 Okay, so let us continue with our discussion of basic open sets for the Zariski topology. So let me again recall you start with polynomial g in n variables and you look at the complement of the 0 locus of g and call it T g it is called the basic open set defined by g and any open set can be written as finite union of such basic open sets and then we define the ring of functions on this basic open set to be functions of this form namely these are the polynomial functions multiplied by you know inverting powers of g okay which makes sense because g does not vanish on that locus okay. But then the fact is that this basic open set D g is actually itself isomorphic to an affine variety and I have not defined what an isomorphism of affine varieties is but I am trying at least I will try to give you the isomorphism at least at the level of topological spaces and this is how we do it we take an affine n space over k with the usual Zariski topology and look at the 0 locus of g this is a hyper surface okay if of course if g is irreducible it is a hyper surface if g is not irreducible then it is a union of hyper surfaces okay and it will be a union of hyper surfaces which will be the irreducible components. There will be hyper surfaces corresponding to the irreducible components of g which occur in the factorization of g okay and then the complement of this hyper surface is the affine open set is the basic open set D g okay and this D g can be thought of as a close subset as a close sub variety of an irreducible close subset of a larger affine space namely an affine space of dimension 1 more and that is done in the following way you look at the 0 set of gy-1 where y is the extra variable that you add to get the ring of functions on this affine space of 1 dimension more and since this polynomial gy-1 is irreducible the ideal generated by gy-1 is prime and the 0 set therefore the 0 locus of that polynomial gy-1 is a hyper surface okay and it is a close subset of this n plus 1 dimensional affine space and it is a irreducible close subset and it is ring of functions is defined to be the ring of polynomials on the ambient affine space divided by the ideal of functions that vanish on that hyper surface which is just the ideal generated by gy-1 and the fact is that we have a bijective map from this basic open set here in an and this irreducible close subset this hyper surface in an plus 1 so what is happening is that a basic open set in affine n space is being identified with a hyper surface in affine n plus 1 space okay. So it may look at at first it may look a little confusing because here it is open and there it is closed okay but you must remember that the affine spaces are of different dimensions and you must also remember that the ambient affine spaces are of different dimensions but the dimensions of these two spaces of course they match okay see the dimension of a hyper surface is always one time is always one less than the dimension of the affine space. So here it is n plus 1 dimensional affine space and its dimension is one less so it is dimension n okay and this is something that I have not told you about but this is something that I will try to explain to you the dimension of an open set and a non-empty open set is essentially the same as a dimension of I mean it can be the dimension of a non-empty open set can be defined as the topological dimension okay and the fact is that the dimension of this will also the dimension of this will also be n okay and that coincides with the dimension of this. In fact what will happen is that you know dimension can be defined for any topological space okay it is defined to be the maximal length of you know a chain of irreducible closed subsets properly contained in one another each one properly contained in the next and provided you start with you start indexing with 0 and then you take the max length of maximal the maximal such chain okay essentially you should take the length of the maximal such chain and take away one from that okay. So you can define the dimension of a topological space in that sense you can define the dimension of any subset of any topological space and it will turn out that the dimension of subset is the same as the dimension of the closure of that subset by going to the closure the dimensions is not going to change okay and therefore you know if you believe that statement the dimension of d of G will be the same as the dimension of the closure of dg but then the closure of dfg will be the whole affine space because d fg is know non-empty open subset of the affine space and you know any non-empty open subset is irreducible and dense in particular it is dense. So its closure will be the whole affine space and by going to the closure you change the dimension therefore the dimension of this is the same as the dimension of affine space and the dimension of affine space is n, okay. So this is n dimensional, this is also n dimensional, so dimensions match okay and I told you it is a matter of exercise, it is a good exercise for you to check that this map is actually a homeomorph. In fact as I was trying to point out in the last lecture let me say the following thing An plus, so Ank sits inside An plus 1k as it sits inside as a subset which is given by the 0 set of y. So the 0 set of y is it is a hyper surface defined by y which means you are looking at all the points where the y coordinate vanishes and all the points where y coordinate vanishes will give you the copy of it will give you this An, okay and the fact is that so this is the identification okay, so An is identified with 0 set of y in An plus 1, okay and mind you this means that you are thinking of An as a, An is a hyper surface in An plus 1 and in this case we call it a hyper plane if you want, okay it is a hyper surface in An plus 1 because it is a 0 set of a single polynomial, okay where the set of points where the y coordinate vanishes is precisely a copy of An, right and well what is happening is that you also have a projection this is you have a projection map from An plus 1 into An and the projection map is the map that takes the An plus 1 coordinates and forgets the last coordinate, okay and the statement is that you take this projection map and restrict it to this closed subset then that gives an isomorphism with this open subset, okay. So projection