connections between Grothendieck's and Serre's duality - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T22:53:29Z http://mathoverflow.net/feeds/question/99428 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99428/connections-between-grothendiecks-and-serres-duality connections between Grothendieck's and Serre's duality emmy 2012-06-13T10:55:17Z 2013-02-08T15:49:30Z <p>Hi, I would like to show that if $f: X \rightarrow Y=Spec \, \mathbb{C}$, where $X$ is a nonsingular complex projective variety, is the projection to a point, then the complex <code>$f^! \mathcal{O}_Y$</code>, appearing in Grothendieck's duality, is the dualizing sheaf for $X$. Let <code>$F$</code> be a coherent sheaf on <code>$X$</code>. Starting from <code>$Hom_{\mathcal{O}_X}(F, f^! \mathcal{O}_Y) \simeq Hom_{{O}_Y}(Rf_* F, \mathcal{O}_Y)$</code> applying the cohomology functor $H^i$ we obtain <code>$Ext^i(F, f^! \mathcal{O}_Y) \simeq Ext^i(Rf_* F, \mathcal{O}_Y).$</code> Using Yoneda's Formula, the right term becomes <code>$Hom^i_{D(Y)}(Rf_* F, \mathcal{O}_Y) \cong Hom_{D(Y)}(Rf_* F, \mathcal{O}_Y[i]) \cong Hom(\widetilde{H^1(X,F)}[-i], \mathcal{O}_Y),$</code> where, for the last isomorphism, I use Theorem 8.5 from Hartshorne'a Algebraic Geometry, p.251. Now, by Corollary 5.5 pag.151 from Hartshorne, and remembering that <code>$\Gamma(Y, \mathcal{O}_Y)= \mathbb{C}$</code>, the last term is equal to <code>$Hom(H^1(X,F)[-i], \mathbb{C}) = H^{1-i}(X,F)'$</code>. Now, we have <code>$Hom^i_{D(X)}(F, f^! \mathcal{O}_Y) \cong H^{1-i}(X,F)'$</code> and, shifting by <code>$(-n+1-i)$</code>, <code>$Hom(F, f^! \mathcal{O_Y}[-n+1]) \cong H^n(X,F)'$</code>. So, <code>$f^!(\mathcal{O}_Y)[-n+1]$</code> is a dualizing sheaf for <code>$X$</code>. But we know that, for a nonsingular projective variety the (unique) dualizing sheaf is the canonical sheaf <code>$\omega$</code>. Thus, we must have <code>$f^!\mathcal{O}_Y[-n+1] \cong \omega$</code>, then <code>$f^! \mathcal{O}_Y = \omega[n-1]$</code>.</p> <p>I should obtain $f^! \mathcal{O}_Y = \omega[n]$... what's wrong? Thank you</p> http://mathoverflow.net/questions/99428/connections-between-grothendiecks-and-serres-duality/121164#121164 Answer by Leo Alonso for connections between Grothendieck's and Serre's duality Leo Alonso 2013-02-08T08:53:51Z 2013-02-08T15:49:30Z <p>The proof that if $f \colon X \to Y$ is proper and smooth then $f^!\mathcal{O}_Y \cong \Omega^n_F[n]$ ($n=$ dim. rel.$(f)$) assuming only the existence of $f^!$ follows from theorem 3 in</p> <p>Verdier, Jean-Louis Base change for twisted inverse image of coherent sheaves. 1969 Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) pp. 393–408 Oxford Univ. Press, London.</p> <p>The proof relies on the "fundamental local isomorphism" and several formal properties of $f^!$. Hope it helps.</p>