Definition 101.4.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
101.4 Separation axioms
Let $\mathcal{X} = [U/R]$ be a presentation of an algebraic stack. Then the properties of the diagonal of $\mathcal{X}$ over $S$, are the properties of the morphism $j : R \to U \times _ S U$. For example, if $\mathcal{X} = [S/G]$ for some smooth group $G$ in algebraic spaces over $S$ then $j$ is the structure morphism $G \to S$. Hence the diagonal is not automatically separated itself (contrary to what happens in the case of schemes and algebraic spaces). To say that $[S/G]$ is quasi-separated over $S$ should certainly imply that $G \to S$ is quasi-compact, but we hesitate to say that $[S/G]$ is quasi-separated over $S$ without also requiring the morphism $G \to S$ to be quasi-separated. In other words, requiring the diagonal morphism to be quasi-compact does not really agree with our intuition for a “quasi-separated algebraic stack”, and we should also require the diagonal itself to be quasi-separated.
What about “separated algebraic stacks”? We have seen in Morphisms of Spaces, Lemma 67.40.9 that an algebraic space is separated if and only if the diagonal is proper. This is the condition that is usually used to define separated algebraic stacks too. In the example $[S/G] \to S$ above this means that $G \to S$ is a proper group scheme. This means algebraic stacks of the form $[\mathop{\mathrm{Spec}}(k)/E]$ are proper over $k$ where $E$ is an elliptic curve over $k$ (insert future reference here). In certain situations it may be more natural to assume the diagonal is finite.
In this definition we are using that $\Delta _ f$ is representable by algebraic spaces and we are using Properties of Stacks, Section 100.3 to make sense out of imposing conditions on $\Delta _ f$. We note that these definitions do not conflict with the already existing notions if $f$ is representable by algebraic spaces, see Lemmas 101.3.6 and 101.3.5. There is an interesting way to characterize these conditions by looking at higher diagonals, see Lemma 101.6.5.
Definition 101.4.2. Let $\mathcal{X}$ be an algebraic stack over the base scheme $S$. Denote $p : \mathcal{X} \to S$ the structure morphism.
We say $\mathcal{X}$ is DM over $S$ if $p : \mathcal{X} \to S$ is DM.
We say $\mathcal{X}$ is quasi-DM over $S$ if $p : \mathcal{X} \to S$ is quasi-DM.
We say $\mathcal{X}$ is separated over $S$ if $p : \mathcal{X} \to S$ is separated.
We say $\mathcal{X}$ is quasi-separated over $S$ if $p : \mathcal{X} \to S$ is quasi-separated.
We say $\mathcal{X}$ is DM if $\mathcal{X}$ is DM3 over $\mathop{\mathrm{Spec}}(\mathbf{Z})$.
We say $\mathcal{X}$ is quasi-DM if $\mathcal{X}$ is quasi-DM over $\mathop{\mathrm{Spec}}(\mathbf{Z})$.
We say $\mathcal{X}$ is separated if $\mathcal{X}$ is separated over $\mathop{\mathrm{Spec}}(\mathbf{Z})$.
We say $\mathcal{X}$ is quasi-separated if $\mathcal{X}$ is quasi-separated over $\mathop{\mathrm{Spec}}(\mathbf{Z})$.
In the last 4 definitions we view $\mathcal{X}$ as an algebraic stack over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ via Algebraic Stacks, Definition 94.19.2.
Thus in each case we have an absolute notion and a notion relative to our given base scheme (mention of which is usually suppressed by our abuse of notation introduced in Properties of Stacks, Section 100.2). We will see that (1) $\Leftrightarrow $ (5) and (2) $\Leftrightarrow $ (6) in Lemma 101.4.13. We spend some time proving some standard results on these notions.
Lemma 101.4.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
If $f$ is separated, then $f$ is quasi-separated.
If $f$ is DM, then $f$ is quasi-DM.
If $f$ is representable by algebraic spaces, then $f$ is DM.
Proof. To see (1) note that a proper morphism of algebraic spaces is quasi-compact and quasi-separated, see Morphisms of Spaces, Definition 67.40.1. To see (2) note that an unramified morphism of algebraic spaces is locally quasi-finite, see Morphisms of Spaces, Lemma 67.38.7. Finally (3) follows from Lemma 101.3.4. $\square$
Lemma 101.4.4. All of the separation axioms listed in Definition 101.4.1 are stable under base change.
