 Alright so what we are going to do in this lecture at the next is you know try to look at the Zariski topology but not in terms of closed sets but in terms of open sets. See if you recall that any topology and a topological space is specified by either giving a collection of subsets which is called close I mean called closed sets which satisfy the axioms for closed sets for a topology or equivalently it is given by a collection of subsets called open sets which satisfy the axioms for open sets required of a topology and you know the axioms for closed sets and the axioms of open for open sets are just got they are equivalent to one another by using the the properties of complement taking the complement of a set in a larger set so De Morgan's laws for example so usually when we do classical analysis or topology for example if you study Euclidean space n dimensional real space then the topology is given only by using open sets and the open sets are given by thought to be given by unions of you know open discs or open balls ok which turn out to be open intervals if you are in one dimension ok. Of course if it is two dimension then these are open discs and so on but so the approach is by specifying open sets but in the Zariski topology our approach has been by specifying closed sets ok called the algebraic sets and these closed sets were given by the set of the sets of common zeros of a bunch of polynomials in the right number of variables ok the number of variables should be equal to the number of copies of the field that you are taking ok. Now what I want to do is to now shift the focus and get into study of open sets ok in the Zariski topology so so what we have is so the so open sets in the Zariski topology so this is what we want to get an idea about ok. So you know k of course is an algebraically closed field and if you want as usually you can think of k to be the set of complex numbers field of complex numbers which is algebraically closed and then you are looking at an k this is just kn with the Zariski topology and of course the Zariski topology comes by looking at the polynomial ring in n variables and looking at common zero loci of a bunch of polynomials and calling that common zero locus that particular subset of an as a closed subset and then taking all possible subsets like that we closed sets and that is how you get Zariski topology ok. Now of course we have defined what a variety is a closed sub variety of affine space ok is supposed to be by definition a closed subset which is irreducible right you have done that. So now let I start with an open set here and see how it looks like so let u inside an be an open set ok. So by of course assume that u is non-empty ok and also that u is not the full space which are both open of course ok because the null set is closed the full space which is its complement is open and since the full space is closed the null set which is its complement is open so these two are both open sets they are the trivial open sets ok but you are looking at non-strivial open set proper non-empty proper open subset ok. And of course what you must understand is you must remember that any such open set u is very special in the sense that topologically it is dense ok and it is also irreducible as a topological space. So that is because we have I have told you that the irreducibility of a topological space forces that every non-empty open subset of that topological space continues to be irreducible and also is dense ok and since an itself is reducible ok because it corresponds to the 0 ideal ok which is prime and the 0 ideal in the polynomial ring is prime because the polynomial ring is an integral to line ok. So an corresponds to the whole affine space corresponds to the prime ideal name is 0 ideal therefore it is irreducible we proved that a closed subset in an is irreducible if and only if the ideal that it corresponds to ok name be the ideal of functions that vanish on the closed subset is actually a prime ideal ok that is the translation of irreducibility which is a geometric property into the ring theoretic property of primeness in the polynomial ring ok. So an is irreducible and therefore any non-empty open set is both irreducible and dense ok. So recall that u is irreducible and dense so this is something very special ok does not happen for example in the usual topology ok. For example if you take the topology of the real line given by open sets given by open intervals and or if you take the topology of the plane or 2 copies of r the real plane and give the topology to be given by open sets which are unions of open discs then you can see that a non-empty open set need not be it need not be irreducible it need not be dense ok but this is very special for the Zariski topology. Now the compliment of u which is An-u is a closed set that is by definition and what is a closed set for the Zariski topology it is of the firm Z of i where i is an ideal in the polynomial and mind you this ideal cannot be the unit ideal ok because if this is the unit ideal then the zero set of that will be the null set and then the compliment of u will be the null set and that will mean that u is the full space which is not the case and mind you this ideal cannot be the zero ideal also because if it is a zero ideal then the zero set of the zero ideal is the full space and that means that the compliment of u in the full space is the full space and that will force the set u, open set u to be the null set. So our assumptions tell you that I is a proper ideal which is not 0 okay. I should say I is a proper ideal not equal to 0 okay. Now you see now let us go back and look at it carefully I mean let us go back and recall the fact that this ideal has to be finitely generated. You see we have Hilbert's basis theorem which says that if R is a commutative ring with 1 which is noetherian then any polynomial ring infinitely many variables over R is also noetherian okay which means that and the noetherian property