 This lecture is part of an online algebraic geometry course about schemes and we will be giving the definition of a scheme and the first simplest examples of it and just give some historical background. So schemes were introduced by Grotendick in the 1950s and at that time there was a lot of experimentation going on with the foundations of algebraic geometry so André Vey and Seyre and Chevalet and Grotendick were all trying out different possible generalizations of algebraic variety and Grotendick's notion of scheme won out by sort of process of evolution it out competed all the other notions because it was more useful and more powerful. Incidentally the name scheme appears to have been introduced by Chevalet for a slightly different concept and was borrowed by Grotendick. So I draw a sort of family tree of some of the many attempts people made to generalize projective varieties so we have projective varieties which were what people studied in classical algebraic geometry. André Vey extended this to the concept of an abstract variety which was made by gluing together affine varieties. Schemes are a sort of generalization of this where you glue together things more general than affine varieties. Schemes in turn are a special case of locally ringed spaces and locally ringed space includes several other concepts for instance smooth manifolds are locally ringed spaces so a complex manifold and so are topological manifold. So locally ringed spaces are very general sort of geometric object that includes almost every other geometric object you've come across. Locally ringed spaces can in turn be generalized to locally ringed toposes or possibly topoi. And there are other generalizations of schemes for instance these can be generalized to algebraic spaces which in turn can be generalized to various forms of stacks. Stacks are notoriously difficult to understand the definition of color once said the study of stacks is recommended to people who would have been flagellants and former lives which I think is a pretty good summary of them. I've tried to learn the definition of a stack half a dozen times and probably forgot it a day later every time. Anyway so this course is mostly going to be about schemes rather than any of these other generalizations. So what is a scheme? Well a scheme is a locally ringed space that is locally isomorphic to an affine scheme. Well that's not much use as a definition because I haven't told you what a locally ringed space is and I haven't told you what an affine scheme is so I better define these. So what's a locally ringed space? Well let's first do a ringed space it's just a space x plus a sheaf of rings and there are loads of examples of this we can take say a smooth manifold plus the sheaf of smooth functions and as usual you can vary that for a complex manifold or a topological manifold or an algebraic variety plus a sheaf of regular functions and so on. So ringed spaces are quite common. We actually want to work with locally ringed space so locally ringed space is a ringed space where all the stalks are local rings. So you remember if we've got a sheaf then we can take the stalk of the sheaf at any point so if you've got a sheaf of rings we get some rings at every point so a local ring has a unique maximal ideal and the unique maximal ideal sort of corresponds to functions vanishing at the point. So in all the examples we had the ringed space of where the sheaf was functions on the manifold the stalk at any point is going to be functions defined near the point and it will have a local ring this will be a local ring whose maximal ideal is just functions vanishing at a point and the reason this is a maximal ideal is if f is not zero at p then f has an inverse in a neighborhood of the point p. So of course if f is non zero at a point p f might not have an inverse because it might be zero somewhere else but if you're allowed to restrict to a small neighborhood of p then in that neighborhood you can invert f. So all these examples like smooth manifolds and all the rest of them are in fact locally ringed spaces because if you've got a smooth function that's non zero at a point it has an inverse in some neighborhood at that point so that the set of all germs of smooth functions at a point is indeed a local ring. So it's a locally ringed space that sort of captures the concept of some sort of space with functions on it. Now there are lots of examples of locally ringed spaces where they don't really come from a space with a function taking values in a field but if they're locally ringed spaces they still sort of behave as if they were like that. So that's what a locally ringed space is. Well I should explain the difference between a ringed space and a locally ringed space just by giving an example of a ringed space that is not locally ringed and this is quite easy just take x to be any topological any reasonable topological space and take the sheaf o of x to be sorry define a sheaf on x where o of u is just equal to so define the pre-sheaf where o of u is equal to r for a fixed ring r and then we take the corresponding sheaf of this this is the sheaf the constant sheaf of r so o of u is going to be functions from u to r where you give r the discrete topology then the local ring at any point p in x is just isomorphic to r so if r is not a local ring then this will be a ringed space that is not a locally ringed space and this sort of example doesn't really correspond very nicely to a geometric object with with functions on it so so we exclude those when we insist that things should be locally ringed spaces um okay so that's given some idea of what a locally ringed space is now we should define an affine scheme well an affine scheme is just one that is isomorphic to the spectrum of a ring r so um all rings are commutative by the way since this is algebraic geometry so what is the spectrum of a ring r well the spectrum is it's going to be a locally ringed space so in order to define it i need to tell you what its points