Formal smoothness of path algebras and connections

Question

Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if $$ \Omega^1_kA = \operatorname{Ker}(\mathrm{mult}\colon A\otimes_k A\to A) $$ is a finitely generated projective $A^e$-module.

According to 19.2 (4) of Ginzburg's Lectures on Noncommutative Geometry (arxiv), $A$ is always formally smooth over $k$. Therefore there exists (by Lemma 18.4.4 of loc. cit.) a connection of the form $\nabla\colon\Omega^1_kA\to\Omega^1_kA^e\otimes_{A^e}\Omega^1_kA$.

Question. How can we obtain an explicitly defined connection? Or, equivalently, is there a reasonable way to realise $\Omega^1_kA$ as a summand of some free $A^e$-module?

For example, if $Q$ has only one vertex, then $A$ is isomorphic to the tensor algebra $T(V)$ over some $k$-vector space $V$, and $\Omega^1_kA$ is already a free $A^e$-module of rank $\operatorname{dim}V$. How about $Q = (\bullet\leftrightarrows\bullet)$?

Any comments or references are appreciated.

Answer 1

It helps to realize that a quiver algebra is in fact a tensor algebra. (I'm assuming that $Q$ has finitely many vertices and arrows.) Let $S \subseteq kQ$ be the span of the vertices, and let $V \subseteq kQ$ be the span of the arrows. Then $V$ is an $(S,S)$-bimodule, and we have an isomorphism of $k$-algebras $$kQ\cong T_S(V).$$

Notice that $S \cong k^n$ where $n$ is the number of vertices in $Q$, and thus is a semisimple $k$-algebra. It's even separable, meaning that $S^e$ is also semisimple and thus all $k$-central bimodules over $S$ are projective as $S^e$-modules. At this point, we can find a proof that such an algebra is formally smooth in Proposition 5.3 of the paper

Cuntz, Joachim; Quillen, Daniel, Algebra extensions and nonsingularity, J. Am. Math. Soc. 8, No. 2, 251-289 (1995). ZBL0838.19001.

which is my usual go-to source on formal smoothness.

This doesn't quite give the explicit argument that you were asking for. Maybe you would be happier using Corollary 2.10 of Cuntz-Quillen. This shows that for our tensor algebra $T := T_S(V)$, there is a short exact sequence $$0 \to T \otimes_S \Omega^1 S \otimes_S T \to \Omega^1 T \to T \otimes_S V \otimes_S T \to 0.$$ We can rewrite this in terms of enveloping algebras as $$0 \to T^e \otimes_{S^e} \Omega^1 S \to \Omega^1 T \to T^e \otimes_{S^e} V \to 0.$$ Since $S^e$ is semisimple, the $S^e$-modules $\Omega^1 S$ and $V$ are projective, so that the base extension to $T^e$ yields projective modules. So the sequence above splits and $\Omega^1 T$ is a projective $T^e$-module as desired.