one of the definitions of the noetherian property is that every ideal is finitely generated. Therefore a field K is always noetherian because it contains only 2 ideals namely the full field as unit ideal and the 0 ideal therefore a field is always noetherian and therefore a polynomial ring infinitely many variables over a field is also an noetherian ring okay that is this is because of Hilbert's basis theorem or M E Noether's theorem and therefore this ideal I is finitely generated okay. So let I be generated by f1 etc up to fm okay so let I have finitely many generators f1 through fm okay then what is then what is Z of i then Z of i is going to be just Z of f1 through fm and that is that is which is actually equal to you know intersection i equal to 1 to M Z of f5 okay. So the Z of i is a set of points in the affine space which are common zeros for each of the polynomials f i okay i running through 1 to M and the common zero locus is just gotten by taking the zero locus of each of the sets in intersecting okay. Now what does this tell you about U so what will U be U will be the complement of Z i so it will be the complement of this and by D Morgan's laws this is the union i equal to 1 to M of a and k minus Z of f5 okay. So you get this you get this expression which expresses any non-empty non-trivial open sets set as a union of open sets of this type but the open sets of this type are special in fact they are the building blocks for the Zariski topology for the open sets mind you Z of f5 is a hypersurface okay it is essentially hypersurface because it is defined by a single equation okay it is defined by a single equation of course for example we assume that if you assume that say f i is actually irreducible okay then Z of f i is a hypersurface okay it is defined by a single equation and what this is this is the complement of a hypersurface okay this is the complement of the hypersurface what is this locus this is the locus where the particular function f i does not vanish it is the complement of the locus where f i vanishes Z of f i is a locus of points where f i vanishes and this is the complement of that locus okay and so it is a complement if for example the f i is irreducible then this is actually a complement of a hypersurface the hypersurface defined by f i and these sets are very special they are called they are they will turn out to be the basic open sets okay so a set of the form an-Z of G is called a basic open set open set and is denoted by D of G so this is the notation D of G, D of G is a locus where G does not vanish it is a complement of Z of G which is a locus where G vanishes okay and this is this is such a such sets DG are called basic open sets. Now the what we have just seen above is tells you that any open set any non-trivial open set is a finite union of basic open sets okay so any open set is finite union of such basic open sets okay. Now you see there is a there is if you have gone through a first course in topology there is a statement I mean there is a notion called what is meant by a basic open a collection of basic open sets. A collection of subsets of topological space is called a collection of basic open sets if this is a collection of open sets such that any other open set can be written as union of open sets in this collection so the condition for collection of subsets to be basic open sets is that they should be a collection of open sets and any open set should be written as union of such basic open sets and you can see that in that sense also any open set is writable as a union in fact it is a finite union we get more any open set is in fact not just a union of basic open sets but it is a finite union of basic open sets okay and the other beautiful thing is if you take the you know if you take the intersection of basic open sets okay that will also continue to be if you take finitely many basic open sets and take their intersection that will continue to be a basic open set okay. So note also that d of g1 intersection d of gm or if you want let me not use the same m I can be something else gr is actually d of g1 product gr okay this is quite clear because you see the d of taking the intersection of all the d of gis is trying to look at those points where none of the gis vanish and none of the gis vanish at a point if and only if the product does not vanish at that point okay. So you see these basic open sets have the property that you take finitely many of them and intersect them the resulting subset is again a basic open subset of the same time okay they are closed under finite intersections and their finite unions give you all possible open sets okay. So this justifies the terminology basic open set from the topological point of view okay. So what all this tells you is that the open sets for the Zariski topology are built up by simply taking finite unions of basic open sets where basic open sets are of this form the basic open set is just given by the locus of non vanishing of a single polynomial okay fine. Now that is not the that is just the beginning of the story in fact the whole philosophy in the most sophisticated form of algebraic geometry is that not only do these basic open sets define the most sophisticated possible object called a scheme in algebraic geometry okay but the fact is even the functions are built by looking at the functions on such small pieces. So you see by now I think you should have noticed that our focus has started shifting to looking into rings of functions okay see the last lectures what we did was we assigned to every affine variety it is coordinate ring the affine coordinate ring of functions which is simply the polynomial ring which is a ring of functions on the affine space in which the closed sub variety sits divided by the prime ideal given by the ideal of vanishing of that closed of that irreducible closed subset okay. So this is what we called as the affine coordinate ring of a affine variety okay and I told you that some in I gave