Abstract
We discuss vanishing of cohomology of finite modules over Cohen-Macaulay local rings $(R, \mathfrak m)$. Special attention is given to the case when the modules are annihilated by $\mathfrak m^2$. (Note that if $\mathfrak m^3=0$, then we can assume the modules satisfy this condition.) In this case we obtain effective versions of conjectures of Auslander-Reiten and Tachikawa.
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