Let $k$ be a field and $f: A \rightarrow B$ be a surjective ring morphism between smooth Noetherian $k$-algebras. By smooth I mean that the module of Kahler Differentials $\Omega_{A|k}$ is a projective $A$-module (likewise with B).
Why must each $Ker(f_{f^{-1}[\mathfrak{m}]})$ be generated by a regular sequence, where $\mathfrak{m}$ is a maximal ideal in B?
The argument seems to boil down to two things:
1) Notherian + Smooth $\Rightarrow$ Regular (Without deviating from my definition os smoothness, I don't see how this follows)
More importantly tho
2) The kernel of a morphism between regular local rings is generated by a regular sequence.
(I've looked at some theory on this but it always seems to take a big detour and passes through more general objects like (CIs) which I am neither comfortable with, nor need. My main issue would be to kind a simple, possibly bland proof of this fact).
Thanks in advance! :)
You can find the thing about kernels in Section Tag 00NN. Namely, any minimal system of generators of the maximal ideal is a regular sequence and the kernel of a surjection of regular local rings can be generated by the first few elements of such a minimal system of generators.
For results on smoothness one can look at Section Tag 00TZ which gives an overview of results on smooth ring maps. A quick argument that smooth over a field implies regular when the ground field $k$ is algebraically closed and the residue field is also $K$ can be found somehwere in Hartshorne Chapter I.