## 10.155 Japanese rings

In this section we being to discuss finiteness of integral closure.

reference
Definition 10.155.1. Let $R$ be a domain with field of fractions $K$.

We say $R$ is *N-1* if the integral closure of $R$ in $K$ is a finite $R$-module.

We say $R$ is *N-2* or *Japanese* if for any finite extension $K \subset L$ of fields the integral closure of $R$ in $L$ is finite over $R$.

The main interest in these notions is for Noetherian rings, but here is a non-Noetherian example.

Example 10.155.2. Let $k$ be a field. The domain $R = k[x_1, x_2, x_3, \ldots ]$ is N-2, but not Noetherian. The reason is the following. Suppose that $R \subset L$ and the field $L$ is a finite extension of the fraction field of $R$. Then there exists an integer $n$ such that $L$ comes from a finite extension $k(x_1, \ldots , x_ n) \subset L_0$ by adjoining the (transcendental) elements $x_{n + 1}, x_{n + 2}$, etc. Let $S_0$ be the integral closure of $k[x_1, \ldots , x_ n]$ in $L_0$. By Proposition 10.156.16 below it is true that $S_0$ is finite over $k[x_1, \ldots , x_ n]$. Moreover, the integral closure of $R$ in $L$ is $S = S_0[x_{n + 1}, x_{n + 2}, \ldots ]$ (use Lemma 10.36.8) and hence finite over $R$. The same argument works for $R = \mathbf{Z}[x_1, x_2, x_3, \ldots ]$.

Lemma 10.155.3. Let $R$ be a domain. If $R$ is N-1 then so is any localization of $R$. Same for N-2.

**Proof.**
These statements hold because taking integral closure commutes with localization, see Lemma 10.35.11.
$\square$

Lemma 10.155.4. Let $R$ be a domain. Let $f_1, \ldots , f_ n \in R$ generate the unit ideal. If each domain $R_{f_ i}$ is N-1 then so is $R$. Same for N-2.

**Proof.**
Assume $R_{f_ i}$ is N-2 (or N-1). Let $L$ be a finite extension of the fraction field of $R$ (equal to the fraction field in the N-1 case). Let $S$ be the integral closure of $R$ in $L$. By Lemma 10.35.11 we see that $S_{f_ i}$ is the integral closure of $R_{f_ i}$ in $L$. Hence $S_{f_ i}$ is finite over $R_{f_ i}$ by assumption. Thus $S$ is finite over $R$ by Lemma 10.22.2.
$\square$

Lemma 10.155.5. Let $R$ be a domain. Let $R \subset S$ be a quasi-finite extension of domains (for example finite). Assume $R$ is N-2 and Noetherian. Then $S$ is N-2.

**Proof.**
Let $L/K$ be the induced extension of fraction fields. Note that this is a finite field extension (for example by Lemma 10.121.2 (2) applied to the fibre $S \otimes _ R K$, and the definition of a quasi-finite ring map). Let $S'$ be the integral closure of $R$ in $S$. Then $S'$ is contained in the integral closure of $R$ in $L$ which is finite over $R$ by assumption. As $R$ is Noetherian this implies $S'$ is finite over $R$. By Lemma 10.122.14 there exist elements $g_1, \ldots , g_ n \in S'$ such that $S'_{g_ i} \cong S_{g_ i}$ and such that $g_1, \ldots , g_ n$ generate the unit ideal in $S$. Hence it suffices to show that $S'$ is N-2 by Lemmas 10.155.3 and 10.155.4. Thus we have reduced to the case where $S$ is finite over $R$.

Assume $R \subset S$ with hypotheses as in the lemma and moreover that $S$ is finite over $R$. Let $M$ be a finite field extension of the fraction field of $S$. Then $M$ is also a finite field extension of $K$ and we conclude that the integral closure $T$ of $R$ in $M$ is finite over $R$. By Lemma 10.35.16 we see that $T$ is also the integral closure of $S$ in $M$ and we win by Lemma 10.35.15.
$\square$

Lemma 10.155.6. Let $R$ be a Noetherian domain. If $R[z, z^{-1}]$ is N-1, then so is $R$.

