Commutation of pushforward and pullback along cartesian squares

Question

I am sure this is well-known, but I cannot find a reference. Consider a cartesian square $$\require{AMScd} \begin{CD} P @>{v}>> X\\ @V{g}VV @VV{f}V \\ Z @>{u}>> Y \end{CD}$$ where all morphisms are morphisms of algebraic varieties over a field. It is known (e.g. Harthorne, III.9.3) that when $u$ is flat and $f$ is proper, there is a natural isomorphism $$u^\ast f_\ast\overset{\sim}{\longrightarrow} g_\ast v^\ast$$ of functors $Coh_X\to Coh_Z$.

Question. Is there also an isomorphism $v_\ast g^\ast\cong f^\ast u_\ast$ of functors $QCoh_Z\to QCoh_X$?

Assuming $Z$ is proper (and $Y$ separated), we can restrict attention to coherent sheaves. I am actually interested in the case when $u$ is \'etale and $f$ is a closed immersion, but I do not think this specialization is of any real help.

Thank you in advance!

Answer 1

I think what you are saying is (almost) true: it works in the derived category.

Below, for a scheme map $\alpha$, we denote $\alpha^*$ to be the derived functor $\mathsf{L}\alpha^*$ and denote $\alpha_*$ to be the derived functor $\mathsf{R}\alpha_*$. Also, $\mathbf{D}_\mathsf{qc}(-)$ denotes the derived category of sheaves of modules with quasi-coherent cohomology.

We adopt your notation, but generalize a bit: let $X,Y,P,Z$ be quasi-separated schemes, and suppose they fit into a commutative square $\sigma$: $$\begin{CD} P @>v>> X\\ @VgVV \sigma @VVfV\\ Z @>u>> Y \end{CD}$$ where all maps are quasi-compact and quasi-separated.

We say $\sigma$ is

• independent if the functorial map $$u^*f_* \overset{\sim}{\longrightarrow} g_*v^*$$ of functors $\mathbf{D}_\mathsf{qc}(X) \to \mathbf{D}_\mathsf{qc}(Z)$ is an isomorphism;
• $'$-independent if the functorial map $$f^*u_* \overset{\sim}{\longrightarrow} v_*g^*$$ of functors $\mathbf{D}_\mathsf{qc}(Z) \to \mathbf{D}_\mathsf{qc}(X)$ is an isomorphism;
• tor-independent if $\sigma$ is cartesian and for all pairs of points $z \in Z$, $x \in X$ such that $y := u(z) = f(x)$, $$\operatorname{Tor}_i^{\mathcal{O}_{Y,y}}(\mathcal{O}_{Z,z},\mathcal{O}_{X,x}) = 0, \quad \text{for all}\ i > 0.$$

In this situation, the following theorem holds:

Theorem [Lipman, 3.10.3]. The three independence conditions above are equivalent.

The proof is a bit complicated, but the idea is you want to reduce to the affine case after showing the independence conditions above are local. Lipman then proceeds by showing that these three independence conditions are equivalent to being Künneth-independent, whose definition is a bit harder to state.

Answer 2

The condition that $f$ be proper is necessary to ensure that the direct image of a coherent sheaf is coherent.

Under more general conditions, the direct image of a quasicoherent sheaf is quasicoherent.

The standard reference for all this is, no doubt, EGA; EGA I, Prop. 9.2.1, Cor. 9.2.2 show that the image sheaf of a quasicoherent sheaf is quasicoherent. Ravi Vakil has a lovely set of notes on foundations of algebraic geometry in his webpage, based on EGA, that you may find useful (but it won't carry more than EGA on this topic).

The case of a closed immersion $f$ is especially easy: you simply deal with the affine case, where $f_*\tilde{M}$ is simply viewing $M$ as an $A$-module instead of an $A/I$-module. There is an equivalence of quasicoherent modules that are isomorphic to $f_*F$ for $F$ on $X$ with those on $Y$ which are supported on $X$. Thus pulling back by $A\to B$ is the same as pulling back via $A/I \to B/IB$ and then pushing forward. The notation used is: $I$ is the ideal of a closed subscheme $X$ of $Spec(A)=Y$ (It suffices to reduce to the affine case for $f$, that's why the above settles the proof of equivalence you are seeking)

That's how one settles the desired equivalence in your case, with $u$ flat and $f$ a closed immersion in the case of quasicoherent sheaves.

In fact, with the generality you ask, one may use that every quasicoherent sheaf on a Noetherian scheme is a direct limit of quasicoherent subsheaves in order to obtain a proof of equivalence for quasicoherent sheaves, see e.g. these notes (based on EGA I.9, of course).

https://math.berkeley.edu/~mhaiman/math256-fall13-spring14/EGAI-9.pdf