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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.

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  • $\begingroup$ 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? $\endgroup$
    – S. Carnahan ♦
    Commented Jan 18, 2015 at 1:41

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