Proof. Let $f : \mathcal{X} \to \mathcal{Y}$ and $\mathcal{Y}' \to \mathcal{Y}$ be morphisms of algebraic stacks. Let $f' : \mathcal{Y}' \times _\mathcal {Y} \mathcal{X} \to \mathcal{Y}'$ be the base change of $f$ by $\mathcal{Y}' \to \mathcal{Y}$. Then $\Delta _{f'}$ is the base change of $\Delta _ f$ by the morphism $\mathcal{X}' \times _{\mathcal{Y}'} \mathcal{X}' \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$, see Categories, Lemma 4.31.14. By the results of Properties of Stacks, Section 100.3 each of the properties of the diagonal used in Definition 101.4.1 is stable under base change. Hence the lemma is true. $\square$
Lemma 101.4.5. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $W \to \mathcal{Y}$ be a surjective, flat, and locally of finite presentation where $W$ is an algebraic space. If the base change $W \times _\mathcal {Y} \mathcal{X} \to W$ has one of the separation properties of Definition 101.4.1 then so does $f$.
Proof. Denote $g : W \times _\mathcal {Y} \mathcal{X} \to W$ the base change. Then $\Delta _ g$ is the base change of $\Delta _ f$ by the morphism $q : W \times _\mathcal {Y} (\mathcal{X} \times _\mathcal {Y} \mathcal{X}) \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$. Since $q$ is the base change of $W \to \mathcal{Y}$ we see that $q$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Hence the result follows from Properties of Stacks, Lemma 100.3.4. $\square$
Lemma 101.4.6. Let $S$ be a scheme. The property of being quasi-DM over $S$, quasi-separated over $S$, or separated over $S$ (see Definition 101.4.2) is stable under change of base scheme, see Algebraic Stacks, Definition 94.19.3.
Proof. Follows immediately from Lemma 101.4.4. $\square$
Lemma 101.4.7. Let $f : \mathcal{X} \to \mathcal{Z}$, $g : \mathcal{Y} \to \mathcal{Z}$ and $\mathcal{Z} \to \mathcal{T}$ be morphisms of algebraic stacks. Consider the induced morphism $i : \mathcal{X} \times _\mathcal {Z} \mathcal{Y} \to \mathcal{X} \times _\mathcal {T} \mathcal{Y}$. Then
$i$ is representable by algebraic spaces and locally of finite type,
if $\Delta _{\mathcal{Z}/\mathcal{T}}$ is quasi-separated, then $i$ is quasi-separated,
if $\Delta _{\mathcal{Z}/\mathcal{T}}$ is separated, then $i$ is separated,
if $\mathcal{Z} \to \mathcal{T}$ is DM, then $i$ is unramified,
if $\mathcal{Z} \to \mathcal{T}$ is quasi-DM, then $i$ is locally quasi-finite,
if $\mathcal{Z} \to \mathcal{T}$ is separated, then $i$ is proper, and
if $\mathcal{Z} \to \mathcal{T}$ is quasi-separated, then $i$ is quasi-compact and quasi-separated.
Proof. The following diagram
\[ \xymatrix{ \mathcal{X} \times _\mathcal {Z} \mathcal{Y} \ar[r]_ i \ar[d] & \mathcal{X} \times _\mathcal {T} \mathcal{Y} \ar[d] \\ \mathcal{Z} \ar[r]^-{\Delta _{\mathcal{Z}/\mathcal{T}}} \ar[r] & \mathcal{Z} \times _\mathcal {T} \mathcal{Z} } \]
is a $2$-fibre product diagram, see Categories, Lemma 4.31.13. Hence $i$ is the base change of the diagonal morphism $\Delta _{\mathcal{Z}/\mathcal{T}}$. Thus the lemma follows from Lemma 101.3.3, and the material in Properties of Stacks, Section 100.3. $\square$
Lemma 101.4.8. Let $\mathcal{T}$ be an algebraic stack. Let $g : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks over $\mathcal{T}$. Consider the graph $i : \mathcal{X} \to \mathcal{X} \times _\mathcal {T} \mathcal{Y}$ of $g$. Then
$i$ is representable by algebraic spaces and locally of finite type,
if $\mathcal{Y} \to \mathcal{T}$ is DM, then $i$ is unramified,
if $\mathcal{Y} \to \mathcal{T}$ is quasi-DM, then $i$ is locally quasi-finite,
if $\mathcal{Y} \to \mathcal{T}$ is separated, then $i$ is proper, and
if $\mathcal{Y} \to \mathcal{T}$ is quasi-separated, then $i$ is quasi-compact and quasi-separated.
