Annihilation of cohomology and decompositions of derived categories

TL;DR

The paper characterizes when an element in the center of a coherent ring annihilates certain Ext groups, linking this to a specific decomposition of the bounded derived category and discussing implications for derived dimension finiteness.

Contribution

It establishes a precise criterion connecting central element annihilation of Ext groups with a decomposition of the derived category, advancing understanding of derived category structures.

Findings

Characterization of Ext annihilation via derived category decomposition

Link between central elements and n-fold extensions in derived categories

Applications to finiteness of derived category dimension

Abstract

It is proved that an element $r$ in the center of a coherent ring $\Lambda$ annihilates $\mathrm{Ext}^{n}_{\Lambda}(M,N)$, for some positive integer $n$ and all finitely presented $\Lambda$-modules $M$ and $N$, if and only if the bounded derived category of $\Lambda$ is an extension of the subcategory consisting of complexes annihilated by $r$ and those obtained as $n$-fold extensions of $\Lambda$. This has applications to finiteness of dimension of derived categories.