are and what the topology is and what the sheaf of rings on it is so the points are the prime ideals of r the topology well for any f in r we can consider set df which is the set of primes not containing f if we think of the primes as being points and this is informally the points where f is non-zero so um it's not really the points where f isn't zero because f isn't really a function on on this but um in in in simple cases like the um that like the spectrum of continuous functions on a topological manifold this is exactly the set of points where f is non-zero so so it's plausible these are open and these form uh not all open sets but these form the base for the topology now we have to define the sheaf um and so for every open set i've got to tell you um i've got to tell you a ring well i'm not going to tell you the ring for every open set so i'm going to call the sheaf o and i'm only going to tell you what o is for df and o of df is just the ring r with f inverted and a key point here is that i don't really have to tell you what the sheaf is on all open sets because if i've told you what a sheaf is on a base for the topology that actually determines um this everywhere so key point is we ignore open sets not of the form um df and this is really a key way of understanding affine schemes if you look at most books on affine schemes they spend a lot of time worrying about what happens to open sets not of this form and the best thing to do with open sets of this form is just to ignore them completely that they're most of the time they're just not important and it's not really necessary to say anything about them because the ring on them is determined by the sheaf property so this is just r where you force f to be invertible and by the way it's not obvious that this is a sheaf um that's something we have to define that's something we have to prove later um um you can define arbitrary open sets or arbitrary closed sets so an arbitrary closed set of of this is you take any subset of the ring and look at the set of primes disjoint from that subset and that gives you arbitrary closed sets but as i said arbitrary closed sets aren't really particularly interesting or useful so now we should have a few examples of affine schemes um so first of all let's take r to be a field then the points of spectrum of r well it's a field so it's only got one maximal ideal so it just has one point the apology well there's not i mean there's nothing you can say about a topology of a one point set and the sheaf um we just need to say what the ring of this point is and that's obviously just r so nothing terribly exciting is going on if r is a field so now let's look at r being the integer z now the points are the prime ideals which are first of all the maximal ideals which just correspond to primes and the ideal zero terminology for prime ideals and maximal ideals is a bit messed up so the the maximal ideals of the integers correspond to primes and the prime ideals correspond to primes together with zero i'm sorry there's too late to do anything about this terminal terminological mix up so what are the open sets or df there's just the set of primes not containing an integer f so df is either empty or it's that the whole of spectrum of z minus a finite number of non-zero primes and this is a bit weird because you see that this point here is not closed so points of the spectrum of a ring don't have to be closed it's not only non-house dwarf which we already saw when we were looking at the zariski topology but points don't even need to be closed so it's not even a t one space so you can draw a picture of this as far as well you can't really draw a picture of it because since it's non-house dwarf and points aren't closed you can't really embed it in euclidean space best you can do is you try and picture it like this you picture the points that the prime points on a line as tending to the limits and that this limit is the ideal naught and a typical open set kind of looks something like this so it will contain zero and all but a finite number of primes unless it's the empty open set so this is what an open set looks like and then we should ask what are the what what do the rings of the sheaf look like so o of df is just the integers of the form m over f to the k so for instance this open set here there will be all rational numbers that only have powers of two and five in the denominator we can also ask what are the local rings so the local ring at the point naught is just the set of all rational numbers and the local ring at the prime p not equal to zero is usually denoted by z of p which is all numbers m over n with n not divisible by p so this is a fairly complete description of of the spectrum of z we've described its points open sets local rings and and the and the sheaf of ideals in a completely explicit form and now we can do the ring of polynomials over complex numbers and this is very very similar to the ring z this is because both this ring and the ring z are examples of principal ideal domains so the points are the ideal naught and the irreducible polynomials of the form x minus alpha for alpha in c so the points look like the complex numbers together with a pointed infinity and the topology just as for the integers the open sets are either the empty set or the whole spectrum minus a finite number of points and we can work out the sets so this so this is going to be a df which is the points the form x minus alpha where f of alpha is non-zero so in other words for each polynomial it has an open set just corresponding to the points where it's non-zero if we identify the prime ideal x minus alpha with the point alpha of the complex numbers and we should say what is the odf well this is just all rational functions um g over h where h is can be taken to be a power of f so these are rational functions which have no poles on the set d of f so they're allowed to have poles at this finite number of points but not elsewhere um of course o of the sorry now we do the local