# connections between Grothendieck's and Serre's duality

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

• I'm too lazy to check through the whole thing, but certainly your identification $Rf_*F= \widetilde{H^1(X,F)}$ isn't right. Jun 13, 2012 at 11:26
• emmy, how do you define the dualizing sheaf for $X$? Do you just want to show it satisfies the usual Serre duality? Jun 13, 2012 at 13:38
• If you just want to find the mistake in the argument above, then you could try specialising it to the case where $X$ is a point and $n=0$ :-) because I reckon that even in this degenerate case $\omega$ is non-zero (I guess it's the structure sheaf). Jun 13, 2012 at 21:26
• emmy, that's right, but how are you getting $H^1$? You should get $\widetilde{R \Gamma(X, F) }$. Jun 13, 2012 at 23:06
• emmy, no, you definitely don't get that. $R\Gamma(X,F)$ is a complex (an object in the derived category). You get $h^1(R \Gamma(X, F)) = H^1(X,F)$. In particular, one of the cohomologies of that complex is what you want. Jun 14, 2012 at 12:06

## 1 Answer

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.