connections between Grothendieck's and Serre's duality

2012-06-13T10:55:17Z

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 $f^! \mathcal{O}_Y$ , appearing in Grothendieck's duality, is the dualizing sheaf for $X$. Let $F$ be a coherent sheaf on $X$ . Starting from $ Hom_{\mathcal{O}_X}(F, f^! \mathcal{O}_Y) \simeq Hom_{{O}_Y}(Rf_* F, \mathcal{O}_Y) $ applying the cohomology functor $ H^i $ we obtain $Ext^i(F, f^! \mathcal{O}_Y) \simeq Ext^i(Rf_* F, \mathcal{O}_Y).$ Using Yoneda's Formula, the right term becomes $Hom^i_{D(Y)}(Rf_* F, \mathcal{O}_Y) \cong Hom_{D(Y)}(Rf_* F, \mathcal{O}_Y[i]) \cong Hom(\widetilde{H^1(X,F)}[-i], \mathcal{O}_Y),$ 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 $\Gamma(Y, \mathcal{O}_Y)= \mathbb{C}$ , the last term is equal to $Hom(H^1(X,F)[-i], \mathbb{C}) = H^{1-i}(X,F)'$ . Now, we have $Hom^i_{D(X)}(F, f^! \mathcal{O}_Y) \cong H^{1-i}(X,F)'$ and, shifting by $(-n+1-i)$ , $Hom(F, f^! \mathcal{O_Y}[-n+1]) \cong H^n(X,F)'$ . So, $f^!(\mathcal{O}_Y)[-n+1]$ is a dualizing sheaf for $X$ . But we know that, for a nonsingular projective variety the (unique) dualizing sheaf is the canonical sheaf $\omega$ . Thus, we must have $f^!\mathcal{O}_Y[-n+1] \cong \omega$ , then $f^! \mathcal{O}_Y = \omega[n-1]$ .

I should obtain $f^! \mathcal{O}_Y = \omega[n]$... what's wrong? Thank you

Answer by Leo Alonso for connections between Grothendieck's and Serre's duality

2013-02-08T08:53:51Z

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

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.

The proof relies on the "fundamental local isomorphism" and several formal properties of $f^!$. Hope it helps.

connections between Grothendieck's and Serre's duality - MathOverflow

connections between Grothendieck's and Serre's duality

Answer by Leo Alonso for connections between Grothendieck's and Serre's duality