Grothendieck Duality via Diagonally Supported Sheaves

Following a formula found in the paper of Avramov, Iyengar, Lipman, and Nayak (2010) and ideas of Neeman and Khusyairi, we indicate that Grothendieck duality for finite tor-amplitude maps can be developed from scratch via the formula $f^! := \delta^*\pi_1^{\times}f^*$. Our strategy centers on the subcategory $\Gamma_{\Delta}(\mathrm{QCoh}(X \times X))$ of quasicoherent sheaves on $X \times X$ supported on the diagonal. By exclusively using this subcategory instead of the full category $\mathrm{QCoh}(X \times X)$ we give systematic categorical proofs of results in Grothendieck duality and reprove many formulas found in Neeman (2018). We also relate some results in Grothendieck duality with properties of the sheaf of (derived) Grothendieck differential operators.