Hilbert syzygy theorem
This article gives the statement, and possibly proof, of a commutative unital ring property satisfying a commutative unital ring metaproperty
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Contents
Name
This result is termed the Hilbert syzygy theorem.
Statement
Verbal statement
Minimal resolutions over a multivariate polynomial ring over a field of finitely generated graded modules have length bounded by the number of variables.
Symbolic statement
Let be a polynomial ring over a field and a finitely generated graded module. Then there exists an exact sequence with degree 0 maps
where the are free modules.
Proof
Proof idea
The key tools of the proof are the symmetry of Tor, and the Koszul complex.
The idea is that the Koszul complex of has length and hence for . Using this symmetry, one can also compute the Tor groups by tensoring a free resolution of by . In particular, taking a minimal free resolution (it is easy to see that minimal free resolutions exist) of , the differentials become 0 upon tensoring with by definition of minimal resolution. Hence it follows that for .
Remarks
We are assuming that has the canonical grading, i.e., that the degree part is the vector space generated by the monomials of degree .