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I believe there are no exterior algebras in sight. To see this, let us think through the $1$-categorical case carefully. We have a commutative ring $R$ and the ordinary category of $R$-modules $\mathr …

In "A Survey of Elliptic Cohomology", Section 3.2, Lurie asserts that the preorientations of the additive formal group $\widehat{\mathbf G}_a$ over $\mathbf Z$ are classified by the $\mathbb E_\infty$ …

The presentation of the formal moduli problems story in Gaitsgory-Rosenblyum A Study in Derived Algebraic Geometry, Vol 2 may be what you are looking for. We review it here (in the case over $\mathrm{ …

It is instructive to look at the simplest case of Proj: that of a free module, i.e. the projective space. Lurie works these out for us quite carefully in his Spectral Algebraic Geometry tome. Project …

Okay, I think I may have figured out how B) $\Rightarrow$ A). Because nobody else has given an answer yet, I'm spelling it out (I hope that's not considered bad form). And if what I wrote doesn't make …

Let $\mathbb G$ be an abelian vatiety over an $\mathbb E_\infty$-ring $A$. That is to say, it consists of an abelian group object in the $\infty$-category of relative schemes $\mathbb G\to \operatorna …