The Stacks project

Lemma 49.11.7. Let $f : Y \to X$ be a morphism of schemes. If $f$ satisfies the equivalent conditions of Lemma 49.11.1 then for every $x \in X$ there exist a $d$ and a commutative diagram

\[ \xymatrix{ Y \ar[d] & V \ar[d] \ar[l] \ar[r] & V_ d \ar[d] \ar[r] & Y_ d \ar[d]^{\pi _ d}\\ X & U \ar[l] \ar[r] & U_ d \ar[r] & X_ d } \]

with the following properties

  1. $U \subset X$ is open, $x \in U$, and $V = f^{-1}(U)$,

  2. $\pi _ d : Y_ d \to X_ d$ is as in Example 49.11.2,

  3. $U_ d \subset X_ d$ is as in Lemma 49.11.3 and $V_ d = \pi _ d^{-1}(U_ d) \subset Y_ d$,

  4. where the middle square is cartesian.

Proof. Choose an affine open neighbourhood $U = \mathop{\mathrm{Spec}}(A) \subset X$ of $x$. Write $V = f^{-1}(U) = \mathop{\mathrm{Spec}}(B)$. Then $B$ is a finite locally free $A$-module and the inclusion $A \subset B$ is a locally direct summand. Thus after shrinking $U$ we can choose a basis $1 = e_1, e_2, \ldots , e_ d$ of $B$ as an $A$-module. 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 $\alpha _{ij}^ l \in A$. By construction $V \cong \mathop{\mathrm{Spec}}(A) \times _{X_ d} Y_ d$. By the definition of $U_ d$ we see that $\mathop{\mathrm{Spec}}(A) \to X_ d$ factors through $U_ d$. This finishes the proof. $\square$

Comments (2)

Comment #7809 by Peng Du on

I think in condition (1), it needs add that U=Spec(A)⊂X is an (affine) open neighbourhood of x.

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