# quotient a scheme by a stratified vector bundle

Let $k$ be a field.

Let $X$ be a $k$-scheme of finite type, normal and integral. We consider $f,g:R\rightarrow X$ an equivalence relation, surjective and such that it is a stratified vector bundle, i.e. there exists a decomposition of $X$ into locally closed subschemes $W_{i}$, such that over $W_{i}$, it is a trivial vector bundle of dimension $n_{i}$.

The $n_{i}$'s don't have to be the same.

Is there a nice category ( schemes, stacks, derived stacks) where we can form the quotient $[X/R]$?

Of course, we can do it as a fppf sheaf, but we would like extra structure.

• The expression "it is a stratified vector bundle" is still unclear. My guess is that $R \rightrightarrows X$ is the action groupoid of a sheaf of abelian groups that, when base changed to $W_i$, is isomorphic to a vector bundle. Is this correct? – S. Carnahan Jan 18 '15 at 1:41