rings well the local ring at the point zero is just the field of all rational functions in x the local ring at x minus alpha is just rational functions with no pole at alpha so we should think of this as being the analog of or of the local ring of the integers where we have all rational numbers with n not divisible by p so say n not divisible by a prime p just corresponds to saying the denominator g over h this so the denominator h of g over h is not divisible by x minus alpha in other words it's got no pole at alpha um now i will i will answer the question why is spec of r called the spectrum it turns out this is actually related to the spectrum in physics so you remember the spectrum of some some element is a set of line frequencies that it emits or absorbs light at um so first of all the spectrum of an atom turns out to be related to the spectrum of a linear operator on Hilbert's space where the spectrum just means a set of eigenvalues so the spectrum isn't exactly the eigenvalues of the operator it's something like the difference set of differences of eigenvalues but whatever it's it's very similar next we noticed that the um spectrum um so that so the um the eigenvalues of a matrix a um correspond to the maximal ideals of the ring um c of a that's sort of obvious because the the set of all polynomials over a matrix a is just c of x modulo the minimum polynomial of a and the roots of the minimum polynomial of its eigenvalues so already we're seeing the spectrum of an atom is sort of vaguely related to the set of maximal ideals of a certain ring um and you can also do this instead of taking one matrix a you can set collect a set of commuting operators a and then the joint spectrum of these operators is just the maximal ideals of the ring they generate we can also take x to be um some sort let's say compact house store space and take r to be the continuous real valued functions on x sometimes denoted by c of x then we see the points of x correspond to the maximal ideals maybe i should say closed maximal ideals of the ring r because for each point you can just take the maximal ideal of functions vanishing at that point and the topology is given by setting df um is the points where f is not zero in other words the point such that f is not contained in that maximal ideal so if you've if you've got a the ring of continuous functions on the compact house store space if you sort of rather carelessly lose the house store space for some reason you can reconstruct it from r just by taking the space to be the maximal ideals and the topology to be given by this so so the compact house store space x and this these rings here which are certain sorts of commutative um bannock algebras are essentially equivalent um so from this you can just say the maximal ideals of any commutative ring form a topological space where you just define the topology in this way you see this construction works if you replace r by any commutative ring you can just take its maximal ideals and define the topology like this so this idea is is used a lot in analysis in order to convert compact house store spaces into bannock algebras well that says you know we should be defining the spectrum of a ring to be a set of maximal ideals why so the final question is why do we use prime ideals not maximal ideals so people used to use maximal ideals and this sort of works fine and algebraic geometry and analysis but doesn't work very well more generally the problem is as follows suppose we're given f r to s where these are rings we would like to define a map from the spectrum of s to the spectrum of r and it's very obvious how to do that if you take a maximal ideal in s you can just take its inverse image the trouble is if this is maximal this is need not be maximal for example if you just take the integers to the rational numbers and take the ideal zero here then its inverse image is the ideal not of z which is not maximal now it just happens that if you work with algebraic varieties or compact house store spaces then in these cases it turns out that in practice the inverse image of a maximal ideal under reasonable homomorphisms is indeed maximal which is why people could get away with using maximal ideals at first but if you want to work with more general rings this this really breaks down so what can you do well let's analyze what the problem is so that we see how to fix it so maximal ideal m is just the same as saying that r over m is a field now if you've got a map from r to s and a maximal ideal m of s so we get s over m then r over f minus one of m will be a subring of this so we have a subring here so r over f minus one m is a subring of a field well the problem is a subring of a field need not be a field so it's not a field in general it is an integral domain meaning it's got no zero devices and a subring of an integral domain is indeed an integral domain so if we instead of using maximal ideals we'll use ideals such that r over m is an integral domain then we're okay well a prime ideal so p is a prime ideal of a ring r if and only if r over p is an integral domain so if we work with prime ideals rather than maximal ideals then f the minus one of a prime ideal is always a prime ideal and in some sense prime ideals are the smallest collection of ideals bigger than maximal ideals that always have this property the inverse image of a prime ideal is as a prime ideal so this is why growth index switched from maximal ideals to prime ideals it's so that you could get a nice functor or a contravariant functor from rings to the spectrum of a ring whenever we have a homomorphism between rings we'll see that we get a homomorphism between their spectrum and by the way the single biggest problem in algebraic geometry is that the map from f to r goes in the opposite direction when you so instead of going from r to s the induced map goes from the spectrum of s to the spectrum of r and this is an endless source of confusion so in the next lecture we'll give some more examples of schemes