Since $\mathcal{X}$ is DM, there exists an étale atlas $p: W \to \mathcal{X}$ with $W$ a scheme. Choose a geometric point $w \in W$ mapping to $x$. The fiber $W \times_\mathcal{X} W$ over the diagonal is an étale equivalence relation, and the stabilizer group at $w$ is the finite group $G_x$.

By the theory of étale groupoids, we can find an étale neighborhood $(V, v)$ of $(W, w)$ such that $V \times_\mathcal{X} V \cong G_x \times V$ étale-locally near $v$. Shrinking $V$ (and possibly replacing by a henselian local scheme), we get $[V/G_x] \xrightarrow{\sim} \mathcal{X} \times_X U'$ for some étale neighborhood $U'$ of $\pi(x)$.

Taking $U$ to be an affine $G_x$-invariant neighborhood of $v$ in $V$ gives the desired étale-local description.

We verify the coarse moduli space axioms:

Universality: Let $f: [U/G] \to Y$ be a morphism to an algebraic space $Y$. The composite $U \to [U/G] \to Y$ is $G$-invariant, hence factors through a $G$-invariant morphism $U \to Y$. Since $U \to U/G$ is the categorical quotient in the category of affine schemes, the map factors uniquely as $U \to U/G \to Y$.

To extend this to the non-affine case, we use that $Y$ is a sheaf for the étale topology and that $U \to U/G$ is a categorical quotient for the étale topology on the category of algebraic spaces.

Geometric bijectivity: For an algebraically closed field $k$, we have

since over algebraically closed fields, $G$-orbits on $k$-points are exactly the $k$-points of the GIT quotient (by Hilbert's theorem on invariants). These sets are naturally in bijection.

Structure sheaf: The map $A^G \to A$ induces $\mathcal{O}_{U/G} \to \pi_*\mathcal{O}_{[U/G]}$, and the latter equals $(\pi_*\mathcal{O}_U)^G = A^G$ (since $G$ is finite and acts on $A$). Thus $\pi_*\mathcal{O}_{[U/G]} = \mathcal{O}_{U/G}$.

Step 3a: Choose an étale cover. For each geometric point $x$ of $\mathcal{X}$, choose an étale neighborhood $[U_x / G_x] \to \mathcal{X}$ as in Step 1. By quasi-compactness of $\mathcal{X}$, finitely many of these cover: $\{[U_i/G_i] \to \mathcal{X}\}_{i=1}^n$.

Step 3b: Form local coarse moduli spaces. For each $i$, set $X_i = U_i / G_i = \mathrm{Spec}(A_i^{G_i})$. By Step 2, $[U_i/G_i] \to X_i$ is a coarse moduli map.

Step 3c: Construct gluing data. For each pair $(i, j)$, consider the fiber product $\mathcal{U}_{ij} = [U_i/G_i] \times_\mathcal{X} [U_j/G_j]$. This is an étale equivalence relation on $\coprod_i [U_i/G_i]$. The coarse moduli space functor, applied to $\mathcal{U}_{ij}$, gives an étale equivalence relation on $\coprod_i X_i$.

Specifically, define $R_{ij}$ as the coarse moduli space of $\mathcal{U}_{ij}$. The projections $R_{ij} \to X_i$ and $R_{ij} \to X_j$ are étale (because the original projections $\mathcal{U}_{ij} \to [U_i/G_i]$ are étale, and the coarse moduli space functor preserves étaleness in appropriate situations).

Step 3d: Verify the cocycle condition. On triple overlaps, the transitivity of the equivalence relation on $\mathcal{X}$ descends to transitivity for the equivalence relation $R$ on $\coprod_i X_i$.

Step 3e: Form the quotient. The algebraic space $X$ is defined as the quotient

where $R = \coprod_{i,j} R_{ij}$. Since $R \rightrightarrows \coprod_i X_i$ is an étale equivalence relation on schemes, the quotient $X$ is an algebraic space.

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

Étale-locally: On each $[U_i/G_i]$, the morphism $f|_{[U_i/G_i]}: [U_i/G_i] \to Y$ factors uniquely through $X_i = U_i/G_i$ (by Step 2). This gives maps $g_i: X_i \to Y$.

Compatibility: On overlaps, the two factorizations $g_i|_{R_{ij}}$ and $g_j|_{R_{ij}}$ must agree. This follows from the uniqueness of the factorization on $\mathcal{U}_{ij}$: the coarse moduli map for $\mathcal{U}_{ij}$ is $R_{ij}$, and $f|_{\mathcal{U}_{ij}}$ factors uniquely through $R_{ij}$.

Gluing: The compatible family $\{g_i: X_i \to Y\}$ descends to a unique morphism $g: X \to Y$ with $f = g \circ \pi$.

We verify properness using the valuative criterion.

