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Tag 0395

Chapter 10: Commutative Algebra > Section 10.147: Unramified ring maps

Proposition 10.147.9. Let $R \to S$ be a ring map. Let $\mathfrak q \subset S$ be a prime. If $R \to S$ is unramified at $\mathfrak q$, then there exist

  1. a $g \in S$, $g \not \in \mathfrak q$,
  2. a standard étale ring map $R \to S'$, and
  3. a surjective $R$-algebra map $S' \to S_g$.

Proof. This proof is the ''same'' as the proof of Proposition 10.141.16. The proof is a little roundabout and there may be ways to shorten it.

Step 1. By Definition 10.147.1 there exists a $g \in S$, $g \not \in \mathfrak q$ such that $R \to S_g$ is unramified. Thus we may assume that $S$ is unramified over $R$.

Step 2. By Lemma 10.147.3 there exists an unramified ring map $R_0 \to S_0$ with $R_0$ of finite type over $\mathbf{Z}$, and a ring map $R_0 \to R$ such that $S$ is a quotient of $R \otimes_{R_0} S_0$. Denote $\mathfrak q_0$ the prime of $S_0$ corresponding to $\mathfrak q$. If we show the result for $(R_0 \to S_0, \mathfrak q_0)$ then the result follows for $(R \to S, \mathfrak q)$ by base change. Hence we may assume that $R$ is Noetherian.

Step 3. Note that $R \to S$ is quasi-finite by Lemma 10.147.6. By Lemma 10.122.15 there exists a finite ring map $R \to S'$, an $R$-algebra map $S' \to S$, an element $g' \in S'$ such that $g' \not \in \mathfrak q$ such that $S' \to S$ induces an isomorphism $S'_{g'} \cong S_{g'}$. (Note that $S'$ may not unramified over $R$.) Thus we may assume that (a) $R$ is Noetherian, (b) $R \to S$ is finite and (c) $R \to S$ is unramified at $\mathfrak q$ (but no longer necessarily unramified at all primes).

Step 4. Let $\mathfrak p \subset R$ be the prime corresponding to $\mathfrak q$. Consider the fibre ring $S \otimes_R \kappa(\mathfrak p)$. This is a finite algebra over $\kappa(\mathfrak p)$. Hence it is Artinian (see Lemma 10.52.2) and so a finite product of local rings $$ S \otimes_R \kappa(\mathfrak p) = \prod\nolimits_{i = 1}^n A_i $$ see Proposition 10.59.6. One of the factors, say $A_1$, is the local ring $S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}$ which is isomorphic to $\kappa(\mathfrak q)$, see Lemma 10.147.5. The other factors correspond to the other primes, say $\mathfrak q_2, \ldots, \mathfrak q_n$ of $S$ lying over $\mathfrak p$.

Step 5. We may choose a nonzero element $\alpha \in \kappa(\mathfrak q)$ which generates the finite separable field extension $\kappa(\mathfrak p) \subset \kappa(\mathfrak q)$ (so even if the field extension is trivial we do not allow $\alpha = 0$). Note that for any $\lambda \in \kappa(\mathfrak p)^*$ the element $\lambda \alpha$ also generates $\kappa(\mathfrak q)$ over $\kappa(\mathfrak p)$. Consider the element $$ \overline{t} = (\alpha, 0, \ldots, 0) \in \prod\nolimits_{i = 1}^n A_i = S \otimes_R \kappa(\mathfrak p). $$ After possibly replacing $\alpha$ by $\lambda \alpha$ as above we may assume that $\overline{t}$ is the image of $t \in S$. Let $I \subset R[x]$ be the kernel of the $R$-algebra map $R[x] \to S$ which maps $x$ to $t$. Set $S' = R[x]/I$, so $S' \subset S$. Here is a diagram $$ \xymatrix{ R[x] \ar[r] & S' \ar[r] & S \\ R \ar[u] \ar[ru] \ar[rru] & & } $$ By construction the primes $\mathfrak q_j$, $j \geq 2$ of $S$ all lie over the prime $(\mathfrak p, x)$ of $R[x]$, whereas the prime $\mathfrak q$ lies over a different prime of $R[x]$ because $\alpha \not = 0$.

