Definition 10.58.1. Let $(R, \mathfrak m)$ be a local Noetherian ring. An ideal $I \subset R$ such that $\sqrt{I} = \mathfrak m$ is called *an ideal of definition of $R$*.

## 10.58 Noetherian local rings

In all of this section $(R, \mathfrak m, \kappa )$ is a Noetherian local ring. We develop some theory on Hilbert functions of modules in this section. Let $M$ be a finite $R$-module. We define the *Hilbert function* of $M$ to be the function

defined for all integers $n \geq 0$. Another important invariant is the function

defined for all integers $n \geq 0$. Note that we have by Lemma 10.51.3 that

There is a variant of this construction which uses an ideal of definition.

Let $I \subset R$ be an ideal of definition. Because $R$ is Noetherian this means that $\mathfrak m^ r \subset I$ for some $r$, see Lemma 10.31.5. Hence any finite $R$-module annihilated by a power of $I$ has a finite length, see Lemma 10.51.8. Thus it makes sense to define

for all $n \geq 0$. Again we have that

Lemma 10.58.2. Suppose that $M' \subset M$ are finite $R$-modules with finite length quotient. Then there exists a constants $c_1, c_2$ such that for all $n \geq c_2$ we have

**Proof.**
Since $M/M'$ has finite length there is a $c_2 \geq 0$ such that $I^{c_2}M \subset M'$. Let $c_1 = \text{length}_ R(M/M')$. For $n \geq c_2$ we have

On the other hand, since $I^{c_2}M \subset M'$, we have $I^ nM \subset I^{n - c_2}M'$ for $n \geq c_2$. Thus for $n \geq c_2$ we get

which finishes the proof. $\square$

Lemma 10.58.3. Suppose that $0 \to M' \to M \to M'' \to 0$ is a short exact sequence of finite $R$-modules. Then there exists a submodule $N \subset M'$ with finite colength $l$ and $c \geq 0$ such that

and

for all $n \geq c$.

**Proof.**
Note that $M/I^ nM \to M''/I^ nM''$ is surjective with kernel $M' / M' \cap I^ nM$. By the Artin-Rees Lemma 10.50.2 there exists a constant $c$ such that $M' \cap I^ nM = I^{n - c}(M' \cap I^ cM)$. Denote $N = M' \cap I^ cM$. Note that $I^ c M' \subset N \subset M'$. Hence $\text{length}_ R(M' / M' \cap I^ nM) = \text{length}_ R(M'/N) + \text{length}_ R(N/I^{n - c}N)$ for $n \geq c$. From the short exact sequence

and additivity of lengths (Lemma 10.51.3) we obtain the equality

for $n \geq c$. We have $\varphi _{I, M}(n) = \chi _{I, M}(n) - \chi _{I, M}(n - 1)$ and similarly for the modules $M''$ and $N$. Hence we get $\varphi _{I, M}(n) = \varphi _{I, M''}(n) + \varphi _{I, N}(n-c)$ for $n \geq c$. $\square$

Lemma 10.58.4. Suppose that $I$, $I'$ are two ideals of definition for the Noetherian local ring $R$. Let $M$ be a finite $R$-module. There exists a constant $a$ such that $\chi _{I, M}(n) \leq \chi _{I', M}(an)$ for $n \geq 1$.

**Proof.**
There exists an integer $c$ such that $(I')^ c \subset I$. Hence we get a surjection $M/(I')^{c(n + 1)}M \to M/I^{n + 1}M$. Whence the result with $a = c + 1$.
$\square$

Proposition 10.58.5. Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module. Let $I \subset R$ be an ideal of definition. The Hilbert function $\varphi _{I, M}$ and the function $\chi _{I, M}$ are numerical polynomials.

**Proof.**
Consider the graded ring $S = R/I \oplus I/I^2 \oplus I^2/I^3 \oplus \ldots = \bigoplus _{d \geq 0} I^ d/I^{d + 1}$. Consider the graded $S$-module $N = M/IM \oplus IM/I^2M \oplus \ldots = \bigoplus _{d \geq 0} I^ dM/I^{d + 1}M$. This pair $(S, N)$ satisfies the hypotheses of Proposition 10.57.7. Hence the result for $\varphi _{I, M}$ follows from that proposition and Lemma 10.54.1. The result for $\chi _{I, M}$ follows from this and Lemma 10.57.5.
$\square$

Definition 10.58.6. Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module. The *Hilbert polynomial* of $M$ over $R$ is the element $P(t) \in \mathbf{Q}[t]$ such that $P(n) = \varphi _ M(n)$ for $n \gg 0$.

