# lift elements in affine singular variety.

Let $B$ a $k$-algebra of finite type and $\mathcal{O}=k[[t]]$ and $X=Spec(B)$.

Let $X(\mathcal{O})$. We know that by Elkik's theorem that if we have a section $y\in X(\mathcal{O}/\pi^{2d+1})$ such that th valuation of the Jacobian ideal at $y$ is less or equal than $d$ then , the map lifts and the lifting is congruent to $y$ mod $t^{d+1}$.

So if we consider $X(\mathcal{O})^{\leq d}$ the open subset where the valuation of the jacobian ideal is less or equal than $d$, do we have some kind of formal smoothness for the map:

$X(\mathcal{O})^{\leq d}\rightarrow S(\mathcal{O}/\pi^{d+1}\mathcal{O})$ ?

where $S(\mathcal{O}/\pi^{d+1}\mathcal{O})$ is the schematic image of $\pi_{d}:X(\mathcal{O}/\pi^{2d+1})\rightarrow X(\mathcal{O}/\pi^{d+1})$

N.B: As X is singular,$\pi_{d}$ is not nevessarily surjective.

-