Algebraic mapping cone
From Manifold Atlas
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− | == Definition | + | == 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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$$ | $$ | ||
− | Every book on homological algebra contains this definition. A good source is for example \cite{Weibel1994}. | + | Every book on homological algebra contains this definition, except for the sign conventions in the differentials! |
+ | For example, it is possible to have | ||
+ | $$ | ||
+ | \partial_{Cone(f)}= | ||
+ | \left( \begin{array}{cc} | ||
+ | \partial_E & 0 \\ | ||
+ | (-)^kf & \partial_F | ||
+ | \end{array} \right) : Cone(f)_k\rightarrow Cone(f)_{k-1}. | ||
+ | $$ | ||
+ | |||
+ | |||
+ | |||
+ | A good source is for example \cite{Weibel1994}. | ||
</wikitex> | </wikitex> | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} |
Latest revision as of 10:52, 1 October 2012
Definition
Let be a map of chain complexes. Define the algebraic mapping cone of as a chain complex given in degree by
with differential
Every book on homological algebra contains this definition, except for the sign conventions in the differentials! For example, it is possible to have
A good source is for example [Weibel1994].
References
- [Weibel1994] C. A. Weibel, An introduction to homological algebra, Cambridge University Press, 1994. MR1269324 (95f:18001) Zbl 0834.18001