Hilbert syzygy theorem

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This article gives the statement, and possibly proof, of a commutative unital ring property satisfying a commutative unital ring metaproperty
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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 A = k[x_1, \dots, x_n] be a polynomial ring over a field k and M a finitely generated graded A-module. Then there exists an exact sequence with degree 0 maps

 0 \to F_n \to F_{n-1} \to \cdots \to F_1 \to F_0 \to M \to 0

where the F_i 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 k has length n and hence \mathrm{Tor}^A_i(k,M) = \mathrm{Tor}^A_i(M,k) = 0 for i>n. Using this symmetry, one can also compute the Tor groups by tensoring a free resolution of M by k. In particular, taking a minimal free resolution (it is easy to see that minimal free resolutions exist) F_\bullet of M, the differentials become 0 upon tensoring with k by definition of minimal resolution. Hence it follows that F_i = \mathrm{Tor}^A_i(k,M) = 0 for i>0.

Remarks

We are assuming that A has the canonical grading, i.e., that the degree i part is the vector space generated by the monomials of degree i.