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Let $X$ and $Y$ be Artin algebraic stacks, and $f:X\to Y$ a morphism. I am interested in a pullback morphism $Lf^\ast : D^-_{qcoh}(Y)\to D^-_{qcoh}(X)$.

In his paper Sheaves on Artin stacks, Martin Olsson uses the lisse-etale site to define $D_{qcoh}(X)$ and $D_{qcoh}(Y)$, and remarks that because $f:X\to Y$ does not induce a morphism of lisse-etale topoi, he can define $Lf^\ast$ only on a category of projective systems of derived category objects.

Question: Is this the best one can expect in the world of Artin stacks, or can we do better, for example, by replacing the lisse-etale site with the big etale site. I would hope so, because $f$ does induce a morphism of big etale topoi.

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I don't know about others, but I am having trouble reading your question because of something you have done with the math typesetting here. Can you rewrite it so that it displays correctly? (Or am I the only one with a problem reading this?) –  Michael A Warren Sep 28 '11 at 2:32
I'm sorry, it displays perfectly on my browser, so I don't know what to fix. –  euklid345 Sep 28 '11 at 4:32
Weird, now it also shows up fine on mine (same computer, same browser) as well. –  Michael A Warren Sep 28 '11 at 13:22
Well, I changed all the fancy mathfrak fonts to roman. I guess that helped. –  euklid345 Sep 28 '11 at 18:55

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