Derived pullback of the coarse moduli morphism

Question

Let $f: \mathcal{X}\to X$ be a morphism from a smooth DM-stack $\mathcal{X}$ to its coarse moduli space $X$. Assume that $X$ is also smooth. Is it true that $Lf^*$ is fully faithful and induces an equivalence of $D^b(coh(X))$ with an admissible subcategory of $D^b(coh(\mathcal{X}))$?

Answer 1

If both $\mathcal{X}$ and $X$ are locally Noetherian and regular, then $f$ is flat. Then $Lf^*$ is the usual pullback $f^*$. If $\mathcal{X}$ is tame, then the natural transformation $$\theta:\text{Id} \Rightarrow f_*f^*,$$ is a natural isomorphism. However, when $\mathcal{X}$ is not tame, this can fail. For instance, let $k$ be a field of characteristic $p>0$, let $(\mathbb{Z}/p\mathbb{Z})_k$ denote the usual finite, étale $k$-group scheme whose underlying group of $k$-points is $\mathbb{Z}/p\mathbb{Z}$, and consider the Artin-Schreier action, $$ \mu:(\mathbb{Z}/p\mathbb{Z})_k\times_k \mathbb{P}^1_k \to \mathbb{P}^1_k, \ a\cdot[s,t] = [s+at,t]. $$ Let $q:\mathbb{P}^1_k \to \mathcal{X}$ be the associated quotient stack, which is a smooth Deligne-Mumford stack. Consider the $(\mathbb{Z}/p\mathbb{Z})_k$-invariant $k$-morphism, $$ F:\mathbb{P}^1_k \to \mathbb{P}^1_k, \ F([s,t]) = [s^p-st^{p-1},t^p]. $$ This is the uniform categorical quotient in the category of $k$-schemes. Denote the target by $X$. Thus $F$ factors through a $1$-morphism of stacks, $$ f:\mathcal{X} \to X, $$ and this is a coarse moduli space of $\mathcal{X}$. But now consider the fiber over the closed point $\infty = [1,0]$ in $X$. The scheme-theoretic fiber of $F$ is the closed subscheme $$Z(t^p) \cong \text{Spec}\ k[(t/s)]/\langle (t/s)^p \rangle.$$ Moreover, the induced action of $(\mathbb{Z}/p\mathbb{Z})_k$ on $Z(t^p)$ by $k$-morphisms is $$ \mu_\infty: (\mathbb{Z}/p\mathbb{Z})_k \times_k \text{Spec}\ k[(t/s)]/\langle (t/s)^p \rangle \to \text{Spec}\ k[(t/s)]/\langle (t/s)^p \rangle,$$ $$ \mu_\infty(a)^*(t/s) = (t/s)/(1+a(t/s)) = (t/s)(1-a(t/s)+a^2(t/s)^2 + \dots + (-a)^{p-2}(t/s)^{p-2}), $$ for every $a\in \mathbb{Z}/p\mathbb{Z} \subset k$.
In particular, the $\mathbb{Z}/p\mathbb{Z}$-invariant $k$-subspace of $k[(t/s)]/\langle (t/s)^p \rangle$ is spanned by $1$ and $(t/s)^{p-1}$. Since the invariant subspace is not $1$-dimensional, for the skyscraper sheaf $\kappa(\infty)$ on $X$ supported at $\infty$, the natural transformation $$ \theta_{\kappa(\infty)} : \kappa(\infty) \to f_*f^*\kappa(\infty), $$ is not an isomorphism.