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 \(H_*(C) = 0\).

\envlist
  • 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:

Link to Diagram

Adjunctions

\todo[inline]{todo}

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*}