 Okay so let me continue with what we were doing in the previous lecture, so let me quickly recall we start with the commutative ring R with unit element 1 and we define the prime spectrum which corresponds to the prime ideals of R being thought of as points of the prime spectrum spec R and then we define the Zariski topology on the prime spectrum which comes out of ideals of R, given an ideal of R we define a closed subset defined by that ideal namely it is all those it consists of all those points in the prime spectrum which correspond to prime ideals which contain the given ideal, okay and sets like this form closed sets for a topology called the Zariski topology, we satisfy the conditions the axioms of closed sets, okay and this is called the Zariski topology on the prime spectrum of R and the formation of the spectrum is actually a functor it is a what is called a contra variant functor, okay. So it is a contra variant functor from the category of rings commutative rings with one the objects are commutative rings with one and the morphisms are ring home morphisms which take one to one and it is a contra variant functor from this category to the category of topological spaces namely it I did not it associates to every R as spec R which is a topological space and to every arrow every ring home morphism it associates an arrow in the continuous map in the reverse direction in for the corresponding topological space is given by this prime spectrum, okay this reversal of the arrow when you apply the spec is what is meant by its contra variants, okay and it is a functor alright of course if you do not know the definition of functor you can always look it up the idea of a functor is something like a function it is a generalization of a function the idea is it has to go from what is called one category to another category and the category is supposed to consist of objects and morphisms maps between objects and of course these maps have to preserve the objects have some structure the maps have to preserve the structure. So for example we are all familiar with so many categories for example category of sets is a category for which the objects are sets and the maps are just maps of sets the morphisms of the category are just maps of sets we are also familiar with the category of groups for which the objects are groups and the maps are the group home morphisms similarly we have the category of rings where the objects are rings and the morphisms of the category are the arrows in the category are ring home morphisms, okay and for example you can also talk about the category of topological spaces where the objects are topological spaces and the morphisms are continuous maps between topological spaces similarly you also can talk about you know category of modules over a ring and so on and so forth, okay. So the fact is that a functor from one category to another is that something that does the following thing given an object in the source category it gives you an object in the target category given a morphism in the source category between two objects it gives you a morphism between the corresponding objects in the target category and the only thing is that the arrow in the target category may be in the reverse direction if it is in the reverse direction it is called a contra variant functor if it is in the same direction it is called a covariant functor. Of course another example of you know a category is the category of top I mean of vector spaces over a field the objects are all vector spaces over a field and the maps are the linear maps that is also category. So speck is contra variant functor from the category of rings commutative rings with one the objects are commutative rings with one and the morphisms are ring home morphisms which carry one to one and from this category speck is contra variant functor into the category of topological spaces namely the category for which the objects are topological spaces and the morphisms or the maps are the continuous maps. So what it does is that to every ring R commutative ring R with one it associates speck R which is a topological space given the Zariski topology and to every arrow in the category of rings namely to every ring home morphism which takes one to one it associates continuous map in the reverse direction of the corresponding topological spaces. So it is a contra variant functor okay and the point that I want to recall is that the if you take the close subset Z of I that itself is a speckterum and it is actually identify with the speckterum of the quotient of R by I okay and if you look at the open subset that corresponds to the compliment of the ideal generated by a single element that also is a speckterum that can be identified with the speckterum of the ring localized at that element okay. So roughly the point is that the close subsets correspond to quotients okay and open subsets given by compliment of a single function or a single element the 0 locus of a single element they correspond to localizations by that element okay. So the fact is that just like you know in the usual algebraic geometry when you first started out on this side we put you know we put affine space and then we put close subsets of affine space and then we had a arrow reversing equivalence that went from close subsets here to radical ideals