## 40.6 Properties of groupoids

Let $(U, R, s, t, c)$ be a groupoid scheme. The idea behind the results in this section is that $s: R \to U$ is a base change of the morphism $U \to [U/R]$ (see Diagram (40.1.0.1). Hence the local properties of $s : R \to U$ should reflect local properties of the morphism $U \to [U/R]$. This doesn't work, because $[U/R]$ is not always an algebraic stack, and hence we cannot speak of geometric or algebraic properties of $U \to [U/R]$. But it turns out that we can make some of it work without even referring to the quotient stack at all.

Here is a first example of such a result. The open $W \subset U'$ found in the lemma is roughly speaking the locus where the morphism $U' \to [U/R]$ has property $\mathcal{P}$.

Lemma 40.6.1. Let $S$ be a scheme. Let $(U, R, s, t, c, e, i)$ be a groupoid over $S$. Let $g : U' \to U$ be a morphism of schemes. Denote $h$ the composition

$\xymatrix{ h : U' \times _{g, U, t} R \ar[r]_-{\text{pr}_1} & R \ar[r]_ s & U. }$

Let $\mathcal{P}, \mathcal{Q}, \mathcal{R}$ be properties of morphisms of schemes. Assume

1. $\mathcal{R} \Rightarrow \mathcal{Q}$,

2. $\mathcal{Q}$ is preserved under base change and composition,

3. for any morphism $f : X \to Y$ which has $\mathcal{Q}$ there exists a largest open $W(\mathcal{P}, f) \subset X$ such that $f|_{W(\mathcal{P}, f)}$ has $\mathcal{P}$, and

4. for any morphism $f : X \to Y$ which has $\mathcal{Q}$, and any morphism $Y' \to Y$ which has $\mathcal{R}$ we have $Y' \times _ Y W(\mathcal{P}, f) = W(\mathcal{P}, f')$, where $f' : X_{Y'} \to Y'$ is the base change of $f$.

If $s, t$ have $\mathcal{R}$ and $g$ has $\mathcal{Q}$, then there exists an open subscheme $W \subset U'$ such that $W \times _{g, U, t} R = W(\mathcal{P}, h)$.

Proof. Note that the following diagram is commutative

$\xymatrix{ U' \times _{g, U, t} R \times _{t, U, t} R \ar[rr]_-{\text{pr}_{12}} \ar@<1ex>[d]^-{\text{pr}_{02}} \ar@<-1ex>[d]_-{\text{pr}_{01}} & & R \times _{t, U, t} R \ar@<1ex>[d]^-{\text{pr}_1} \ar@<-1ex>[d]_-{\text{pr}_0} \\ U' \times _{g, U, t} R \ar[rr]^{\text{pr}_1} & & R }$

with both squares cartesian (this uses that the two maps $t \circ \text{pr}_ i : R \times _{t, U, t} R \to U$ are equal). Combining this with the properties of diagram (40.3.0.2) we get a commutative diagram

$\xymatrix{ U' \times _{g, U, t} R \times _{t, U, t} R \ar[rr]_-{c \circ (i, 1)} \ar@<1ex>[d]^-{\text{pr}_{02}} \ar@<-1ex>[d]_-{\text{pr}_{01}} & & R \ar@<1ex>[d]^-{s} \ar@<-1ex>[d]_-{t} \\ U' \times _{g, U, t} R \ar[rr]^ h & & U }$

where both squares are cartesian.

Assume $s, t$ have $\mathcal{R}$ and $g$ has $\mathcal{Q}$. Then $h$ has $\mathcal{Q}$ as a composition of $s$ (which has $\mathcal{R}$ hence $\mathcal{Q}$) and a base change of $g$ (which has $\mathcal{Q}$). Thus $W(\mathcal{P}, h) \subset U' \times _{g, U, t} R$ exists. By our assumptions we have $\text{pr}_{01}^{-1}(W(\mathcal{P}, h)) = \text{pr}_{02}^{-1}(W(\mathcal{P}, h))$ since both are the largest open on which $c \circ (i, 1)$ has $\mathcal{P}$. Note that the projection $U' \times _{g, U, t} R \to U'$ has a section, namely $\sigma : U' \to U' \times _{g, U, t} R$, $u' \mapsto (u', e(g(u')))$. Also via the isomorphism

$(U' \times _{g, U, t} R) \times _{U'} (U' \times _{g, U, t} R) = U' \times _{g, U, t} R \times _{t, U, t} R$

the two projections of the left hand side to $U' \times _{g, U, t} R$ agree with the morphisms $\text{pr}_{01}$ and $\text{pr}_{02}$ on the right hand side. Since $\text{pr}_{01}^{-1}(W(\mathcal{P}, h)) = \text{pr}_{02}^{-1}(W(\mathcal{P}, h))$ we conclude that $W(\mathcal{P}, h)$ is the inverse image of a subset of $U$, which is necessarily the open set $W = \sigma ^{-1}(W(\mathcal{P}, h))$. $\square$

Remark 40.6.2. Warning: Lemma 40.6.1 should be used with care. For example, it applies to $\mathcal{P}=$“flat”, $\mathcal{Q}=$“empty”, and $\mathcal{R}=$“flat and locally of finite presentation”. But given a morphism of schemes $f : X \to Y$ the largest open $W \subset X$ such that $f|_ W$ is flat is not the set of points where $f$ is flat!

