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 fi eld. By studying the relative S$\_2$-condition and base change properties, valuable results are proved for $\mathbb Q$-Gorenstein morphisms, which include the in finitesimal criteria, the valuative criterion, and $\mathbb Q$-Gorenstein re finements.