Minimal resolution

Definition

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.

Metaproperties

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 $n$.