The Stacks project

Lemma 39.23.8. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume

  1. $U = \mathop{\mathrm{Spec}}(A)$, and $R = \mathop{\mathrm{Spec}}(B)$ are affine, and

  2. there exist elements $x_ i \in A$, $i \in I$ such that $B = \bigoplus _{i \in I} s^\sharp (A)t^\sharp (x_ i)$.

Then $A = \bigoplus _{i\in I} Cx_ i$, and $B \cong A \otimes _ C A$ where $C \subset A$ is the $R$-invariant functions on $U$ as in (39.23.0.1).

Proof. During this proof we will write $s, t : A \to B$ instead of $s^\sharp , t^\sharp $, and similarly $c : B \to B \otimes _{s, A, t} B$. We write $p_0 : B \to B \otimes _{s, A, t} B$, $b \mapsto b \otimes 1$ and $p_1 : B \to B \otimes _{s, A, t} B$, $b \mapsto 1 \otimes b$. By Lemma 39.13.5 and the definition of $C$ we have the following commutative diagram

\[ \xymatrix{ B \otimes _{s, A, t} B & B \ar@<-1ex>[l]_-c \ar@<1ex>[l]^-{p_0} & A \ar[l]^ t \\ B \ar[u]^{p_1} & A \ar@<-1ex>[l]_ s \ar@<1ex>[l]^ t \ar[u]_ s & C \ar[u] \ar[l] } \]

Moreover the tow left squares are cocartesian in the category of rings, and the top row is isomorphic to the diagram

\[ \xymatrix{ B \otimes _{t, A, t} B & B \ar@<-1ex>[l]_-{p_1} \ar@<1ex>[l]^-{p_0} & A \ar[l]^ t } \]

which is an equalizer diagram according to Descent, Lemma 35.3.6 because condition (2) implies in particular that $s$ (and hence also then isomorphic arrow $t$) is faithfully flat. The lower row is an equalizer diagram by definition of $C$. We can use the $x_ i$ and get a commutative diagram

\[ \xymatrix{ B \otimes _{s, A, t} B & B \ar@<-1ex>[l]_-c \ar@<1ex>[l]^-{p_0} & A \ar[l]^ t \\ \bigoplus _{i \in I} B x_ i \ar[u]^{p_1} & \bigoplus _{i \in I} A x_ i \ar@<-1ex>[l]_ s \ar@<1ex>[l]^ t \ar[u]_ s & \bigoplus _{i \in I} C x_ i \ar[u] \ar[l] } \]

where in the right vertical arrow we map $x_ i$ to $x_ i$, in the middle vertical arrow we map $x_ i$ to $t(x_ i)$ and in the left vertical arrow we map $x_ i$ to $c(t(x_ i)) = t(x_ i) \otimes 1 = p_0(t(x_ i))$ (equality by the commutativity of the top part of the diagram in Lemma 39.13.4). Then the diagram commutes. Moreover the middle vertical arrow is an isomorphism by assumption. Since the left two squares are cocartesian we conclude that also the left vertical arrow is an isomorphism. On the other hand, the horizontal rows are exact (i.e., they are equalizers). Hence we conclude that also the right vertical arrow is an isomorphism. $\square$

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