 So, in this algebraic geometry video, we will cover the definition of amorphism of varieties and show how to make algebraic sets and varieties into a category. So suppose we've got two varieties, maybe quasi-projective varieties, x and y. We want to define what is meant by a regular map from f to y. Well, first of all, f should be a function from the points of x to the points of y and it should somehow, we want to say it's a nice function. So how do we do that? Well, we know what nice functions to the field k are, so they're just the regular functions that we defined earlier. Now, if we've got a nice function from x to y, then its composition with a regular function to k should also be a regular function to k. And this is more or less the definition of amorphism of varieties. It's one that preserves regular functions. We've got to be a little bit more careful. So suppose we take any open set contained in y, so u is going to be open in y, then we have this map f minus one u, which is open in x. Now we have the concept of regular functions on u and the inverse image of u. So suppose we've got a regular function g here, then we have f restricted to this map here. I'm going to say f is called amorphism if for any g open u in y and any regular g from u to k, the composition f dot g, that really means the restriction of f2 f minus one u is regular on f minus one u. And it's not difficult to check that the composition of two regular maps is also sorry of two morphisms, it's also amorphism. And so the several quasi-projective varieties, the quasi-projective varieties together with these morphisms defined above form a category as discussed a couple of lectures ago. I should have a bit of a warning here that there's a sort of slight technical problem. This is amorphism of ringed spaces. Well what's a ringed space? A ringed space is just a topological space with a sheaf of rings. And as we mentioned last lecture, the regular functions on open sets form what is essentially a sheaf of rings. Well don't need to worry too much about that yet. Now when we define schemes, schemes will again be ringed spaces. However, this definition of amorphism fails for schemes. This is not the, for arbitrary schemes, you need to use something called amorphism of locally ringed spaces. So this is not the same as amorphism of locally ringed spaces which we'll define later. Now for quasi-projective varieties, this doesn't matter, it turns out that the quasi-projective varieties, morphisms of ringed spaces are essentially the same as morphisms of locally ringed spaces and we don't need to worry about this technical point and we can get away with this rather cheap definition. But when we do schemes later on, you have to be aware that the analog of this definition of morphism definitely fails. Anyway, morphisms of varieties are very similar to morphisms of other things we get in topology. So let's have some examples of ringed spaces. Well, we can have topological spaces. Here we make these into ringed spaces by just taking continuous functions on every open set and it's pretty obvious that continuous functions satisfy all the definitions I wrote down for a sheaf earlier. Then we can have C1 manifolds. These are differentiable manifolds where you only ask for things to be differentiable once. Technically speaking, you want the derivative to be continuous. Then you can look at manifolds where you allow two derivatives and so on and then you can go all the way up to smooth manifolds where you insist that they should have a smooth structure or the ringed space structure is given by taking all smooth functions on every open set. Beyond that, we have analytic manifolds. So these are analytic, these are smooth and beyond that, we get algebraic varieties, at least over the complex numbers or over the reals. So we've got a whole collection of different sorts of ringed spaces and roughly speaking, these are inclusions. So if you've got a topological, I guess that should be a topological manifold not a topological space. Sorry, I've got all these inclusions going the wrong way around. One of us getting a bit confused. So all these are inclusions. So if you've got a smooth algebraic variety, which we'll explain later, over C, it's an analytic manifold and analytic manifolds are smooth manifolds and smooth manifolds are C2 manifolds and so on. What is going on is that as you go in this direction, things get floppier and floppier. So topological manifolds are very floppy. You can bend them very easily and algebraic varieties are very rigid. They're very hard to bend and that there's a sort of major change in behavior at this point here. So analytic manifolds behave in a very similar way to algebraic varieties in some sense. You can't really bend them at all. If you remember from complex analysis, there's a phenomenon called analytic continuation where if you define a function near one point, it's sort of automatically defined near other points. In other words, the function is very rigid and you can't modify it anywhere. On the other hand for smooth functions, it's easy to find functions that are zero somewhere and then non-zero somewhere else and smooth. So you can bend the function very easily. So the category of algebraic varieties is really very similar to things like the category of smooth manifolds or the category of topological manifolds. The main difference is that you