Commutative algebra Homological methods Theorem

Hilbert–Burch theorem

Builds on Noetherian ring Ideal Projective resolution Determinantal ideal Depth Zero divisor

Statement

Let $R$ be a commutative Noetherian ring, let $I$ be an ideal of $R$ , and let $n \ge 1$ . Suppose that $0 \to R^n \xrightarrow{A} R^{n+1} \xrightarrow{B} R \to R/I \to 0$ is a free resolution of $R/I$ , where $A$ is an $(n+1) \times n$ matrix. For $i = 1, \ldots, n+1$ , let $\Delta_i$ be the determinant of the matrix obtained from $A$ by deleting row $i$ . Let $I_n(A) = (\Delta_1, \ldots, \Delta_{n+1})$ be the ideal of maximal minors. Then:

There is a non-zero divisor $a \in R$ such that the entries of $B$ are $a \Delta_1, \ldots, a \Delta_{n+1}$ up to alternating signs; in particular $I = a I_n(A)$ .
If the depth of $I$ on $R$ is at least $2$ , then $a$ can be taken to be $1$ , so $I = I_n(A)$ .
Conversely, let $A$ be any $(n+1) \times n$ matrix over $R$ such that the depth of $I_n(A)$ on $R$ is at least $2$ . Let $B$ be the row of the signed maximal minors of $A$ . Then the complex above is a free resolution of $R/I_n(A)$ .

Remarks

The proof compares the resolution with the complex built from the signed minors. That complex is exact by the Buchsbaum–Eisenbud acyclicity criterion when the minors generate an ideal of depth at least $2$ . Example: for $R = K[x, y]$ and $A = (-y, x)^T$ , the maximal minors are $x$ and $-y$ , and $0 \to R \xrightarrow{A} R^2 \xrightarrow{(x \;\; y)} R \to R/(x, y) \to 0$ is a free resolution. A classical application: the ideal of the twisted cubic curve in $K[x_0, x_1, x_2, x_3]$ is generated by the three $2 \times 2$ minors of a $3 \times 2$ matrix of linear forms. The theorem describes all perfect ideals of depth $2$ , that is, all ideals $I$ of depth $2$ with $\operatorname{pd}_R R/I = 2$ .

Going deeper

For a student

The theorem is the first case of the structure theory of finite free resolutions. It explains, for example, why the ideal of an arithmetically Cohen–Macaulay subscheme of codimension $2$ in a projective space is generated by the maximal minors of a matrix. The analogous result for Gorenstein ideals of depth $3$ is the Buchsbaum–Eisenbud structure theorem with Pfaffians.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Depth concept
Determinantal ideal concept
Ideal concept
Noetherian ring concept
Projective resolution concept
Zero divisor concept
Hilbert–Burch theorem statement

Tag	`AG4FF4DNYQ`
URI	`m2://AG4FF4DNYQ`
Kind	statement, theorem
Writing	W0 stub
MSC	13D02, 13C40
Status	This copy of the site has no build data. Run `m2 status AG4FF4DNYQ`.
About	Projective resolution, Determinantal ideal
Uses	Noetherian ring, Ideal, Projective resolution, Determinantal ideal, Depth, Zero divisor
Same as	Wikidata Q5761240 Wikipedia Hilbert–Burch theorem
Changed	2 November 2026

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