---
tag: AG4FF4DNYQ
kind: statement
role: theorem
name: Hilbert–Burch theorem
msc: [13D02, 13C40]
uses: [0670T59HGE, 096Z9MP2ME, HK2AA1YSQ2, SXD7X0WQH1, M2RADBFW99, 0NNRDW10RZ]
about: [HK2AA1YSQ2, SXD7X0WQH1]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert–Burch_theorem
same_as:
  wikidata: Q5761240
  wikipedia: Hilbert–Burch theorem
---

## Statement

Let $R$ be a commutative [[Noetherian ring|0670T59HGE]], let $I$ be an [[ideal|096Z9MP2ME]] 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|HK2AA1YSQ2]] 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|SXD7X0WQH1]].
Then:

1. There is a non-[[zero divisor|0NNRDW10RZ]] $a \in R$ such that the entries of $B$ are $a \Delta_1, \ldots, a \Delta_{n+1}$ up to alternating signs; in particular $I = a I_n(A)$.
2. If the [[depth|M2RADBFW99]] of $I$ on $R$ is at least $2$, then $a$ can be taken to be $1$, so $I = I_n(A)$.
3. 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 $\operatorname{pd}_R R/I = 2$.

## Explanations

### L1 human

If a quotient ring $R/I$ has a free resolution of length $2$, then the whole resolution is controlled by one matrix.
The generators of $I$ are, up to a common factor, the maximal minors of that matrix.
So these ideals are determined by a single rectangular matrix.

### L2 human

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.