**Proof.**
Let $R'$ be the integral closure of $R$ in its field of fractions $K$. Let $S'$ be the integral closure of $R[z, z^{-1}]$ in its field of fractions. Clearly $R' \subset S'$. Since $K[z, z^{-1}]$ is a normal domain we see that $S' \subset K[z, z^{-1}]$. Suppose that $f_1, \ldots , f_ n \in S'$ generate $S'$ as $R[z, z^{-1}]$-module. Say $f_ i = \sum a_{ij}z^ j$ (finite sum), with $a_{ij} \in K$. For any $x \in R'$ we can write

\[ x = \sum h_ i f_ i \]

with $h_ i \in R[z, z^{-1}]$. Thus we see that $R'$ is contained in the finite $R$-submodule $\sum Ra_{ij} \subset K$. Since $R$ is Noetherian we conclude that $R'$ is a finite $R$-module.
$\square$

Lemma 10.155.7. Let $R$ be a Noetherian domain, and let $R \subset S$ be a finite extension of domains. If $S$ is N-1, then so is $R$. If $S$ is N-2, then so is $R$.

**Proof.**
Omitted. (Hint: Integral closures of $R$ in extension fields are contained in integral closures of $S$ in extension fields.)
$\square$

Lemma 10.155.8. Let $R$ be a Noetherian normal domain with fraction field $K$. Let $K \subset L$ be a finite separable field extension. Then the integral closure of $R$ in $L$ is finite over $R$.

**Proof.**
Consider the trace pairing (Fields, Definition 9.20.6)

\[ L \times L \longrightarrow K, \quad (x, y) \longmapsto \langle x, y\rangle := \text{Trace}_{L/K}(xy). \]

Since $L/K$ is separable this is nondegenerate (Fields, Lemma 9.20.7). Moreover, if $x \in L$ is integral over $R$, then $\text{Trace}_{L/K}(x)$ is in $R$. This is true because the minimal polynomial of $x$ over $K$ has coefficients in $R$ (Lemma 10.37.6) and because $\text{Trace}_{L/K}(x)$ is an integer multiple of one of these coefficients (Fields, Lemma 9.20.3). Pick $x_1, \ldots , x_ n \in L$ which are integral over $R$ and which form a $K$-basis of $L$. Then the integral closure $S \subset L$ is contained in the $R$-module

\[ M = \{ y \in L \mid \langle x_ i, y\rangle \in R, \ i = 1, \ldots , n\} \]

By linear algebra we see that $M \cong R^{\oplus n}$ as an $R$-module. Hence $S \subset R^{\oplus n}$ is a finitely generated $R$-module as $R$ is Noetherian.
$\square$

Example 10.155.9. Lemma 10.155.8 does not work if the ring is not Noetherian. For example consider the action of $G = \{ +1, -1\} $ on $A = \mathbf{C}[x_1, x_2, x_3, \ldots ]$ where $-1$ acts by mapping $x_ i$ to $-x_ i$. The invariant ring $R = A^ G$ is the $\mathbf{C}$-algebra generated by all $x_ ix_ j$. Hence $R \subset A$ is not finite. But $R$ is a normal domain with fraction field $K = L^ G$ the $G$-invariants in the fraction field $L$ of $A$. And clearly $A$ is the integral closure of $R$ in $L$.

The following lemma can sometimes be used as a substitute for Lemma 10.155.8 in case of purely inseparable extensions.

Lemma 10.155.10. Let $R$ be a Noetherian normal domain with fraction field $K$ of characteristic $p > 0$. Let $a \in K$ be an element such that there exists a derivation $D : R \to R$ with $D(a) \not= 0$. Then the integral closure of $R$ in $L = K[x]/(x^ p - a)$ is finite over $R$.

