 So this lecture is part of an online commutative algebra course and we'll be about functions On the spectrum of a ring so Let's start by looking at an example Suppose we look at the ring of continuous functions on a topological space X so X is compact House dwarf and this just means the continue real-valued functions on X So the elements of the ring of the ring are it's called the string are our functions on X and we saw earlier that X is more or less the spectrum of the ring are it's really the Spectrum of maximal ideals of our And you remember that are had some rather funny-looking prime ideals, which we will kind of ignore and This makes it easy to visualize the ring are because we can just think of the space X We think of R as being functions on X and the problem is can we do Something similar for any are In other words, we want to The elements of R should be something to do with functions on the spectrum of R and Well, the first question is what what should these functions take values in? So, so let's have a look here. So here the functions on X are taking values in the real numbers. Well, what are the real numbers? Well, if you take a maximal ideal of C of X then the maximal ideal kind of corresponds to a point of X. So this is just functions F with F of X equals zero and The function F is taking values in the field C of X over M of X of course in this case the field just happens to be isomorphic to the reals and we can do this for We can do something similar to this for any ring R. So so so now let's take R to be any ring Then for each prime ideal in the spectrum of R We get Well, we don't get a field we get an integral domain R over P And we can think of an element F and R as a function from the spectrum of R to Well, what we're doing is for each point of the spectrum of R. It's taking values in the The integral domain R over P at each point So let's see what happens so for example suppose we take R to be the Ring of polynomials over the complex numbers and then the maximal ideals Adjust of the form X minus alpha. So this sort of corresponds to the Complex number alpha in some sense and if we've got some function in in R For example, we might take F of X equals X squared Then we can look at the spectrum of Of C of X and the spectrum of C of X is more or less the complex line So there's the ideal X which kind of corresponds to point zero and the ideal X minus one corresponds to the point one and X minus pi corresponds to the point pi and so on and then for each of these points We've got a fiber Which is C of X Modulo whatever the ideal of that point is and all these quotients how to be isomorphic to the complex numbers Well, there's one slight exception because you've also got the ideal zero and Here we just get C of X rather than C of X over C. So if we think of Function F of X as such as X squared then Here it takes the value zero and here it takes the value one and here it takes the value pi and so on so This is really just a rather complicated roundabout way of saying we're drawing the graph of the function F So that sort of works except there's something funny going on at the generic point because this is no longer the complex numbers But we'll put that problem aside for the moment and Let's let's just take a look at what happens when R is the integer Z Well in the previous example All these fields Corresponding to the maximal ideals were all isomorphic to the complex numbers So we just had a function from the spectrum to the complex numbers If we look at Z things are a little bit more complicated because now we've got the prime ideals two Two three five and so on and if we look at the corresponding rings Z over two this now is two point zero and one for a Z over three now as three points Zero one two zero over five as Five point zero one two three four. So this is the integers mod five and if we look at a typical Element F in Z we want to think of this as being a function So let's suppose F is Eight say and now I'm going to think of eight as being a function from the spectrum of Z To finite fields so it's value at two is going to be zero because eight mod two is zero And it's value at three well eight mod three is two and eight mod five is three So so we can think of eight as being a function with a graph that looks like this so it's a bit of a funny graph because It's not a graph from the spectrum Z to a fixed space It's a graph from the spectrum Z to a varying space and again at the prime zero It's not quite clear what's going on. I mean we could take the Quotient my zero and then we would just get the Z and if we want to make this into a field We could take Q so we'd have a Q up here and it would kind of go through the point eight at Q So anyway, we can sort of try and picture Functions of rings as being elements of rings as being functions from the spectrum of a ring to some funny varying field Well that we don't only have a problem with this generic point here, but there's also another problem um so we've got a map from R to functions and we can ask is this Injective because If we want to represent elements of R as functions, then a non zero element of R should Minimum correspond to a function that isn't zero everywhere Well, when is when is the function corresponding to an element of f of our injectives? So suppose f goes to Sorry, when is it zero suppose f goes to zero in all the fields And r over p of I guess this is integral domain. So take the field r over p and Turn it into its field of quotients well F maps to zero in r over p is equivalent to saying that f is in the prime ideal p so f maps to zero in all These integral domains r over p means that f is in the intersection of all prime ideals So what is the intersection of all prime ideals? Well for the integers the intersection is just zero. So that's okay but Sometimes there are elements in the intersection of all prime ideals for example if a squared is not This implies a is in all primes Because if a times a is in a prime then a or a is in that prime Similarly if a to the n equals naught for some n greater than or equal to One this implies a is in all primes so We see a nil radical of the ring Which is the radical of zero which is just the set of a such that a to the n equals naught for some n Is contained in the intersection of all primes? so If r has a non-zero nil radical then we can't represent those elements as functions We can ask is there anything else in the intersection of all primes the answer is no in fact the nil radical Equals the intersection of all primes to see this we remember we had this this useful lemma last time which said that if we had a multiplicative subset disjoint from an ideal It means we can find a prime with P contain the ideal I and P disjoint from the multiplicative subset S so suppose a is not in The nil radical of zero then we take the multiplicative subset to be one a a squared a to the n and so on and we take the ideal I to be zero and we notice that S Intersection I is empty because no power of a is zero so we can find Some prime P with P Contains I which is a vacuous condition and P intersection S is is empty so No power So so a is not in this prime ideal P. So a is not in the intersection of all primes Anyway, we certainly get rings turning up in algebraic geometry number