\begin{center}{\sc \large Nilpotent elements in 
Hochschild cohomology }

\medskip

Karin Erdmann



\begin{abstract}
\noindent We study the algebra
$A=K\langle x, y\rangle /(x^2, y^2, (xy)^k+q(yx)^k)$ 
over the field $K$ where $k\geq 1$, and where
$0\neq q \in K$.  
We determine a minimal projective bimodule resolution of $A$.
In the case when  $q$ is not a root of unity, we compute 
its Hochschild cohomology. In particular we
show that for $n\geq 3$, the $n$-th part $HH^n(A)$ has dimension
$k-1$ if char$(K)$ does not divide $k$. We also show that
every element in $HH^n(A)$ for $n\geq 1$ is nilpotent.
This is motivated by the problem of understanding why the
finite generation condition (Fg) fails which is needed to ensure
existence of support varieties.

\bigskip

\textbf{MR Subject Classification:} 16E40, 16S80, 20C20, 16G20.

\noindent
\textbf{Keywords:} Algebra of dihedral
type, socle deformation, bimodule resolution, 
Hochschild cohomology, nilpotent elements
in Hochschild cohomology.
\end{abstract}
\end{center}