Hilbert–Burch theorem

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. Hilbert (1890) proved a version of this theorem for polynomial rings, and Burch (1968, p. 944) proved a more general version. Several other authors later rediscovered and published variations of this theorem. Eisenbud (1995, theorem 20.15) gives a statement and proof.

Statement

If R is a local ring with an ideal I and

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

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 ${\displaystyle \operatorname {Fitt} _{1}I}$ of I, i.e., the ideal generated by the determinants of the minors of size m of the matrix of f.

References

• Burch, Lindsay (1968), "On ideals of finite homological dimension in local rings", Proc. Cambridge Philos. Soc., 64: 941–948, doi:10.1017/S0305004100043620, ISSN 0008-1981, MR 0229634, Zbl 0172.32302
• Eisenbud, David (1995), Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, ISBN 3-540-94268-8, MR 1322960, Zbl 0819.13001
• Eisenbud, David (2005), The Geometry of Syzygies. A second course in commutative algebra and algebraic geometry, Graduate Texts in Mathematics, 229, New York, NY: Springer-Verlag, ISBN 0-387-22215-4, Zbl 1066.14001
• Hilbert, David (1890), "Ueber die Theorie der algebraischen Formen", Mathematische Annalen (in German), 36 (4): 473–534, doi:10.1007/BF01208503, ISSN 0025-5831, JFM 22.0133.01