Useful Facts

Notational conventions:

• Finite direct products: $$\bigoplus$$
• Cohomological indexing: $$C^i, {{\partial}}^i$$
• Homological indexing: $$C_i, {{\partial}}_i$$
• Right-derived functors $$R^iF$$.
• Come from left-exact functors
• Require injective resolutions
• Extend to the right: $$0 \to F(A) \to F(B) \to F(C) \to L_1 F(A) \cdots$$
• Left-derived functors $$L_i F$$.
• Come from right-exact functors
• Require projective resolutions
• Extend to the left: $$\cdots L_1F(C) \to F(A) \to F(B) \to F(C) \to 0$$
• Colimits:
• Examples: coproducts, direct limits, cokernels, initial objects, pushouts
• Commute with left adjoints, i.e. $$L(\colim F_i) = \colim LF_i$$.
• 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
• On limits:
• Limits commute with limits, and colimits commute with colimits.

• Generally, limits do not commute with colimits.

• In $${\mathsf{Set}}$$, filtered colimits commute with finite limits.

• In $${\mathsf{Ab}}$$, direct colimits commute with finite limits. Inverse limits do not generally commute with finite colimits.

• On adjoints:
• Left adjoints are right-exact with left-derived functors. Right adjoints are left-exact with right-derived functors.

• 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.

• Left adjoints are right-exact with left-derived functors. Right adjoints are left-exact with right-derived functors.

• Left adjoints commute with colimits: $$L( \colim F) = \colim (L\circ F)$$

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}{\hbox{-}}\mathsf{R} } { \mathsf{Mod}{\hbox{-}}\mathsf{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 $$F: {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \to {\mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}}$$ be the forgetful functor, then there are adjunctions \begin{align*} \adjunction{F}{ \mathop{\mathrm{Hom}}_{\mathbb{Z}}(R, {-})} {{\mathsf{R}{\hbox{-}}\mathsf{Mod}} } {{\mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}} } \\ \\ \adjunction{R\otimes_{\mathbb{Z}}{-}}{F}{ {\mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}} }{ {\mathsf{R}{\hbox{-}}\mathsf{Mod}} } .\end{align*}

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