 This algebraic geometry video will be about regular functions on varieties and their open subsets. So we recall that an affine variety in, affine variety Y in n-dimensional affine space is defined by some ideal I. And the coordinate ring of Y is the polynomial ring. We think of this as being polynomial functions on affine space restricted to Y. And this coordinate ring is going to be the ring of regular functions on Y. So for affine varieties, or for that matter, for closed subsets of affine space, regular functions are just restrictions of polynomials. Fine. More generally, we want to define regular functions whenever U is an open subset of an affine variety Y. So open subsets of affine varieties are called quasi-affine varieties. As I said, quasi is usually a sign that someone messed up the terminology, but anyway. So some examples, we might take U is equal to A1, one-dimensional space with the origin root. And as we mentioned earlier, U will actually turn out to be isomorphic to the hyperbola of points where X, Y is equal to 1 in A2. Because the X coordinates of a hyperbola are exactly non-zero values of K. So you can map U to the hyperbola by taking X to X, and it's inverse and vice versa. So this quasi-affine variety is really just an affine variety in a slightly disguised form. Another example is A2, the orange move. We will see later that this is not isomorphic to any affine variety. So this is an example of a quasi-affine variety that isn't affine. So quasi-affine varieties are definitely more general sorts of objects. So now we want to think about what do we want the regular functions on U to be? So if we take U to be A1 minus the origin, we want X to the minus 1 to be regular on U. It's a perfectly good function. You can define an algebraic way. And it's obviously not the restriction of a polynomial. I mean, the polynomials on here would just be K of X. So we don't want the regular functions on U to be K of X. We want the ring of regular functions to be K of X, X to minus 1. So we need to think of a slightly different definition. So we define a regular function on U. Here now we're taking U to be an open subset of an affine variety. It's a function, all functions for the next, most of the rest of this lecture, functions to the field K, by the way. It's a function that is locally regular, all points P of the quasi affine variety U. Well, we haven't yet said what locally regular means. Locally regular means that if our function is F, then F is equal to G over H in some neighborhood of P, where G and H are polynomials. In other words, they're just elements of this polynomial ring restricted to U. And H is not zero at the point P. So a regular function sort of looks locally like a rational function, except the denominator must be non-zero. G and H, by the way, may depend on the point P. So X to minus 1 would now be a regular function on A1 minus 0 because we're allowed to invert by X because X is non-zero everywhere here. So roughly speaking, regular functions mean you're allowed to divide by polynomials so long as the polynomial is non-zero at all the points we're interested in. Well, we have a bit of a problem here because we seem to have defined Wittler in two ways. So we have the following problem. Suppose Y is affine. Then we can ask, are regular functions regular? Well, the problem you see is we've defined regular in two ways. So here we have defined to be regular if they're polynomials on affine space. And here, this means regular at each point. So we should check these are compatible. So this is a sort of local regularity condition. So this says they're locally regular. And this says they're globally regular in some sense. So we want to show that saying something is regular using this local condition is the same as saying it's regular using this global condition. So let's assume that Y is a union of open subsets. So each UI is open. Here Y is a variety on Y as usual with values in K. And suppose it's regular on all UIs. So if F is regular at every point, then we can cover Y by open subsets such that F is regular on each open subset. Well, this means F is equal to GI over HI on UI, where HI is not zero on UI. Where the GI, the HI are of course just both polynomials. And the question is, is F a polynomial? And this isn't made obvious since we've defined F to be locally given by rational functions. Well, in order to show this, we observe that one is equal to A1H1 plus A2H2 and so on, modulo I for some polynomials, AI. And this has to make sense all the definite number of these AI should be zero. The reason for this is that Y is covered by all these sets UI. Well, this means that no point of Y is outside all the UIs. This means that no maximal ideal of KX1 to XN containing I contains all the HI's because the maximal ideals containing I are the same as the points of I by the wheat and all still and that's and saying that a point is not outside all the UIs. The same as saying that the corresponding maximal ideal contains all these HI's because UI is the place where I is none zero. So since no maximal ideal of K, if there's no maximal ideal containing I and all these HI's, this means the ideal generated by I and all the HI's must be the unit ideal. So the ideal generated by H1H2 and so on and I is the ideal generated by one. In other words, one can be written as a linear combination of the HI's. Now we're going to use this relation to construct a polynomial F. Well, so, well, we've got a function F and we want to write F as a polynomial. So since one is equal to A1H1 plus A2H2 and so on, modulo I, we multiply by F and we find F is equal to A1H1F plus A2H2F and so on. Again, mod I and now we know that H1F is equal to G1. So this is equal to A1G1 plus A2G2 and so on, mod I. So this suggests the definition of F, so just 5 plus A2G2 and so on. And now we should just check that defining