**Proof.**
After replacing $x$ by $fx$ and $a$ by $f^ pa$ for some $f \in R$ we may assume $a \in R$. Hence also $D(a) \in R$. We will show by induction on $i \leq p - 1$ that if

\[ y = a_0 + a_1x + \ldots + a_ i x^ i,\quad a_ j \in K \]

is integral over $R$, then $D(a)^ i a_ j \in R$. Thus the integral closure is contained in the finite $R$-module with basis $D(a)^{-p + 1}x^ j$, $j = 0, \ldots , p - 1$. Since $R$ is Noetherian this proves the lemma.

If $i = 0$, then $y = a_0$ is integral over $R$ if and only if $a_0 \in R$ and the statement is true. Suppose the statement holds for some $i < p - 1$ and suppose that

\[ y = a_0 + a_1x + \ldots + a_{i + 1} x^{i + 1},\quad a_ j \in K \]

is integral over $R$. Then

\[ y^ p = a_0^ p + a_1^ p a + \ldots + a_{i + 1}^ pa^{i + 1} \]

is an element of $R$ (as it is in $K$ and integral over $R$). Applying $D$ we obtain

\[ (a_1^ p + 2a_2^ p a + \ldots + (i + 1)a_{i + 1}^ p a^ i)D(a) \]

is in $R$. Hence it follows that

\[ D(a)a_1 + 2D(a) a_2 x + \ldots + (i + 1)D(a) a_{i + 1} x^ i \]

is integral over $R$. By induction we find $D(a)^{i + 1}a_ j \in R$ for $j = 1, \ldots , i + 1$. (Here we use that $1, \ldots , i + 1$ are invertible.) Hence $D(a)^{i + 1}a_0$ is also in $R$ because it is the difference of $y$ and $\sum _{j > 0} D(a)^{i + 1}a_ jx^ j$ which are integral over $R$ (since $x$ is integral over $R$ as $a \in R$).
$\square$

Lemma 10.155.11. A Noetherian domain of characteristic zero is N-1 if and only if it is N-2 (i.e., Japanese).

**Proof.**
This is clear from Lemma 10.155.8 since every field extension in characteristic zero is separable.
$\square$

Lemma 10.155.12. Let $R$ be a Noetherian domain with fraction field $K$ of characteristic $p > 0$. Then $R$ is N-2 if and only if for every finite purely inseparable extension $K \subset L$ the integral closure of $R$ in $L$ is finite over $R$.

**Proof.**
Assume the integral closure of $R$ in every finite purely inseparable field extension of $K$ is finite. Let $K \subset L$ be any finite extension. We have to show the integral closure of $R$ in $L$ is finite over $R$. Choose a finite normal field extension $K \subset M$ containing $L$. As $R$ is Noetherian it suffices to show that the integral closure of $R$ in $M$ is finite over $R$. By Fields, Lemma 9.27.3 there exists a subextension $K \subset M_{insep} \subset M$ such that $M_{insep}/K$ is purely inseparable, and $M/M_{insep}$ is separable. By assumption the integral closure $R'$ of $R$ in $M_{insep}$ is finite over $R$. By Lemma 10.155.8 the integral closure $R''$ of $R'$ in $M$ is finite over $R'$. Then $R''$ is finite over $R$ by Lemma 10.7.3. Since $R''$ is also the integral closure of $R$ in $M$ (see Lemma 10.35.16) we win.
$\square$

Lemma 10.155.13. Let $R$ be a Noetherian domain. If $R$ is N-1 then $R[x]$ is N-1. If $R$ is N-2 then $R[x]$ is N-2.

**Proof.**
Assume $R$ is N-1. Let $R'$ be the integral closure of $R$ which is finite over $R$. Hence also $R'[x]$ is finite over $R[x]$. The ring $R'[x]$ is normal (see Lemma 10.36.8), hence N-1. This proves the first assertion.