restricted to Z of gy minus 1 from Z of gy minus 1 to dg is a homeomorphism and you know you will see this later and in fact an isomorphism of varieties this is something that we will see later because I have to define what isomorphism of varieties is but then if you believe this then it is and also believe the fact that you know an isomorphism of varieties has to correspond to an isomorphism of their coordinate rings namely rings of functions okay then it will tell you that the rings of functions on this and the rings of functions on this have to be the same so they will tell you the ring of functions on this has to be isomorphic to this but that is also isomorphic to this because of commutative algebra. So it will tell you that ring of functions on dg is this it is correct to define the ring of functions on dg to be this. So a nice to see that in a very simple case what you can do is that you can simply take you can just look at the plane a2 and then you can look at the rectangular hyperbola which is given by the 0 set of xy minus 1 this is x axis this is the y axis okay this x axis actually corresponds to an a1 which is sitting inside a2 by given by the equation y equal to 0 okay and if you take the projection if you take the projection onto the the first coordinate that is you forget the last coordinate okay then what you will get is you will get under the projection the image of this rectangular hyperbola will be the complement of the origin in a1 and that is precisely the affine open set dx the complement of the point where x equal to 0. So projection restricted to z of xy minus 1 from z of xy minus 1 to dx which is a1 minus a1 minus the origin is an isomorphism. So this is a very simple diagram that you can always remember that tells you what what is happening more generally. So I have taken n equal to 1 and there is a1 sitting as x axis inside a2 okay and a1 is sitting as y equal to 0 just like this an is sitting as y equal to 0 in an plus 1 and then you know this projection from an plus 1 to an which is in this case projection from a2 to a1 and this projection is simply given by forgetting the last coordinate which is y that means projection onto the x axis okay and under this projection the 0 set of xy minus the 0 set of gy minus 1 goes to dg for me now g is x. So 0 set of xy minus 1 which is the rectangular hyperbola it projects onto the complement of the origin because that is the only point that will not that will be left out okay and the complement of the origin is of course dx it is a set of all points where x does not vanish and that is the complement of the origin and you get a isomorphism like this okay. And the fact that this is an isomorphism varieties is a geometric fact and what is the corresponding translation of this fact into commutative algebra it is just a statement it is just a statement that if you take the fine coordinate ring of z of xy minus 1 which is k xy by xy minus 1 that is isomorphic to k x localized at x okay which is defined to be the affine coordinate ring of the ring of functions on dx okay. So that tells you I mean that gives you picture in the simplest possible case okay as to what is happening and what is happening here the same thing is happening here right fine. Now so I have given you you know I have given you two lines of justification or two lines to convince you that this definition is correct okay so let me recall them one is that the functions on the open set basic open set dg have to allow in evaluation of negative powers of g okay which is sensible enough because g does not vanish and when a function does not vanish its reciprocal is also valid as a function should be valid as a function okay so natural that you should be able to invert g and if you invert g or actually localizing at g and this is the ring of functions localize at g so that is one justification the other justification is d of g is also an affine variety it is isomorphic to an affine variety for which the ring of functions is this which is also isomorphic to this from the sense of commutative algebra so this is another justification now I will give you yet another justification and this is the justification essentially that it goes on the that goes along the lines of the fact that you must have an isomorphism of affine varieties as equivalent to an isomorphism of their affine co-ordinate rings okay and so let me recall something from the previous lecture okay so you see we did the following thing what we did was we put on one side you know we put affine varieties okay and on the other side we put affine co-ordinate rings which are just co-ordinate rings of rings of functions of affine varieties okay I mean they are always called as co-ordinate rings because they essentially are based on the variables which are thought of as co-ordinates giving co-ordinates on the ambient affine space and the word affine is used because they are all affine varieties okay they are all considered inside affine space right and so what is the definition here the definition here is my original definition on this side was an irreducible closed subset of affine space so you know if I started with affine space a and k well I would end up with affine co-ordinate ring a so you know this direction the association is given by a of okay so a of a n is just the polynomial ring in n variables okay and if you give me a irreducible closed subset z of i or z of p for p a prime ideal in the polynomial ring if you take the 0 set of p okay that will give me a irreducible closed subset of a n okay so this is irreducible closed and we call called such an irreducible closed subset as an affine variety a closed irreducible closed subset of some a n and for it the corresponding affine co-ordinate ring or ring of functions was defined to be a of z of p is equal to the co-ordinate ring the ring of functions on the ambient affine space the larger affine space divided by the ideal of that variety which is just p okay where of course p is ideal of z of p okay this how we define now you see and in fact I also told you that this fact that this is an irreducible closed sub variety is reflected by showing that from here to here you have a quotient because it is just quotient by the prime ideal p okay and I was trying to tell you that you know you have a more general picture that is on this side whatever