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\title{Lecture 5-31: Review, part 3}
\date{May 31, 2024}
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The final exam will take place on Monday, June 3, from 8:30 to 10:20 (note earlier start time) in the usual room.  You should bring your own paper; you are allowed one page of handwritten notes on both sides.  Only material in Dummit and Foote will be covered.
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The Zariski topology is generalized to the setting of an arbitrary commutative ring $R$ (in practice, usually Noetherian) as follows.  The \alert{(prime) spectrum} Spec~$R$ of $R$ is the set of its prime ideals; this is the underlying topological space.  The closed sets are then the set $\mathcal Z(I)$ of prime ideals containing a fixed ideal $I$ (or equivalently a set of generators of $I$).  The Nullstellensatz in this setting asserts the existence of inverse inclusion-reversing bijections between the sets of radical ideals of $R$ and of nonempty closed subsets of Spec~$R$; here a radical ideal $I$ corresponds to the set $\mathcal Z(I)$ of prime ideal containing it while a closed subset $C$ of Spec~$R$ corresponds to the intersection $\mathcal I(C)$ of the prime ideals in $C$.
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Any closed subset of Spec~$R$, like any closed subset of affine $n$-space $\mathbf A^n=k^n$, is then uniquely the union of finitely many irreducible closed subsets, none of them contained in any other; the irreducible closed subsets correspond under the Nullstellensatz to the prime ideals.  The points of Spec~$R$ corresponding to maximal ideals $M$ are then exactly the ones such that the singleton set $\{M\}$ is closed; they are not surprisingly called \alert{closed points}.  The point $\{0\}$, whose closure is all of Spec~$R$, is called \alert{generic}.
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 Thus if $R=P_n=k[x_1,\ldots,x_n]$, where $k$ is an algebraically closed field, then the points of Spec~$R$ correspond to the subvarieties of $\mathbf A^n$, with the closure of the point corresponding to $V$ containing the point corresponding to $W$ if and only if $W\subset V$ .  In general one attempts to understand Spec~$R$ by relating it to Spec~$S$ for $S$ a suitable subring of $R$, exploiting the continuous map from Spec~$R$ to Spec~$S$ given by contraction (sending a prime ideal $P$ to $Q=P\cap S$).  For example, if $R=\mathbb Z[i]$, the ring of Gaussian integers, then we take $S=\mathbb Z$.  The nonzero ideals in Spec~$R$ are then of three types.  If $p\in\mathbb Z$ is prime with $p\equiv3\mod4$, then $Rp$ is also prime in $R$; here the fiber over the principal ideal $(p)\subset S$ is a single point.  If $p\equiv1\mod 4$, then $Rp$ is the product of two distinct prime ideals $R(a+bi)$ and $R(a-bi)$, where $a,b\in\mathbb Z$ satisfy $a^2+b^2=p$; here the fiber over $(p)\subset S$ has two points.  Finally, the exceptional prime $2\in\mathbb Z$ is such that $2R=(R(1+i))^2$, the square of a prime ideal.  The fiber over $(2)\subset S$ again consists of just one point.
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The ring $\mathbb Z[i]$ is a particularly nice example of a \alert{Dedekind domain}, that is, an integrally closed integral domain of Krull dimension one.  More generally, the integral closure $\mathcal O_K$ of $\mathbb Z$ in any finite Galois extension $K$ of $\mathbb Q$ is also a Dedekind domain.  In a Dedekind domain any nonzero ideal $I$ is a finite product $P_1\cdots P_m$ of prime ideals, where this product is unique up to reordering the factors.  What makes $\mathbb Z[i]$ (or more generally any PID) especially nice even among Dedekind domains is that it is a UFD; that is, any nonzero, nonunit element is a finite product of primes, unique up to multiplying the factors by units and reordering them.  On the other hand, $\mathbb Z[x]$ is not quite a Dedekind domain, since it admits nonzero nonmaximal prime ideals (such as the one generated by 2).  Nor, as you saw in homework, is $\mathbb Z[\sqrt{-7}]$ a Dedekind domain, since it fails to be integrally closed.
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A very special (but very important) class of Dedekind domains) consists of the local ones, called DVRs (standing for \alert{discrete valuation rings}).  These are PIDs having a unique prime element $x$ up to multiplication by units.  In any such ring the only nonzero ideals are the principal ones $(x^n)$ generated by powers of $x$,  The two simplest examples of such rings are the localization $\mathbb Z_{(p)}$ of $\mathbb Z$ by a nonzero prime ideal $(p)$ and the power series ring $k[[x]]$ in one variable $x$ over a field $k$.
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Good luck and have a good summer!  
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