Definition 106.8.1. A morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks is said to be formally smooth if it is formally smooth on objects as a $1$-morphism in categories fibred in groupoids as explained in Criteria for Representability, Section 97.6.
106.8 Formally smooth morphisms
In this section we introduce the notion of a formally smooth morphism $\mathcal{X} \to \mathcal{Y}$ of algebraic stacks. Such a morphism is characterized by the property that $T$-valued points of $\mathcal{X}$ lift to infinitesimal thickenings of $T$ provided $T$ is affine. The main result is that a morphism which is formally smooth and locally of finite presentation is smooth, see Lemma 106.8.7. It turns out that this criterion is often easier to use than the Jacobian criterion.
We translate the condition of the definition into the language we are currently using (see Properties of Stacks, Section 100.2). Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Consider a $2$-commutative solid diagram
106.8.1.1
\begin{equation} \label{stacks-more-morphisms-equation-diagram} \vcenter { \xymatrix{ T \ar[r]_-x \ar[d]_ i & \mathcal{X} \ar[d]^ f \\ T' \ar[r]^-y \ar@{..>}[ru] & \mathcal{Y} } } \end{equation}
where $i : T \to T'$ is a first order thickening of affine schemes. Let
\[ \gamma : y \circ i \longrightarrow f \circ x \]
be a $2$-morphism witnessing the $2$-commutativity of the diagram. (Notation as in Categories, Sections 4.28 and 4.29.) Given (106.8.1.1) and $\gamma $ a dotted arrow is a triple $(x', \alpha , \beta )$ consisting of a morphism $x' : T' \to \mathcal{X}$ and $2$-arrows $\alpha : x' \circ i \to x$, $\beta : y \to f \circ x'$ such that $\gamma = (\text{id}_ f \star \alpha ) \circ (\beta \star \text{id}_ i)$, in other words such that
\[ \xymatrix{ & f \circ x' \circ i \ar[rd]^{\text{id}_ f \star \alpha } \\ y \circ i \ar[ru]^{\beta \star \text{id}_ i} \ar[rr]^\gamma & & f \circ x } \]
is commutative. A morphism of dotted arrows $(x'_1, \alpha _1, \beta _1) \to (x'_2, \alpha _2, \beta _2)$ is a $2$-arrow $\theta : x'_1 \to x'_2$ such that $\alpha _1 = \alpha _2 \circ (\theta \star \text{id}_ i)$ and $\beta _2 = (\text{id}_ f \star \theta ) \circ \beta _1$.
The category of dotted arrows just described is a special case of Categories, Definition 4.44.1.
Lemma 106.8.2. A morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks is formally smooth (Definition 106.8.1) if and only if for every diagram (106.8.1.1) and $\gamma $ the category of dotted arrows is nonempty.
Proof. Translation between different languages omitted. $\square$
Lemma 106.8.3. The base change of a formally smooth morphism of algebraic stacks by any morphism of algebraic stacks is formally smooth.
Proof. Follows from Categories, Lemma 4.44.2 and the definition. $\square$
Lemma 106.8.4. The composition of formally smooth morphisms of algebraic stacks is formally smooth.
Proof. Follows from Categories, Lemma 4.44.3 and the definition. $\square$
Lemma 106.8.5. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. Then the following are equivalent
$f$ is formally smooth,
for every scheme $T$ and morphism $T \to \mathcal{Y}$ the morphism $\mathcal{X} \times _\mathcal {Y} T \to T$ is formally smooth as a morphism of algebraic spaces.
Proof. Follows from Categories, Lemma 4.44.2 and the definition. $\square$
Lemma 106.8.6. Let $T \to T'$ be a first order thickening of affine schemes. Let $\mathcal{X}'$ be an algebraic stack over $T'$ whose structure morphism $\mathcal{X}' \to T'$ is smooth. Let $x : T \to \mathcal{X}'$ be a morphism over $T'$. Then there exists a morphsm $x' : T' \to \mathcal{X}'$ over $T'$ with $x'|_ T = x$.
Proof. We may apply the result of Lemma 106.7.4. Thus it suffices to construct a smooth surjective morphism $W' \to T'$ with $W'$ affine such that $x|_{T \times _{W'} T'}$ lifts to $W'$. (We urge the reader to find their own proof of this fact using the analogous result for algebraic spaces already established.) We choose a scheme $U'$ and a surjective smooth morphism $U' \to \mathcal{X}'$. Observe that $U' \to T'$ is smooth and that the projection $T \times _{\mathcal{X}'} U' \to T$ is surjective smooth. Choose an affine scheme $W$ and an étale morphism $W \to T \times _{\mathcal{X}'} U'$ such that $W \to T$ is surjective. Then $W \to T$ is a smooth morphism of affine schemes. After replacing $W$ by a disjoint union of principal affine opens, we may assume there exists a smooth morphism of affines $W' \to T'$ such that $W = T \times _{T'} W'$, see Algebra, Lemma 10.137.20. By More on Morphisms of Spaces, Lemma 76.19.6 we can find a morphism $W' \to U'$ over $T'$ lifting the given morphism $W \to U'$. This finishes the proof. $\square$
The following lemma is the main result of this section. It implies, combined with Limits of Stacks, Proposition 102.3.8, that we can recognize whether a morphism of algebraic stacks $f : \mathcal{X} \to \mathcal{Y}$ is smooth in terms of “simple” properties of the $1$-morphism of stacks in groupoids $\mathcal{X} \to \mathcal{Y}$.
Lemma 106.8.7 (Infinitesimal lifting criterion). Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent:
The morphism $f$ is smooth.
The morphism $f$ is locally of finite presentation and formally smooth.
Proof. Assume $f$ is smooth. Then $f$ is locally of finite presentation by Morphisms of Stacks, Lemma 101.33.5. Hence it suffices given a diagram (106.8.1.1) and a $\gamma : y \circ i \to f \circ x$ to find a dotted arrow (see Lemma 106.8.2). Forming fibre products we obtain
\[ \xymatrix{ T \ar[d] \ar[r] & T' \times _\mathcal {Y} \mathcal{X} \ar[d] \ar[r] & \mathcal{X} \ar[d] \\ T' \ar[r] & T' \ar[r] & \mathcal{Y} } \]
Thus we see it is sufficient to find a dotted arrow in the left square. Since $T' \times _\mathcal {Y} \mathcal{X} \to T'$ is smooth (Morphisms of Stacks, Lemma 101.33.3) existence of a dotted arrow in the left square is guaranteed by Lemma 106.8.6.
Conversely, suppose that $f$ is locally of finite presentation and formally smooth. Choose a scheme $U$ and a surjective smooth morphism $U \to \mathcal{X}$. Then $a : U \to \mathcal{X}$ and $b : U \to \mathcal{Y}$ are representable by algebraic spaces and locally of finite presentation (use Morphisms of Stacks, Lemma 101.27.2 and the fact seen above that a smooth morphism is locally of finite presentation). We will apply the general principle of Algebraic Stacks, Lemma 94.10.9 with as input the equivalence of More on Morphisms of Spaces, Lemma 76.19.6 and simultaneously use the translation of Criteria for Representability, Lemma 97.6.3. We first apply this to $a$ to see that $a$ is formally smooth on objects. Next, we use that $f$ is formally smooth on objects by assumption (see Lemma 106.8.2) and Criteria for Representability, Lemma 97.6.2 to see that $b = f \circ a$ is formally smooth on objects. Then we apply the principle once more to conclude that $b$ is smooth. This means that $f$ is smooth by the definition of smoothness for morphisms of algebraic stacks and the proof is complete. $\square$
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