Some questions on derived pull-back and push-forward functors of proper birational morphism of Noetherian quasi-separated schemes

Question

Let $f: X \to Y$ be a proper birational morphism of Noetherian quasi-separated schemes. We have the derived pull-back $Lf^*: D(QCoh(Y))\to D(QCoh(X))$ (https://stacks.math.columbia.edu/tag/06YI) and similarly the derived push-forward $Rf_*: D(QCoh(X))\to D(QCoh(Y))$ functor. I have the following questions:

(1) Is $Rf_*$ a right-adjoint to $Lf^*$ ?

(2) Is $Lf^* \mathcal O_Y\cong O_X$ ?

(3) Does $Rf_*$ map $D(Coh(X))$ to $D(Coh(Y))$ and $D^b(Coh(X))$ to $D^b(Coh(Y))$ ?

(4) Does $Lf^*$ map $D(Coh(Y))$ to $D(Coh(X))$ and $D^b(Coh(Y))$ to $D^b(Coh(X))$ ?

Perhaps all these are standard and well-known, in which case, I would highly appreciate some references as well. Thank you.

Answer 1

Let $f \colon X \to Y$ be a morphism between locally ringed spaces. Then

(1) Is $Rf_*$ a right-adjoint to $Lf^*$?

Yes [L] Proposition (3.2.3), after Spaltenstein, 1988.

(2) Is $Lf^* \mathcal O_Y\cong O_X$?

Obvious: $\mathcal O_Y$ is $\mathcal O_Y$-flat therefore

$$ Lf^* \mathcal O_Y \cong f^* \mathcal O_Y \cong O_X $$

(4) Does $Lf^*$ map $D(Coh(Y))$ to $D(Coh(X))$?

And $D^b(Coh(Y))$ to $D^b(Coh(X))$?

For the first question, assuming that $X$ and $Y$ and Noetherian schemes this is a local question and thus it follows form the fact that a finitely presented module keeps finitely presented after extending scalars. To preserve boundedness you have to assume that $f$ is of finite flat dimension aka finite Tor-dimension.

Now assume $f$ is a proper map of Noetherian schemes

(3) Does $Rf_*$ map $D(Coh(X))$ to $D(Coh(Y))$ and $D^b(Coh(X))$ to $D^b(Coh(Y))$?

Yes. A general discussion even without Noetherian hypothesis is in [L], section 4.3. For a proof in the case considered see [GW] Theorem 23.17.

References

[GW] Görtz, U., Wedhorn, T.; Algebraic Geometry II: Cohomology of Schemes. Springer 2023.

[L] Lipman, Joseph: Notes on derived functors and Grothendieck duality. Foundations of Grothendieck duality for diagrams of schemes, 1–259, Lecture Notes in Math., 1960, Springer, Berlin, 2009.