on this side on this side we put radical ideals inside the polynomial ring in as many variables as the dimension of affine space we started with and there is a similar thing that is going on here. So what you can do is actually put on this side you can just put close subsets of speck R on this side on that side you can put ideals so if you want I can put ideals of R. So again you have a there is a Z map like this which takes any ideal to Z of I which is a close subset in speck R and then there is a map like this which is script I which takes it to any subset T it takes it to I of T okay and what is I of T? I of T is so T is a subset of speck R so it is a certain collection of points which correspond to certain prime ideals and I of T will simply be the intersection of all those prime ideals. So I of T will be defined to be equal to the intersection of all those P such that the point corresponding to P is in T okay and again you will see that here close subsets if you take the set of close subsets that corresponds to that is a bijective correspondence with radical ideals on this side that is ideals which are equal to their radical and this will also be arrow reversing sorry this will be inclusion reversing namely larger the ideal smaller the close subset larger the close subset smaller the ideal in particular the ideal that corresponds to the unit ideal that will give rise to the null set the ideal which corresponds to 0 will give you the whole space okay and you have statements there are statements here which correspond to the kind of statements that we got in the case of affine space okay. So here we got it between close subsets of an and ideals of the polynomial ring in n variables over k we have similar statements here so here also you have you know if you have statements like z of I1 is equal to z of I2 if and only if well rad I1 is equal to rad I2 okay you have statement like this then you have also the statement that z of I of T is just T bar the closure of the set T and you also have the null shell and that is in its very ring theoretic form which is I of z of I script I of z of capital I is rad I this is the mind you this statement analogous the analogous statement in the case for the affine varieties was a very deep statement it was a null shell as such but it is true in this sense for any commutative ring okay it is already there it is God given. So you have this very beautiful thing happening you start with the you start with the commutative ring R with 1 then there is automatically this beautiful picture which on one side translates from ideals to close subsets okay and in fact as I told you it is more this is at the level of R but if you want to go outside of R and you want to consider other rings gotten from R like for example you want to consider quotient rings of R and you want to consider localizations of R then you will end up with going to the close the quotients of R will correspond to the again close sets okay and the localizations of R by single elements will correspond to the close the complements the open subsets given by complements of those close subsets corresponding to that single element okay the ideals generated by a single element. So you get very beautiful you know on this is you may call this as a geometric side you can call this as a commutative algebraic side but the beautiful thing here is that the geometric side has been cooked up from the algebraic side the space spec R itself has been cooked up from R I mean there is no there is no in analogy here when you compare it with here we started with polynomial ring in n variables and that was being thought of as acting on as functions on the affine n space okay we already had a space here but so you had a space namely kn and you had the polynomials on that space and using these polynomials you define the Zariski topology here and you got all these nice correspondences and equivalences equivalences but in this case what you are doing is you are starting with the ring okay you are cooking up a space using that ring namely the spectrum and then you still get this beautiful correspondence all these properties which are just very very analogous to what you got in this case okay. So this is at the first point you should therefore see feel that you know you should be able to catch the you should be able to catch the space from the ring of functions okay so I will tell you what the arrow what is this max spec okay it is very simple so what you do is that you give me anything on this side namely you give me a finitely generated K algebra which is an integral to me okay for example say this one something like this then of course this is also commutative ring with one alright take each spectrum okay take spec of this the spec of this will contain all prime ideals okay but what you do is you take a smaller subset namely you take the max spec and what is max spec it is not prime ideals but it is maximal ideals so you know every maximal ideal is prime and the converse is not true so you restrict to only the subset of maximal ideals okay that will be a subset of the prime spectrum the maximal spectrum is denoted max spec it is a subset of the prime spectrum which is given by spec and since a subset of a topological space automatically gets induced topology what will happen is that max spec