Hilbert–Burch theorem

{{short description|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. {{harvs|txt|last=Hilbert|authorlink=David Hilbert|year=1890}} proved a version of this theorem for polynomial rings, and {{harvs|txt|last=Burch|year1=1968|loc=p. 944}} proved a more general version. Several other authors later rediscovered and published variations of this theorem. {{harvtxt|Eisenbud|1995|loc=theorem 20.15}} gives a statement and proof.

Statement

If R is a local ring with an ideal I and

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

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 $\operatorname{Fitt}_1 I$ of I, i.e., the ideal generated by the determinants of the minors of size m of the matrix of f.

References

{{Citation | last1=Burch | first1=Lindsay | title=On ideals of finite homological dimension in local rings | doi=10.1017/S0305004100043620 | mr=0229634 | year=1968 | journal=Proc. Cambridge Philos. Soc. | volume=64 | issue=4 | pages=941–948 | zbl=0172.32302 | s2cid=123231429 | issn=0008-1981 }}
{{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra. With a view toward algebraic geometry | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=3-540-94268-8 | mr=1322960 | year=1995 | volume=150 | zbl=0819.13001 }}
{{citation |authorlink=David Eisenbud |first=David |last=Eisenbud |title=The Geometry of Syzygies. A second course in commutative algebra and algebraic geometry | series=Graduate Texts in Mathematics | volume=229 | year=2005 |location=New York, NY |publisher=Springer-Verlag | isbn=0-387-22215-4 | zbl=1066.14001 }}
{{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | title=Ueber die Theorie der algebraischen Formen | language=German | doi=10.1007/BF01208503 | year=1890 | journal=Mathematische Annalen | issn=0025-5831 | volume=36 | issue=4 | pages=473–534 | jfm=22.0133.01 | s2cid=179177713 }}

Category:Commutative algebra

Category:Theorems in algebra