Let $A \to B$ be a surjective homomorphism between (unital) noetherian commutative rings with the same Krull dimension. Is the kernel of this map nilpotent ?
Thanks to Makoto Kato and Martin Brandenburg, it seems that the answer to the question is trivially false.
Now assume $A$ is a quotient formal power serie rings over a DVR, say $\mathbb{Z}_p$ the ring of $p$-adic integers. Can we say something ?
Best Answer
Geometrically, you are asking whether a closed immersion $\operatorname {Spec} B\hookrightarrow \operatorname {Spec} A$ of affine schemes of the same dimension is surjective.
This can be seen to be false (algebraic geometry is a visual art!): just inject the $x$-axis of the plane $\mathbb A^2$ into the union of the coordinate axes of that plane $\mathbb A^2$.
Translating back to algebra, the above counterexample corresponds to the the surjective algebra morphism $A=k[X,Y]/(X\cdot Y)\to B=k[X]$ sending $\bar X\mapsto X$ and $\bar Y\mapsto 0$ .
Edit
The translation between algebra and geometry is based on the following results:
(i) A surjective ring morphism $\phi:A\to B$ gives rise to a closed immersion $f:\operatorname {Spec} B\hookrightarrow \operatorname {Spec} A$ of affine schemes
(ii) The image $\operatorname {Image}(f)$ of $f$ is the closed subset $V(\operatorname {ker}(\phi))$ .
(iii) The equality of $\operatorname {Image}(f) =V(\operatorname {ker}(\phi))$ with $\operatorname {Spec} A=V(0)$ holds if and only if $\operatorname {ker}(\phi)\subset \sqrt {(0)}=\operatorname {Nil}(A)$.
New Edit
If $A$ is a domain with finite Krull dimension, the closed immersion $\operatorname {Spec} B\hookrightarrow \operatorname {Spec} A$ is necessarily surjective since a strict closed subset $F\subsetneq \operatorname {Spec} A$ must have dimension smaller than that of $ \operatorname {Spec} A$.
Since $Nil(A)=0$ for a domain, the translation above implies that $\operatorname {ker}(\phi)\subset \operatorname {Nil}(A)=0$ , i.e. that $\phi$ is injective (as well as surjective).
In other words, a surjective morphism $A\to B$ of rings of the same finite Krull dimension is an isomorphism as soon as $A$ is a domain.