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

quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

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 * Local government, a form of public administration, usually the lowest tier of administrat ...

or graded

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

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, by keeping some of the main properties of free modules. Various equivalent characterizat ...

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 (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...

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 Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...

''I'' and :

0 \rightarrow R^m\stackrel R^n \rightarrow R \rightarrow R/I\rightarrow 0

is a free resolution of the ''R''-

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...

''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 describe the obstructions to generating the module by a given number of elements. They were introduced by . Definition If ''M'' is a finitely generate ...

\operatorname_1 I

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

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...

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

References

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