put you have a different choice of rings on each open set where the rings determine the functions you're interested in. For instance, in algebraic varieties, we're sort of many interested in functions that are rational functions that are very restricted. Whereas for smooth manifolds, we allow a much larger collection of smooth functions. Now I mentioned earlier that there was the concept of a locally-ringed space as well as a ringed space. It turns out that all of these are examples of locally-ringed spaces. I'll explain what a locally-ringed space is. Well, what we need to do that is to talk about the local ring at a point, p, in some object x. So x is going to be a topological space together with a sheaf of rings. So for each open set, u of x, we have a ring, I call this o of u. And furthermore, if we have open sets u contained in v, then we get a restriction map from o of v to o of u. And notice, by the way, that we have a map from u to v, but the restriction map goes in the other direction from regular functions on v to regular functions on o. So how do we find the local ring of a point? Well, the local ring at p is informally functions defined near p. Well, you have to be a little bit careful about this. So more precisely, an element of the local ring is given by an open set u such that p is contained in u. And secondly, a function f defined on u. So if we're given an open set u and a function f in o of u, then this defines an element of the local ring. However, we need to put an equivalence relation on these. So f and u, so I suppose we've got some function on u and g on v are considered to be the same if we can find w contained in u intersection v with p and w. So f is equal to g on w. So here we've got the open set u, the open set v, and here we've got some much smaller set w. And here's our point p. So in other words, we consider two functions to be the same if they're the same on some neighborhood. So if we take the set of all functions defined on some open neighborhood of p and put this equivalence relation on them, then it's not difficult to check we get a ring. So the set of equivalence classes is a ring called the local ring at p. And furthermore, this is actually a local ring, at least in the cases we are doing the thinking of. So in the cases above, this is indeed a local ring. So you remember a local ring is one that has a unique maximal ideal. Well, what's the maximal ideal? The maximal ideal is equal to functions vanishing at p. And it's pretty obvious what is meant by function vanishing at p. So all of these examples we had earlier, like topological spaces, smooth manifolds, analytic manifolds. In all these cases, if we do this construction, we get a local ring. And if this happens, the ring space is called a locally ringed space. And it turns out that a locally ringed space is a very good way of describing what a geometric object is. So all these things like topological manifolds and algebraic varieties are all special cases of locally ringed spaces. And the locally ringed spaces differ in what sort of rings you allow for the ring of functions on you. So let's give a few examples. So there's an example we've mentioned several times before. We can take the hyperbola x, y equals 1, which is a subset of the plane. And we can look at the points a1 minus 0, which is a subset of the line. And now we can define morphisms in both directions. So I can define a morphism in this direction taking x, y to x. And I can define another morphism in the other direction taking x to x, x to the minus 1. And you notice that x to the minus 1 is indeed a regular function on here. And you can easily check these maps at inverses of each other. So these two objects are isomorphic in the category of quasi-projective varieties. So even though this isn't closed in a one, this is closed, they're still isomorphic as quasi-projective varieties. Another example, let's take the curve y squared equals x cubed. So here we get a sort of curve with a cusp. And there's a map from the affine line to the curve y squared equals x cubed, which takes a point t to t squared t cubed. There's obviously the square of that is equal to the cube of that. And you can check that this is continuous. And furthermore, it's a bijection. So as a map of the underlying sets, this is an isomorphism. However, it's even also its inverse is continuous. So it's actually a homeomorphism of topological spaces. However, it's not an isomorphism of varieties. So it's an isomorphism for the underlying topological spaces, but it's not an isomorphism of the corresponding varieties, even though it's, I should say, sorry, I forgot to mention, it's also a morphism of varieties. So the fact that something is a morphism of varieties and also an isomorphism of the underlying topological spaces does not imply it's an isomorphism of varieties. This is kind of intuitively obvious because this curve here has some sort of weird singular point and A1 doesn't. So there's no way to define a regular map from this curve back to A1. We could try defining t to be y over x, but y over x is not regular at x equals zero, and there's no way to make it regular. Okay, so next we want to show what are the morphisms to an affine variety. And in particular, we want to show that morphisms between affine varieties correspond exactly to homomorphisms of their coordinate rings. So we will do that next lecture.