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# Homology, Homotopy and Applications

## Volume 16 (2014)

### Number 2

### Annihilation of cohomology and decompositions of derived categories

Pages: 231 – 237

DOI: http://dx.doi.org/10.4310/HHA.2014.v16.n2.a12

#### Authors

#### 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.

#### Keywords

cohomology annihilator, derived category, projective class

#### 2010 Mathematics Subject Classification

16E30, 16E35, 18G25

Published 30 November 2014