Let $A$ be a $\mathbb{N}$-graded, locally finite $\mathbb{k}$-algebra, $\mathbb{k}$ being a field, $A = \bigoplus_{n \geq 0} A^n$, each $A^i$ being finitely-dimensional as a $\mathbb{k}$ vector space. Assume also that $A$ is connected, i.e., $A^0 = \mathbb{k}$.
Let $A^\ast$ denote its graded dual, i.e., $A^\ast = \bigoplus_{n \geq 0} \mathrm{Hom}_{\mathbb{k}}(A^n, \mathbb{k})$, which is a connected, graded $\mathbb{k}$-coalgebra.
Starting from some computations involving the (co)bar complexes for Hochschild (co)homology, I came up with the following question:
What is the connection between $\mathrm{Tor}_\bullet^A(\mathbb{k},\mathbb{k})$ and $\mathrm{Ext}^\bullet_{A^\ast} (\mathbb{k},\mathbb{k})$?
Computing them using the normalized (co)bar complexes, one obtains, in both cases, bigraded structures, $\mathrm{Ext}^\bullet_{A^\ast} (\mathbb{k}, \mathbb{k})$ being the Yoneda algebra of $A^\ast$ and $\mathrm{Tor}_\bullet^A(\mathbb{k}, \mathbb{k})$ being a coalgebra, obtained in a “dual” way as for the Yoneda algebra.
It would be great to have a duality between these structures, but I don't know how to approach an attempt of proof.
Thank you.