Step 6. Denote $\mathfrak q' \subset S'$ the prime of $S'$ corresponding to $\mathfrak q$. By the above $\mathfrak q$ is the only prime of $S$ lying over $\mathfrak q'$. Thus we see that $S_{\mathfrak q} = S_{\mathfrak q'}$, see Lemma 10.40.11 (we have going up for $S' \to S$ by Lemma 10.35.22 since $S' \to S$ is finite as $R \to S$ is finite). It follows that $S'_{\mathfrak q'} \to S_{\mathfrak q}$ is finite and injective as the localization of the finite injective ring map $S' \to S$. Consider the maps of local rings $$ R_{\mathfrak p} \to S'_{\mathfrak q'} \to S_{\mathfrak q} $$ The second map is finite and injective. We have $S_{\mathfrak q}/\mathfrak pS_{\mathfrak q} = \kappa(\mathfrak q)$, see Lemma 10.147.5. Hence a fortiori $S_{\mathfrak q}/\mathfrak q'S_{\mathfrak q} = \kappa(\mathfrak q)$. Since $$ \kappa(\mathfrak p) \subset \kappa(\mathfrak q') \subset \kappa(\mathfrak q) $$ and since $\alpha$ is in the image of $\kappa(\mathfrak q')$ in $\kappa(\mathfrak q)$ we conclude that $\kappa(\mathfrak q') = \kappa(\mathfrak q)$. Hence by Nakayama's Lemma 10.19.1 applied to the $S'_{\mathfrak q'}$-module map $S'_{\mathfrak q'} \to S_{\mathfrak q}$, the map $S'_{\mathfrak q'} \to S_{\mathfrak q}$ is surjective. In other words, $S'_{\mathfrak q'} \cong S_{\mathfrak q}$.

Step 7. By Lemma 10.125.7 there exist $g \in S$, $g \not \in \mathfrak q$ and $g' \in S'$, $g' \not \in \mathfrak q'$ such that $S'_{g'} \cong S_g$. As $R$ is Noetherian the ring $S'$ is finite over $R$ because it is an $R$-submodule of the finite $R$-module $S$. Hence after replacing $S$ by $S'$ we may assume that (a) $R$ is Noetherian, (b) $S$ finite over $R$, (c) $S$ is unramified over $R$ at $\mathfrak q$, and (d) $S = R[x]/I$.

Step 8. Consider the ring $S \otimes_R \kappa(\mathfrak p) = \kappa(\mathfrak p)[x]/\overline{I}$ where $\overline{I} = I \cdot \kappa(\mathfrak p)[x]$ is the ideal generated by $I$ in $\kappa(\mathfrak p)[x]$. As $\kappa(\mathfrak p)[x]$ is a PID we know that $\overline{I} = (\overline{h})$ for some monic $\overline{h} \in \kappa(\mathfrak p)$. After replacing $\overline{h}$ by $\lambda \cdot \overline{h}$ for some $\lambda \in \kappa(\mathfrak p)$ we may assume that $\overline{h}$ is the image of some $h \in R[x]$. (The problem is that we do not know if we may choose $h$ monic.) Also, as in Step 4 we know that $S \otimes_R \kappa(\mathfrak p) = A_1 \times \ldots \times A_n$ with $A_1 = \kappa(\mathfrak q)$ a finite separable extension of $\kappa(\mathfrak p)$ and $A_2, \ldots, A_n$ local. This implies that $$ \overline{h} = \overline{h}_1 \overline{h}_2^{e_2} \ldots \overline{h}_n^{e_n} $$ for certain pairwise coprime irreducible monic polynomials $\overline{h}_i \in \kappa(\mathfrak p)[x]$ and certain $e_2, \ldots, e_n \geq 1$. Here the numbering is chosen so that $A_i = \kappa(\mathfrak p)[x]/(\overline{h}_i^{e_i})$ as $\kappa(\mathfrak p)[x]$-algebras. Note that $\overline{h}_1$ is the minimal polynomial of $\alpha \in \kappa(\mathfrak q)$ and hence is a separable polynomial (its derivative is prime to itself).

