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Let $f:X \to Y$ be a morphism of reasonable schemes (qcqs). Let $f^*: D(Y) \to D(X)$ be the pullback defined on the derived unbounded categories of quasi-coherent sheaves.

Question: When does $f^*$ commute with arbitrary products?

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    $\begingroup$ This happens iff $f^*$ is a right adjoint. I think that the go-to answer is that it happens if $f$ is smooth. I am not sure if you can say anything in other situations. $\endgroup$
    – Denis Nardin
    Commented Feb 3, 2017 at 1:01
  • $\begingroup$ @DenisNardin I was under the impression infinite products don't even commute with restriction to open subsets. $\endgroup$
    – Saal Hardali
    Commented Feb 3, 2017 at 9:32
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    $\begingroup$ @DenisNardin For instance i'm pretty sure a counterexample is the sheaf $\prod_n k[t]/t^n k[t]$ pulled back along $\mathbb{A}^1 - 0$ $\endgroup$
    – Saal Hardali
    Commented Feb 3, 2017 at 11:43
  • $\begingroup$ You might also be interested in arxiv.org/abs/1501.01999, where $f^*$ having a left adjoint is called a Grothendieck--Neeman duality situation, see theorem 1.7. This is closely related to the answer of Marc. $\endgroup$
    – pbelmans
    Commented Feb 4, 2017 at 8:12

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The functor $f^*: D(Y) \to D(X)$ preserves limits if $Y$ is noetherian and $f$ is proper flat with Gorenstein fibers, for example if $f$ is smooth and proper. Under these assumptions: $f^!$ is right adjoint to $f_*$ by properness, $f^!(\mathcal F)=f^!(\mathcal O_Y)\otimes f^*(\mathcal F)$ by finite tor-amplitude, and $f^!(\mathcal O_Y)$ is invertible by Gorenstein. Hence, $f^*$ is right adjoint to $f_*(f^!\mathcal O_Y\otimes -)$.

See http://stacks.math.columbia.edu/tag/0C02 for a discussion of Gorenstein morphisms, particularly Lemma 45.43.8, and section 7 in https://arxiv.org/pdf/1406.7599v1.pdf for the relevant properties of $f^!$.

ETA According to pbelmans' comment above, the Gorenstein assumption is not actually needed: if $Y$ is arbitrary and $f$ is proper, pseudo-coherent, and of finite tor-amplitude, then $f^*$ is right adjoint to $f_*(f^!\mathcal O_Y\otimes -)$. Combine Corollary 6.4.2.7 in http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf and Theorem 1.7 in https://arxiv.org/abs/1501.01999.

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