Grothendieck Duality and D-modules via Diagonally Supported Sheaves

Abstract

We study the !-pullback functor and the theory of (relative) D-modules along morphisms of qcqs schemes p_X : X → S which are almost of finite presentation and finite tor-amplitude. Key to our approach is the category of quasicoherent sheaves on X ×S X supported on the diagonal. In particular we indicate that “reduction formulae” can be used as foundations for the theory of Grothendieck Duality. We also set-up the theory of D-modules from scratch using this approach and show that in cases of overlap, it agrees with classical definitions using embeddings or the de Rham stack.