For the second assertion, by Lemma 10.155.7 it suffices to show that $R'[x]$ is N-2. In other words we may and do assume that $R$ is a normal N-2 domain. In characteristic zero we are done by Lemma 10.155.11. In characteristic $p > 0$ we have to show that the integral closure of $R[x]$ is finite in any finite purely inseparable extension of $L/K(x)$ where $K$ is the fraction field of $R$. There exists a finite purely inseparable field extension $L'/K$ and $q = p^ e$ such that $L \subset L'(x^{1/q})$; some details omitted. As $R[x]$ is Noetherian it suffices to show that the integral closure of $R[x]$ in $L'(x^{1/q})$ is finite over $R[x]$. And this integral closure is equal to $R'[x^{1/q}]$ with $R \subset R' \subset L'$ the integral closure of $R$ in $L'$. Since $R$ is N-2 we see that $R'$ is finite over $R$ and hence $R'[x^{1/q}]$ is finite over $R[x]$.
$\square$

Lemma 10.155.14. Let $R$ be a Noetherian domain. If there exists an $f \in R$ such that $R_ f$ is normal then

\[ U = \{ \mathfrak p \in \mathop{\mathrm{Spec}}(R) \mid R_{\mathfrak p} \text{ is normal}\} \]

is open in $\mathop{\mathrm{Spec}}(R)$.

**Proof.**
It is clear that the standard open $D(f)$ is contained in $U$. By Serre's criterion Lemma 10.151.4 we see that $\mathfrak p \not\in U$ implies that for some $\mathfrak q \subset \mathfrak p$ we have either

Case I: $\text{depth}(R_{\mathfrak q}) < 2$ and $\dim (R_{\mathfrak q}) \geq 2$, and

Case II: $R_{\mathfrak q}$ is not regular and $\dim (R_{\mathfrak q}) = 1$.

This in particular also means that $R_{\mathfrak q}$ is not normal, and hence $f \in \mathfrak q$. In case I we see that $\text{depth}(R_{\mathfrak q}) = \text{depth}(R_{\mathfrak q}/fR_{\mathfrak q}) + 1$. Hence such a prime $\mathfrak q$ is the same thing as an embedded associated prime of $R/fR$. In case II $\mathfrak q$ is an associated prime of $R/fR$ of height 1. Thus there is a finite set $E$ of such primes $\mathfrak q$ (see Lemma 10.62.5) and

\[ \mathop{\mathrm{Spec}}(R) \setminus U = \bigcup \nolimits _{\mathfrak q \in E} V(\mathfrak q) \]

as desired.
$\square$

Lemma 10.155.15. Let $R$ be a Noetherian domain. Assume

there exists a nonzero $f \in R$ such that $R_ f$ is normal, and

for every maximal ideal $\mathfrak m \subset R$ the local ring $R_{\mathfrak m}$ is N-1.

Then $R$ is N-1.

**Proof.**
Let $K$ be the fraction field of $R$. Suppose that $R \subset R' \subset K$ is a finite extension of $R$ contained in $K$. Note that $R_ f = R'_ f$ since $R_ f$ is already normal. Hence by Lemma 10.155.14 the set of primes $\mathfrak p' \in \mathop{\mathrm{Spec}}(R')$ with $R'_{\mathfrak p'}$ non-normal is closed in $\mathop{\mathrm{Spec}}(R')$. Since $\mathop{\mathrm{Spec}}(R') \to \mathop{\mathrm{Spec}}(R)$ is closed the image of this set is closed in $\mathop{\mathrm{Spec}}(R)$. For such a ring $R'$ denote $Z_{R'} \subset \mathop{\mathrm{Spec}}(R)$ this image.

Pick a maximal ideal $\mathfrak m \subset R$. Let $R_{\mathfrak m} \subset R_{\mathfrak m}'$ be the integral closure of the local ring in $K$. By assumption this is a finite ring extension. By Lemma 10.35.11 we can find finitely many elements $r_1, \ldots , r_ n \in K$ integral over $R$ such that $R_{\mathfrak m}'$ is generated by $r_1, \ldots , r_ n$ over $R_{\mathfrak m}$. Let $R' = R[x_1, \ldots , x_ n] \subset K$. With this choice it is clear that $\mathfrak m \not\in Z_{R'}$.

