finite if only finitely many of the modules involved are non-zero. The length of a finite resolution is the maximum index labeling a nonzero module in the finite resolution.

Free, projective, injective, and flat resolutions

In many circumstances conditions are imposed on the modules resolving the given module . For example, a free resolution of a module is a left resolution in which all the modules are free -modules. Likewise, projective and flat resolutions are left resolutions

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