Derived equivalences for chain complexes with support

TL;DR

This paper establishes derived equivalences for chain complexes with support in abelian categories, providing new insights into their structure and applications in K-theory, especially for complexes with finite support dimensions.

Contribution

It introduces conditions under which derived equivalences hold for complexes with support, extending previous results to graded modules and coherent sheaves.

Findings

Derived equivalence between certain subcategories of chain complexes.

Application to complexes with finite support dimensions in algebraic geometry.

Descriptions of homotopy fibers in hermitian K-theory.

Abstract

For a Serre subcategory $\mathscr L$ and a resolving subcategory $\mathscr A$ of an abelian category, we show that the derived equivalence $D^b(\overline{\mathscr A} \cap \mathscr L) \simeq D^b_{\mathscr L}(\mathscr A)$ holds under certain conditions. We apply this to obtain derived equivalences in the contexts of chain complexes of graded modules or coherent sheaves, with finite $\mathscr A$-dimension, supported on closed sets having eventually finite $\mathscr A$-dimension. Using this, we obtain descriptions of the homotopy fibers in (hermitian) K theory of the restriction maps to certain open sets.