Derived pullback of quasi-coherent complexes between algebraic stacks - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T04:06:55Z http://mathoverflow.net/feeds/question/76583 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76583/derived-pullback-of-quasi-coherent-complexes-between-algebraic-stacks Derived pullback of quasi-coherent complexes between algebraic stacks euklid345 2011-09-28T00:29:11Z 2011-09-28T04:36:11Z <p>Let $X$ and $Y$ be Artin algebraic stacks, and $f:X\to Y$ a morphism. I am interested in a pullback morphism <code>$Lf^\ast : D^-_{qcoh}(Y)\to D^-_{qcoh}(X)$</code>. </p> <p>In his paper <a href="http://www.reference-global.com/doi/abs/10.1515/CRELLE.2007.012" rel="nofollow">Sheaves on Artin stacks</a>, 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. </p> <p><strong>Question:</strong> 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.</p>