you an indication in the last couple of lectures that the proper I mean the association of a variety to its affine coordinate ring is a equivalence of categories it can also be thought of as a bijection okay. So the set of isomorphism classes of varieties is bijective to the set of isomorphism classes of affine coordinate rings okay and the affine coordinate rings are of course given by abstractly they are defined as finitely generated K-algebra which are integral domains okay. So the moral of the story is I told you that you know the whole affine variety is completely controlled by its ring of functions which is in line with the statement of Felix Klein that this geometry of the space is controlled by the functions on the space. So the whole point about algebraic geometry in going from the geometric side to the commutative algebra side is to completely is to associate to the space its ring of functions okay. So the fact is that not only does not only do these basic open sets form the building blocks of any open sets but also that the very functions on your space they also come by gluing together or putting together okay just like you put these sets together to get an open set the fact is to get a function on that open set you will have to put together functions on the sets like this which you put together to get that open set okay. So the fact is this these are not just basic open sets in the topological sense they are basic open sets even in the function theoretic sense even the functions on these will dictate the functions on the union okay so the whole so what I am trying to tell you is that I am just trying to tell you that it is very important to look at if you want to study functions on open sets okay then you should study functions on basic open sets okay that is what I am trying to tell you. And why do you look at functions on open sets because you will have to look at functions if you want to go to the commutative algebra side in all this time we were looking at functions on irreducible closed subsets which are sub varieties and the functions where the ring of functions where they are corresponding affine coordinate rings which are finitely generated k-algebra which were integral domains okay. Now if you try to do that to open sets the first step you have to start with is try to understand what are the functions on a set like this a basic open set like this okay. So to that so to get into that let me so let me make a definition definition so okay so the definition the ring of functions on DG is namely denoted A of DG is defined to be the polynomial ring which is the ring of functions on the ambient affine space on which in which you are considering D of G localized at G. So look at this definition okay the definition says that the coordinate ring the affine coordinate ring the coordinate ring of functions is this polynomial ring localized at G okay and if you remember if you go back and go back to commutative algebra what this means is this is as I said this is this consists of equivalence classes of the form F by G where F is a polynomial in those many variables and maybe I will have a power of G and M is well greater than what equal to 0 and of course this equivalence class is the D I put a square bracket the square bracket denotes equivalence class and you know F by G power M equivalence class is equal to let me say H by G power L if and only if there exists T such that G power T into G power L F minus G power M H is equal to 0 that is G power L F should be equal to G power M H okay and of course so in all these things I must remind you that you when you localize at a single element in commutative algebra it means that you are taking the multiplicative subset to be at the set containing the powers of this element along with the element 1 okay. So and then it is also important to make sure that the element is not null potent okay the multiplicative subset should not contain 0 and if the multiplicative subset is going to contain powers of G and 1 and 1 is being thought of as 0th power of G if you want okay then no power of G should vanish and the fact is no power of G will vanish because the ring here is an integral domain it has no 0 devices in particular it is reduced it has no null potent so the condition that G is not null potent is not necessary it is automatic okay. So and well so leave alone this probably not so nice looking bunch of equations basically what it says is the set of functions on this locus where G does not vanish is just given by taking the usual polynomial functions and multiplying them with powers of G inverted okay. So this F by G power M can be thought of as F into 1 by G power M which is F into G power M inverted okay and that is a very sensible definition because you see on this locus G does not vanish okay therefore reciprocal of G makes sense so 1 by G on this locus if you evaluate 1 by G it is going to give you a non-zero scale up so 1 by G is certainly a valid function where G does not vanish I mean this is very this is a very standard thing that we come across always if you whenever a function is non-zero then the reciprocal of the function is also a valid function with the same properties of the original function. For example if a function is continuous then if wherever it is non-zero the reciprocal of that function makes sense and that is also continuous similarly if a function is holomorphic where the function is non-zero the reciprocal of that function also becomes holomorphic if a function is differentiable and if you look at the points where it is non-zero okay of course you always assume the points where it is non-zero is an open set which will be true because bear the function the points where the function vanishes will always be a closed set because the functions will be continuous basically and if you have a function with a certain property then at the locus open locus where the function does not vanish the reciprocal of the function will also have the