F in this way does actually have the properties we want and this is pretty trivial because we want to show, we want to check that HIF is equal to GI on UI and this just follows because HIF is equal to HIA1G1 plus HIA2G2 and so on. And it is that HIJ is equal to HJGI because GI over HI and GJ over HJ must be the same on the intersection of the sets UI and UJ. So this is, I'm just like confused, so. And this is just equal to, sorry, now we notice that this is equal to GI A1H1 plus GI A2H2 using this relation here. And this is equal to GI times A1H1 and so on, mod I, which is equal to GI mod I, which is what we wanted to prove. So now we show that HI times F is equal to GI mod I. So the local condition of regularity is the same as the global condition of regularity. Next we can ask you is a quasi-projective. So what does quasi-projective mean? Well quasi-affine means open subset of an affine variety. Quasi-projective means open subset of a projective variety. So we've got four sorts of varieties. We've got affine varieties, which are special cases of quasi-affine. And we've got projective varieties, which are special cases of quasi-projective. And quasi-affine are in fact special cases of quasi-projective varieties because any affine variety is an open subset of a projective variety. So quasi-projective are the most general sort of things we have. And now you notice that any quasi-projective variety is covered by affine subvarieties. Or if you like open subsets of affine subvarieties, it's the open subsets of. So this is because projective space is covered by a finite number of copies of affine space. So any quasi-projective variety you can easily see is covered by a finite number of open subsets of affine varieties. And now we say a function on a quasi-projective variety, which people thought was a shorter name for these things, is regular, if it is locally regular. Where locally regular means for each point has a neighborhood that is an open subset of an affine neighborhood where it is regular. So we find regularity for open subsets of affine varieties. We can extend regularity to all local affine subsets of projective varieties, and this allows us to define regularity of functions on open subsets of projective varieties. Next we notice that if we have a quasi-projective variety, then for each open subset u of y we have a ring sometimes denoted by o of u of regular functions. And these rings are the following property. Suppose u is a union, u1, union, u2, union, u3 and so on of other open subsets, then it has the following two rather obvious properties. First of all, if a function on u is equal to zero on o of ui, where we can just restrict it for all i, then f is zero. In other words, if a function is zero or six other than it's i are functions on ui, and suppose that they're compatible, fi equals fj on ui intersection uj, then we can find f that's regular on u such that f becomes fi on ui. So this is essentially just the condition we showed earlier that if a function is locally regular on an affine set, then it's regular. Now, these are the conditions that you need to define a sheaf. And we will see later when we do schemes that schemes also have a sheaf of rings on them with these two properties. Now for ordinary quasi-projective varieties, it's kind of pointless introducing sheaves because it's just the properties of sheaves are completely obvious because we're just talking about regular functions on the space and regularity is a local property and being a local property is sort of like this. However, when we cover affine schemes, things will be a bit more complicated because the elements of these rings won't correspond in any obvious functions on something. So we will need to start working with sheaves. So let's finish up this lecture with an example. Let's just find regular functions on dimensional project space. Well, we know the regular functions on affine space is just the ring of polynomials. So to find the regular functions on projective space, we cover it with two copies of affine space. You remember if we denote points of the line by x colon y, we can think of these as the points with x not equal zero and these are the points with y not equal zero. So we have a sort of picture like this. Here we've got one copy of A1. And here we've got another copy of A1. And the intersection is A1 minus the origin. So to specify a regular function on projective space, well, we have to give its restriction. Well, here will be a regular function on affine space. So pick regular function on A1 and a regular function on the second A1. So we have to pick a function here. And these functions must be the same on their intersection. So these must be the same on A1 minus zero, which is the intersection of these two one functions here. Well, A1 minus zero. So if we think of this as being the points of this A1 denoted by x, then the regular functions on A1 minus naught. Will be the ring of polynomials kx, x minus one. And these functions will be the ring of functions kx. And so if this is x and this is y, we notice that these regular functions will be the ring ky, which we can identify with k of x minus one. So what we need to do is we need to pick a function that is a polynomial in x and a function that is a polynomial x minus one that become the same in this ring here. So we've got maps from these two rings to that ring, which are these maps here. Well, pretty obvious there's only one-dimensional space of functions in here, which you can produce in both these ways. So f must actually be a constant. So this is the only element of this ring which can be obtained as the restriction of something here and as the restriction of something there. So all functions on the projective line are just constants. Okay, next lecture we will show how to use regular functions in order to define morphisms of varieties.