will also get a topology it will get a Zariski topology induced from the topology on spec okay the beautiful thing is if you take max spec of this along with that topology it will be exactly homeomorphic to Z of B spec okay so in particular what I am saying is if you take max spec of this you will simply get back a fine space up to homeomorphism see we have already seen this in the null-cell sets I have already told you that max spec of this is actually kn because every point in kn corresponds to a maximal ideal in the polynomial ring that is exactly what the null-cell set says so what the null-cell set says is that max spec of this as a set is exactly a fine space the point with coordinates lambda i corresponds to the maximal ideal given by xi-lambda i which generate as xi-lambda i right that is what the null-cell sets says but that is just identification of max spec of the polynomial ring in n variables with an with kn in fact it is just a set theoretic identification but what the claim now is is that if you take max spec of this with the Zariski topology induced from spec then this identification of max spec of kx1 etc with kn will actually be an topological isomorphism a homeomorphism from affine n space to max spec of kx1 etc so it is not so the null-cell sets is not just a set theoretic bijection it is a topological isomorphism so homeomorphism okay so in other words even if I do not have this side I can get back my affine space by simply taking max spec of polynomial ring in n variables so that is a beautiful thing so this whole side I can reconstruct from here by simply applying this max spec and then it is a matter of exercise to check that this followed by this is identity and this followed by this is identity so that these two are actually inverse associations okay and of course when you check that you will have to worry about isomorphisms okay but that can be done okay so in fact to check that these two are inverse of each other in a rough way you can do that checking now but then I need to tell you what are the morphisms on this side so I need to go on and explain morphisms of varieties okay so let me write that d for a finitely generated k algebra k ky1 by p which is an integral domain we have here homeomorphism max spec k ky1 by n mod p to z of p which is considered as a closed subset of an am so maybe I let me let me use m here you have to put m here am okay so the point is a homeomorphism okay and how does one see it let me try to explain that so this is exactly how you go from here to here okay so you know if I start with z of p inside am where p is a prime ideal in polynomial ring over k with m variables which is the affine coordinate ring of functions on am the largest space okay then what is the ring of functions on zp it is the quotient of the ring of functions on am mod p namely it is this so that is what you get when you go from here to here and then when you come back what you will get is max spec of k of y1 etc ym by p and what that statement says is that that is exactly z of p how is that true that is very very simple see what happens is that so what is the proof for this I will just outline that so if you want theorem it is a theorem in its entirety but it is actually corollary because all these are all more and more grandiose versions of the Nulstrelensatz okay I mean which you get by using after putting all these Zariski topology and so on and so forth and using enough commutative algebra okay so these are all various grander and grander versions of the Nulstrelensatz so let me explain this so you know this is Nulstrelensatz from am to max spec k x1 sorry y1 etc ym there is a map like this which is given by you know give me a point lambda 1 etc lambda m that corresponds to the ideal y1 minus lambda 1 and so on ym minus lambda m this is the ideal this is a maximal ideal in k x1 etc k y1 etc ym and then I take the point I put this square bracket to say that I am considering it as a point of the maximal spectrum okay so this is this bijection is simply because of Nulstrelensatz which tells you that you know every ideal of this form is maximal that is true for any field and conversely if your field is algebraically closed every maximal ideal is of this form that is the Nulstrelensatz okay. Now what you do is you go to you go to the subset which corresponds to Z of p okay just think what are going to be the points here which are going to lie here mind you the point which coordinates lambda 1 through lambda m lies in Z of p if and only if every function in p vanishes at each of those points that is the definition you see what is this this is Z of y1 minus lambda 1 dot dot dot ym minus lambda m this is what it is this is just the 0 set of this maximal ideal okay so and you know Z of this is contained in this belongs to Z of p I mean Z of this is a subset of Z of p if and only if p is inside this maximal ideal okay in other words yeah if and only if this maximal ideal contains p that is because you can apply the script i if you apply script i okay to Z of y1 minus lambda 1 etc ym minus lambda 1 contained in Z of p if you apply script i to both sides you will get I of Z of that which is just radical of that which is same as that maximal ideal contains I of Z of p which will be radical of p which is p because radical of p is p itself