same property so you should expect that 1 by g should be good enough only thing is it is not a polynomial but it is a reciprocal of a polynomial and then you see therefore the functions on this basic open set are given by actually rational functions rational function is a quotient of two polynomials. The point is that the denominator polynomial is always some power of g of course it may be a honest polynomial m can be 0 okay or it could have powers of g in the denominator okay now this is the definition now what we need to do is I need to convince you that I need to convince you about two things I need to convince you about that this definition is correct in by looking at it in another way and that involves trying to tell you that this d of g which is an open subset of an okay is actually also an affine variety there is another avatar of this d of g which makes it an affine variety it in fact becomes a close sub variety of an plus 1 okay and this fact is you would have already encountered in commutative algebra when you looked at properties of localization so you know so the fact is a of d of g is itself an affine variety this is geometric fact okay I will explain why that is correct okay and this is a geometric fact the corresponding fact in commutative algebra is that so let me write that the corresponding fact in terms of commutative algebra is k of x1 etc xn localized at g is actually equal to or maybe so let me write it here let me rub that out k x1 to xn localized at g localization is actually equal to k x1 through xn then add one more extra variable y modulo gy-1 this is the so the fact that the the fact that these basic of these basic open sets are actually affine varieties okay affine varieties means they must be close subsets of affine space irreducible close subsets of affine space what you must understand is they are in the affine space in which you have started considering them they are not close they are open mind you d of g is being considered in an and in an d of g is an open set it is a it is a non-trivial open set okay so it cannot it cannot be a close set because you know it is irreducible so if it is close then it has to be the whole space okay so because it is if it is close it is also dense so it has to be equal to if it is close it has to be equal to its closure and its closure has to be the whole space so it will have to be equal to the whole space so if you consider d of g in an it is certainly pukka open set it is not a close set but the fact is that in a higher dimensional affine space namely a n plus 1 it sits as a close subset okay so and this is this this geometric fact in the corresponding commutative algebraic factor related as we will see and the fact is that in that bigger affine space this is the affine coordinate ring of that subset which is the this which is given by the coordinate ring of the full space divided by the ideal the prime ideal that corresponds to that subset that close subset so let me explain this so what you do is so let us try to understand this so here is geometric segment here is the corresponding segment in algebraic terms and of course when I write here maybe I should say isomorphic okay this is an isomorphism as k algebras if you want okay so I have written equal to but actually it is strictly speaking I should say they are isomorphic to each other okay and of course that isomorphism comes because of universal properties the isomorphism in one direction the isomorphism consists of two homomorphisms which turn out to be inverses of each other the homomorphism from this side comes because of the universal property of localization the homomorphism from this side comes because of the universal property of the polynomial ring okay so that is how that isomorphism comes. So let me do the following thing so what you do is so here is so here is let me draw a diagram it is not a very nice diagram but anyway let me draw it so here is my an and in an in the space an which will have several coordinates I mean several dimensions that I am drawing something like a three dimensional thing and then g the locus of g will correspond to hyper surface so this is z of g okay so I am just drawing a diagram that will help you to think but it is not accurately correct and it is a complement of z of g which is d of g it is everything outside the hyper surface okay. Now what you do is you take and you see here my here the ring of functions is a of an k which is k of x1 through xn and g is of course is a non constant polynomial if you want you can take g to be reducible but it does not matter then what you do is you look at an plus 1 so here is so in an plus 1 there is one extra coordinate alright and well I am not going to really draw a picture but I am going to do the following thing I am going to look at the 0 set of gy minus 1 inside an plus 1 okay so what does that mean I am that means I am taking the affine coordinate ring of an plus 1 to be just k x1 through xn so I am taking the same n variables plus I am going to add another variable y so this is now n plus 1 variables okay and the way I have written it since I have put an extra variable this an is actually sitting as an n dimensional plane inside this n plus 1 dimensional space which is so this an is actually sitting inside this an plus 1 okay and it is the locus given by y equal to 0 when I put y equal to 0 then I get the I cut down by one dimension okay and the corresponding sub variety that I get here is actually this an that is what it means to take these variables to be x1 through xn okay and if you look at the polynomial gy minus 1 gy minus 1 is an irreducible polynomial okay this is something that you can check the polynomial gy minus 1 is irreducible alright and therefore the ideal that it generates is a prime ideal alright the ideal that it generates is a prime ideal and therefore the 0 set of that ideal is an irreducible close sub variety alright so this is this certainly is an affine variety this certainly