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This is probably pretty basic, but as I said before I'm just beginning my way in the language of stacks.

Say you have an etale cover X->Y of stacks (in the etale site). Is there a standard way to define the degree of this cover? Here's my intuition: if X and Y are schemes, we can look etale locally and then this cover is Yoneda-trivial in the sense of http://front.math.ucdavis.edu/0902.3464 , meaning that etale locally it is just a disjoint unions of (d many) pancakes. Can we do this generally? Is there some "connectivity" conditions on Y for this to work? Is there a different valid definition for degree of an etale cover of stacks?

Yoneda-Triviality

I figured since nobody answered so far, maybe I should write down what a possible Yoneda-triviality condition could mean for stacks:

Def: Call f:X->Y (stacks) Yoneda-trivial if there exists a set of sections of f, S, such that the natural map Y(Z)xS->X(Z) is an isomorphism (or maybe a bijection?) for any connected scheme Z.

But I'm still clinging to the hope that there's a completely different definition out there that I'm just not aware of.

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3 Answers 3

up vote 4 down vote accepted

There is a notion of degree of an integral morphism of DM stacks, and I think it is in Vistoli's "Intersection theory ... " paper here. Its unrelated to Yoneda triviality.

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Make a base change $f':Y'\to Y$ so that $f':X'=X\times_Y Y' \to Y'$ is a morphism of schemes. That should be possible if $f:X\to Y$ is representable. Then define $\deg(f):=\deg(f')$.

Most, if not all, properties of stacks and morphisms between them are defined by descent, i.e. by making a base change to schemes. If the property (P) is stable by further base changes, then it can be so defined, and is independent of the base change $Y'\to Y$.

In your case, you should ask: is the degree of an etale morphism of schemes well-defined? It is well defined locally on the base, but not globally (what if $Y$ is disconnected?). So clearly you need a few natural assumptions on $X\to Y$.

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I think the definition of VA is equivalent to the one found in Katz-Mazur –  Arno Kret Mar 7 '10 at 12:45

The question depends on which maps you want to call "etale". If one thinks that the property of $f:X \to Y$ being etale should be etale local on $X,$ then etale morphisms doesn't need to be representable; for instance $BG \to pt$ is etale and epic, if $G$ is a finite group, and its degree is....$1/|G|?$ Anyway, you can rule them out by always considering representable etale surjections.

Also, I guess degree should better be defined for representable finite etale coverings, rather than just representable etale surjections. Imagine the disjoint union of two open sets covering a space: the degree over their overlap is 2, and is 1 elsewhere. This is not a locally constant function on the space.

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