Section1.7Derived Functors, Proofs
Example1.7.2
\(\Rmod\) has enough projectives.Warning: The category of abelian sheaves on a topological space does not have enough projectives in general.
Lemma1.7.3
Projective resolutions exist in \(\cA\) if \(\cA\) has enough projectives.Proof
Theorem1.7.4Comparison Theorem
Let \(\epsilon \colon P \to M\) and \(\eta \colon Q \to N\) be two projective resolutions and let \(f\colon M \to N\) then there exists a lift \(\tilde{f} \colon P \to Q\) (a chain map) unique up to chain homotopy. \[ \xymatrix{ P_1 \ar[d] & Q_1\ar[d]\\ P_0 \ar[d] & Q_0\ar[d]\\ M \ar[r]^{f} & N\\ } \]Proof
Corollary1.7.5
Projective resolutions are well defined up to chain homotopy equivalence and so derived functors are well defined.Proof
Corollary1.7.6
\(L_iF\) are functors.Lemma1.7.7Horseshoe Lemma
Given a short exact sequence \[ A^1 \to A^2 \to A^3 \] and projective resolutions \(P^1 \to A^1\) and \(P^3 \to A^3\) there exists a projective resolution \(P^2\) of \(A^2\) with \(P^2_i = P^1_i \oplus P^3_i\) and the inclusion and projection maps lift. So we have the following situation \[ \xymatrix{ &P_1^1 \ar[d] & P_1^2 \ar[d]&P_1^3\ar[d]&\\ &P_0^1 \ar[d]_{\epsilon^1} & P_0^2 \ar[d]_{\epsilon^2}&P_0^3\ar[d]_{\epsilon^3}&\\ 0 \ar[r]& A^1 \ar[r]_i & A^2\ar[r]_p &A^3 \ar[r]&0 } \]Proof
Corollary1.7.8
A short exact sequence \(0 \to A \to B \to C \to 0\) in \(\cA\) gives a long exact sequence of left derived functors \[ \to L_2FC \to L_1FA \to L_1 FB \to L_1FC\to FA \to FB \to FC \to 0. \]Proof
Proposition1.7.9
The boundary map \(\dd\) is natural, i.e. given \[ \xymatrix{ A^1 \ar[r]\ar[d]^{f^1} & A^2 \ar[r]\ar[d]^{f^2} & A^2 \ar[d]^{f^3} \\ B^1 \ar[r] & B^2 \ar[r] & B^3 \\ } \] we have lifts \(\dd\circ L_i f_3 = L_{i-1} f_1 \circ \dd\).Proof
Note that there is no extra work needed to do all of this for right derived functors.