you have there is a reflection of that here this is a geometric picture this is a commutative algebraic picture and I told you that the equivalence comes because of an arrow that is going in this direction and I told you that arrow that arrow is actually max spec it is given by max spec okay. So what I told you was well you give me a general affine co-ordinate ring so what is the definition of general affine co-ordinate ring a general affine co-ordinate ring is defined to be something like this it is a finitely generated k algebra which is a integral domain namely it is a polynomial ring in n variables in any number of variables not necessarily in any number of but finite number of variables divided by a prime ideal why do I want divided by a prime ideal because I want an integral domain and why finitely many variables because I want a finitely generated k algebra so on this side the more general definition of affine co-ordinate ring will be finitely generated k algebras which are integral domains okay. So you know so let me write that here if you take a finitely generated k algebras there is enough space here so let me write it in the next line finitely generated k algebra that are integral domains okay this is what you should put on this side in general and what will that be that will be some k of well y1 etc ym modulo some well some p which is a prime ideal okay and what does this come from this comes from affine variety that is something that you can very easily see what is the affine variety you take you take max spec of k y1 etc ym this is nothing but am okay this is what we saw in the last lecture I told you that this is a what I proved was this is a isomorphism as topological spaces okay but the and I told you that that is only half the story in fact it is an isomorphism of even varieties but that is something that I will keep telling you but that is something that I will justify later because I have not defined what isomorphism is on this side okay but just take it for that. So if you believe that then this is an isomorphism of varieties okay but mind you at the set theoretic level this statement is just the Nuhl-Scheven cells that corresponding to a point with coordinates lambda i you are associating the maximal ideal given by xi-lambda i generated by xi-lambda i and that is the Nuhl-Scheven cells that every maximal ideal is of this form okay and my and let me recall that max spec was supposed to be the maximal ideals in this ring and the which is subset of the full spectrum called the prime spectrum which consists of the prime ideals and that spectrum itself has had a Zariski topology and therefore the maximal spectrum which is a subspace of that subset of that topological space got an induced topology and with respect to that induced topology this identification became not just a bijective map but it became actually a homeomorphism of topological spaces and the most than the strongest statement is that this is fact an isomorphism of varieties okay. Now in this what you do is you look at max spec of the of this quotient k of y1 ym mod p then you know this sits inside as a closed subset here and that closed subset actually corresponds to that is the identification of Z of p which is identified with this this diagram commutes so this the very identification that associates to every point of am the maximal ideal in the polynomial ring corresponding maximal ideal a unique maximal ideal in polynomial in m variables will associate to every maximal ideal here I mean to every point here a maximal ideal of the polynomial ring which contains the ideal p because ideal the maximal ideals in the quotient are precisely the maximal ideals in the parent ring which contain the kernel okay that is the correspondence so this is what is happening in this case okay this is what we saw in the last lecture. Now what I am going to do is I am going to slightly modify see just like I am modifying the objects on this side I am not saying that they are affine coordinate rings when I say they are affine coordinate rings it means that I am already starting with something here and taking its affine coordinate ring but instead of that if I want to independently define it on this side I simply define it like this finitely generated k algebra that are integral domains okay so in particular what happens is that these guys something like this does come from here okay but of course the way I defined it it is a completely the definition as completely commutative algebraic it has got no geometry in it okay I am not making any reference to this side I am not saying that these are affine coordinate they are the rings of functions of some affine varieties but they turn out to be okay similarly on this side what I am going to do is I am going to follow a variety affine variety if it is isomorphic to an affine variety okay now that is again in a way like bringing the question but I am not defined what a general variety is okay but let us assume that for the moment you nively accept it meant by an isomorphism of varieties and then you say that any variety which is isomorphic to an affine variety should be also be called an affine variety suppose you make that definition then the beautiful thing is that if you look at in an if you look at this open subset given by dg okay the basic open subset given by dg mind you this is not a close of variety of an it is not a close of variety of an then this also turns out to be an affine variety okay and I told you roughly the story as I have done there is that this is isomorphic to the set z of gy-1 in a bigger affine space space of dimension one more okay and there is a projection like this and under this projection which forgets the last coordinate this is identified with this okay that is what I have that is what I have explained here that is exactly what I have explained here okay. Now if you believe that then it tells you that even open subsets of basic open subsets should also be thought of as affine varieties because they are isomorphic to affine varieties okay so if you go by this then what I should get on this side is a of d of g and a of d of g is well I have defined it as k x1 etc xn localized at g and well there is no problem with this isomorphism because this isomorphism in principle should also give an isomorphism of rings and that isomorphism is already