Motivation commutative algebra I'm starting to learn commutative algebra. I heard before that a reason to learn it is because it unifies the language of algebraic geometry and algebraic number theory. Are there any examples of problems in algebraic geometry and algebraic number theory, that share the same notions in commutative algebra? It just seems to me. bit disconnected. I've been reading Atiyah and Eisenbud. I know a little bit of the language of AG, but don't know how the DVR and Dedekind domains relate to any AG. Any help or references would be appreciated.
 A: Dedekind domains is a good example of a notion that can be thought both arithmetically and geometrically. They arise naturally in algebraic number theory as rings of integers of number fields, but how do they arise in algebraic geometry?
Recall that an affine variety is fully determined by its coordinate ring. If $X$ is an affine variety over an algebraically closed field $k$, then $X$ is a smooth irreducible curve iff its coordinate ring $k[X]$ is a Dedekind domain (and not a field). In fact:


*Coordinate rings are always noetherian, so we can forget about this condition.

*$X$ is irreducible iff $k[X]$ is an integral domain.

*$\dim X = \dim k[X]$, so $X$ is a curve iff $k[X]$ is a 1-dimensional ring. In our case $k[X]$ is domain, so this is equivalent to saying that every nonzero prime ideal of $k[X]$ is maximal.

*$X$ is smooth iff for every $P \in X$, $k[X]_{\mathfrak{m}_P}$ is a regular local ring. For 1-dimensional noetherian local domains (our case) this is equivalent to $k[X]_{\mathfrak{m}_P}$ being integrally closed for every $P$ (see Proposition 9.2 in Atiyah-Macdonald). Now, since $k$ is algebraically closed every maximal ideal of $k[X]$ is of the form $\mathfrak{m}_P$. Thus the condition is equivalent to $k[X]_\mathfrak{m}$ being integrally closed for every maximal ideal $\mathfrak{m}$ of $k[X]$. It is a property of domains that if this happens, then the domain itself is integrally closed (check https://proofwiki.org/wiki/Integrally_Closed_is_Local_Property). Therefore, $X$ is smooth iff $k[X]$ is integrally closed.

A: The basic geometric object in algebraic geometry is a scheme, which is an honestly geometric thing, and one can think of these intuitively as a space that's completely determined by its "functions" on it. In the case of algebraic varieties, we can realise these are schemes with certain nice properties, which make them seem "more geometric", but its good to view any scheme as a geometric thing.
However if you look up a definition of a scheme it might not be clear why this is a helpful perspective, so we will recall the main, motivating theorem, that in the category of schemes we have a nice, full subcategory of affine schemes, and the category of affine schemes is $equivalent$ to the opposite category of commutative rings. So one loses no information in this interpretation, and illustrates the geometry behind many ring theoretic ideas.
In no particular order, let $R$ be a commutative ring:
Elements of $R$ are "functions" on this space, which we call $Spec(R)$.
Ideals are collections of functions that vanish on (closed) subspaces $Y$ of $Spec(R)$, and the canonical map $R\rightarrow R/I$ is just restricting functions to this subspace.
Intersection of ideals is "union" of closed subspaces, sum of ideals is the intersection.
Prime ideals are "points" of $Spec(R)$, where we interpret a "point" to be a subspace on which functions can vanish that can't be broken down further.
Localisation is looking at the functions on open subsets of your space, and the localisation map $R\rightarrow S^{-1}R$ is the restriction map.
Idempotents are functions which are $1$ on a connected component, $0$ off a connected component of $Spec(R)$.
Dedekind domains correspond to functions on spaces that are one dimensional, nonsingular, where every function is (up to units) determined by its finite set of zeros and poles.
DVR's correspond to tiny pieces of one dimensional, nonsingular spaces, where there's only a single point for which the function can be zero or not.
These are just the easy interpretations, by the opposite category perspective, everything can (and should) be interpreted geometrically, once one accepts that $Ring^{op}$ is a category of geometric objects.
As an example of this perspective being useful, lets consider the Chinese remainder theorem, if $I,J$ are ideals of $R$ such that $I+J=(1)$, then \begin{equation}R/(I\cap J)\cong R/I\times R/J\end{equation}
If $I+J=(1)$, then the subspaces $Z,Y$ that $I$ and $J$ vanish on are disjoint, so the functions on their union $Z\sqcup Y$, given by $R/I\cap J$, should be products of functions on each piece separately, which is exactly $R/I\times R/J$.
Unfortunately, I'm not sure of a good reference that spells out this basic idea in detail, I think Eisenbud's geometry of Schemes should be okay for it, and the exercises of Hartshorne $II$,$1$ also cover some of this. Any exercise in Atiyah Macdonald that deals with the $Spec$ functor would also give some intuition. I think however that this is more of a mindset thing, a lot of which can be figured out yourself just by going "all in" on the idea of rings being functions on spaces, and trying to view everything through this lens.
