New enhancements of derived categories of coherent sheaves and applications

TL;DR

This paper develops new enhancements for derived categories of coherent sheaves, enabling better understanding of their properties and applications in relating algebraic and geometric smoothness, as well as properness.

Contribution

It introduces novel enhancements for derived categories and subcategories, facilitating the translation of Fourier-Mukai functors and linking homological smoothness to geometric smoothness.

Findings

Established new criteria for homological smoothness of derived categories.

Characterized properness of categories in geometric terms.

Connected algebraic properties with geometric smoothness.

Abstract

We introduce new enhancements for the bounded derived category $D^b(Coh(X))$ of coherent sheaves on a suitable scheme $X$ and for its subcategory $Perf(X)$ of perfect complexes. They are used for translating Fourier-Mukai functors to functors between derived categories of dg algebras, for relating homological smoothness of $Perf(X)$ to geometric smoothness of $X,$ and for proving homological smoothness of $D^b(Coh(X)).$ Moreover, we characterize properness of $Perf(X)$ and $D^b(Coh(X))$ geometrically.