What is the upper shriek in Grothendieck duality in the non-proper case? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T02:54:32Z http://mathoverflow.net/feeds/question/103820 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103820/what-is-the-upper-shriek-in-grothendieck-duality-in-the-non-proper-case What is the upper shriek in Grothendieck duality in the non-proper case? Akhil Mathew 2012-08-02T22:12:06Z 2012-08-02T22:49:46Z <p>I'm trying to learn a little about Grothendieck duality. One version of the theorem states that if $f: X \to Y$ is a proper morphism of schemes, then the induced functor on derived categories $f_*: D^+(\mathrm{QCoh}(X)) \to D^+(\mathrm{QCoh}(Y))$ has a right adjoint $f^!$ (and under nice hypotheses, these will preserve the subcategories with coherent cohomology). The existence of an adjoint can be proved via adjoint functor arguments, even without assuming $f$ proper; this was done, e.g., by Neeman. (The point is that a triangulated functor between nice triangulated categories (or, stable $\infty$-categories) which preserves coproducts is a left adjoint.) </p> <p>However, in trying to identify $f^{!}$, we might want to be able to localize on $X$ and $Y$, and thus deal with the non-proper case. My understanding is that the upper-shriek functor $f^{!}$ there is not supposed to be the right adjoint to $f_{\ast}$: for example, for an open immersion it should be the upper-star $f^*$. </p> <p>In the topological version, one can define a $f_{!}$ functor for sheaves (push-forward with compact support) and $f^!$ is the right adjoint to $f_{!}$. Is there any "functorial" way to interpret $f^{!}$ when $f$ is not proper? </p> http://mathoverflow.net/questions/103820/what-is-the-upper-shriek-in-grothendieck-duality-in-the-non-proper-case/103821#103821 Answer by Jacob Bell for What is the upper shriek in Grothendieck duality in the non-proper case? Jacob Bell 2012-08-02T22:49:46Z 2012-08-02T22:49:46Z <p>I don't have an answer, but maybe these <a href="http://www.google.co.uk/url?sa=t&amp;rct=j&amp;q=lipman%20duality&amp;source=web&amp;cd=1&amp;ved=0CF8QFjAA&amp;url=http%3A%2F%2Fwww.math.purdue.edu%2F~lipman%2FDuality.pdf&amp;ei=UgEbUOjkCoPR0QXb5ICwCQ&amp;usg=AFQjCNHa4qdoLEbh-LfoYLDr5h5r7KJ9ag&amp;sig2=ueCVTqW8rd03uIsnOvndyw" rel="nofollow">notes by Lipman</a> help.</p> <p>From what I understand the upper pling functor $f^!$ is a sort of Frankenstein, definitely for etale maps it's given by ordinary pullback. More generally for smooth maps you have to tensor with the relative canonical bundle.</p> <p>The only description I know of is via dualising complexes, but that's perhaps not categorical enough for what you want.</p>