Let $\Lambda$ be a finite-dimensional associative algebra. We can think of this as an $A_\infty$-algebra with vanishing $m_i$ for $i \ne 2$ and consider $A_\infty$-modules $M$ over it. A list of structure maps and axioms are given in Section 1.1 of http://arxiv.org/abs/0705.3948v1

I'm interested in the special case when $\Lambda$ is an exterior algebra and $M = Ext^\bullet(N,k)$ for a finitely generated module $N$ over the symmetric algebra (Koszul dual to $\Lambda$). If I understand correctly, then $M$ can be endowed with the structure of an $A_\infty$-module in the above sense. But I think that there should be extra "shuffle" relations imposed on the structure maps coming from the fact that $\Lambda$ is graded-commutative (the analogy is with $C_\infty$-algebras, which are $A_\infty$-algebras with the additional property that the $m_n$ maps vanish on all "shuffles" like in Section 3 of http://arxiv.org/abs/0811.1655v1).

Is this correct? And is there a reference for what the definition of a "$C_\infty$-module" or what these extra relations are?


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