Remark 40.6.3. Notwithstanding the warning in Remark 40.6.2 there are some cases where Lemma 40.6.1 can be used without causing too much ambiguity. We give a list. In each case we omit the verification of assumptions (1) and (2) and we give references which imply (3) and (4). Here is the list:

1. $\mathcal{Q} = \mathcal{R} =$“locally of finite type”, and $\mathcal{P} =$“relative dimension $\leq d$”. See Morphisms, Definition 29.29.1 and Morphisms, Lemmas 29.28.4 and 29.28.3.

2. $\mathcal{Q} = \mathcal{R} =$“locally of finite type”, and $\mathcal{P} =$“locally quasi-finite”. This is the case $d = 0$ of the previous item, see Morphisms, Lemma 29.29.5.

3. $\mathcal{Q} = \mathcal{R} =$“locally of finite type”, and $\mathcal{P} =$“unramified”. See Morphisms, Lemmas 29.35.3 and 29.35.15.

What is interesting about the cases listed above is that we do not need to assume that $s, t$ are flat to get a conclusion about the locus where the morphism $h$ has property $\mathcal{P}$. We continue the list:

1. $\mathcal{Q} =$“locally of finite presentation”, $\mathcal{R} =$“flat and locally of finite presentation”, and $\mathcal{P} =$“flat”. See More on Morphisms, Theorem 37.15.1 and Lemma 37.15.2.

2. $\mathcal{Q} =$“locally of finite presentation”, $\mathcal{R} =$“flat and locally of finite presentation”, and $\mathcal{P}=$“Cohen-Macaulay”. See More on Morphisms, Definition 37.20.1 and More on Morphisms, Lemmas 37.20.6 and 37.20.7.

3. $\mathcal{Q} =$“locally of finite presentation”, $\mathcal{R} =$“flat and locally of finite presentation”, and $\mathcal{P}=$“syntomic” use Morphisms, Lemma 29.30.12 (the locus is automatically open).

4. $\mathcal{Q} =$“locally of finite presentation”, $\mathcal{R} =$“flat and locally of finite presentation”, and $\mathcal{P}=$“smooth”. See Morphisms, Lemma 29.34.15 (the locus is automatically open).

5. $\mathcal{Q} =$“locally of finite presentation”, $\mathcal{R} =$“flat and locally of finite presentation”, and $\mathcal{P}=$“étale”. See Morphisms, Lemma 29.36.17 (the locus is automatically open).

Here is the second result. The $R$-invariant open $W \subset U$ should be thought of as the inverse image of the largest open of $[U/R]$ over which the morphism $U \to [U/R]$ has property $\mathcal{P}$.

Lemma 40.6.4. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $\tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\}$1. Let $\mathcal{P}$ be a property of morphisms of schemes which is $\tau$-local on the target (Descent, Definition 35.19.1). Assume $\{ s : R \to U\}$ and $\{ t : R \to U\}$ are coverings for the $\tau$-topology. Let $W \subset U$ be the maximal open subscheme such that $s|_{s^{-1}(W)} : s^{-1}(W) \to W$ has property $\mathcal{P}$. Then $W$ is $R$-invariant, see Groupoids, Definition 39.19.1.

Proof. The existence and properties of the open $W \subset U$ are described in Descent, Lemma 35.19.3. In Diagram (40.3.0.1) let $W_1 \subset R$ be the maximal open subscheme over which the morphism $\text{pr}_1 : R \times _{s, U, t} R \to R$ has property $\mathcal{P}$. It follows from the aforementioned Descent, Lemma 35.19.3 and the assumption that $\{ s : R \to U\}$ and $\{ t : R \to U\}$ are coverings for the $\tau$-topology that $t^{-1}(W) = W_1 = s^{-1}(W)$ as desired. $\square$

Lemma 40.6.5. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $G \to U$ be its stabilizer group scheme. Let $\tau \in \{ fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\}$. Let $\mathcal{P}$ be a property of morphisms which is $\tau$-local on the target. Assume $\{ s : R \to U\}$ and $\{ t : R \to U\}$ are coverings for the $\tau$-topology. Let $W \subset U$ be the maximal open subscheme such that $G_ W \to W$ has property $\mathcal{P}$. Then $W$ is $R$-invariant (see Groupoids, Definition 39.19.1).

Proof. The existence and properties of the open $W \subset U$ are described in Descent, Lemma 35.19.3. The morphism

$G \times _{U, t} R \longrightarrow R \times _{s, U} G, \quad (g, r) \longmapsto (r, r^{-1} \circ g \circ r)$

is an isomorphism over $R$ (where $\circ$ denotes composition in the groupoid). Hence $s^{-1}(W) = t^{-1}(W)$ by the properties of $W$ proved in the aforementioned Descent, Lemma 35.19.3. $\square$

[1] The fact that $fpqc$ is missing is not a typo.

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