Proof. This is a special case of Lemma 101.4.7 applied to the morphism $\mathcal{X} = \mathcal{X} \times _\mathcal {Y} \mathcal{Y} \to \mathcal{X} \times _\mathcal {T} \mathcal{Y}$. $\square$
Lemma 101.4.9. Let $f : \mathcal{X} \to \mathcal{T}$ be a morphism of algebraic stacks. Let $s : \mathcal{T} \to \mathcal{X}$ be a morphism such that $f \circ s$ is $2$-isomorphic to $\text{id}_\mathcal {T}$. Then
$s$ is representable by algebraic spaces and locally of finite type,
if $f$ is DM, then $s$ is unramified,
if $f$ is quasi-DM, then $s$ is locally quasi-finite,
if $f$ is separated, then $s$ is proper, and
if $f$ is quasi-separated, then $s$ is quasi-compact and quasi-separated.
Proof. This is a special case of Lemma 101.4.8 applied to $g = s$ and $\mathcal{Y} = \mathcal{T}$ in which case $i : \mathcal{T} \to \mathcal{T} \times _\mathcal {T} \mathcal{X}$ is $2$-isomorphic to $s$. $\square$
Lemma 101.4.10. All of the separation axioms listed in Definition 101.4.1 are stable under composition of morphisms.
Proof. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks to which the axiom in question applies. The diagonal $\Delta _{\mathcal{X}/\mathcal{Z}}$ is the composition
\[ \mathcal{X} \longrightarrow \mathcal{X} \times _\mathcal {Y} \mathcal{X} \longrightarrow \mathcal{X} \times _\mathcal {Z} \mathcal{X}. \]
Our separation axiom is defined by requiring the diagonal to have some property $\mathcal{P}$. By Lemma 101.4.7 above we see that the second arrow also has this property. Hence the lemma follows since the composition of morphisms which are representable by algebraic spaces with property $\mathcal{P}$ also is a morphism with property $\mathcal{P}$, see our general discussion in Properties of Stacks, Section 100.3 and Morphisms of Spaces, Lemmas 67.38.3, 67.27.3, 67.40.4, 67.8.5, and 67.4.8. $\square$
Lemma 101.4.11. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks over the base scheme $S$.
If $\mathcal{Y}$ is DM over $S$ and $f$ is DM, then $\mathcal{X}$ is DM over $S$.
If $\mathcal{Y}$ is quasi-DM over $S$ and $f$ is quasi-DM, then $\mathcal{X}$ is quasi-DM over $S$.
If $\mathcal{Y}$ is separated over $S$ and $f$ is separated, then $\mathcal{X}$ is separated over $S$.
If $\mathcal{Y}$ is quasi-separated over $S$ and $f$ is quasi-separated, then $\mathcal{X}$ is quasi-separated over $S$.
If $\mathcal{Y}$ is DM and $f$ is DM, then $\mathcal{X}$ is DM.
If $\mathcal{Y}$ is quasi-DM and $f$ is quasi-DM, then $\mathcal{X}$ is quasi-DM.
If $\mathcal{Y}$ is separated and $f$ is separated, then $\mathcal{X}$ is separated.
If $\mathcal{Y}$ is quasi-separated and $f$ is quasi-separated, then $\mathcal{X}$ is quasi-separated.
Proof. Parts (1), (2), (3), and (4) follow immediately from Lemma 101.4.10 and Definition 101.4.2. For (5), (6), (7), and (8) think of $\mathcal{X}$ and $\mathcal{Y}$ as algebraic stacks over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ and apply Lemma 101.4.10. Details omitted. $\square$
The following lemma is a bit different to the analogue for algebraic spaces. To compare take a look at Morphisms of Spaces, Lemma 67.4.10.
Lemma 101.4.12. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks.
If $g \circ f$ is DM then so is $f$.
If $g \circ f$ is quasi-DM then so is $f$.
If $g \circ f$ is separated and $\Delta _ g$ is separated, then $f$ is separated.
If $g \circ f$ is quasi-separated and $\Delta _ g$ is quasi-separated, then $f$ is quasi-separated.
