$F$-finite schemes have a dualizing complex

Abstract

In this paper we show that any Noetherian $F$-finite scheme has a dualizing complex $\ddot{I}{‰}_X^{\bullet }$ with the property that for all finite type maps $f:X\to Y$ between $F$-finite Noetherian schemes there is a canonical isomorphism $\ddot{I}{‰}_X^{\bullet }\stackrel{\cong }{\to }{f}^{!}\ddot{I}{‰}_Y^{\bullet }$ in $D_{coh}^b(X)$. This, in particular, applies to the Frobenius morphism $F:X\to X$ so that we obtain a canonical isomorphism $\ddot{I}{‰}_X^{\bullet }\stackrel{\cong }{\to }{F}^{!}\ddot{I}{‰}_X^{\bullet }$. To prove this, we rely on a result of Gabber that every Noetherian $F$-finite ring is a quotient of a regular ring, from which it follows that every $F$-finite Noetherian scheme has a (potentially non-canonical) dualizing complex. To make this canonical, we identify the dualizing complex of any $F$-finite Noetherian scheme as a unit of an alternate symmetric monoidal structure on $D_{coh}^b(X)$ we call the $!$-tensor product. We also sketch an alternate approach to finding this canonical dualizing complex following the more classical approach to Grothendieck duality.

59 pages, comments welcome