 So this lecture is part of an online graduate course on commutative algebra and will be about dedicated domains. So I will start with an example. So the standard example of something that is not unique factorization domain is z of root minus five. So this consists of all integers, all numbers that form n plus n root minus five with m and n integers. And an example of an unique factorization is two times three is equal to one plus root minus five times one minus root minus five and these are irreducible and these are irreducible and they're not units times each other so this is two completely different factorizations of six. And Kummer found a way of fixing this using ideals. Well that's actually not true, Kummer didn't use ideals, he used something a little bit different called ideal elements and ideals were only introduced later but I'm going to ignore this and pretend that ideals were invented by Kummer. So in terms of ideals you can factor the ideal two as a product of two ideals like this and similarly you can factor three as a product of two ideals and you can also factor these two ideals here in a similar way so you can check one plus root minus five is just equal to two one plus root minus five times three one plus root minus five and finally one minus root minus five factors in pretty much the same way. And now if you call these ideals let's call them call this one a and call this one b and call this one c and this one d then you find that ab times cd is equal to ac times bd and this corresponds to the factorization two times three is one plus root minus five times one minus root minus five and although this factorization isn't unique you can see in terms of ideals it is unique up to order it's just saying abcd is equal to acbd. So what Kummer showed is that every none zero ideal is a product of primes and I say prime I mean prime ideals in a unique way unique way means unique up to order of course and this condition here is more or less the definition of a dedicated domain actually this isn't usually used as the definition of a dedicated domain because it's a little bit difficult to work with it's not terribly obvious whether or not a ring satisfies this condition. So people usually use a slightly different definition of a dedicated domain this condition for integral domains turns out to be equivalent to the union of the following three conditions first of all the ring is notarian and secondly every none zero prime is maximal and thirdly it's integrally closed so you remember integrally closed means you take the ring embed it into its quotient field of quotients k and every element of k that's integral over r in other words satisfies an equation of form x to the n plus r in minus one x to the n minus one plus r zero is already in r. I should say you being dedicated domain is equivalent to these are three separate conditions and they all have to hold so these conditions certainly aren't equivalent or anything like that. There are two standard ways of getting examples of dedicated domains and the first is the integers of an algebraic number field. So a typical example of these are first for the integers themselves from a dedicated domain they're just the integers of q and the example we just had the integers of z root minus 5 so you can think of this as being zx modulo x squared plus 5 and this is the algebraic this is the integral closure of z in the ring q of root in the field q of root minus 5. Another standard example are the coordinate rings of affine algebraic curves. A typical example of this might be just the ring of polynomials over over a field that's the coordinate ring of the affine line. Another example might be the coordinate ring of say an elliptic curve which would look something like this. So here this is a corresponding elliptic curve if you draw a quick picture of it it sort of looks something like this and you see both of these are rather similar so you know the integers kind of behave rather like the ring of polynomials in one variable they're both Euclidean domains and pretty much anything you can do with one has an analog for the other and similarly this ring here you can think of as being sort of analogous to this ring here you see in each of them you're starting with this ring and adjoining the square root of something in one case you're joining the square root of minus 5 in the other case you're joining the square root of x cubed minus x so so there's a sort of close analogy between these two rings here. So it's easier to understand these three conditions if you look at them geometrically I mean if you look at algebraically these these seem to be three completely random conditions you put in a ring and it's not entirely clear why you should put more together. So if you look at some sort of geometric dedicated domain such as this one let's look at what these three conditions mean first of all we have this condition that it's notarian well in algebraic geometry the notarian condition means roughly not weird so most of the examples of rings you come across at least in basic algebraic geometry are notarian and rings that aren't notarian behave rather strangely in algebraic geometry at least they used to I mean none notarian rings are becoming more and more important in algebraic geometry but I weren't worried too much about that. What about the condition that nonzero primes are maximal well geometrically this corresponds to the dimension of the your algebraic set being at most one so you remember the dimension is the length of the longest possible chain of maximal ideas like this and if every nonzero ideal is maximal the biggest length you can possibly