mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Hilbert–Burch theorem describes the structure of some free resolutions of a

quotient In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...

of a

local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States Arts, entertainment, and media * ''Local'' (comics), a limited series comic book by Bria ...

or graded ring in the case that the quotient has

projective dimension In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Various equivalent characterizatio ...

2. proved a version of this theorem for

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...

s, 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 :

0 \rightarrow R^m\stackrel 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 In commutative algebra, the Fitting ideals of a finitely generated module over a commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative ring ...

\operatorname_1 I

of ''I'', i.e., the ideal generated by the

determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...

s of the minors of size ''m'' of the matrix of ''f''.

References

* * * * Commutative algebra Theorems in algebra {{commutative-algebra-stub