because p is prime this is just I am using the Nulstrelensatz now. So what this will tell you that is that this point belongs to this if and only if this is if and only if by the Nulstrelensatz the maximal ideal y1 minus lambda 1 etc ym minus lambda m contains p okay and what does this mean this means that this is in Z of p as far as the Zariski topology on the spectrum is concerned so it is an element of max spec k of y1 etc ym mod p see what is max a maximal ideal which contains p the set of maximal ideals which contain p are precisely the set of maximal ideals in the quotient by p the set of maximal ideals of a quotient of a ring by an ideal is precisely the set of ideals of the ring which contain that ideal okay okay there is a correspondence between a ring the ideals in the quotient of a ring by an ideal and the original ring and what is the correspondence an ideal in the quotient corresponds to an ideal which contains the kernel and under this correspondence prime ideals go to prime ideals maximal ideals go to maximal ideals it is an inclusion preserving correspondence so the maximal ideals in k y1 etc ym mod p are precisely the maximal ideals in k y1 etc ym which contain p okay so that will tell you that so this will tell you this will imply the proof okay so the so you know if you want to complete this diagram what is happening is that here I have max spec of k y1 etc ym which is identified with a m by the null shell and such and you know from this to this there is a quotient map and there is map in this direction is the max spec of the canonical quotient map see there is a canonical quotient map from k y1 etc ym to its quotient by p if you apply spec to that you will get the arrows will get reversed you will get a map from max spec of the quotient to the max spec of the parent of the quotient and it is this map which is a closed remersion namely it is an identification of this with a closed subset of this what is that closed subset with which this is being identified that closed subset is this that of p that is what it means this diagram commends this is the null shell and such and this is also derived from the null shell and such so it is actually the null shell and such for affine space you restrict it to a closed subset that is exactly the statement of this theorem it is simply null shell and such you know restricted to a closed subset nothing more than that okay. So that tells you how you get this bijection okay now to make to show that is a homeum of some okay you will have to show that this is a continuous map in both directions okay now that is something that you can very easily check okay so for example let us show that this map the identification of due to the null shell and such okay. So the homeomorphism I am talking about is this one okay let us first show that this bijective map which is the one null shell and such is actually homeomorphism let us show that that is also pretty easy to see okay so let me do that first so it is very clear that I have this comparative diagram now I have this comparative diagram alright because of what I explained so you have this bijection you have this bijection this bijection is actually coming from here from this and this bijection is the null shell and such I will just show that this is a homeomorphism okay once I show this is the homeomorphism this is the closed subset this is the closed subset okay and it will follow immediately that this is also a homeomorphism okay. So to show that this is a homeomorphism that the map let me write this map AM 2 max spec K of Y1 extra YM is a homeomorphism let me first show so I will first show that this arrow is a homeomorphism okay then I will show this arrow is a homeomorphism I will deduce that this arrow is a homeomorphism and I am done okay so how do I show something is a homeomorphism I will just have to show that you know it is already a bijective map one way of showing that it is homeomorphism is to show that both the map and its inverse are closed okay. So you know the condition for a map to be continuous is that the inverse image of every open set is open and that is also equally that is also equivalent to requiring that the inverse image of every closed set is closed okay and therefore to check that this is a homeomorphism it is enough to show that this map this arrow is closed and it is also which means it maps a closed set to a closed set and to also show that the reverse arrow is also closed because it is already a bijection okay. So let me give this some name let me use I and let me call the inverse map in this direction as small z which is I inverse okay and the reason is why I want to use I is because if you give me a point lambda 1 through lambda m what I get on this side is actually the ideal of that point which is y1-lambda 1 through y1-lambda m considered as a point of the maximal spectrum which is subset to the prime spectrum okay and the map from this direction is give me a maximal ideal I am just looking at the zero set for that maximal ideal which is a single point. The inverse sets actually says that the zero sets of maximal ideals are exactly the single point sets okay so that is the reason for this notation I and z. So it is enough to show to show that both I and z are closed maps it is enough to show that they are closed maps. So in other