there from commutative algebra you have an isomorphism of this with k of x1 etc xn y you add the extra variable y and divide by gy-1. These two are of course isomorphic okay that is something that we have seen there okay and by the way I should tell you that this trick of looking at g which is an affine open as a hypersurface in a affine space of one dimension more is called the Rabinovich trick it is a trick of inverting g okay it is just trying to say that see it is a beautiful thing what it tells in commutative algebra is localization by a single element is a quotient see k x1 through xn localized at g means you are inverting a single element g it is a localization okay whereas what you have here is the quotient of polynomial ring in one variable more by a suitable ideal and these two rings are the same so it says that it does not say every localization is a quotient what it says is localization is at a single element is always a quotient okay and that the geometric content of that is that any basic open set is actually an affine variety and that is the reason why people call sets of this form as basic affine open that is the word they use they also add the adjective affine they say basic affine open because it is not just a basic open set it is actually an affine variety in its own right under this identification. Now you know if you want to believe things what you should expect is if I take max spec from here to here I should get back my dg okay so if I take max spec of this localization I must get dg if everything is correct okay if everything fits into the picture properly and that is the case okay so in fact if you apply max spec of max spec to this you do get dg and how is that true that is just because of some basic facts from commutative algebra which you must have come across in the first course in commutative algebra namely that namely the following let R be a so I recall let R be a commutative ring with 1 let S in R be a multiplicative subset let this be a multiplicative subset so that means that S is a subset of R which contains 1 okay and it is closed under multiplication and it does not contain any nilpotent elements okay it does not contain 0 in particular right and then we have the localization S inverse R this is the localization of R at S what it means is just invert S okay and sometimes people also write it as RS inverse okay and the reason why we write it as RS inverse is actually because this is isomorphic to R of you take the polynomial ring in R in as many variables as there are elements of S okay and then you go modulo S times XS minus 1 the ideal generated by all these S times XS minus 1 where S is in S that is what it is I mean the point is that I want to invert how do I invert S how do I invert an element what I do is I add a variable and then I add a variable XS X sub small s corresponding to the element small s and then I kill small s times XS minus 1 because when I kill this what I am trying to do is I am in the quotient ring S times XS will be equal to 1 and when a product of elements is 1 it means that each of these elements is a unit so therefore S has become a unit that means I have inverted S okay this is what is happening and this is a this is again something that you would have in a course in commutative algebra it is very easy to verify okay again the map from here to here comes because of the universal property of localization the map from here to here will come because of universal property of the polynomial ring okay. Now okay now you see the point is what are the ideals in spec inverse R I mean what is the what are the elements of spec of S inverse R what are the ideals in S inverse so the fact is that the every ideal in S inverse R is of the form S inverse of I where I in R is an ideal okay so every ideal in the localization is given by localization of an ideal in the original ring okay. Now what is localization of an ideal it is just you take the ideal and invert elements that you have to invert okay so localization of an ideal is particular case of even more general thing namely it is localization of a module so in fact if M is an R module then you can also make sense of S inverse M S inverse M is simply the localized module S inverse M will become a module over S inverse R and S inverse M will just be a module such that you are allowed to multiply not only by ring elements you are allowed to also divide by elements of the multiplicative set which is equivalent of multiplying by their reciprocals which exist because they are units in the localized free okay. So this is a fact from community algebra that every ideal is of this form and in fact if you want a non-trivial ideal then this I if you recall this ideal I should not meet S okay this ideal I should not meet S and further every prime ideal in S inverse R also will be localization of a prime ideal in R that does not meet S so this characterization of ideals in the localization coming as localization of ideals in the original ring it will not only hold for ideals it will hold for prime ideals it will also hold for radical ideals okay this is a fact but what we are interested is in the fact that it holds for prime ideals okay. So the moral of the story is that if you if you take if we take S to be the subset 1 g g square where g is a non-unit and is not input not not not nilpotent then we usually denote S inverse R as R of R g this is a notation sometimes you also write R of 1 by g R g inverse or R of 1 by g in keeping with this notation R of S inverse okay and what are the ideals in R g they will be the ideals in R localized at g okay and in particular the ideals should not contain g so the prime ideals in R g will be precisely the prime ideals in R which do not contain g okay and now if you apply this to if you apply it to this localization what will tell you will tell you that the max spec of this ring the localization of the polynomial ring in n variables at a single polynomial g the maximal spectrum of that namely the prime I mean the set of maximal ideals there the set of maximal ideals will be precisely those maximal ideals in the original ring it will be in correspondence with the maximal ideals in the original ring which do not contain g okay. So what all this will tell you is that max spec of k x1 through xn localized at g will is can be identified in a