Notational conventions:
- Finite direct products: \(\bigoplus\)
- Cohomological indexing: \(C^i, {{\partial}}^i\)
- Homological indexing: \(C_i, {{\partial}}_i\)
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Right-derived functors \(R^iF\).
- Come from left-exact functors
- Require injective resolutions
- Extend to the right: \(0 \to F(A) \to F(B) \to F(C) \to L_1 F(A) \cdots\)
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Left-derived functors \(L_i F\).
- Come from right-exact functors
- Require projective resolutions
- Extend to the left: \(\cdots L_1F(C) \to F(A) \to F(B) \to F(C) \to 0\)
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Colimits:
- Examples: coproducts, direct limits, cokernels, initial objects, pushouts
- Commute with left adjoints, i.e. \(L(\colim F_i) = \colim LF_i\).
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Examples of limits:
- Products, inverse limits, kernels, terminal objects, pullbacks
- Commute with right adjoints. i.e. \(R(\colim F_i) = \colim RF_i\).
A chain complex \(C\) is acyclic if and only if \(H_*(C) = 0\).
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Free \(\implies\) projective \(\implies\) flat \(\implies\) torsionfree (for finitely-generated \(R{\hbox{-}}\)modules)
- Over \(R\) a PID: free \(\iff\) torsionfree
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On limits:
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Limits commute with limits, and colimits commute with colimits.
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Generally, limits do not commute with colimits.
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In \({\mathsf{Set}}\), filtered colimits commute with finite limits.
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In \({\mathsf{Ab}}\), direct colimits commute with finite limits. Inverse limits do not generally commute with finite colimits.
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On adjoints:
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Left adjoints are right-exact with left-derived functors. Right adjoints are left-exact with right-derived functors.
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Left adjoints commute with colimits: \(L( \colim F) = \colim (L\circ F)\) In \({\mathsf{Ab}}\), direct colimits commute with finite limits. Inverse limits do not generally commute with finite colimits.
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Left adjoints are right-exact with left-derived functors. Right adjoints are left-exact with right-derived functors.
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Left adjoints commute with colimits: \(L( \colim F) = \colim (L\circ F)\)
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Universal Properties
If \(f: G\to K\) and \(H{~\trianglelefteq~}G\) (so that \(G/H\) is defined), then the map \(f\) descends to the quotient if and only if \(H \subseteq \ker(f)\).
The kernel \(\ker f\) of a morphism \(f\) can be characterized as a cartesian square, and the cokernel \(\operatorname{coker}f\) as a cocartesian square:
Adjunctions
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For a fixed \(M\in ({R}, {S}){\hbox{-}}\mathsf{biMod}\), there is an adjunction \begin{align*} \adjunction{ {-}\otimes_R M }{\mathop{\mathrm{Hom}}_S(M, {-})}{ {\mathsf{Mod}}_{R} } { {\mathsf{Mod}}_{S} } ,\end{align*} so for \(Y \in ({A}, {R}){\hbox{-}}\mathsf{biMod}\) and \(Z \in ({B}, {S}){\hbox{-}}\mathsf{biMod}\), there is a (natural) isomorphism in \(({B}, {A}){\hbox{-}}\mathsf{biMod}\): \begin{align*} \mathop{\mathrm{Hom}}_S(X \otimes_R M, Z) \xrightarrow{\sim} \mathop{\mathrm{Hom}}_R( X, \mathop{\mathrm{Hom}}_S(M, Z) ) .\end{align*}
Let $F: {}{R}{\mathsf{Mod}} \to {}{{\mathbf{Z}}}{\mathsf{Mod}} $ be the forgetful functor, then there are adjunctions \begin{align*} \adjunction{F}{ \mathop{\mathrm{Hom}}_{\mathbf{Z}}(R, {-})} { {}_{R}{\mathsf{Mod}} } { {}_{{\mathbf{Z}}}{\mathsf{Mod}} } \\ \\ \adjunction{R\otimes_{\mathbf{Z}}{-}}{F}{ {}_{{\mathbf{Z}}}{\mathsf{Mod}} }{ {}_{R}{\mathsf{Mod}} } .\end{align*}