g = extend(D, C, f, p)
g = extend(D, C, f)
Let $C$ be a chain complex such that each term is a free module. Let $D$ be a chain complex which is exact at the $k$-th term for all $k > j$. Given a map of modules $f \colon C_i \to D_j$ such that the image of $f \circ \operatorname{dd}^C_{i+1}$ is contained in the image of $\operatorname{dd}^D_{j+1}$, this method constructs a morphism of chain complexes $g \colon C \to D$ of degree $j-i$ such that $g_i = f$.
$\phantom{WWWW} \begin{array}{cccccc} 0 & \!\!\leftarrow\!\! & C_{i} & \!\!\leftarrow\!\! & C_{i+1} & \!\!\leftarrow\!\! & C_{i+2} & \dotsb \\ & & \downarrow \, {\scriptstyle f} & & \downarrow \, {\scriptstyle g_{i+1}} && \downarrow \, {\scriptstyle g_{i+2}} \\ 0 & \!\!\leftarrow\!\! & D_{j} & \!\!\leftarrow\!\! & D_{j+1} & \!\!\leftarrow\!\! & D_{j+2} & \dotsb \\ \end{array} $
A map between modules extends to a map between their free resolutions.
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Extension of maps to complexes is also useful in constructing a free resolution of a linked ideal.
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Inspired by a yonedaMap computation, we extend a map of modules to a map between free resolutions having homological degree $-1$.
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The source of this document is in Complexes/ChainComplexMapDoc.m2:3594:0.