words what does that mean it is enough to show that the map is said to be closed if every image of a closed set is closed okay and it is equivalent to even showing that they are open maps but we always work only with closed sets because that is how the Zariski topology is defined it is we specify the Zariski topology the topology is specified by only defining closed sets okay so we always try to do all the checking with respect to closed sets. So you see so let us start with a closed set here and let us see what it is image there is so I start with a closed set f inside am then you see what is f, f is z of some ideal I where you know I inside k y1 etc ym is an ideal this is the definition of a closed set in the Zariski topology. Now what is I of f? I of f is look at this it is a set of all maximal ideals so let me write I of f is just I of x where x belongs to f by definition this is the image of a set under a map and so this will be equal to I of how does the point so you know this x in f will have coordinates so you know it will be I of lambda 1 etc lambda m where lambda 1 through lambda m is a point of f so this will be but I of this is supposed to be the maximal ideal corresponding to this point considered as a point of the maximal spectrum is a subset of the prime spectrum. So what I will get is I will get the maximal ideal at y1 minus lambda 1 and so on ym minus lambda m such that lambda 1 etc lambda m belong to f this is what I will get okay but what is f? What is f? f is by definition f is z of I we will have to use that and we will probably apply the nulseless so you know lambda 1 through lambda m belongs to f if and only if the singleton point lambda 1 through lambda m is a subset of set f that is if and only if you know I can take ideal of a subset so I of lambda 1 lambda m contains I of f okay because you know when you apply script I when you apply script I the inclusion is reversed but this is but what is I of a point it is just y1 minus lambda 1 etc ym minus lambda m and what is I of f it is just I of z of capital I which is rad I okay it is just rad I. So what you going to get is so that so the condition becomes if and only if the y1 the ideal y1 minus lambda 1 etc ym minus lambda m contains rad I okay but you see this is equivalent to saying that this contains I okay because you know because if an ideal contains I if one ideal contains the other then the radical of that ideal will contain the radical of the other okay so this is equivalent to saying that y1 minus lambda 1 etc ym minus lambda m contains I it is okay of course if y1 minus if this ideal contains rad I it also contains I because rad I contains I conversely if this contains I then it will also contain rad I because you know what is rad I it is all those elements some power of which is in I so if some power of an element is in I then that power of an element is in this but this is a maximal ideal so it is prime so that element has to be in this so these two are equivalent because the thing on the left side is a maximal ideal which is actually prime it is a primeness okay so but what does this tell you but this is if and only if the point corresponding to this y1 minus lambda 1 etc ym minus lambda m this the point corresponding to the spectrum is in Z of I because this is how the Z of I is defined in the spectrum in the spectrum of the ring okay so in other words what is this this is just Z of I and so I of F is Z of I which is a close subset so you know I should say it is Z of I intersection all the maximal ideal which is closed in max spectrum okay mind you if I take Z of I in the spectrum it will consist of all those prime ideals which contain I but I want I am looking at only maximal ideals which contain I so I will get Z of I intersection the max spec and what I want you to understand is there is a small there is a small element of confusion this Z of I is the 0 set of I in if you want I will put a subscript here I will put am okay this is 0 set of the ideal I in am alright whereas this 0 set of is the 0 set in spec R so there is a difference between this Zi and that Zi okay so let me do that so this is so I need to write that let me put SP sorry maybe I will write it a little larger so that you know you see the difference so here is this is Z in am K of I okay and this is Z I will put Z sub R of I okay where Z sub R of I this is being considered in spec R where R is of course K Y1 etc Yn okay so and this Z is Z sub this is Z sub am so let me correct that also okay so the 0 sets are being considered in different Zariski topologies one is a Zariski topology and affine space the other is a Zariski topology and this prime spectrum but the beautiful thing is it is the same ideal you start you start with the closed set here that is given by an ideal the same ideal that comes on that side when you take its image that makes that image of that a closed set it is the same ideal okay so that tells you that I is a closed map okay so this is the reason for that last equality so this implies that I is a closed map and in principle one should be able to reverse this whole argument and show that I inverse which is Z is also a closed map so how will you show that that is also pretty easy so what you do is conversely let T be a closed set in this in max spec K Y1 through Ym okay so what does