Step 9. Let $m \in I$ be a monic element; such an element exists because the ring extension $R \to R[x]/I$ is finite hence integral. Denote $\overline{m}$ the image in $\kappa(\mathfrak p)[x]$. We may factor $$ \overline{m} = \overline{k} \overline{h}_1^{d_1} \overline{h}_2^{d_2} \ldots \overline{h}_n^{d_n} $$ for some $d_1 \geq 1$, $d_j \geq e_j$, $j = 2, \ldots, n$ and $\overline{k} \in \kappa(\mathfrak p)[x]$ prime to all the $\overline{h}_i$. Set $f = m^l + h$ where $l \deg(m) > \deg(h)$, and $l \geq 2$. Then $f$ is monic as a polynomial over $R$. Also, the image $\overline{f}$ of $f$ in $\kappa(\mathfrak p)[x]$ factors as $$ \overline{f} = \overline{h}_1 \overline{h}_2^{e_2} \ldots \overline{h}_n^{e_n} + \overline{k}^l \overline{h}_1^{ld_1} \overline{h}_2^{ld_2} \ldots \overline{h}_n^{ld_n} = \overline{h}_1(\overline{h}_2^{e_2} \ldots \overline{h}_n^{e_n} + \overline{k}^l \overline{h}_1^{ld_1 - 1} \overline{h}_2^{ld_2} \ldots \overline{h}_n^{ld_n}) = \overline{h}_1 \overline{w} $$ with $\overline{w}$ a polynomial relatively prime to $\overline{h}_1$. Set $g = f'$ (the derivative with respect to $x$).

Step 10. The ring map $R[x] \to S = R[x]/I$ has the properties: (1) it maps $f$ to zero, and (2) it maps $g$ to an element of $S \setminus \mathfrak q$. The first assertion is clear since $f$ is an element of $I$. For the second assertion we just have to show that $g$ does not map to zero in $\kappa(\mathfrak q) = \kappa(\mathfrak p)[x]/(\overline{h}_1)$. The image of $g$ in $\kappa(\mathfrak p)[x]$ is the derivative of $\overline{f}$. Thus (2) is clear because $$ \overline{g} = \frac{\text{d}\overline{f}}{\text{d}x} = \overline{w}\frac{\text{d}\overline{h}_1}{\text{d}x} + \overline{h}_1\frac{\text{d}\overline{w}}{\text{d}x}, $$ $\overline{w}$ is prime to $\overline{h}_1$ and $\overline{h}_1$ is separable.

Step 11. We conclude that $\varphi : R[x]/(f) \to S$ is a surjective ring map, $R[x]_g/(f)$ is étale over $R$ (because it is standard étale, see Lemma 10.141.14) and $\varphi(g) \not \in \mathfrak q$. Thus the map $(R[x]/(f))_g \to S_{\varphi(g)}$ is the desired surjection. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 39517–39526 (see updates for more information).

    \begin{proposition}
    \label{proposition-unramified-locally-standard}
    Let $R \to S$ be a ring map. Let $\mathfrak q \subset S$ be a prime.
    If $R \to S$ is unramified at $\mathfrak q$, then there exist
    \begin{enumerate}
    \item a $g \in S$, $g \not \in \mathfrak q$,
    \item a standard \'etale ring map $R \to S'$, and
    \item a surjective $R$-algebra map $S' \to S_g$.
    \end{enumerate}
    \end{proposition}
    
    \begin{proof}
    This proof is the ``same'' as the proof of
    Proposition \ref{proposition-etale-locally-standard}.
    The proof is a little roundabout and there may be ways to
    shorten it.
    
    \medskip\noindent
    Step 1. By Definition \ref{definition-unramified}
    there exists a $g \in S$, $g \not \in \mathfrak q$
    such that $R \to S_g$ is unramified. Thus we may assume that $S$ is
    unramified over $R$.
    