Proof. Consider the factorization
\[ \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X} \to \mathcal{X} \times _\mathcal {Z} \mathcal{X} \]
of the diagonal morphism of $g \circ f$. Both morphisms are representable by algebraic spaces, see Lemmas 101.3.3 and 101.4.7. Hence for any scheme $T$ and morphism $T \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ we get morphisms of algebraic spaces
\[ A = \mathcal{X} \times _{(\mathcal{X} \times _\mathcal {Z} \mathcal{X})} T \longrightarrow B = (\mathcal{X} \times _\mathcal {Y} \mathcal{X}) \times _{(\mathcal{X} \times _\mathcal {Z} \mathcal{X})} T \longrightarrow T. \]
If $g \circ f$ is DM (resp. quasi-DM), then the composition $A \to T$ is unramified (resp. locally quasi-finite). Hence (1) (resp. (2)) follows on applying Morphisms of Spaces, Lemma 67.38.11 (resp. Morphisms of Spaces, Lemma 67.27.8). This proves (1) and (2).
Proof of (4). Assume $g \circ f$ is quasi-separated and $\Delta _ g$ is quasi-separated. Consider the factorization
\[ \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X} \to \mathcal{X} \times _\mathcal {Z} \mathcal{X} \]
of the diagonal morphism of $g \circ f$. Both morphisms are representable by algebraic spaces and the second one is quasi-separated, see Lemmas 101.3.3 and 101.4.7. Hence for any scheme $T$ and morphism $T \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ we get morphisms of algebraic spaces
\[ A = \mathcal{X} \times _{(\mathcal{X} \times _\mathcal {Z} \mathcal{X})} T \longrightarrow B = (\mathcal{X} \times _\mathcal {Y} \mathcal{X}) \times _{(\mathcal{X} \times _\mathcal {Z} \mathcal{X})} T \longrightarrow T \]
such that $B \to T$ is quasi-separated. The composition $A \to T$ is quasi-compact and quasi-separated as we have assumed that $g \circ f$ is quasi-separated. Hence $A \to B$ is quasi-separated by Morphisms of Spaces, Lemma 67.4.10. And $A \to B$ is quasi-compact by Morphisms of Spaces, Lemma 67.8.9. Thus $f$ is quasi-separated.
Proof of (3). Assume $g \circ f$ is separated and $\Delta _ g$ is separated. Consider the factorization
\[ \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X} \to \mathcal{X} \times _\mathcal {Z} \mathcal{X} \]
of the diagonal morphism of $g \circ f$. Both morphisms are representable by algebraic spaces and the second one is separated, see Lemmas 101.3.3 and 101.4.7. Hence for any scheme $T$ and morphism $T \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ we get morphisms of algebraic spaces
\[ A = \mathcal{X} \times _{(\mathcal{X} \times _\mathcal {Z} \mathcal{X})} T \longrightarrow B = (\mathcal{X} \times _\mathcal {Y} \mathcal{X}) \times _{(\mathcal{X} \times _\mathcal {Z} \mathcal{X})} T \longrightarrow T \]
such that $B \to T$ is separated. The composition $A \to T$ is proper as we have assumed that $g \circ f$ is quasi-separated. Hence $A \to B$ is proper by Morphisms of Spaces, Lemma 67.40.6 which means that $f$ is separated. $\square$
Lemma 101.4.13. Let $\mathcal{X}$ be an algebraic stack over the base scheme $S$.
$\mathcal{X}$ is DM $\Leftrightarrow $ $\mathcal{X}$ is DM over $S$.
$\mathcal{X}$ is quasi-DM $\Leftrightarrow $ $\mathcal{X}$ is quasi-DM over $S$.
If $\mathcal{X}$ is separated, then $\mathcal{X}$ is separated over $S$.
If $\mathcal{X}$ is quasi-separated, then $\mathcal{X}$ is quasi-separated over $S$.
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks over the base scheme $S$.
If $\mathcal{X}$ is DM over $S$, then $f$ is DM.
If $\mathcal{X}$ is quasi-DM over $S$, then $f$ is quasi-DM.
If $\mathcal{X}$ is separated over $S$ and $\Delta _{\mathcal{Y}/S}$ is separated, then $f$ is separated.
If $\mathcal{X}$ is quasi-separated over $S$ and $\Delta _{\mathcal{Y}/S}$ is quasi-separated, then $f$ is quasi-separated.
