Let $f: X\longrightarrow Y$ be a morphism of algebraic stacks, and assume I know the Krull dimension of the ring of global functions on $Y$ to be $n$. Assume that the stack $X$ has "stacky" dimension $k$ with $k < n$, then it is known that this doesn't necessarily imply that there exists a non-zero function in $\text{Ker}(f^*\mathcal{O}(Y)\longrightarrow \mathcal{O}(X))$.
An example should look something like $\text{Spec}\mathbb{C}/\mathbb{G}_m\longrightarrow \text{Spec}\mathbb{C}$. Then $\text{Spec}\mathbb{C}$ has $\mathbb{C}$ as its ring of global functions, which has Krull dimension 0, while the stacky dimension of $\text{Spec}\mathbb{C}/\mathbb{G}_m$ is minus one. However, the map on rings of global sections is a $\mathbb{C}$ morphism, and hence injective.
It seems like the failure of the previous example is in the "inertia", in the sense that the stack $\text{Spec}\mathbb{C}/\mathbb{G}_m$ is the groupoid whose objects are a single point $*$, and whose morphisms are all automorphisms, indexed by $\mathbb{G}_m$. Being one dimensional, it seems that the dimension of the inertia comes into play in the dimension constraints for the existence of a non-trivial global function on $Y$ vanishing on the image of $X$. More precisely, we require something like $n > \dim X + \dim\text{Inertia}$.
In the case where $X$ is an algebraic stack, we have a nice notion of an inertia stack $I_X$ fibred over $X$, whose objects are pairs $(x, g)$, where $x$ is an object of $X$ and $g\in \text{Aut}(x)$. My question is if it is possible to turn this intuition into a formal mathematical theorem, which says something like: under the above assumptions, assume $\dim I_X < n$, then there exists a non-trivial section of the sheaf $\text{Ker}(f^*\mathcal{O}(Y)\longrightarrow \mathcal{O}(X))$.
Thanks in advance!
