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97.18 Strong Rim-Schlessinger

In the rest of this chapter the following strictly stronger version of the Rim-Schlessinger conditions will play an important role.

Definition 97.18.1. Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. We say $\mathcal{X}$ satisfies condition (RS*) if given a fibre product diagram

\[ \xymatrix{ B' \ar[r] & B \\ A' = A \times _ B B' \ar[u] \ar[r] & A \ar[u] } \]

of $S$-algebras, with $B' \to B$ surjective with square zero kernel, the functor of fibre categories

\[ \mathcal{X}_{\mathop{\mathrm{Spec}}(A')} \longrightarrow \mathcal{X}_{\mathop{\mathrm{Spec}}(A)} \times _{\mathcal{X}_{\mathop{\mathrm{Spec}}(B)}} \mathcal{X}_{\mathop{\mathrm{Spec}}(B')} \]

is an equivalence of categories.

We make some observations: with $A \to B \leftarrow B'$ as in Definition 97.18.1

  1. we have $\mathop{\mathrm{Spec}}(A') = \mathop{\mathrm{Spec}}(A) \amalg _{\mathop{\mathrm{Spec}}(B)} \mathop{\mathrm{Spec}}(B')$ in the category of schemes, see More on Morphisms, Lemma 37.14.3, and

  2. if $\mathcal{X}$ is an algebraic stack, then $\mathcal{X}$ satisfies (RS*) by Lemma 97.18.2.

If $S$ is locally Noetherian, then

  1. if $A$, $B$, $B'$ are of finite type over $S$ and $B$ is finite over $A$, then $A'$ is of finite type over $S$1, and

  2. if $\mathcal{X}$ satisfies (RS*), then $\mathcal{X}$ satisfies (RS) because (RS) covers exactly those cases of (RS*) where $A$, $B$, $B'$ are Artinian local.

Lemma 97.18.2. Let $\mathcal{X}$ be an algebraic stack over a base $S$. Then $\mathcal{X}$ satisfies (RS*).

Proof. This is implied by Lemma 97.4.1, see remarks following Definition 97.18.1. $\square$

Lemma 97.18.3. Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$ satisfy (RS*), then so does $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$.

Proof. The proof is exactly the same as the proof of Lemma 97.5.3. $\square$

[1] If $\mathop{\mathrm{Spec}}(A)$ maps into an affine open of $S$ this follows from More on Algebra, Lemma 15.5.1. The general case follows using More on Algebra, Lemma 15.5.3.

Comments (2)

Comment #5509 by Martin Olsson on

Is observation (2) the same as Lemma 0CXP?


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