Masnevets: multivariate polynomial ring link

2009-01-03T19:12:10Z

multivariate polynomial ring link

← Older revision		Revision as of 19:12, 3 January 2009
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	===Verbal statement===		===Verbal statement===
	Minimal resolutions over a [[polynomial ring over a field]] of finitely generated [[graded module]]s have length bounded by the number of variables.		Minimal resolutions over a [[multivariate polynomial ring over a field]] of finitely generated [[graded module]]s have length bounded by the number of variables.

	===Symbolic statement===		===Symbolic statement===

Masnevets: New page: {{curing metaproperty satisfaction}} ==Name== This result is termed the '''Hilbert syzygy theorem'''. ==Statement== ===Verbal statement=== Minimal resolutions over a [[polynomial ring ...

2009-01-03T19:11:14Z

New page: {{curing metaproperty satisfaction}} ==Name== This result is termed the '''Hilbert syzygy theorem'''. ==Statement== ===Verbal statement=== Minimal resolutions over a [[polynomial ring ...

New page

{{curing metaproperty satisfaction}}

==Name==

This result is termed the '''Hilbert syzygy theorem'''.

==Statement==

===Verbal statement===
Minimal resolutions over a [[polynomial ring over a field]] of finitely generated [[graded module]]s have length bounded by the number of variables.

===Symbolic statement===

Let <math>A = k[x_1, \dots, x_n]</math> be a polynomial ring over a field <math>k</math> and <math>M</math> a finitely generated graded <math>A</math>-module. Then there exists an exact sequence with degree 0 maps

::<math> 0 \to F_n \to F_{n-1} \to \cdots \to F_1 \to F_0 \to M \to 0 </math>

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

==Remarks==

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

Hilbert syzygy theorem - Revision history

Masnevets: multivariate polynomial ring link

Masnevets: New page: {{curing metaproperty satisfaction}} ==Name== This result is termed the '''Hilbert syzygy theorem'''. ==Statement== ===Verbal statement=== Minimal resolutions over a [[polynomial ring ...