Grothendieck-Verdier duality without the noetherian condition

Question

The Grothendieck-Verdier duality: $$ Rf_*\big(R\mathcal{H}\textit{om}_X^\bullet(\mathcal{E}^\bullet,f^!\mathcal{F}^\bullet)\big) \cong R\mathcal{H}\textit{om}^\bullet_Y(Rf_*\mathcal{E}^\bullet,\mathcal{G}^\bullet) $$ is known to hold for $f:X\to Y$ being a proper map of noetherian schemes.

Is there a way to get rid of this requirement on schemes to be noetherian, possibly by putting some extra conditions on the morphism $f$?

For example: let $X$ be smooth (projective in necessary) and consider the projection to the second factor $f:X\times U\to U$, where $U$ is some affine scheme. Does one have the duality in this setting?

Edit: After reading about the topic in Neeman's and Lipman's work, I have not managed to find an explicit construction of the right adjoint $f^\times$. In what cases is the explicit construction of $f^\times$ known? What would it be in the above example?

Answer 1

Does one have the duality in this setting?

Yes, we do have duality in a very general setting. Your question is equivalent to asking for the existence of a right adjoint to the derived pushforward functor $\mathbf{R}f_*\colon \mathbf{D}_{qc}(X) \to \mathbf{D}_{qc}(Y)$. It turns out that it exists for any morphism $f\colon X \to Y$ of qcqs schemes. Look at tag 0A9D for more details.

P. S. It is not a good idea to call this adjoint functor by $f^!$ unless $f$ is proper. Usually, another functor is denoted by $f^!$ that actually differs from the right adjoint $f^\times$ (SP denotes this functor by $a(f)$).

UPD: If $Y$ is qcqs and $f\colon X \to Y$ is proper and smooth of (pure) relative dimension $d$, then $f^!(-)\cong f^{\times}(-)\cong \Omega^d_{X/Y}[d]\otimes_{\mathcal O_X} f^*(-)$. Look at tag 0BRT and tag 0A9U for a proof in the noetherian case. The proof in the non-noetherian setup is essentially the same. Just note that noetherianness of $Y$ is used only to show that $\mathbf{R}f_*$ preserves perfect objects. However, it is true without this assumption as it is briefly explained in Example 2.2 of the paper Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor.