freeResolution M
A free resolution of a module $M$ is a complex $ F_0 \leftarrow F_1 \leftarrow F_2 \leftarrow \ldots$ of free modules, which is acyclic: the cokernel of the map to $F_0$ is $M$ and the complex is exact at all other locations.
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Giving an ideal as the input produces a free resolution not of the module I, but of the module R^1/I.
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Over a quotient ring, free resolutions are often infinite. Use the optional argument LengthLimit to obtain part of the resolution.
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The object freeResolution is a method function with options.