---
title: "Worksheet 23 - Derivations, Hilbert–Samuel Multiplicity, and Krull Dimension"
stable_id: br-ak-2025-2026-w23
language: en
upstream_title: "Kurs:Algebraische Kurven (Osnabrück 2025-2026)/Arbeitsblatt 23"
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# Worksheet 23 {#br-ak-2025-2026-w23}

## Practice exercises {#br-ak-2025-2026-w23-practice}

<!-- upstream_entity: Polynomring/Partielle Ableitung/Derivation/Aufgabe -->

### Exercise 23.1 {#br-ak-2025-2026-w23-ex-01}

Let $K$ be a field, $K[X_1,\ldots,X_n]$ the polynomial ring over $K$, and

$$
\frac{\partial}{\partial X_1}
$$

formal partial differentiation with respect to $X_1$, that is, the map

$$
\begin{aligned}
K[X_1,\ldots,X_n]&\longrightarrow K[X_1,\ldots,X_n],\\
f&\longmapsto\frac{\partial f}{\partial X_1}.
\end{aligned}
$$

Show that this map is a $K$-derivation.

<!-- upstream_entity: Polynomring/Maximales Ideal/Potenzen/Restklassenbasis/Aufgabe -->

### Exercise 23.2 {#br-ak-2025-2026-w23-ex-02}

Consider the maximal ideal

$$
\mathfrak m=(X_1,\ldots,X_n)
\subseteq K[X_1,\ldots,X_n]
$$

in the polynomial ring over a field $K$, together with its powers
$\mathfrak m^d$. Show that the monomials

$$
X_1^{\nu_1}\cdots X_n^{\nu_n},
\qquad \sum_{i=1}^n\nu_i<d,
$$

form a $K$-basis of the quotient ring

$$
K[X_1,\ldots,X_n]/\mathfrak m^d.
$$

<!-- upstream_entity: Achsenkreuz/R mod m^n/Basis und Hilbert Funktion/Berechne/Aufgabe -->

### Exercise 23.3 {#br-ak-2025-2026-w23-ex-03}

Consider the union of the coordinate axes

$$
V(xy)\subseteq\mathbb A_K^2
$$

and the local ring $R$ at the origin, with maximal ideal $\mathfrak m$.
Describe explicitly a $K$-basis of the quotient rings $R/\mathfrak m^n$
and determine their dimensions.

<!-- upstream_entity: Polynomring/2/Multiplizität/Multiplikation/Aufgabe -->

### Exercise 23.4 ★ {#br-ak-2025-2026-w23-ex-04}

Let

$$
F=F_m+\cdots+F_d\in K[X,Y]
$$

be the homogeneous decomposition of a polynomial, with $m\le d$ and
$F_m\ne0$, and let $\mathfrak m=(X,Y)$. Show that, for every $n\ge m$, the
multiplication map

$$
\begin{aligned}
K[X,Y]&\longrightarrow K[X,Y],\\
G&\longmapsto FG
\end{aligned}
$$

induces a well-defined injective homomorphism of $K[X,Y]$-modules

$$
K[X,Y]/\mathfrak m^{n-m}
\longrightarrow
K[X,Y]/\mathfrak m^n.
$$

*Edition note -- clarification of the source hypothesis:* Here $F_m$ is
the lowest nonzero homogeneous component. The source leaves $F_m\ne0$
implicit, but it is required for injectivity and is used in the source
argument and the corrected solution below.

<!-- upstream_entity: Numerisches Monoid/N ab e/Hilbert-Funktion/Aufgabe -->

### Exercise 23.5 ★ {#br-ak-2025-2026-w23-ex-05}

Let $e\in\mathbb N_+$ and

$$
M:=\{0\}\cup\mathbb N_{\ge e}\subseteq\mathbb N.
$$

1. Determine $nM_+$ for $n\in\mathbb N_+$.
2. Determine $\#(M\setminus nM_+)$.
3. Let $K$ be a field and set

   $$
   R=K[M]_{\mathfrak m},
   \qquad
   \mathfrak m=K[M_+]\subseteq K[M].
   $$

   Determine $\dim_K(R/\mathfrak m^n)$.

<!-- upstream_entity: Numerisches Monoid/Abschätzungen für Multiplizität und Differenzanzahl/4,9/bis n ist 6/Aufgabe -->

### Exercise 23.6 {#br-ak-2025-2026-w23-ex-06}

For the numerical monoid $M\subseteq\mathbb N$ generated by $4$ and $9$,
compute the quantities appearing in the bounds of Lemma 23.8 for $n\le6$.