this implies T is actually max spec K Y1 through Ym intersected with a closed subset of the bigger space which is a spectrum of K Y1 through Ym okay and closed subset in that is the 0 set of an ideal so it will be Z sub R of some J where J inside R is an ideal okay and what is this so this so what is Z sub R of J it is all those prime ideals in spec R which contain J if I intersect with the maximal spectrum I will get all those maximal ideals in spec R which contain J so this will be the set of all maximal ideals in this which by the Null seven sets is of the form R of the form X Y1-lambda1 Ym-lambdam these are all those maximal ideals such that the corresponding maximal ideal Y1-lambda1 dot dot dot Ym-lambdam contains J okay and this is what T is you can see very clearly that Z of T is just small Z of T is Z sub AM of J so it is closed so let me write that have very little space so let me write it anyway then small Z of T is just EZ in AM of AM sub K of J and which is closed so what is Z of T is going to be you know all those points of AM which corresponds to maximal ideals which contain J that means it is going to be all the points of Z of J in affine space okay so Z of T will just be Z of AM K of the ideal J so what this tells you is that the inverse map of I named small I which is small Z that is also closed therefore this is the homeomorphism okay so the moral of the story is therefore that this lower arrow which is the Null seven sets which says that every maximal ideal corresponds to a point okay that identification is actually a homeomorphism of topology spaces and now it is a matter of routine checking which I want you to do to show that you know under this homeomorphism okay if you see you already know that this diagram commutes okay you know that this is a subset here this is a subset there and you know that this bijection restricts to this bijection this is a closed set this is a closed set okay a continuous map restricted to a subset will be continuous for that subset with the induced topology so what it will tell you is that this the arrow in this direction will be continuous okay and because this is a closed subset here that is a closed subset there and similarly the reverse arrow will also be continuous so that will also tell you that this is also a homeomorphism so it is not just a bijection so that will prove the theorem that you have a homeomorphism between Z of P and max spec and of course this Z of P is Z in Am okay so moral of the story is that if I start with Z of P here in Am I take its affine coordinate ring I get this quotient ring if I again take its max spec what I get is something that is homeomorphic to Z of P okay I get back Z of P so it tells you that if I go like this and come back I get up get something that is not exactly the same as this but something that is isomorphic isomorphic in the sense on this side at least we have been able to prove that it is a topological isomorphism but in fact we are going to prove we are going to define what is meant by a morphism of varieties and we will show that this map this map and for that matter this map this map you can also think of it as in the most general sense you can even think of it as a morphism of affine varieties isomorphism of affine varieties okay so the moral of the story is if I start with Z of P then I go to the affine coordinate ring then if I come by applying this A and then if I come back by applying max spec what I get is not just something that is homeomorphic Z of P in fact you will get something that is isomorphic to Z of P isomorphic as affine varieties and what this isomorphism means it should be an invertible morphism so I should define what is meant by a morphism of affine varieties and that is something that I will do so I am saying that still the picture is not over that is on this side what we have got in this theorem is just a homeomorphism but it is not just a homeomorphism it is an isomorphism of varieties okay and that is the story if you go from here to there and come back okay and if you go from if you go this way okay you will also have to show that I mean the same kind of argument should convince you that if I start with this okay and if I take max spec of this okay then max spec of this can be identified with this because of literally the same argument and then if you take the affine coordinate ring of that I will get something that isomorphic to this okay only thing is that the yi is instead of calling the variables as y1 through ym I might call them as if you want you know say t1 to tm or some other names I could give but in any case I will get exactly a polynomial ring in m variables divided by a prime ideal in that polynomial ring which will go to this prime ideal under an isomorphism between this polynomial ring in the y's and the other polynomial ring that I started that I am thinking of okay. So up to so the point is if you go start from here go there and come back what you will get is an object up to isomorphism here isomorphism varieties and conversely we start with an object here go there and come back what you will get here is an isomorphism of rings okay so that is what happens okay. So I will stop here and probably in the next lecture we will try to understand what happens for an open set and try to understand how to define morphisms.