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Let $A=kG$ be a group algebra with a finite group $G$ and a field $k$. Let $T^i(M,N)= \underline{Hom_A}(\Omega^i(M),N)$ be the $i$-th Tate cohomology group. Note $T^i(M,N)= Ext_A^i(M,N)$ in case $i \geq 1$.

Question: Do we have $dim(T^i(M,M)) \geq dim(Ext_A^1(M,M))$?

This is true in case $A$ is representation-finite.

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    $\begingroup$ No, the Quaternion group $Q$ of order 8 has $\dim \hat{H}^1(Q,\mathbb{F}_2) =2$ and $\dim \hat{H}^n(Q,\mathbb{F}_2) =1$ for $n$ a multiple of 4. $\endgroup$
    – tj_
    Commented Nov 14, 2019 at 17:52

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