## Krull dimension {#br-ak-2025-2026-w23-krull}

Some of the following exercises use the Krull dimension of a commutative
ring. Since our main interest is in curves, which correspond to
one-dimensional rings, we shall not develop a systematic dimension theory.

<!-- upstream_entity: Kommutative Ringtheorie/Primidealkette/Krulldimension/Definition -->

### Definition: prime ideal chains and Krull dimension {#br-ak-2025-2026-w23-def-01}

Let $R$ be a commutative ring. A chain of prime ideals

$$
\mathfrak p_0
\subset
\mathfrak p_1
\subset
\cdots
\subset
\mathfrak p_n
$$

is called a *prime ideal chain of length $n$*. Thus we count the inclusions,
not the prime ideals in the chain. The *dimension*, or *Krull dimension*,
of $R$ is the supremum of all lengths of prime ideal chains, denoted by

$$
\dim(R).
$$

<!-- upstream_entity: Krulldimension/Hauptidealbereich, kein Körper/Krulldimension 1/Aufgabe -->

### Exercise 23.7 {#br-ak-2025-2026-w23-ex-07}

Let $R$ be a principal ideal domain that is not a field. Show that its Krull
dimension is one.

## Exercises for submission {#br-ak-2025-2026-w23-submitted}

<!-- upstream_entity: Numerisches Monoid/Abschätzungen für Multiplizität und Differenzanzahl/5,8,11/bis n ist 5/Aufgabe -->

### Exercise 23.8 (4 points) {#br-ak-2025-2026-w23-ex-08}

For the monoid $M\subseteq\mathbb N$ generated by $5,8,11$, compute the
quantities appearing in the bounds of Lemma 23.8 for $n\le5$.

<!-- upstream_entity: Kommutative Ringtheorie/Ideal mit nur einem einzigen maximalen Oberideal/Restklassenring direkt und nach Lokalisierung/Aufgabe -->

### Exercise 23.9 (3 points) {#br-ak-2025-2026-w23-ex-09}

Let $\mathfrak a\subseteq R$ be an ideal in a commutative ring, and suppose
that the only prime ideal containing $\mathfrak a$ is a maximal ideal
$\mathfrak m$. Show that

$$
R/\mathfrak a
\cong
R_{\mathfrak m}/\mathfrak aR_{\mathfrak m}.
$$

Deduce that, for a maximal ideal $\mathfrak m$ in a Noetherian commutative
ring, there is an isomorphism

$$
R/\mathfrak m^n
\cong
R_{\mathfrak m}/\mathfrak m^nR_{\mathfrak m}
$$

for every $n$.

<!-- upstream_entity: Krulldimension/Algebraisch abgeschlossener Körper/Affine Ebene ist zweidimensional/Aufgabe -->

### Exercise 23.10 (5 points) {#br-ak-2025-2026-w23-ex-10}

Let $K$ be an algebraically closed field and $R=K[X,Y]$ the polynomial ring
in two variables. Show that $R$ has Krull dimension two.

<!-- upstream_entity: Krulldimension/Noethersch/Charakterisierung von nulldimensional/Fakt/Beweis/Aufgabe -->

### Exercise 23.11 (5 points) {#br-ak-2025-2026-w23-ex-11}

Let $R$ be a Noetherian commutative ring. Show that the following statements
are equivalent.

1. $R$ has Krull dimension $0$.
2. $R$ is an Artinian ring.
3. $R$ has finitely many prime ideals, all of which are maximal.
4. There is $n\in\mathbb N$ such that

   $$
   (\mathfrak mR_{\mathfrak m})^n=0
   $$

   for every maximal ideal $\mathfrak m$.
5. The reduction $R_{\mathrm{red}}=R/\sqrt{(0)}$ is a finite product of
   fields.

*Edition note -- correction to the source statement:* In item (4), the
source writes $\mathfrak m^n=0$ in $R$ for every maximal ideal. This is not
equivalent to the other four items: for example, in $R=K\times K$ both
maximal ideals are idempotent rather than nilpotent, although $R$ is
zero-dimensional and Artinian. The edition replaces this with the uniform
local formulation above, which correctly characterises zero-dimensional
Noetherian rings. The “product” in item (5) is also explicitly stated to be
*finite*, as forced by the Noetherian/Artinian condition.

<!-- upstream_entity: Krulldimension/Dimension des Polynomringes ist mindestens eins größer/Aufgabe -->

### Exercise 23.12 (3 points) {#br-ak-2025-2026-w23-ex-12}

Let $R$ be a commutative ring of finite Krull dimension $d$. Show that the
Krull dimension of the polynomial ring $R[X]$ is at least $d+1$.

*Source remark:* Over a Noetherian base ring, passing to the polynomial ring
increases the dimension by exactly one; this stronger result is harder to
prove.