Possible inequality of krull dimension of local injection of Noetherian local domains

Question

If $(A, \mathfrak{m}) \hookrightarrow (B, \mathfrak{n})$ is a local injection of Noetherian local domains, do we necessarily have $\dim B \geq \dim A$?

Answer 1

As suggested in the comments, this is false, with counterexample $k[x,y]_{(x,y)}\to k[t,x]_{(x)}$ by $x\mapsto x$, $y\mapsto tx$. Geometrically what's happening here is that we're looking at the map $Bl_{(0,0)}\Bbb A^2\to\Bbb A^2$, which is a birational map of integral schemes but contracts a positive-dimensional subvariety to a point, and considering the induced map on local rings from the generic point of the exceptional divisor to the blown-up point.

As any birational map of integral schemes $f:X\to Y$ gives injections $\mathcal{O}_{Y,f(x)}\to\mathcal{O}_{X,x}$, we have that our map is local and an injection. For a variety $Z$ over a field $k$, we have $\dim \mathcal{O}_{Z,z}=\operatorname{codim} \overline{\{z\}}$, so the local ring of the generic point of the exceptional divisor is one-dimensional while the local ring of the blown-up point is two-dimensional, contradicting your desired conclusion.