Lemma 49.11.4. With notation as in Example 49.11.2 and $U_ d$ as in Lemma 49.11.3 then $U_ d$ is smooth over $\mathop{\mathrm{Spec}}(\mathbf{Z})$.

**Proof.**
Let us use More on Morphisms, Lemma 37.12.1 to show that $U_ d \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is smooth. Namely, suppose that $\mathop{\mathrm{Spec}}(A) \to U_ d$ is a morphism and $A' \to A$ is a small extension. Then $B = A \otimes _{A_ d} B_ d$ is a finite free $A$-algebra which is syntomic over $A$ (by construction of $U_ d$). By Smoothing Ring Maps, Proposition 16.3.2 there exists a syntomic ring map $A' \to B'$ such that $B \cong B' \otimes _{A'} A$. Set $e'_1 = 1 \in B'$. For $1 < i \leq d$ choose lifts $e'_ i \in B'$ of the elements $1 \otimes e_ i \in A \otimes _{A_ d} B_ d = B$. Then $e'_1, \ldots , e'_ d$ is a basis for $B'$ over $A'$ (for example see Algebra, Lemma 10.101.1). Thus we can write $e'_ i e'_ j = \sum \alpha _{ij}^ l e'_ l$ for unique elements $\alpha _{ij}^ l \in A'$ which satisfy the relations $\sum _ l \alpha _{ij}^ l \alpha _{lk}^ m = \sum _ l \alpha _{il}^ m \alpha _{jk}^ l$ and $\alpha _{ij}^ k = \alpha _{ji}^ k$ and $\alpha _{i1}^ j - \delta _{ij}$ in $A'$. This determines a morphism $\mathop{\mathrm{Spec}}(A') \to X_ d$ by sending $a_{ij}^ l \in A_ d$ to $\alpha _{ij}^ l \in A'$. This morphism agrees with the given morphism $\mathop{\mathrm{Spec}}(A) \to U_ d$. Since $\mathop{\mathrm{Spec}}(A')$ and $\mathop{\mathrm{Spec}}(A)$ have the same underlying topological space, we see that we obtain the desired lift $\mathop{\mathrm{Spec}}(A') \to U_ d$ and we conclude that $U_ d$ is smooth over $\mathbf{Z}$.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)