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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This result is termed the Hilbert syzygy theorem.
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.
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 .
We are assuming that has the canonical grading, i.e., that the degree part is the vector space generated by the monomials of degree .