get is not contained in some maximal ideal so this is a chain of length one so so none zero primes are maximal just means dimension at most one. Next we have the integrally closed condition well geometrically this corresponds to the condition being normal and normal implies in particular that singularities have co-dimension at least two it's slightly stronger than saying that singularities have co-dimension at least two but that will do for the moment now if the dimension is at most one and singularities have co-dimension at least two then this just means everything is non-singular so so the condition about being integrally closed more or less says that the Dedekind domain is corresponding to something non-singular so so Dedekind domains correspond very closely to non-singular curves and here you have to interpret the word curve in a fairly general sense because things like the integers don't quite correspond to a curve in the classical sense but you can think of them as being almost a curve the point is that they're one-dimensional things that don't have singularities so now let's look at this unique factorization from the geometric point of view so here we have let's just take this elliptic curve k of x y over y squared minus x cubed plus x which looks a bit like this and there's an obvious example of non-unique factorization because y squared is equal to x x minus one times x plus one and here we have two completely different factorizations so this you can either fact this is y squared or as x times x minus one times x plus one and now let's look at the zeros of all these so the zeros of y look like this so so so so here are the here are the points where y equals north on the curve and the points where x equals one look like this and the points where x equals zero look like this there's a double double zero there and the points where x equals minus one I guess that's barely visible use a different color so x equals minus one correspond to these two points here and now y squared equals x times x minus one times x plus one means if you take two copies of these orange points it's the same as taking the green points and the blue points and whatever this color is added up some sort of pink I think and now we can find ideal the ideals of functions vanishing at the individual point so the functions vanishing at this point here is just the ideal y y plus one and similarly the ideal y x is just functions vanishing at this point in the ideal y times x minus y x minus one corresponds to this point here now you find the ideal y is equal to y x plus one y x y x minus one which just says the zeros of y are this zero and this zero and this zero incidentally checking the product of these three ideals as y is not actually completely trivial it's not obvious at first sight how you can get y as an element of this product similarly x is equal to y x squared x minus one is equal to y x minus one squared and x plus one is equal to y x plus one squared so we find that the none unique factorization y squared equals x x minus one x plus one just corresponds to this ideals this ideal squared being equal to this ideal times this ideal times this ideal which as you can see just says both sides it's just this squared and this squared and this squared so we do indeed get unique factorization of ideals you see that unique factorization of ideals is a very simple geometric meaning it's basically just saying a union of points with multiplicities can be written as a union of single points in an essentially unique way and in fact we can get the following dictionary between algebra and geometry for derekine domains so for derekine domains ideals correspond geometrically to things called divisors which are formally sum over i of n i times p i where p i is a point on your curve and all but a finite number of the n i have to be zero and these should be non-negative integers well let's guess these correspond to positive divisors then prime ideals just correspond to points p i that's prime ideals that are none zero of course and an element f of the ring correspond to the zeros of sorry corresponds to a function on the curve and the map from an element to the ideal generated by f just corresponds to taking a function f to the zeros of the function unique factorization is corresponds to the basically the trivial geometric fact that any sum of points with multiplicities can be written as a sum of points with multiplicities in some sense you I mean you're just saying a divisor is a linear integral combination of points that's that's just the same as saying any ideal can be written as a product of of prime ideals so so the unique factorization conditions for derekine domains is very natural geometrically now I'm going to give some examples of things that are not quite derekine them dedicated domains so first of all remember there were three conditions that had to be notarian primes were maximal and they had to be intimately closed so we can drop each of these three conditions and see what happens so first let's look at examples that are not notarian and I'm going to give two examples I'm going to give a sort of algebraic example and a geometric example so for the geometric example you might take the ring of power series in one variable well this is a derekine domain and it's contained in the power series in say x the half which is contained in the power series and say x the sixth and so on you can keep going like this and what you get is the ring of preser series which something like you would think of them as being something like power series except the exponents don't necessarily have to be integral and this is certainly not