Hilbert–Burch theorem

Template:Short description In mathematics, the Hilbert–Burch theorem describes the structure of some free resolutions of a quotient of a local or graded ring in the case that the quotient has projective dimension 2. Template:Harvs proved a version of this theorem for polynomial rings, and Template:Harvs proved a more general version. Several other authors later rediscovered and published variations of this theorem. Template:Harvtxt gives a statement and proof.

Statement

If R is a local ring with an ideal I and

$0\to R^m\stackrel{f}{\to }{R}^n\to R\to R/I\to 0$

is a free resolution of the R-module R/I, then m = n – 1 and the ideal I is aJ where a is a regular element of R and J, a depth-2 ideal, is the first Fitting ideal ${\mathrm{Fitt}}_{1}I$ of I, i.e., the ideal generated by the determinants of the minors of size m of the matrix of f.

References

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