bijective correspondence with the set of all m in max spec of the original ring of the polynomial ring such that m does not contain g this is what we will get so let me erase a little bit of this so that I will get some more space to write. So I will have x1 through xn max spec of this with m g0 in m okay so let me draw a line like this right this is what it is so the maximal spectrum of this localization will actually be all those points which correspond to maximal ideals in the original ring of which this is the localization such that m should not contain g okay. Now you see what you must understand is that let me see whether I am saying that correctly so the ideals in the localization will correspond to the localization of ideals in the original ring that do not meet the multiplicative subset so this is correct. So this m will meet this s if and only if some power of g is in m and that is the same as saying g is itself in m because m is a maximal ideals so it is prime so this so the statement is right okay but then what you must understand is what is this set see if m is not if g is not in m it means that the point m is in dg so this is actually dg what is dg dg corresponds to points in the affine space where g does not vanish okay but what does a point in affine space correspond to it corresponds to a maximal ideal okay and the fact that g does not vanish at that point means that g should not belong to that maximal ideal so this set is the same as dg okay this set is exactly the same as dg. So the moral of the story is that if you apply max spec to this you end up getting dg which also tells you therefore that this is the correct coordinate ring if you take afdfg and then apply max spec you get back dg that is the third justification for the definition of afdfg to be the localization at g okay so these are these are three justifications as to why this definition is correct okay but then all this is only to get you feel of how things are going it helps you to understand that you can make sense of the ring of functions see because my aim is to go from the geometry side to the algebra side my aim is always to associate rings of functions and from the commutative algebra side if I want to go to the geometry side I will always look at the maximal spectrum that is how we are doing it. So on the geometric side the open sets are also important so if you give me an open set I need to know what are the ring of functions what is the ring of functions on that open set okay. Now that is the question that to answer that question first of all you break that question down into two pieces first you realize that any open set is a finite union of basic open sets and then the idea is that if you know what are the functions on the basic open sets then using this you can get an idea of what will be the functions on open sets okay so that is why all this is important right. So I hope that convinces you that this definition is valid and that these basic open sets are actually basic affine open sets in fact they are affine varieties they can be identified as affine varieties in an affine space of one dimension higher okay fine so there is one more thing that I need to tell you and this has got to do with a version of compactness the usual compactness which comes for free in the case of Zariski topology and that is the reason why it is called it is not called compactness in Zariski topology it is called quasi compactness that just comes for free and then you can expect that you can expect that as it is the case that the corresponding notion for compactness in algebraic geometry is very different it is called completeness or properness okay so let me explain this. See in the usual topology what is this what is the definition of compactness of a subset you have topological space you have a subset when do you say the subset is compact so the definition goes the most general definition goes by using open covers the definition is that if you have an open cover of that subset then out of that open cover only a finite sub cover will suffice that is if you are given a collection of open subsets whose union contains this given subset which is supposed to be compact then from that collection of open subsets you can just take only finitely many whose union will also contain that subset which is supposed to have the property of compactness. So the condition of for compactness in topology is every open cover has a finite sub cover admits a finite sub cover okay now the beautiful thing in algebraic geometry is that this is there for free okay so what happens in algebraic geometry is that you give me a collection of you give me any open cover okay of a subset then a finite sub cover will always be enough okay and the reason is as follows you give me an open cover the union of all the elements in the open cover will be an open set okay the union of all the elements in that open cover will be an open set but you we have already seen that any such open set is a finite union of basic affine opens okay and therefore what will happen is that this open set is a is a it is expressible also as a finite union of basic affine opens. Now you intersect each of these finitely many affine opens by that open cover okay and use the fact that an open subset of a basic open set is again a basic open set okay and therefore what will happen is that see essentially the fact that any open set can be covered by finitely many basic open sets will tell you that any open cover admits a finite sub cover okay and therefore the moral of the story is that you get any open cover admitting a finite sub cover very trivially in the Zariski topology okay and for that reason this property is not called compactness okay but it is called quasi compactness. So let me write that the so fact the Zariski topology is quasi compact okay that is any open cover admits a finite sub cover whose union is the same as union of the original open cover okay and this is this fact just follows from the fact that any open set is a finite union of basic affine open sets okay. So I will there is also way of looking at this from the commutative algebra point of view which I think you would have come across in an earlier course in commutative algebra but nevertheless I will try to I will try to recollect that okay I will do that in the next lecture. So let us stop here.