notarian because for instance we can have the ideal i which is equal to x x the half x the sixth and so on so it's generated by all of these and we notice for example that i squared is actually equal to i so we certainly don't get any sort of unique factorization of ideals you can do a similar thing algebraically if you take the p adding integers you might add p to the half p to the sixth p to the one over 24 and so on and you get something rather similar there are other examples of things that are almost but not quite derekine domains for instance the ring of holomorphic functions and this very nearly is a unique factorization domain because if you've got a function f you can write it as a product of via strice factors so it's it sort of has unique factorization except that this product may have to be infinite for instance the via strice factors might look like you might get things like one minus c over a one but then you might have something like one minus z over a two times x of z over a two where via strice discovered you you in order to get convergence you have to put in various cleverly chosen fudge factors anyway so holomorphic functions come pretty close to being a derekine domain except they're not notary and only have unique factorization if you allow infinite products which is of course not really an algebraic operation next example is something that is not integrally closed and as before i'll give an algebraic example and a geometric example so the algebraic or arithmetic example is something like z of root minus three and this is not integrally closed because um as the the element one plus root minus three over two is integral over this ring and not contained in it and it doesn't have unique factorization of ideals for example two one plus root minus three squared is equal to two times two one plus root minus three so here we've got two different factorizations of an ideal into a product of prime ideals a corresponding geometric example might be something with a cusp for example we might take the ring of polynomials in two variables and quotient out by y squared minus x cubed so this looks something like this it's a curve and as you see it's got a singular point here which corresponds to the fact that this ring isn't integrally closed and let's check it's not integrally closed well if we put t equals y over x then we find we can identify this ring with k with the ring of polynomials and t squared and t cubed which is contained in the ring of polynomials in one variable and you should think of this ring as being contained in its integral closure like this corresponds to this ring being contained in its integral closure here and again you can see that the the ideals don't behave terribly nicely here for example t squared t cubed squared is equal to t squared t cubed times the ideal t squared and this funny factorization have a non-unique factorization corresponds to this non-unique factorization in both cases we're sort of getting i squared equals i times j for two different ideals i and j so so something is clearly going wrong um so finally we have some examples where non-zero primes need not be maximal and again we'll have an algebraic or arithmetic example and a geometric example and you remember this condition said that the ring is sort of one-dimensional in some sense so all we need to do is to take something that's kind of two-dimensional and an arithmetic example of a two-dimensional ring is just the ring of polynomials over z and you notice the ideals of this can be really quite hairy for example we can have various sorts of ideals like eight four x x squared and there's some sort of primer ideal but it's not a product of prime ideals and similarly we could just take a ring of polynomials and two variables corresponding to the affine plane and again the ideals are complicated for example we can have ideals that look like x cubed x squared y y squared and you know you can't write all ideals as a product of prime ideals what we get instead for both of these is the laska-nerta theorem which says that every ideal is an intersection of primary ideals and this is a little bit weaker than saying it's a product of prime ideals I mean primary ideals are more general than prime powers and intersections are a little bit more complicated than taking products of ideals incident you notice that both of these are in fact unique factorization domains that even though then even though you can't write ideals as products of prime ideals in unique way you can still write the elements as product of irreducible elements so the relation between dedicated domains and unique factorization domains is neither of them are contained in the other if you've got a principal ideal domain then this is a unique factorization domain and a dedicated domain but here we've got some examples of unique factorization domains that are not dedicated domains and of course we had examples of dedicated domains like z root minus five that are not unique factorization domains so in general these are not the same incident there is a common generalization of these which are called curl domains but I'm not going to talk about curl domains because growth and it's strongly disapproved of curl domains he said they were just a silly attempt to to make something containing both unique factorization domains and dedicated domains even though unique factorization domains and dedicated domains are really totally different ideas so I'm not going to go against growth and Dick's opinion okay the next lecture on dedicated domains will be I'll be talking about equivalents of various conditions for something to be a dedicated domain