Grothendieck duality and Q-Gorenstein morphisms

TL;DR

This paper introduces and studies the concepts of Q-Gorenstein schemes and morphisms using dualizing complexes, extending previous notions and establishing key properties and criteria for these objects in algebraic geometry.

Contribution

It defines Q-Gorenstein schemes and morphisms via dualizing complexes, generalizing existing notions and providing foundational results on their properties and criteria.

Findings

Established criteria for Q-Gorenstein morphisms

Proved base change and S2-condition results

Extended notions to all known Q-Gorenstein varieties

Abstract

The notions of $\mathbb Q$-Gorenstein scheme and of $\mathbb Q$-Gorenstein morphism are introduced for locally Noetherian schemes by dualizing complexes and (relative) canonical sheaves. These cover all the previously known notions of $\mathbb Q$-Gorenstein algebraic variety and of $\mathbb Q$-Gorenstein deformation satisfying Koll\'ar condition, over a field. By studies on relative $\mathbf S_{2}$-condition and base change properties, valuable results are proved for $\mathbb Q$-Gorenstein morphisms, which include infinitesimal criterion, valuative criterion, $\mathbb Q$-Gorenstein refinement, and so forth.