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

Statement

If R is a local ring with an ideal I and
is a free resolution of the R-module R/I, then m = n – 1 and the ideal I is aJ where a is a non zero divisor of R and J is the depth 2 ideal generated by the determinants of the minors of size m of the matrix of the map from R^m to Rⁿ.