theory that have nil potent elements So we need to know how do we represent these as functions? Well, we can do this We consider F in R is is a function From the spectrum of R to The local rings are P instead of Modulo P So we're using localization rather than the quotient. So you remember this is we get this from R by inverting all elements not in P So when does F have image? Zero in our localized at P well, this happens when F s equals naught for some S not in P. So this was the condition for number to be zero in a localization It must be killed by something in the multiplicative set which are exactly the things not in P So suppose F has image naught in all the local rings are P for all primes P in the spectrum of R This means F is killed by some SP not in P for any prime P And now let's look at the annihilator of F, which is the set of X such that F of X equals zero So we see this is an ideal and we also see that the annihilator of F is Not contained in P for any prime P Well, in particular, it's not contained in any maximal ideals. So the annihilator of F Must be the whole ring R so contains one So one times F equals zero which I think you can see implies that F equals zero so If instead of looking at the quotient fit fit the quotient integral domains of R We look at the localizations then we can get something which catches the nil potent elements of R So so we think of R is a function With domain So we think of an element F of R as a function with domain the spectrum of R and at Any element P in the spectrum of R It takes values in the local ring F of P So if the ring has no nil potents then instead of this local ring F of P We can we can look at the integral domain R of P modulo P Which is an integral domain or which take its quotient field and simplify things But in general we have to use this local ring Well local rings aren't quite as easy as fields to deal with but they're still a lot easier than general rings. So so We sort of simplified the ring instead of thinking of a function as being an element of an orbit Instead of thinking of F as being an element of an arbitrary ring we can think of as being a function taking values in local rings and we can study each of these local rings separately and Hopefully obtain information about the ring from that So There's another property of C of the continuous functions on a compact house or space we should look at suppose That U is open and contained in X Then we can look at O of U which is the continuous Functions on you and this O is a sort of calligraphic Letter O Which is usually used in sheath theory we're about to construct a sheath associated to any ring and We're first going to look at this in order to get motivation for this And so what properties does O of U have? So properties Well, first of all if U is contained in V then For any continuous function on V. There's a sort of restriction map to O of U Second property is a sort of pre sheath called a pre sheath property Suppose U is covered by open sets U1 U2 and so on possibly infinite number of them Then if a function on U is not on all the UI Then F must be the zero function on the whole review That's just saying if a function is zero at every on every element of an open cover Then it's zero on the whole set and that's kind of obvious for functions And the third property is the sheath property So again, we assume U is the union of UI So let's just assume it's the union of U1 U2 and U3 Now suppose we've got functions Fi on the open set UI so F is a function on UI and suppose that Fi and Fj have the same image in O of Ui Intersection Uj So we can think of that meaning the function on U1 and the function on U2 Agree on the intersection of U1 and U2 then we can find F a function on U with restrictions Fi on in O of Ui So this is saying we can define a function locally on an open cover By defining it on each open set and as long as the functions We specify these open sets are compatible. This defines a function on the whole set U So for continuous functions on a topological space the sheath property and the pre-sheath property are obvious Well, what we want to do is define analogs of these for a ring So given a Ring up ring R with the spectrum Spectrum of R We want to define rings O of U for U open in Spectrum of R behaving like this What do I mean by behaving like this? It means as if O of U is the nice Functions on the open set U and it's not quite obvious how to do this because you remember functions on the spectrum of R are a little bit funny that the The space they're taking values in keeps changing whenever you change a point of R So it's not entirely clear what you mean by a nice function on an open set U Well, we will just look at At open sets of The form U fi. So you remember this is the set of primes such that fi is not in the prime P And you can ask what about other open sets and What we do with other open sets is we ignore them It turns out that 90% 99% of the time the only open subsets of the spectrum of R You are interested in of these special open sets Which are sort of the places where some function is none zero informally speaking and The other open sets are not too difficult to deal with they're basically just an irrelevant complication So What should O of U of F be Well, let's sort of draw a picture of what's going on. So here we've got the spectrum of R Which we think was being some sort of space and we've got some sort of function fi and The function Fi might sort of vanish at the spectrum of R. So this is the zeros of F by which we mean Primes P such that F is contained in the prime P So if we're thinking of F as being a function, these could be the If we're thinking of F as being a function to fields, these would be where the function vanishes So so the saying this is the zeros of F shouldn't be taken too seriously and U of F is the complement of this So we think of the zeros of F as being some sort of Codimension one space of the spectrum of R and the open set U of F is the complement of this this hyper surface And we want to know what are the nice functions on U of F Well, what nice functions can we think of? Well, first of all the elements of the ring are ought to be nice functions on this and secondly How about one over F because F is none zero on U of F. So we ought to be able to invert it So this suggests we just define the nice functions the nice functions O of U of F to be R together with the Inverse of F. So this is the localization of the ring R Um, so what we have done is we have defined What is called a sheaf we have defined a sheaf of rings on The spectrum of R and what the sheaf of ring consists of is a map taking each open set U of F I to the localization are the F I Inverted so this is a map from nice open sets of the spectrum R to rings And now we want this to behave like the map taking an open set of X to continuous functions on X And as we pointed out the the properties of this map were the Restriction property and the pre-sheaf property and the sheaf property So what we're going to do next lecture is to check that this way of assigning rings to open sets does indeed behave as if As if this assignment was just taking the nice functions on an open set This will allow us to think of a ring as being much closer to functions on the spectrum of the ring