    \medskip\noindent
    Step 2. By Lemma \ref{lemma-unramified}
    there exists an unramified ring map $R_0 \to S_0$
    with $R_0$ of finite type over $\mathbf{Z}$, and a ring map
    $R_0 \to R$ such that $S$ is a quotient of $R \otimes_{R_0} S_0$. Denote
    $\mathfrak q_0$ the prime of $S_0$ corresponding to $\mathfrak q$.
    If we show the result for $(R_0 \to S_0, \mathfrak q_0)$ then the
    result follows for $(R \to S, \mathfrak q)$ by base change. Hence
    we may assume that $R$ is Noetherian.
    
    \medskip\noindent
    Step 3.
    Note that $R \to S$ is quasi-finite by
    Lemma \ref{lemma-unramified-quasi-finite}.
    By Lemma \ref{lemma-quasi-finite-open-integral-closure}
    there exists a finite ring map $R \to S'$, an $R$-algebra map
    $S' \to S$, an element $g' \in S'$ such that
    $g' \not \in \mathfrak q$ such that $S' \to S$ induces
    an isomorphism $S'_{g'} \cong S_{g'}$.
    (Note that $S'$ may not unramified over $R$.)
    Thus we may assume that (a) $R$ is Noetherian, (b) $R \to S$ is finite
    and (c) $R \to S$ is unramified at $\mathfrak q$
    (but no longer necessarily unramified at all primes).
    
    \medskip\noindent
    Step 4. Let $\mathfrak p \subset R$ be the prime corresponding
    to $\mathfrak q$. Consider the fibre ring
    $S \otimes_R \kappa(\mathfrak p)$. This is a finite algebra over
    $\kappa(\mathfrak p)$. Hence it is Artinian
    (see Lemma \ref{lemma-finite-dimensional-algebra}) and
    so a finite product of local rings
    $$
    S \otimes_R \kappa(\mathfrak p) = \prod\nolimits_{i = 1}^n A_i
    $$
    see Proposition \ref{proposition-dimension-zero-ring}. One of the factors,
    say $A_1$, is the local ring $S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}$
    which is isomorphic to $\kappa(\mathfrak q)$,
    see Lemma \ref{lemma-unramified-at-prime}. The other factors correspond to
    the other primes, say $\mathfrak q_2, \ldots, \mathfrak q_n$ of
    $S$ lying over $\mathfrak p$.
    
    \medskip\noindent
    Step 5. We may choose a nonzero element $\alpha \in \kappa(\mathfrak q)$ which
    generates the finite separable field extension
    $\kappa(\mathfrak p) \subset \kappa(\mathfrak q)$ (so even if the
    field extension is trivial we do not allow $\alpha = 0$).
    Note that for any $\lambda \in \kappa(\mathfrak p)^*$ the
    element $\lambda \alpha$ also generates $\kappa(\mathfrak q)$
    over $\kappa(\mathfrak p)$. Consider the element
    $$
    \overline{t} =
    (\alpha, 0, \ldots, 0) \in
    \prod\nolimits_{i = 1}^n A_i =
    S \otimes_R \kappa(\mathfrak p).
    $$
    After possibly replacing $\alpha$ by $\lambda \alpha$ as above
    we may assume that $\overline{t}$ is the image of $t \in S$.
    Let $I \subset R[x]$ be the kernel of the $R$-algebra
    map $R[x] \to S$ which maps $x$ to $t$. Set $S' = R[x]/I$,
    so $S' \subset S$. Here is a diagram
    $$
    \xymatrix{
    R[x] \ar[r] & S' \ar[r] & S \\
    R \ar[u] \ar[ru] \ar[rru] & &
    }
    $$
    By construction the primes $\mathfrak q_j$, $j \geq 2$ of $S$ all
    lie over the prime $(\mathfrak p, x)$ of $R[x]$, whereas
    the prime $\mathfrak q$ lies over a different prime of $R[x]$
    because $\alpha \not = 0$.
    
