Grothendieck Duality and $\mathbb{Q}$-Gorenstein Morphisms

Abstract

The notions of a $\mathbb{Q}$-Gorenstein scheme and a $\mathbb{Q}$-Gorenstein morphism are introduced for locally Noetherian schemes by dualizing complexes and (relative) canonical sheaves. These cover all the previously known notions of a $\mathbb{Q}$-Gorenstein algebraic variety and a $\mathbb{Q}$-Gorenstein deformation satisfying the Kollar condition, over a field. By studying the relative S${}_2$-condition and base change properties, valuable results are proved for $\mathbb{Q}$-Gorenstein morphisms, which include the infinitesimal criteria, the valuative criterion, and $\mathbb{Q}$-Gorenstein refinements.