complexes up to quasi-isomorphism. We will reinterpret derived functors as ways of lifting functors to derived categories.

Remark1.8.1
If we add simply inverses of quasi-isomorphisms we get nasty stuff!
Definition1.8.2Homotopy categories
Let the homotopy category \(K(\cA)\) of \(\cA\) have objects the objects of \(\Ch(\cA)\) and morphism the chain homotopy classes of chain maps. We can add boundedness conditions to our categories. So we let \(\Ch_{+}(\cA)\) be only those chain complexes \(A\) with \(A_n = 0\) when \(n \lt \lt 0\), these are bounded below chain complexes. Similarly we define \(\Ch_-(\cA)\) and \(\Ch_b(\cA) = \Ch_-(\cA) \cap \Ch_+(\cA)\). We also define \(\Ch^+(\cA)\) etc. for cochain complexes. Finally we define \(K^+(\cA)\) etc. in the obvious way.
Definition1.8.3Localisations of categories
Given a category \(\cC\) and a class of morphisms \(S\) define the localisation of \(\cC\) at \(S\) to be a category \(\cC[S^{-1}]\) with a functor \(\cC \xrightarrow{Q} \cC[S^{-1}]\) such that \(Q\) sends any \(s\in S\) to an isomorphism, and also such that \(Q\) is universal with respect to having this property. If \(\cC \xrightarrow{R} \cB\) sends \(S\) to isomorphisms then there exists some \(P\) so that we have \[ \xymatrix{ \cC \ar[rr]^R\ar[dr]_Q &&\cB\\ &\cC[S^{-1}] \ar[ur]_{P}& } \]
Definition1.8.4Derived categories
Let \(D(\cA)\), the derived category of \(\cA\), be the localisation of \(K(\cA)\) at the quasi-isomorphisms. Similarly define the usual suspects \(D^b = K^b(\cA)[\text{quasi-isomorphisms}]\), etc.

Although we didn't prove this we should note that we can write morphisms in \(D(\cA)\) as \[ A \xleftarrow{\sim} A' \xrightarrow{f} B \] with \(f\in \Hom_{K(\cA)}(A', B)\) and \(q\in \Hom_{K(\cA)}(A', A)\).

Remark1.8.6
\(D^b(\cA)\) is equivalent to the subcategory of \(D(\cA)\) with cohomology in bounded degrees.
Example1.8.7
Let \(X\) be a scheme, \(\Coh(X)\) the abelian category of coherent sheaves on \(X\). Then the derived category of \(X\) is defined to be \(D^b(X) = D^b(\Coh(X))\).

Note that \(D(\cA)\) is an additive, but not necessarily abelian category.