    \medskip\noindent
    Step 6. Denote $\mathfrak q' \subset S'$ the prime of $S'$
    corresponding to $\mathfrak q$. By the above $\mathfrak q$ is
    the only prime of $S$ lying over $\mathfrak q'$. Thus we see that
    $S_{\mathfrak q} = S_{\mathfrak q'}$, see
    Lemma \ref{lemma-unique-prime-over-localize-below} (we have
    going up for $S' \to S$ by Lemma \ref{lemma-integral-going-up}
    since $S' \to S$ is finite as $R \to S$ is finite).
    It follows that $S'_{\mathfrak q'} \to S_{\mathfrak q}$ is finite
    and injective as the localization of the finite injective ring map
    $S' \to S$. Consider the maps of local rings
    $$
    R_{\mathfrak p} \to S'_{\mathfrak q'} \to S_{\mathfrak q}
    $$
    The second map is finite and injective. We have
    $S_{\mathfrak q}/\mathfrak pS_{\mathfrak q} = \kappa(\mathfrak q)$,
    see Lemma \ref{lemma-unramified-at-prime}.
    Hence a fortiori
    $S_{\mathfrak q}/\mathfrak q'S_{\mathfrak q} = \kappa(\mathfrak q)$.
    Since
    $$
    \kappa(\mathfrak p) \subset \kappa(\mathfrak q') \subset \kappa(\mathfrak q)
    $$
    and since $\alpha$ is in the image of $\kappa(\mathfrak q')$ in
    $\kappa(\mathfrak q)$
    we conclude that $\kappa(\mathfrak q') = \kappa(\mathfrak q)$.
    Hence by Nakayama's Lemma \ref{lemma-NAK} applied to the
    $S'_{\mathfrak q'}$-module map $S'_{\mathfrak q'} \to S_{\mathfrak q}$,
    the map $S'_{\mathfrak q'} \to S_{\mathfrak q}$ is surjective.
    In other words,
    $S'_{\mathfrak q'} \cong S_{\mathfrak q}$.
    
    \medskip\noindent
    Step 7. By Lemma \ref{lemma-isomorphic-local-rings} there exist
    $g \in S$, $g \not \in \mathfrak q$ and
    $g' \in S'$, $g' \not \in \mathfrak q'$ such that $S'_{g'} \cong S_g$.
    As $R$ is Noetherian the ring $S'$ is finite over $R$
    because it is an $R$-submodule
    of the finite $R$-module $S$. Hence after replacing $S$ by $S'$ we may
    assume that (a) $R$ is Noetherian, (b) $S$ finite over $R$, (c)
    $S$ is unramified over $R$ at $\mathfrak q$, and (d) $S = R[x]/I$.
    
    \medskip\noindent
    Step 8. Consider the ring
    $S \otimes_R \kappa(\mathfrak p) = \kappa(\mathfrak p)[x]/\overline{I}$
    where $\overline{I} = I \cdot \kappa(\mathfrak p)[x]$ is the ideal generated
    by $I$ in $\kappa(\mathfrak p)[x]$. As $\kappa(\mathfrak p)[x]$ is a PID
    we know that $\overline{I} = (\overline{h})$ for some monic
    $\overline{h} \in \kappa(\mathfrak p)$. After replacing $\overline{h}$
    by $\lambda \cdot \overline{h}$ for some $\lambda \in \kappa(\mathfrak p)$
    we may assume that $\overline{h}$ is the image of some $h \in R[x]$.
    (The problem is that we do not know if we may choose $h$ monic.)
    Also, as in Step 4 we know that
    $S \otimes_R \kappa(\mathfrak p) = A_1 \times \ldots \times A_n$ with
    $A_1 = \kappa(\mathfrak q)$ a finite separable extension of
    $\kappa(\mathfrak p)$ and $A_2, \ldots, A_n$ local. This implies
    that
    $$
    \overline{h} = \overline{h}_1 \overline{h}_2^{e_2} \ldots \overline{h}_n^{e_n}
    $$
    for certain pairwise coprime irreducible monic polynomials
    $\overline{h}_i \in \kappa(\mathfrak p)[x]$ and certain
    $e_2, \ldots, e_n \geq 1$. Here the numbering is chosen so that
    $A_i = \kappa(\mathfrak p)[x]/(\overline{h}_i^{e_i})$ as
    $\kappa(\mathfrak p)[x]$-algebras. Note that $\overline{h}_1$ is
    the minimal polynomial of $\alpha \in \kappa(\mathfrak q)$ and hence
    is a separable polynomial (its derivative is prime to itself).
    
