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) gives a statement and proof.
Statement
If R is a local ring with an ideal I and
- [math]\displaystyle{ 0 \rightarrow R^m\stackrel{f}{\rightarrow} R^n \rightarrow R \rightarrow R/I\rightarrow 0 }[/math]
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 [math]\displaystyle{ \operatorname{Fitt}_1 I }[/math] 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 (4): 941–948, doi:10.1017/S0305004100043620, ISSN 0008-1981
- 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
- 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
- Hilbert, David (1890), "Ueber die Theorie der algebraischen Formen" (in German), Mathematische Annalen 36 (4): 473–534, doi:10.1007/BF01208503, ISSN 0025-5831
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