Algebraic mapping cone

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== Definition ==
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== Definition ==
Let $E \xrightarrow{f} F$ be a map of chain complexes. Define the ''algebraic mapping cone of'' $f$ as a chain complex $Cone(f)$ given in degree $k$ by
Let $E \xrightarrow{f} F$ be a map of chain complexes. Define the ''algebraic mapping cone of'' $f$ as a chain complex $Cone(f)$ given in degree $k$ by
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Revision as of 09:51, 1 October 2012

Definition

Let E \xrightarrow{f} F be a map of chain complexes. Define the algebraic mapping cone of f as a chain complex Cone(f) given in degree k by

\displaystyle  Cone(f)_k=E_{k-1}\oplus F_k

with differential

\displaystyle  \partial_{Cone(f)}= \left( \begin{array}{cc} -\partial_E & 0 \\ f & \partial_F  \end{array} \right) : Cone(f)_k\rightarrow Cone(f)_{k-1}.

Every book on homological algebra contains this definition, except for the sign conventions in the differentials! For example, it is possible to have

\displaystyle  \partial_{Cone(f)}= \left( \begin{array}{cc} \partial_E & 0 \\ (-)^kf & \partial_F  \end{array} \right) : Cone(f)_k\rightarrow Cone(f)_{k-1}.



A good source is for example [Weibel1994].

References

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