## 10.54 Homomorphisms essentially of finite type

Some simple remarks on localizations of finite type ring maps.

Definition 10.54.1. Let $R \to S$ be a ring map.

1. We say that $R \to S$ is essentially of finite type if $S$ is the localization of an $R$-algebra of finite type.

2. We say that $R \to S$ is essentially of finite presentation if $S$ is the localization of an $R$-algebra of finite presentation.

Lemma 10.54.2. The class of ring maps which are essentially of finite type is preserved under composition. Similarly for essentially of finite presentation.

Proof. Omitted. $\square$

Lemma 10.54.3. The class of ring maps which are essentially of finite type is preserved by base change. Similarly for essentially of finite presentation.

Proof. Omitted. $\square$

Lemma 10.54.4. Let $R \to S$ be a ring map. Assume $S$ is an Artinian local ring with maximal ideal $\mathfrak m$. Then

1. $R \to S$ is finite if and only if $R \to S/\mathfrak m$ is finite,

2. $R \to S$ is of finite type if and only if $R \to S/\mathfrak m$ is of finite type.

3. $R \to S$ is essentially of finite type if and only if the composition $R \to S/\mathfrak m$ is essentially of finite type.

Proof. If $R \to S$ is finite, then $R \to S/\mathfrak m$ is finite by Lemma 10.7.3. Conversely, assume $R \to S/\mathfrak m$ is finite. As $S$ has finite length over itself (Lemma 10.53.6) we can choose a filtration

$0 \subset I_1 \subset \ldots \subset I_ n = S$

by ideals such that $I_ i/I_{i - 1} \cong S/\mathfrak m$ as $S$-modules. Thus $S$ has a filtration by $R$-submodules $I_ i$ such that each successive quotient is a finite $R$-module. Thus $S$ is a finite $R$-module by Lemma 10.5.3.

If $R \to S$ is of finite type, then $R \to S/\mathfrak m$ is of finite type by Lemma 10.6.2. Conversely, assume that $R \to S/\mathfrak m$ is of finite type. Choose $f_1, \ldots , f_ n \in S$ which map to generators of $S/\mathfrak m$. Then $A = R[x_1, \ldots , x_ n] \to S$, $x_ i \mapsto f_ i$ is a ring map such that $A \to S/\mathfrak m$ is surjective (in particular finite). Hence $A \to S$ is finite by part (1) and we see that $R \to S$ is of finite type by Lemma 10.6.2.

If $R \to S$ is essentially of finite type, then $R \to S/\mathfrak m$ is essentially of finite type by Lemma 10.54.2. Conversely, assume that $R \to S/\mathfrak m$ is essentially of finite type. Suppose $S/\mathfrak m$ is the localization of $R[x_1, \ldots , x_ n]/I$. Choose $f_1, \ldots , f_ n \in S$ whose congruence classes modulo $\mathfrak m$ correspond to the congruence classes of $x_1, \ldots , x_ n$ modulo $I$. Consider the map $R[x_1, \ldots , x_ n] \to S$, $x_ i \mapsto f_ i$ with kernel $J$. Set $A = R[x_1, \ldots , x_ n]/J \subset S$ and $\mathfrak p = A \cap \mathfrak m$. Note that $A/\mathfrak p \subset S/\mathfrak m$ is equal to the image of $R[x_1, \ldots , x_ n]/I$ in $S/\mathfrak m$. Hence $\kappa (\mathfrak p) = S/\mathfrak m$. Thus $A_\mathfrak p \to S$ is finite by part (1). We conclude that $S$ is essentially of finite type by Lemma 10.54.2. $\square$

The following lemma can be proven using properness of projective space instead of the algebraic argument we give here.

Lemma 10.54.5. Let $\varphi : R \to S$ be essentially of finite type with $R$ and $S$ local (but not necessarily $\varphi$ local). Then there exists an $n$ and a maximal ideal $\mathfrak m \subset R[x_1, \ldots , x_ n]$ lying over $\mathfrak m_ R$ such that $S$ is a localization of a quotient of $R[x_1, \ldots , x_ n]_\mathfrak m$.

Proof. We can write $S$ as a localization of a quotient of $R[x_1, \ldots , x_ n]$. Hence it suffices to prove the lemma in case $S = R[x_1, \ldots , x_ n]_\mathfrak q$ for some prime $\mathfrak q \subset R[x_1, \ldots , x_ n]$. If $\mathfrak q + \mathfrak m_ R R[x_1, \ldots , x_ n] \not= R[x_1, \ldots , x_ n]$ then we can find a maximal ideal $\mathfrak m$ as in the statement of the lemma with $\mathfrak q \subset \mathfrak m$ and the result is clear.

Choose a valuation ring $A \subset \kappa (\mathfrak q)$ which dominates the image of $R \to \kappa (\mathfrak q)$ (Lemma 10.50.2). If the image $\lambda _ i \in \kappa (\mathfrak q)$ of $x_ i$ is contained in $A$, then $\mathfrak q$ is contained in the inverse image of $\mathfrak m_ A$ via $R[x_1, \ldots , x_ n] \to A$ which means we are back in the preceding case. Hence there exists an $i$ such that $\lambda _ i^{-1} \in A$ and such that $\lambda _ j/\lambda _ i \in A$ for all $j = 1, \ldots , n$ (because the value group of $A$ is totally ordered, see Lemma 10.50.12). Then we consider the map

$R[y_0, y_1, \ldots , \hat{y_ i}, \ldots , y_ n] \to R[x_1, \ldots , x_ n]_\mathfrak q,\quad y_0 \mapsto 1/x_ i,\quad y_ j \mapsto x_ j/x_ i$

Let $\mathfrak q' \subset R[y_0, \ldots , \hat{y_ i}, \ldots , y_ n]$ be the inverse image of $\mathfrak q$. Since $y_0 \not\in \mathfrak q'$ it is easy to see that the displayed arrow defines an isomorphism on localizations. On the other hand, the result of the first paragraph applies to $R[y_0, \ldots , \hat{y_ i}, \ldots , y_ n]$ because $y_ j$ maps to an element of $A$. This finishes the proof. $\square$

Comment #673 by Keenan Kidwell on

In the last sentence of the first paragraph of the proof of 07DT, "finite $S$-module" should be "finite $R$-module."

Comment #2257 by Dario Weißmann on

What is the convention for "localization" in this section? Is it localization at a prime or at a multiplicative closed subset? The proof of the last Lemma (10.53.5) seems to suggest the former.

Comment #2262 by on

Localization, when not otherwise specified, just means inverting some multiplicatively closed subset. But if $S$ is a local ring and if $S$ is the localization of a ring $R$, then of course $S$ is the localization of $R$ at a prime.

Comment #2264 by Dario Weißmann on

Thank you for your answer. I think in the proof of Lemma 10.53.5 the map $R[y_0,y_1,\dots,y_n]\to R[x_1,\dots,x_n]_{\mathfrak{q}}$ does not induce an isomorphism on localization as it is not injective ($y_i\mapsto 1$). If we consider the polynomial ring without the $y_i$ it works fine though. Oh, and there is also a missing comma in $R[y_0,y_1,\dots y_n]$.

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