As $\mathop{\mathrm{Spec}}(R)$ is quasi-compact, the above shows that we can find a finite collection $R \subset R'_ i \subset K$ such that $\bigcap Z_{R'_ i} = \emptyset $. Let $R'$ be the subring of $K$ generated by all of these. It is finite over $R$. Also $Z_{R'} = \emptyset $. Namely, every prime $\mathfrak p'$ lies over a prime $\mathfrak p'_ i$ such that $(R'_ i)_{\mathfrak p'_ i}$ is normal. This implies that $R'_{\mathfrak p'} = (R'_ i)_{\mathfrak p'_ i}$ is normal too. Hence $R'$ is normal, in other words $R'$ is the integral closure of $R$ in $K$.
$\square$

reference
Lemma 10.155.16 (Tate). Let $R$ be a ring. Let $x \in R$. Assume

$R$ is a normal Noetherian domain,

$R/xR$ is a domain and N-2,

$R \cong \mathop{\mathrm{lim}}\nolimits _ n R/x^ nR$ is complete with respect to $x$.

Then $R$ is N-2.

**Proof.**
We may assume $x \not= 0$ since otherwise the lemma is trivial. Let $K$ be the fraction field of $R$. If the characteristic of $K$ is zero the lemma follows from (1), see Lemma 10.155.11. Hence we may assume that the characteristic of $K$ is $p > 0$, and we may apply Lemma 10.155.12. Thus given $K \subset L$ be a finite purely inseparable field extension we have to show that the integral closure $S$ of $R$ in $L$ is finite over $R$.

Let $q$ be a power of $p$ such that $L^ q \subset K$. By enlarging $L$ if necessary we may assume there exists an element $y \in L$ such that $y^ q = x$. Since $R \to S$ induces a homeomorphism of spectra (see Lemma 10.45.7) there is a unique prime ideal $\mathfrak q \subset S$ lying over the prime ideal $\mathfrak p = xR$. It is clear that

\[ \mathfrak q = \{ f \in S \mid f^ q \in \mathfrak p\} = yS \]

since $y^ q = x$. Hence $R_{\mathfrak p}$ and $S_{\mathfrak q}$ are discrete valuation rings, see Lemma 10.118.7. By Lemma 10.118.10 we see that $\kappa (\mathfrak p) \subset \kappa (\mathfrak q)$ is a finite field extension. Hence the integral closure $S' \subset \kappa (\mathfrak q)$ of $R/xR$ is finite over $R/xR$ by assumption (2). Since $S/yS \subset S'$ this implies that $S/yS$ is finite over $R$. Note that $S/y^ nS$ has a finite filtration whose subquotients are the modules $y^ iS/y^{i + 1}S \cong S/yS$. Hence we see that each $S/y^ nS$ is finite over $R$. In particular $S/xS$ is finite over $R$. Also, it is clear that $\bigcap x^ nS = (0)$ since an element in the intersection has $q$th power contained in $\bigcap x^ nR = (0)$ (Lemma 10.50.4). Thus we may apply Lemma 10.95.12 to conclude that $S$ is finite over $R$, and we win.
$\square$

Lemma 10.155.17. Let $R$ be a ring. If $R$ is Noetherian, a domain, and N-2, then so is $R[[x]]$.

**Proof.**
Observe that $R[[x]]$ is Noetherian by Lemma 10.30.2. Let $R' \supset R$ be the integral closure of $R$ in its fraction field. Because $R$ is N-2 this is finite over $R$. Hence $R'[[x]]$ is finite over $R[[x]]$. By Lemma 10.36.9 we see that $R'[[x]]$ is a normal domain. Apply Lemma 10.155.16 to the element $x \in R'[[x]]$ to see that $R'[[x]]$ is N-2. Then Lemma 10.155.7 shows that $R[[x]]$ is N-2.
$\square$

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