I'm reading this paper, and definition 2.2.1 (page 5) is the following:
The torsion of a left $A$-covariant derivative $\nabla$ on $\Omega^1A$ is the left $A$-module map $T = \wedge \nabla - d:\Omega^1A\to \Omega^2A$.
Where $\wedge:\Omega^1A\otimes\Omega^1A\to\Omega^2A$ is the exterior product. Then, the following computation is made to show left-functoriality in $A$ (corresponding to standard Riemannian geometry where $T$ is the torsion tensor and functorial over smooth functions): for $a\in A, \phi\in\Omega^1A$,
$$T(a\phi)=\wedge\nabla(a\phi)-d(a\phi)=\wedge(a\nabla\phi)+\wedge(da\otimes\phi)-ad\phi+da\wedge\phi=aT(\phi)$$
I think I'm having a misunderstanding of $\wedge$ as a map, and what $\Omega^2A$ is. My assumption is that $\wedge(a\nabla\phi)=a\wedge\nabla\phi$, $\wedge(da\otimes \phi)=da\wedge\phi$, and that $d(a\phi)=da\wedge\phi+(-1)^{|a|}ad\phi=da\wedge\phi+ad\phi$. In this case, isn't the $+da\wedge\phi$ a mistake in this computation, as it should be minus?
I think also that, in analogy with the classical case, $\Omega^2A$ is basically spanned by things of the form $a(db\wedge dc)$, $a,b,c\in A$, and that $\wedge(da\otimes \phi)=da\otimes \phi$ for $da\neq \phi$ and $0$ otherwise.
Is this correct?