If $f: Y\to Z$ is a finite, surjective morphism of normal integral schemes (of finite type over a field) and $y$ is a prime divisor of $Y$, is then also $z= f(y)$ of codimension 1?
We have an inclusion $\mathcal{O}_{Z,z} \to \mathcal{O}_{Y,y}$ and the latter is a DVR, i.e. it has dimension 1. Does the integrality of both local rings imply that $\operatorname{codim}z =1$?
More generally, is there a relationship between $\dim y, \dim f(z)$ and $\dim f^{-1}(z'), \dim z'$ for $z'\in Z$ arbitrary?
Best Answer
Answer to your first question: We prove the more general statement that if $f: Y \to Z$ is a finite surjective morphism of integral schemes, then $\dim y = \dim f(y)$ (in the case that $Y,Z$ are also of finite type over a field this will also yield that $codim \; y = codim \; f(y)$).
Claim: $\dim Y = \dim Z$, the reason being that if $Spec B$ is any affine open of $Z$, then $f^{-1}(Spec B) = Spec A$ with $B \hookrightarrow A$ being an integral extension hence $dim Spec A = dim Spec B$, now cover $Y$ with affine opens and recall that a scheme has dimension $n$ if and only if it has an affine cover where each affine in the cover has dimension $\le n$ and equality is achieved for at least one, from taking preimages of such a cover of $Z$ we now see that $\dim Z = \dim Y$.
Now we have that the morphism $y \to f^{-1}(z) \to f(y) = z$ is a finite surjective morphism of integral schemes (being the composition of a closed embedding together with a projection from a base change of a finite morphism) hence it follows from the claim that $\dim y = \dim z$ and since codimension is the difference of dimensions for integral schemes of finite type over a field it then follows that $codim y = codim z$.
Answer to your second question: If $f: Y \to Z$ is any morphism of locally Noetherian schemes, and $p \in Y$, $q = f(p)$ then we always have $$(Eq \; 1) \; \; \; \; codim_Y p \le codim_Z q + codim_{f^{-1}(q)}p$$ (here $codim$ of a point is interpeted as the codimension of the closure in the appropriate space). Now for integral schemes of finite type over a field (and more generally for Catenary schemes if I am not mistaken?) this translates to $$ (Eq \; 2) \; \; \; \; \dim Y - \dim \overline{\{p\}} \le \dim Z - \dim \overline{\{q\}} + codim_{f^{-1}(q)}p$$
If the morphism is flat we have equality, and of course if the morphism is finite then the second term on the right hand side can be removed.