Minimal resolution: Difference between revisions
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===Uniqueness=== | |||
Given a fixed module <math>M</math>, minimal resolutions of <math>M</math> are unique up to isomorphism. | Given a fixed module <math>M</math>, minimal resolutions of <math>M</math> are unique up to isomorphism. | ||
If the generalized local ring is a [[polynomial ring over a field]] in <math>n</math> variables, then the [[Hilbert syzygy theorem]] says that the minimal resolution of a finitely generated module has length less than or equal to <math>n</math>. | ===Bounds on length=== | ||
If the generalized local ring is a [[multivariate polynomial ring over a field]] in <math>n</math> variables, then the [[Hilbert syzygy theorem]] says that the minimal resolution of a finitely generated module has length less than or equal to <math>n</math>. | |||
Latest revision as of 19:13, 3 January 2009
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 , minimal resolutions of are unique up to isomorphism.
Bounds on length
If the generalized local ring is a multivariate polynomial ring over a field in variables, then the Hilbert syzygy theorem says that the minimal resolution of a finitely generated module has length less than or equal to .