Exceptional Pullback

This post is going to be a summary of the ideas constructing the exceptional pullback functor $\pi^! \colon \qcoh_Y \to \qcoh_X$ (read "pi shriek") in the case $\pi$ is a finite flat morphism following Vakil's Foundations of Algebraic Geometry. The general construction for $\pi^!$ quite heavily relies on derived categories and so we won't go over it here, but the reader can consult Sheaves on Manifolds by Kashiwara and Schapira.

The purpose is that $\pi^!$ naturally serves as a right adjoint to the pushforward $\pi_*$, which is strange since normally we think of $\pi_*$ as right adjoint to $\pi^*$. Let $\pi \colon X \to Y$ be a finite flat morphism of locally Noetherian schemes, and let $\fang$ be a quasicoherent sheaf on $Y$. Then

is also a quasicoherent sheaf on $Y$ as $\pi_* \calo_X$ is locally finitely presented. To see this, take a local finite presentation on $U \subset Y$

Since $\rHom_U(\cdot, \fang)$ is left-exact,

Since $\qcoh_Y$ is an abelian category, this shows $\rHom_Y(\pi_* \calo_X, \fang)$ is quasicoherent on $Y$. Further, $\rHom_Y(\pi_* \calo_X, \fang)$ has the structure of a $\pi_* \calo_X$-module (by $(r\cdot \varphi)(s) := \varphi(rs)$). We show this gives it the structure of a quasicoherent sheaf on $X$. More generally,

Lemma Suppose $\fanr$ is a quasicoherent sheaf of algebras on $Y$, and $\pi \colon \underline{\spec}_Y \fanr \longrightarrow Y$ is an affine morphism. Then the category of quasicoherent sheaves on $Y$ with the structure of $\fanr$-modules is equivalent to the category of quasicoherent sheaves on $\underline{\spec}_Y \fanr$.

Let $\fanf$ be a quasicoherent sheaf on $Y$ with the structure of a $\fanr$-module. Since everything is affine-local on the target, it suffices to assume $Y = \spec A$ and hence $\pi \colon \underline{\spec}_Y \fanr = \spec R \to \spec A$ for some $A$-algebra $R$. Then $\fanf \simeq \tilde{M}$ as $\calo_{\spec A}$-modules for some $A$-module $M$ by quasicoherence. By assumption, $M$ has the structure of an $R$-module $M_R$, and so we can define the quasicoherent sheaf $\tilde{M_R}$ on $\spec R$. It remains to show this is functorial in $\fanf$, which we omit.

In the above, we take $\fanr = \pi_* \calo_X$ and note that $\underline{\spec}_Y \pi_* \calo_X \simeq X$ since $\pi$ is affine. Since $\rHom_Y(\pi_* \calo_X, \fang)$ is quasicoherent on $Y$ with the structure of a $\pi_* \calo_X$-module, we get a quasicoherent sheaf on $X$. This is the definition of our covariant exceptional pullback functor

We have natural isomorphisms of functors $\qcoh_Y \to \qcoh_Y$

Further, one can check that for $\fanf \in \qcoh_X$ and $\fang \in \qcoh_Y$ we have a natural isomorphism functorial in $\fanf, \fang$

In particular, $(\pi_*, \pi^!)$ form an adjoint pair.

Application to Frobenius Splitting

One nice application appears in characteristic $p > 0$ geometry. Let $F \colon X \to X$ be the absolute Frobenius morphism. A variety $X$ is called Frobenius split if the map

splits as $\calo_X$-modules. Using the exceptional pullback, we can reformulate this condition. Consider

Then $X$ is Frobenius split if and only if there exists a morphism $F^! \calo_X \to \calo_X$ inducing the identity on $\calo_X$ via adjunction. When $X$ is smooth, we can identify

via an extended application of Grothendieck-Serre duality. This gives the classical criterion of $X$ being Frobenius split if and only if there exists a section

whose residue is 1. This perspective relevant to flag and toric varieties, where explicit Frobenius splittings can be constructed.