Proof. Parts (5), (6), (7), and (8) follow immediately from Lemma 101.4.12 and Spaces, Definition 65.13.2. To prove (3) and (4) think of $X$ and $Y$ as algebraic stacks over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ and apply Lemma 101.4.12. Similarly, to prove (1) and (2), think of $\mathcal{X}$ as an algebraic stack over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ consider the morphisms
\[ \mathcal{X} \longrightarrow \mathcal{X} \times _ S \mathcal{X} \longrightarrow \mathcal{X} \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} \mathcal{X} \]
Both arrows are representable by algebraic spaces. The second arrow is unramified and locally quasi-finite as the base change of the immersion $\Delta _{S/\mathbf{Z}}$. Hence the composition is unramified (resp. locally quasi-finite) if and only if the first arrow is unramified (resp. locally quasi-finite), see Morphisms of Spaces, Lemmas 67.38.3 and 67.38.11 (resp. Morphisms of Spaces, Lemmas 67.27.3 and 67.27.8). $\square$
Lemma 101.4.14. Let $\mathcal{X}$ be an algebraic stack. Let $W$ be an algebraic space, and let $f : W \to \mathcal{X}$ be a surjective, flat, locally finitely presented morphism.
If $f$ is unramified (i.e., étale, i.e., $\mathcal{X}$ is Deligne-Mumford), then $\mathcal{X}$ is DM.
If $f$ is locally quasi-finite, then $\mathcal{X}$ is quasi-DM.
Proof. Note that if $f$ is unramified, then it is étale by Morphisms of Spaces, Lemma 67.39.12. This explains the parenthetical remark in (1). Assume $f$ is unramified (resp. locally quasi-finite). We have to show that $\Delta _\mathcal {X} : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is unramified (resp. locally quasi-finite). Note that $W \times W \to \mathcal{X} \times \mathcal{X}$ is also surjective, flat, and locally of finite presentation. Hence it suffices to show that
\[ W \times _{\mathcal{X} \times \mathcal{X}, \Delta _\mathcal {X}} \mathcal{X} = W \times _\mathcal {X} W \longrightarrow W \times W \]
is unramified (resp. locally quasi-finite), see Properties of Stacks, Lemma 100.3.3. By assumption the morphism $\text{pr}_ i : W \times _\mathcal {X} W \to W$ is unramified (resp. locally quasi-finite). Hence the displayed arrow is unramified (resp. locally quasi-finite) by Morphisms of Spaces, Lemma 67.38.11 (resp. Morphisms of Spaces, Lemma 67.27.8). $\square$
Lemma 101.4.15. A monomorphism of algebraic stacks is separated and DM. The same is true for immersions of algebraic stacks.
Proof. If $f : \mathcal{X} \to \mathcal{Y}$ is a monomorphism of algebraic stacks, then $\Delta _ f$ is an isomorphism, see Properties of Stacks, Lemma 100.8.4. Since an isomorphism of algebraic spaces is proper and unramified we see that $f$ is separated and DM. The second assertion follows from the first as an immersion is a monomorphism, see Properties of Stacks, Lemma 100.9.5. $\square$
Lemma 101.4.16. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. Assume the residual gerbe $\mathcal{Z}_ x$ of $\mathcal{X}$ at $x$ exists. If $\mathcal{X}$ is DM, resp. quasi-DM, resp. separated, resp. quasi-separated, then so is $\mathcal{Z}_ x$.
Proof. This is true because $\mathcal{Z}_ x \to \mathcal{X}$ is a monomorphism hence DM and separated by Lemma 101.4.15. Apply Lemma 101.4.11 to conclude. $\square$
[1] The letters DM stand for Deligne-Mumford. If $f$ is DM then given any scheme $T$ and any morphism $T \to \mathcal{Y}$ the fibre product $\mathcal{X}_ T = \mathcal{X} \times _\mathcal {Y} T$ is an algebraic stack over $T$ whose diagonal is unramified, i.e., $\mathcal{X}_ T$ is DM. This implies $\mathcal{X}_ T$ is a Deligne-Mumford stack, see Theorem 101.21.6. In other words a DM morphism is one whose “fibres” are Deligne-Mumford stacks. This hopefully at least motivates the terminology.
[2] If $f$ is quasi-DM, then the “fibres” $\mathcal{X}_ T$ of $\mathcal{X} \to \mathcal{Y}$ are quasi-DM. An algebraic stack $\mathcal{X}$ is quasi-DM exactly if there exists a scheme $U$ and a surjective flat morphism $U \to \mathcal{X}$ of finite presentation which is locally quasi-finite, see Theorem 101.21.3. Note the similarity to being Deligne-Mumford, which is defined in terms of having an étale covering by a scheme.
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