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Given a representation $V$ of a group $G$, we can think of $V$ as a vector bundle over the classifying stack $BG$, and we can define its index $\chi(BG; V)$ to be the dimension of the $G$-invariant part $V^G$ of the representation.

Now let $G = GL(1)$ (or $\mathbb{G}_m$ or $\mathbb{C}^\ast$ if you like). Then let $\phi_n : \mathbb{P}^n \to BG$ be the map corresponding to the bundle $\mathcal{O}(1)$ (or maybe we should take map corresponding to the bundle $\mathcal{O}(-1)$, I always mix it up).

Let $V$ be a representation of $G$. Then we can compute the index (i.e. sheaf cohomology Euler characteristic) of $\phi_n^\ast V$ over $\mathbb{P}^n$. We can show, just by doing the calculation, that for sufficiently large $n$ (how large depends on the representation $V$), this index is a polynomial in $n$. Then, plugging in $n=-1$ into this polynomial, one can show that the result is equal to $\chi(BG; V)$, defined as above.

So, in this fashion, we can recover the index over the stack $BGL(1)$ if we know the index over each $\mathbb{P}^n$ (for sufficiently large $n$).

This result seems to make sense, because at least in topology we have $B\mathbb{C}^\ast \simeq \mathbb{CP}^\infty$, and at least intuitively we can think of $\mathbb{CP}^n$ as "approximating" $\mathbb{CP}^\infty$.

But so far I don't really have a satisfactory explanation for this, other than "it follows by doing the computation" and "it seems to make intuitive sense" as I've explained above. I wonder if there is a better way to see this; does it follow by some deeper facts?; does it follow by some more general theorems? I'm sorry that my question is not very precise.

I would also be interested in any other theorems or results which relate finite dimensional projective spaces with $BGL(1)$. (There are already a few such results in the link in S. Carnahan's comment below.)

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    $\begingroup$ I don't think this is enough to be an answer, but comments 23 (by David Ben-Zvi) and 27 (by "bb") in the following blog post are especially informative: sbseminar.wordpress.com/2008/06/19/whats-a-stack In particular, $\mathbb{A}^1$ homotopy theory gives one rigorous sense in which the classifying stack of $GL(1)$ is approximated by projective spaces: one has a sequence of morphisms whose fibers are increasingly connected. $\endgroup$
    – S. Carnahan ♦
    Commented May 23, 2011 at 5:09
  • $\begingroup$ I've added a bounty in an attempt to attract more answers... $\endgroup$
    – Kevin H. Lin
    Commented May 25, 2011 at 22:40

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The paper arXiv:0808.2785 of Anderson, Griffeth, and Miller uses precisely this approximation technique to prove a certain positivity result in the torus equivariant k-theory of homogeneous spaces (see their prop. 3.1). AGR reference the paper

Daniel Edidin and William Graham, Riemann–Roch for equivariant Chow groups, Duke Math. J. 102 (2000), no. 3, 567–594

for a discussion of the use of this technique in defining equivariant Chow groups.

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  • $\begingroup$ Ekedahl used this technique in "Approximating classifying spaces by smooth projective varieties" (arxiv.org/abs/0905.1538) $\endgroup$
    – Daniel Bergh
    Commented Feb 22, 2016 at 21:19

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