## A Geometric Reason to Care About Discrete Valuation Rings

Perhaps those that have taken an introductory commutative algebra class have come across the profoundly opaque definition of a discrete valuation ring (DVR), thinking to themselves "why have the math gods condemned me?" From a purely algebraic perspective, it is a remarkable that a ring $A$ is a valuation ring if and only if it is a domain so that for any element in the fraction field for $A$, $x$ or $x^{-1}$ is contained in  $A$. Even further, if a valuation ring is Noetherian (and not a field) it is a DVR. Again, algebraically, there are a lot of reasons to care about these rings, but today isn't about algebra. While there are a huge number of equivalences we can speak of, but instead we will make use of the following definition:   Definition:…

## Pullbacks of maximal ideals in a finitely generated k-algebra are again maximal

A few days ago, some classmates and I were thinking about about whether or not morphisms in $\mathrm{maxSpec}(A)$ were well defined. In particular, given a map of $k$-algebras $f:A \to B$, we consider the induced morphism $f^*:\mathrm{maxSpec} A \to \mathrm{maxSpec B}$ given by $X \mapsto f^{-1}(X)$. We all knew that pullbacks of maximal ideals need not be maximal for ring morphisms, which is precisely what led to a great deal of confusion-- thankfully this was all fixed as soon as we were reminded that we were considering morphisms of $k$-algebras. While this was funny, I think the proof for this fact is both an elementary and interesting check up in commutative algebra: Theorem 1: let $A,B$ be finitely generated $k$-algebras. Then the pullback of any maximal ideal in a $k$-algebra homomorphism  is again…