Show that \(\operatorname{Ext}(H, G)\) is a contravariant functor of \(H\) for fixed \(G\), and a covariant functor of \(G\) for fixed \(H\).
Solution
1. Definition of \(\operatorname{Ext}(H, G)\):
1.1. For an abelian group \(H\), choose a free resolution \(0 \to F_1 \xrightarrow{d} F_0 \xrightarrow{\varepsilon} H \to 0\).
every abelian group is a quotient of a free abelian group, and subgroups of free abelian groups are free.
1.2. Applying \(\operatorname{Hom}(-, G)\) yields the cochain complex:\[ 0 \to \operatorname{Hom}(F_0, G) \xrightarrow{d^*} \operatorname{Hom}(F_1, G) \to 0, \]where \(d^*(\varphi) = \varphi \circ d\).
contravariance of the \(\operatorname{Hom}(-, G)\) functor.
1.3. By definition, \(\operatorname{Ext}(H, G) = \operatorname{coker}(d^*) = \operatorname{Hom}(F_1, G) / \operatorname{im}(d^*)\).
definition of Ext for abelian groups.
2. Contravariance of \(\operatorname{Ext}(-, G)\) in the first variable \(H\):
2.1. Let \(\alpha: H \to H'\) be a group homomorphism, and let \(0 \to F_1' \xrightarrow{d'} F_0' \xrightarrow{\varepsilon'} H' \to 0\) be a free resolution of \(H'\).
setup.
2.2. By projectivity of \(F_0\) and \(F_1\), there exists a chain map \((\alpha_0, \alpha_1)\) lifting \(\alpha\):\[ \varepsilon' \circ \alpha_0 = \alpha \circ \varepsilon \quad \text{and} \quad d' \circ \alpha_1 = \alpha_0 \circ d. \]
comparison theorem for projective resolutions.
2.3. Applying \(\operatorname{Hom}(-, G)\) induces dual maps \(\alpha_0^*: \operatorname{Hom}(F_0', G) \to \operatorname{Hom}(F_0, G)\) and \(\alpha_1^*: \operatorname{Hom}(F_1', G) \to \operatorname{Hom}(F_1, G)\) satisfying \(d^* \circ \alpha_0^* = \alpha_1^* \circ (d')^*\).
functoriality of \(\operatorname{Hom}(-, G)\).
2.4. The commutativity implies \(\alpha_1^*(\operatorname{im}((d')^*)) \subseteq \operatorname{im}(d^*)\), so \(\alpha_1^*\) descends to a well-defined homomorphism on quotients:\[ \alpha^*: \operatorname{Ext}(H', G) \longrightarrow \operatorname{Ext}(H, G), \quad [\psi] \mapsto [\psi \circ \alpha_1]. \]
quotient map between cokernels.
2.5. Any two lifts of \(\alpha\) are chain homotopic, so the induced map \(\alpha^*\) is independent of the choice of lift \((\alpha_0, \alpha_1)\).
chain homotopy induces the zero map on homology/cohomology.
2.6. It is direct that \((\operatorname{id}_H)^* = \operatorname{id}_{\operatorname{Ext}(H, G)}\) and \((\beta \circ \alpha)^* = \alpha^* \circ \beta^*\) for \(\beta: H' \to H''\).
composition of chain maps.
2.7. Thus \(\operatorname{Ext}(-, G)\) is a contravariant functor from the category of abelian groups to itself.
4, 5, and 6.
3. Covariance of \(\operatorname{Ext}(H, -)\) in the second variable \(G\):
3.1. Let \(\beta: G \to G'\) be a group homomorphism, and fix a free resolution \(0 \to F_1 \xrightarrow{d} F_0 \to H \to 0\).
setup.
3.2. Post-composition with \(\beta\) defines homomorphisms \(\beta_*: \operatorname{Hom}(F_k, G) \to \operatorname{Hom}(F_k, G')\) by \(\beta_*(\varphi) = \beta \circ \varphi\) for \(k = 0, 1\).
covariance of \(\operatorname{Hom}(F_k, -)\).
3.3. For any \(\varphi \in \operatorname{Hom}(F_0, G)\), \(\beta_*(d^*(\varphi)) = \beta \circ (\varphi \circ d) = (\beta \circ \varphi) \circ d = d^*(\beta_*(\varphi))\).Thus \(\beta_* \circ d^* = d^* \circ \beta_*\).
associativity of map composition.
3.4. This implies \(\beta_*(\operatorname{im}(d^*_G)) \subseteq \operatorname{im}(d^*_{G'})\), so \(\beta_*\) descends to a well-defined homomorphism:\[ \beta_*: \operatorname{Ext}(H, G) \longrightarrow \operatorname{Ext}(H, G'), \quad [\varphi] \mapsto [\beta \circ \varphi]. \]
quotient map between cokernels.
3.5. It is direct that \((\operatorname{id}_G)_* = \operatorname{id}_{\operatorname{Ext}(H, G)}\) and \((\gamma \circ \beta)_* = \gamma_* \circ \beta_*\) for \(\gamma: G' \to G''\).
composition of post-composition maps.
3.6. Thus \(\operatorname{Ext}(H, -)\) is a covariant functor.
4 and 5.
4. Conclusion: \(\operatorname{Ext}(H, G)\) is contravariant in \(H\) and covariant in \(G\).
2 and 3.
Q.E.D.