By Proposition 10.58.5 we see that the Hilbert polynomial exists.

Lemma 10.58.7. Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module.

The degree of the numerical polynomial $\varphi _{I, M}$ is independent of the ideal of definition $I$.

The degree of the numerical polynomial $\chi _{I, M}$ is independent of the ideal of definition $I$.

**Proof.**
Part (2) follows immediately from Lemma 10.58.4. Part (1) follows from (2) because $\varphi _{I, M}(n) = \chi _{I, M}(n) - \chi _{I, M}(n - 1)$ for $n \geq 1$.
$\square$

Definition 10.58.8. Let $R$ be a local Noetherian ring and $M$ a finite $R$-module. We denote *$d(M)$* the element of $\{ -\infty , 0, 1, 2, \ldots \} $ defined as follows:

If $M = 0$ we set $d(M) = -\infty $,

if $M \not= 0$ then $d(M)$ is the degree of the numerical polynomial $\chi _ M$.

If $\mathfrak m^ nM \not= 0$ for all $n$, then we see that $d(M)$ is the degree $+1$ of the Hilbert polynomial of $M$.

Lemma 10.58.9. Let $R$ be a Noetherian local ring. Let $I \subset R$ be an ideal of definition. Let $M$ be a finite $R$-module which does not have finite length. If $M' \subset M$ is a submodule with finite colength, then $\chi _{I, M} - \chi _{I, M'}$ is a polynomial of degree $<$ degree of either polynomial.

**Proof.**
Follows from Lemma 10.58.2 by elementary calculus.
$\square$

Lemma 10.58.10. Let $R$ be a Noetherian local ring. Let $I \subset R$ be an ideal of definition. Let $0 \to M' \to M \to M'' \to 0$ be a short exact sequence of finite $R$-modules. Then

if $M'$ does not have finite length, then $\chi _{I, M} - \chi _{I, M''} - \chi _{I, M'}$ is a numerical polynomial of degree $<$ the degree of $\chi _{I, M'}$,

$\max \{ \deg (\chi _{I, M'}), \deg (\chi _{I, M''}) \} = \deg (\chi _{I, M})$, and

$\max \{ d(M'), d(M'')\} = d(M)$,

**Proof.**
We first prove (1). Let $N \subset M'$ be as in Lemma 10.58.3. By Lemma 10.58.9 the numerical polynomial $\chi _{I, M'} - \chi _{I, N}$ has degree $<$ the common degree of $\chi _{I, M'}$ and $\chi _{I, N}$. By Lemma 10.58.3 the difference

is constant for $n \gg 0$. By elementary calculus the difference $\chi _{I, N}(n) - \chi _{I, N}(n - c)$ has degree $<$ the degree of $\chi _{I, N}$ which is bigger than zero (see above). Putting everything together we obtain (1).

Note that the leading coefficients of $\chi _{I, M'}$ and $\chi _{I, M''}$ are nonnegative. Thus the degree of $\chi _{I, M'} + \chi _{I, M''}$ is equal to the maximum of the degrees. Thus if $M'$ does not have finite length, then (2) follows from (1). If $M'$ does have finite length, then $I^ nM \to I^ nM''$ is an isomorphism for all $n \gg 0$ by Artin-Rees (Lemma 10.50.2). Thus $M/I^ nM \to M''/I^ nM''$ is a surjection with kernel $M'$ for $n \gg 0$ and we see that $\chi _{I, M}(n) - \chi _{I, M''}(n) = \text{length}(M')$ for all $n \gg 0$. Thus (2) holds in this case also.

Proof of (3). This follows from (2) except if one of $M$, $M'$, or $M''$ is zero. We omit the proof in these special cases. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (1)

Comment #679 by Keenan Kidwell on