Minimal resolution

Symbol-free definition
A minimal resolution of a module over a generalized local ring is a graded free resolution (possibly infinite in length) terminating at 0, with the second last member being the given module, such that the differentials of the resolution become 0 after tensoring with the ring modulo its unique homogeneous maximal ideal.

Uniqueness
Given a fixed module $$M$$, minimal resolutions of $$M$$ are unique up to isomorphism.

Bounds on length
If the generalized local ring is a multivariate polynomial ring over a field in $$n$$ variables, then the Hilbert syzygy theorem says that the minimal resolution of a finitely generated module has length less than or equal to $$n$$.