Say $H$ and $K$ are formally smooth group schemes (or, even better, $p$-divisible groups). Can I deduce that that any extension of $H$ by $K$ is formally smooth? It seems like this should be well-known for algebraic groups or group schemes but I can’t seem to find a reference for any of these cases.
I am interested in this because I want to show that any $p$-divisible group $G$ over $A$ (a complete, local, noetherian ring whose residue field is (perfect, maybe?) of characteristic $p$), is formally smooth. I’m aware that it’s difficult to do in general and have seen Messing’s thesis, but I am hoping to avoid any use of the cotangent complex.
I can show that $G^0$ is formally smooth. I think, unless it’s actually a non-trivial thing to show, that I know $G^{et}$ to also be formally smooth. I wonder if it’s actually a formal consequence that $G$ is formally smooth.
I wanted to somehow use that for complete local $k$-algebras $R$, $G(R)$ is an extension of $G^{et}(R)$ by $G^0(R)$, but that doesn’t seem that promising (unless I can perhaps then use some density statement about complete, local algebras that I certainly don’t know either)… I don’t have better ideas or can’t seem to find easy counterexamples.
Thanks in advance!
For completeness, let me say that I’m thinking about $p$-divisible groups as fppf sheaves (I am happy with Tate’s definition too). Also, formal smoothness to me means that for any $k$-algebra $R$ with nilpotent ideal $I$, the map $G(R)\to G(R/I)$ is surjective.