    \medskip\noindent
    Step 9. Let $m \in I$ be a monic element; such an element exists
    because the ring extension $R \to R[x]/I$ is finite hence integral.
    Denote $\overline{m}$ the image in $\kappa(\mathfrak p)[x]$.
    We may factor
    $$
    \overline{m} = \overline{k}
    \overline{h}_1^{d_1} \overline{h}_2^{d_2} \ldots \overline{h}_n^{d_n}
    $$
    for some $d_1 \geq 1$, $d_j \geq e_j$, $j = 2, \ldots, n$ and
    $\overline{k} \in \kappa(\mathfrak p)[x]$ prime to all the $\overline{h}_i$.
    Set $f = m^l + h$ where $l \deg(m) > \deg(h)$, and $l \geq 2$.
    Then $f$ is monic as a polynomial over $R$. Also, the image $\overline{f}$
    of $f$ in $\kappa(\mathfrak p)[x]$ factors as
    $$
    \overline{f} =
    \overline{h}_1 \overline{h}_2^{e_2} \ldots \overline{h}_n^{e_n}
    +
    \overline{k}^l \overline{h}_1^{ld_1} \overline{h}_2^{ld_2}
    \ldots \overline{h}_n^{ld_n}
    =
    \overline{h}_1(\overline{h}_2^{e_2} \ldots \overline{h}_n^{e_n}
    +
    \overline{k}^l
    \overline{h}_1^{ld_1 - 1} \overline{h}_2^{ld_2} \ldots \overline{h}_n^{ld_n})
    = \overline{h}_1 \overline{w}
    $$
    with $\overline{w}$ a polynomial relatively prime to $\overline{h}_1$.
    Set $g = f'$ (the derivative with respect to $x$).
    
    \medskip\noindent
    Step 10. The ring map $R[x] \to S = R[x]/I$ has the properties:
    (1) it maps $f$ to zero, and
    (2) it maps $g$ to an element of $S \setminus \mathfrak q$.
    The first assertion is clear since $f$ is an element of $I$.
    For the second assertion we just have to show that $g$ does
    not map to zero in
    $\kappa(\mathfrak q) = \kappa(\mathfrak p)[x]/(\overline{h}_1)$.
    The image of $g$ in $\kappa(\mathfrak p)[x]$ is the derivative
    of $\overline{f}$. Thus (2) is clear because
    $$
    \overline{g} =
    \frac{\text{d}\overline{f}}{\text{d}x} =
    \overline{w}\frac{\text{d}\overline{h}_1}{\text{d}x} +
    \overline{h}_1\frac{\text{d}\overline{w}}{\text{d}x},
    $$
    $\overline{w}$ is prime to $\overline{h}_1$ and
    $\overline{h}_1$ is separable.
    
    \medskip\noindent
    Step 11.
    We conclude that $\varphi : R[x]/(f) \to S$ is a surjective ring map,
    $R[x]_g/(f)$ is \'etale over $R$ (because it is standard \'etale,
    see Lemma \ref{lemma-standard-etale}) and $\varphi(g) \not \in \mathfrak q$.
    Thus the map $(R[x]/(f))_g \to S_{\varphi(g)}$ is the desired
    surjection.
    \end{proof}

    Comments (1)

    Comment #3274 by Dario WeiƟmann on April 11, 2018 a 5:17 pm UTC

    Typo in step 3: 'Note that $S'$ may not (be) unramified over $R$.'

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