Finite type: $\pi$ is of finite type because étale-locally it is $U_i \to U_i/G_i$, which is finite (hence of finite type).

Separatedness: The diagonal $\mathcal{X} \to \mathcal{X} \times_X \mathcal{X}$ is representable and proper. Since $\mathcal{X}$ is separated, this is a closed immersion, giving separatedness of $\pi$.

Universal closedness: Let $A$ be a DVR with fraction field $K$ and consider a commutative diagram

We need to find a lift $\mathrm{Spec}(A) \to \mathcal{X}$. Étale-locally on $X$, the stack $\mathcal{X}$ is $[U/G]$ and $X = U/G$. The map $\mathrm{Spec}(A) \to U/G$ lifts to $U$ after a finite extension of $A$ (since $U \to U/G$ is finite surjective). The composite $\mathrm{Spec}(A') \to U \to [U/G]$ gives the required lift (after possibly replacing $A$ by a finite extension, which is allowed for properness).

More carefully: the stack $\mathcal{X}$ is proper over $X$ because étale-locally the map $[U/G] \to U/G$ is proper (it is a finite gerbe). A morphism that is étale-locally proper is proper.

Étale-locally on $X$, we have $\mathcal{X} \cong [U/G]$ and $X \cong U/G$. Then

The first equality uses that pushforward from $[U/G]$ to $U/G$ is taking $G$-invariants. The second is the definition of $A^G$. Since this holds étale-locally, it holds globally.

Let $k$ be an algebraically closed field. We need to show $|\mathcal{X}(k)| \to X(k)$ is bijective.

Surjectivity: Let $x \in X(k)$. Étale-locally, $X = U/G$ and $x$ lifts to a $k$-point $u \in U(k)$ (since $k$ is algebraically closed and $U \to U/G$ is surjective). The image of $u$ in $[U/G](k)$ maps to $x$, giving surjectivity.

Injectivity: Let $\xi_1, \xi_2 \in \mathcal{X}(k)$ with $\pi(\xi_1) = \pi(\xi_2) = x \in X(k)$. Étale-locally, $\xi_1, \xi_2$ correspond to $k$-points $u_1, u_2 \in U(k)$ with the same image in $U/G$. This means $u_1$ and $u_2$ lie in the same $G$-orbit: $u_2 = g \cdot u_1$ for some $g \in G$. Therefore $\xi_1 \cong \xi_2$ in $[U/G](k)$ (they are isomorphic as objects of the groupoid), so they represent the same isomorphism class in $|\mathcal{X}(k)|$.

Let $p: W \to \mathcal{X}$ be an étale atlas. Consider the representable morphism $\mathcal{H}om_\mathcal{X}(W, W) \to \mathcal{X}$ whose fiber over a point $\xi \in \mathcal{X}(T)$ is the set of automorphisms of the pullback of the atlas to $\xi$.

Since $\mathcal{X}$ has finite inertia, the groupoid $R = W \times_\mathcal{X} W \rightrightarrows W$ has finite fibers. The key observation: the morphism $R \to W \times_S W$ is finite (because the diagonal of $\mathcal{X}$ is finite, being the composition of the unramified diagonal of a DM stack with the finite inertia map).

Now consider the "symmetrized" version: define $Z = \mathrm{Spec}(\mathrm{Sym}^n(\mathcal{O}_R))$ (informally, the $n$-th symmetric product where $n$ is the degree of $R \to W$). This $Z$ carries a natural finite flat equivalence relation, and the quotient recovers $\mathcal{X}$ étale-locally.

The quotient $U/R$ is an fppf sheaf by descent. The diagonal is representable because $R$ is a scheme (the diagonal is the map $R \to U \times_{U/R} U$, which is an isomorphism by effectivity of the groupoid). The atlas $U \to U/R$ is finite (hence in particular surjective), giving condition (3).

We verify the valuative criterion for separatedness of $X \to S$.

Let $A$ be a DVR with fraction field $K$, and $g_1, g_2: \mathrm{Spec}(A) \to X$ two morphisms agreeing on $\mathrm{Spec}(K)$. We need $g_1 = g_2$.

Pull back to $\mathcal{X}$: the fiber products $\mathcal{X} \times_{X, g_i} \mathrm{Spec}(A)$ are DM stacks proper over $\mathrm{Spec}(A)$ (since $\pi$ is proper). The generic fibers are isomorphic (since $g_1 = g_2$ on $\mathrm{Spec}(K)$).

Since $\mathcal{X}$ is separated, the isomorphism on generic fibers extends to a morphism of the whole stacks over $\mathrm{Spec}(A)$. Applying $\pi$ to both sides, we get $g_1 = g_2$ (using $\pi_*\mathcal{O} = \mathcal{O}$, which implies $\pi$ detects equality of maps from reduced schemes).