is an affine variety in this affine space of dimension one more okay and what is the what is the affine coordinate ring of this affine variety what is the ring of functions on this affine variety it is by definition equal to the polynomial ring of the ambient variety go divided by the ideal of this variety the ideal of this closed subset which is the prime ideal gy minus 1 okay see gy minus 1 is an irreducible polynomial and an irreducible polynomial in a unique factorization domain an irreducible element in a unique factorization domain always generates a prime ideal so this ideal is a prime ideal okay and now the more beautiful thing is that there is in fact you can define a map from here to here which is a bijective map okay the map is given as follows you give me a set of points lambda 1 through lambda n in you give me a set of points here in this open set so what it means is that a set of points in an is in this open set if and only if g does not vanish at this point that means if you plug in lambda 1 through lambda n for x1 through xn in g you should get a non-zero scalar okay and you know what you are going to send it to it is very simple you are simply going to send it to lambda 1 through lambda n and the last coordinate will be 1 by g of lambda 1 through lambda n send it to this point now this point is a point with n plus 1 coordinates the first n coordinates are just as these n coordinates okay and the last coordinate is such that this point satisfies the equation gy-1 equal to 0 because the last coordinate y is 1 by g where g is applied to the first n coordinates okay so you can check that this is a bijective map okay already you know if you take 2 n tuples like this and they go to the same one there okay then they have to be the same here and every point there corresponds to a point here you can get the point here by simply forgetting the last coordinate you restrict to the first n coordinates so what I am saying is that the map from a n plus 1 to a n is the projection projection on to the first n coordinates okay and if you project this you will get dg okay you will get dg that is what is happening now you can do little bit of calculation and show that this map is actually a homeomorphism okay you can try that out you can show that this map is a homeomorphism the fact is that this map is not just a homeomorphism it is actually an isomorphism of varieties okay but that is the fact that we will have to check later on because I still not define for you what is meant by a morphism of varieties okay because an isomorphism of varieties is an invertible morphism of varieties is a morphism of varieties which has an inverse which is also a morphism of varieties okay but just grant that for the moment the fact is that this is an isomorphism of varieties and the philosophy is once you have an isomorphism of varieties that should give rise to an isomorphism of the corresponding affine coordinate rings okay. So if you believe that it will tell you that the affine coordinate ring of this must be isomorphic to the affine coordinate ring of this but what is the affine coordinate ring of this the affine coordinate ring of this is this okay but what is this isomorphic to this isomorphic to the localization of the polynomial ring in n variables at g this isomorphism comes from commutative algebra ok. So if you believe that it is fair to take the ring of functions on this to be this which is what our definition was ok ok. So what you must understand is that if you believe that these two are isomorphic is varieties ok evidence for which is that you can check as an exercise that these bijective map is actually an isomorphism in the topological sense namely a homeomorphism mind you this dg is a subset of an an iso zariski topology this is an open set so it has an induced topology ok and z of gy-1 is a close subset here that also has an induced topology zariski topology and I am saying this bijective map is not just a bijective map it is a homeomorphism it is continuous in both directions ok and that tells you that these two spaces are homeomorphic but it does not stop there they are actually isomorphic as varieties and if they are isomorphic as varieties they are the philosophy is that they are coordinate rings the rings of functions have to be isomorphic therefore you can the ring of functions on this has to be isomorphic to the ring of functions on this which by definition is this by our earlier definition but that is isomorphic to this because of commutative algebra and therefore it is correct in that sense to take the ring of functions on dg to be this ok. So this is a kind of heuristic argument which involves certain things that need that will be checked later ok so this is some justification as to why you should define it like this of course the other justification is where g does not vanish 1 by g and powers of 1 by g will also make sense as functions and therefore most general functions you can write should be of this form ok. And I should also tell you that this geometric picture is actually a translation of this commutative algebraic isomorphism ok this is the coordinate ring of this, this is the coordinate ring of that and the fact that these 2 rings are isomorphic isomorphic as k algebras is actually reflection of the fact that these 2 fellows are isomorphic as varieties ok. So it is commutative algebra this is the commutative algebra statement these 2 being isomorphic as varieties is the algebraic geometric statement and they are just you know each is equivalent to the other provided you make the right definitions alright. So that is to convince you why this definition is correct ok, so I will continue trying to tell you about the points that lie here ok with respect to the null so and so ok. So that that will also give you an idea that this is that will also give you another justification as to why this is the this should be defined as a ring of functions on dg ok, so I will do that in the next lecture.