Let $\mathcal{X}$ be a separated Deligne-Mumford stack of finite type over a Noetherian base $S$ with finite stabilizers. We construct a coarse moduli space $X$ which is a separated algebraic space.

Step 1: Construction of the Coarse Space

Define $X$ as a functor $X: \textbf{Sch}_S^{\text{op}} \to \textbf{Set}$ by: $X(T) = \{\text{morphisms } f: T \to \mathcal{X} \text{ with trivial stabilizers}\}/\text{isomorphism}$

We must show that $X$ is representable by an algebraic space. The key is to prove:

• $X$ is a sheaf in the étale topology
• The diagonal $X \to X \times X$ is representable
• There exists a scheme with a surjective étale morphism to $X$

Step 2: Sheaf Condition

Let $\{T_i \to T\}$ be an étale cover. If $f_i: T_i \to \mathcal{X}$ are morphisms with trivial stabilizers that agree on overlaps $T_i \times_T T_j$, we need to show they glue to $f: T \to \mathcal{X}$.

Since $\mathcal{X}$ is a stack, the $f_i$ glue to a morphism $f: T \to \mathcal{X}$. We must verify that $f$ has trivial stabilizers. This follows from the étale local nature of stabilizers: if stabilizers are trivial after pulling back to an étale cover, they are trivial on the base.

Step 3: Representability of the Diagonal

Given two morphisms $f_1, f_2: T \rightrightarrows X$ (with trivial stabilizers), we must show the equalizer is representable by a scheme. This equalizer consists of points where $f_1$ and $f_2$ agree as morphisms to $\mathcal{X}$.

Since $\mathcal{X}$ has representable diagonal, the equalizer of $f_1, f_2$ viewed as morphisms to $\mathcal{X}$ is a closed subscheme of $T$. The triviality of stabilizers ensures this equalizer is the same whether we work in $\mathcal{X}$ or its coarse space.

Step 4: Étale Cover by a Scheme

Choose an étale atlas $U \to \mathcal{X}$ where $U$ is a scheme. We must show that $U \to X$ is étale and surjective.

Surjectivity: For any field $k$ and $x \in X(k)$, the point $x$ corresponds to a morphism $\text{Spec}(k) \to \mathcal{X}$ with trivial stabilizer. Since $U \to \mathcal{X}$ is surjective, this lifts to $\text{Spec}(k') \to U$ for some field extension $k'/k$. Thus $U \to X$ is surjective on geometric points.

Étaleness: To show $U \to X$ is étale, we must verify it is locally of finite presentation, flat, and unramified. These properties follow from the corresponding properties of $U \to \mathcal{X}$ and the fact that $\mathcal{X} \to X$ is universal among morphisms to algebraic spaces.

Step 5: Universal Property

Let $\phi: \mathcal{X} \to Y$ be a morphism to an algebraic space $Y$. We must show $\phi$ factors uniquely through $X$.

For any scheme $T$ and morphism $f: T \to \mathcal{X}$ with trivial stabilizer, compose to get $T \to Y$. This is compatible with the equivalence relation defining $X$, hence descends to a morphism $X \to Y$. Uniqueness follows from the density of points with trivial stabilizers.

Step 6: Properness

The morphism $\pi: \mathcal{X} \to X$ is proper. To see this, use the valuative criterion: given a commutative diagram with $\text{Spec}(K)$ and $\text{Spec}(R)$ (a DVR), a morphism $\text{Spec}(R) \to X$ and $\text{Spec}(K) \to \mathcal{X}$ lifting the generic point, we must find a unique lift $\text{Spec}(R) \to \mathcal{X}$.

Since $\mathcal{X}$ is separated, such lifts form a closed subspace of the Hom-space, which is proper. The existence and uniqueness follow from the properness of $\mathcal{X}$ over $S$ and the representability of the diagonal.

Step 7: Pushforward of Structure Sheaf

Finally, $\pi_*\mathcal{O}_\mathcal{X} = \mathcal{O}_X$ follows from the fact that sections of $\mathcal{O}_\mathcal{X}$ that are invariant under all automorphisms descend to $X$. Since stabilizers are finite and we work over a base where finite group averaging is possible, invariant functions are precisely those that descend.

This completes the sketch of the Keel-Mori theorem.