I'm reading the article of Braverman and Gaitsgory, Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type, but I'm stuck in a point near the beginning.
Let $A$ be a (positively) graded associative algebra over a field $k$. They define the groupoid of i-th level graded deformation of $A$ to be the category $\mathcal E_i(A)$, whose objects are graded $k[t]/(t^{i+1})$-algebras $B$, free over $k[t]/(t^{i+1})$ with an isomorphism $B / t B \simeq A$ and whose morphisms are isomorphisms of such objects.
This is the classical definition of deformation (of whatever algebro-geometric object) and I have no doubt about it. Moreover, it is clear that reduction modulo $t^i$ defines a functor $\mathcal F_i \colon \mathcal E_{i+1}(A) \to \mathcal E_i(A)$.
Now, they define $\mathcal E(A)$ to be the groupoid of graded algebras $B$ over $k[t]$ (yes, the polynomial ring), free over $k[t]$, together with an isomorphism $B / t B \simeq A$, and they claim the following:
Lemma. Reductions modulo $t^i$ define an equivalence between the category $\mathcal E(A)$ and the inverse limit of the categories $\mathcal E_i(A)$ with respect to the functors $\mathcal F_i$.
This looks false to me: I should be able to prove the statement if we replace the groupoid $\mathcal E(A)$ with the groupoid of deformations over the formal power series ring $k[[t]]$, but I don't see how to pass from a deformation over $\mathrm{Spec}(k[[t]])$ to a deformation over the affine line $\mathrm{Spec}(k[t])$.
Remark 1. They use this property in what follows, because the strategy of the proof is precisely to construct a suitable deformation and then taking its fiber over $1$. However, as far as I understand, what they really do is to show that koszulity and certain conditions imply that the algebra $A$ we are interested in is unobstructed, so that we can construct a deformation over $\mathrm{Spec}(k[[t]])$.
Remark 2. They say that this machinery can be essentially found in the classical work of Gerstenhaber, On the deformation of rings and algebras. I gave it a look, but it doesn't seem to consider deformations which are not infinitesimal, i.e. deformations over the affine line (just over formal power series rings). However, I could be wrong here, for I haven't read the whole article of Gerstenhaber.