Does anybody know a good reference where to study vector bundle stacks? I am interested in a situation of this type: $f:\mathcal{F}\to\mathcal{G}$ is a morphism of stacks which is non-representable but whose fibers have constant dimension. Is this a vector bundle stack?
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1$\begingroup$ I really think you should give more information in your question. Based off of what you have said, I would likely think that your map is flat and not necessarily anything more. Have you checked that $f$ is smooth? Secondly, what exactly do you mean by vector bundle stack? Do your fibers look something like the quotient stack $V/W$ for some finite dimensional vector spaces $V$ and $W$ (for example, the tangent stack is something like this)? $\endgroup$– Mike SkirvinCommented Apr 20, 2011 at 13:54
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The conditions that you list are certainly not enough for something to be a vector bundle stack.
Forget stacks.
Even for varieties, the conditions that you list are not enough to be a vector bundle: you're just asking that the fibers have constant dimension...
So, to your question "Is this a vector bundle stack?", I respond, perhaps disappointingly, "no".
answered Apr 20, 2011 at 16:42
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1$\begingroup$ You might want to rephrase your question, and ask something like "what is a vector bundle stack?". $\endgroup$ Commented Apr 20, 2011 at 16:44