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

Section1.6Derived Functors, Introduction

We fix \(\cA\), and \(\Ch(\cA)\). If we have some right exact functor \(F\) we obtain exact sequences of the form \[FA \to FB \to FC \to 0\] and the question arises, can we extend this exact sequence by placing objects to the left of it?
If \(F\) is exact on short exact sequences of complexes we get a long exact sequence of homology \(H_iFA\). \(F\) is exact on complexes if it is level wise exact, but \(F\) is exact if it is level wise exact. We know \(F\) is exact on split exact sequences. So we can try to force a short exact sequence to be exact by replacing objects by complexes.

Definition1.6.1Projective and injective objects
An object \(M\) is projective if for all epimorphisms \(q\) and maps \(M \xrightarrow{f} B\) there exists a lift making \[ \xymatrix{ & M \ar[d]_f \ar@{-->}[dl]&\\ A \ar[r]^q &B \ar[r] & 0} \] commute. The dual notion is called injective\[ \xymatrix{ & I&\\ 0 \ar[r] &B \ar[u] \ar[r] & A \ar@{-->}[ul]} \]
Example1.6.2
Free modules in \(\Rmod\) are projective. In \(\Mat_n(R)\text{-mod}\) the column vectors \(R^n\) form a projective object. \(\QQ\) is injective in \(\Ab\).

Note that in \(\Rmod\) this shows projectives are exactly summands of free modules.

Definition1.6.4Projective resolutions
A projective resolution\(P_\bullet \xrightarrow{\epsilon} A\) of \(A\) is a non-negative chain complex such that all \(P_i\) are projective and \(\epsilon\) is a quasi-isomorphism. So \(H_i P = 0\) if \(i \gt 0\) and \(H_0 P = A\).
Definition1.6.5Derived functors
The \(i\)th left derived functor\(L_i F(A)\) of a right exact functor \(F\) is defined as \(H_iF(P)\) for some projective resolution \(P\) of \(A\).

Dually we may define injective resolutions \(B \xrightarrow{\sim} I^\bullet\) with \(I \in \Ch^{\ge 0}(\cA)\) and we get right derived functors of a left exact functor, \[ R^i F(B) = H^i (FI ). \] Note \(L_{\lt 0} F(A) = 0\) and \(L_0= fP_0 / FP_1 = F(P_0/P_1) = F(A)\).

Example1.6.6\(\Tor\)
Define \(\Tor^R_i(A, B)\) to be \(L_i(- \otimes_R B)(A)\). Let \(\cA = \Ab\). What is \(\Tor_i(\ZZ/p , B)\)? \[ \xymatrix{\ZZ\ar[d]^p &\\ \ZZ \ar[r]^{\sim} &\ZZ_p} \] is a projective resolution. So \(\Tor_* = H_*(B \xrightarrow{p} B)\) and we have \(\Tor^{\ZZ}_0 (\ZZ/p, B) = B/pB\) and \(\Tor^{\ZZ}_1(\ZZ/p, B) = {}_pB = \{b : pb = 0\}\).
Example1.6.7\(\Ext\)
Define \(\Ext^i_R(A, B)\) to be \(R^i\Hom_R(-, B)(A)\). Injective in \(\Rmod^{\op}\) correspond to projectives in \(\Rmod\). So \(\Ext_{\ZZ}^*i(\ZZ/p\ZZ, B) = H_*(B \xrightarrow{p} B)\) hence \(\Ext^0 (\ZZ/p, B) = {}_pB\) and \(\Ext^1(\ZZ/p, B) = B/pB\).