Following the terminology of
Drozd, Yuriy A., Derived tame and derived wild algebras, Algebra Discrete Math. 2004, No. 1, 57-74 (2004). ZBL1067.16028.
let $A$ and $R$ be algebras over a field $k$. A strict family of $A$-complexes based on $R$ is a complex $P^* \in K^-(A\otimes R^{\rm{op}})$ of finitely generated projective $A-R$-bimodules such that
- For every indecomposable finite dimensional $R$-module $L$, $P^*\otimes_R L$ is indecomposable.
- For every pair of finite dimensional $R$-modules $L$ and $L'$, if $P^* \otimes L \cong P^*\otimes L'$ then $L \cong L'$.
An algebra $A$ is called derived wild if there exists a strict family for every finite dimensional algebra $R$.
My question is: Does anyone have an explicit description of such a strict family for a (preferably simple) derived wild algebra $A$ and some wild algebra $R$, say $R = k\langle x,y \rangle$ for example.
Best Answer
A few such examples are constructed in
Bekkert, Viktor; Drozd, Yuriy; Futorny, Vyacheslav, Derived tame local and two-point algebras, J. Algebra 322, No. 7, 2433-2448 (2009). ZBL1191.16017.