Describes the structure of some free resolutions of a quotient of a local or graded ring
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
0
→
R
m
→
f
R
n
→
R
→
R
/
I
→
0
{\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
Fitt
1
I
{\displaystyle \operatorname {Fitt} _{1}I}
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 , MR 0229634 , S2CID 123231429 , Zbl 0172.32302 Eisenbud, David (1995), Commutative algebra. With a view toward algebraic geometry , Graduate Texts in Mathematics , vol. 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 , vol. 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 36 (4): 473– 534, doi :10.1007/BF01208503 , ISSN 0025-5831 , JFM 22.0133.01 , S2CID 179177713
