Definition 10.59.1. 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 $R$.
10.59 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
\[ \varphi _ M : n \longmapsto \text{length}_ R(\mathfrak m^ nM/{\mathfrak m}^{n + 1}M) \]
defined for all integers $n \geq 0$. Another important invariant is the function
\[ \chi _ M : n \longmapsto \text{length}_ R(M/{\mathfrak m}^{n + 1}M) \]
defined for all integers $n \geq 0$. Note that we have by Lemma 10.52.3 that
\[ \chi _ M(n) = \sum \nolimits _{i = 0}^ n \varphi _ M(i). \]
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 10.32.5. Hence any finite $R$-module annihilated by a power of $I$ has a finite length, see Lemma 10.52.8. Thus it makes sense to define
\[ \varphi _{I, M}(n) = \text{length}_ R(I^ nM/I^{n + 1}M) \quad \text{and}\quad \chi _{I, M}(n) = \text{length}_ R(M/I^{n + 1}M) \]
for all $n \geq 0$. Again we have that
\[ \chi _{I, M}(n) = \sum \nolimits _{i = 0}^ n \varphi _{I, M}(i). \]
Lemma 10.59.2. 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
\[ c_1 + \chi _{I, M'}(n - c_2) \leq \chi _{I, M}(n) \leq c_1 + \chi _{I, M'}(n) \]
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
\begin{eqnarray*} \chi _{I, M}(n) & = & \text{length}_ R(M/I^{n + 1}M) \\ & = & c_1 + \text{length}_ R(M'/I^{n + 1}M) \\ & \leq & c_1 + \text{length}_ R(M'/I^{n + 1}M') \\ & = & c_1 + \chi _{I, M'}(n) \end{eqnarray*}
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
\begin{eqnarray*} \chi _{I, M}(n) & = & \text{length}_ R(M/I^{n + 1}M) \\ & = & c_1 + \text{length}_ R(M'/I^{n + 1}M) \\ & \geq & c_1 + \text{length}_ R(M'/I^{n + 1 - c_2}M') \\ & = & c_1 + \chi _{I, M'}(n - c_2) \end{eqnarray*}
which finishes the proof. $\square$
Lemma 10.59.3. 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 $n \geq c$.
\[ \chi _{I, M}(n) = \chi _{I, M''}(n) + \chi _{I, N}(n - c) + l \]
\[ \varphi _{I, M}(n) = \varphi _{I, M''}(n) + \varphi _{I, N}(n - c) \]
Proof. Note that $M/I^ nM \to M''/I^ nM''$ is surjective with kernel $M' / M' \cap I^ nM$. By the Artin-Rees Lemma 10.51.2 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
\[ 0 \to M' / M' \cap I^ nM \to M/I^ nM \to M''/I^ nM'' \to 0 \]
and additivity of lengths (Lemma 10.52.3) we obtain the equality
\[ \chi _{I, M}(n - 1) = \chi _{I, M''}(n - 1) + \chi _{I, N}(n - c - 1) + \text{length}_ R(M'/N) \]
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 10.59.4. 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 $c \geq 1$ 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 $a = 2c - 1$. $\square$
Proposition 10.59.5. 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 10.58.7. Hence the result for $\varphi _{I, M}$ follows from that proposition and Lemma 10.55.1. The result for $\chi _{I, M}$ follows from this and Lemma 10.58.5. $\square$
Definition 10.59.6. 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 10.59.5 we see that the Hilbert polynomial exists.
Lemma 10.59.7. 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 10.59.4. 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 10.59.8. 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 10.59.9. 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 10.59.2 by elementary calculus. $\square$
Lemma 10.59.10. 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 10.59.3. By Lemma 10.59.9 the numerical polynomial $\chi _{I, M'} - \chi _{I, N}$ has degree $<$ the common degree of $\chi _{I, M'}$ and $\chi _{I, N}$. By Lemma 10.59.3 the difference
\[ \chi _{I, M}(n) - \chi _{I, M''}(n) - \chi _{I, N}(n - c) \]
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 10.51.2). 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. $\square$
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Comment #679 by Keenan Kidwell on