Tags: #todo #todo/learning/definitions Refs: ?
mapping cone
Of chain complexes
For \(f\in \mathsf{Ch}_{ {}_{R}{\mathsf{Mod}} }(A, B)\) a morphism of chain complex of \(R{\hbox{-}}\)modules, the mapping cone complex is \begin{align*} \operatorname{cone}(f) := A[1] \oplus B, \quad d = \begin{bmatrix} d_A & 0 \\ f & d_B \end{bmatrix} \end{align*}
Results
- If \(\operatorname{cone}(f) \simeq 0\), i.e \(\operatorname{cone}(f)\) is an quasi-isomorphism)).
\begin{align*}\begin{tikzcd} A && B \\ \\ & {\operatorname{cone}(f)} \arrow["f", from=1-1, to=1-3] \arrow[from=1-3, to=3-2] \arrow["{[1]}", from=3-2, to=1-1] \end{tikzcd}\end{align*}
