 Okay so we were discussing projective spaces okay so if you recall we have we have projective space projective n space over k which is where k is of course an algebraically closed field and this is of course this thought of as a space of lines in n plus 1 dimensional affine space alright and we have given the Sariski topology on this okay and in fact if you remember the so this so there was a very nice picture one giving the geometric side and the other the algebraic side so for the algebraic side so this is the geometric side and this is algebraic side so this is a picture very similar to the case to the affine case so on the algebraic side you take the so called homogeneous coordinate ring of this projective n space which is defined to be the ring of polynomials in n plus 1 variables and it is customary to start the indexing of the variables from 0 okay and of course you have n plus 1 variables and this is the affine coordinate ring of the projective space above of the affine space above okay. So this projective space is after all affine space the n plus 1 dimensional affine space punctured at the origin a modulo d equivalence which identifies all the points passing through on a line passing through the origin okay all the points on a line passing through the origin in n plus 1 dimensional affine space are identified as a single equivalence class okay and therefore in other words an equivalence class is just a line passing through the origin and therefore this is the space of lines in affine n plus 1 space okay and this is the homogeneous coordinate ring which is the affine coordinate ring of the affine space above alright and what you do is that of course the important structure here is that we are interested in the so called graded structure of this ring this ring so let me write it as S this S is a graded ring in the sense that S is the direct sum of its degree D pieces for D greater than or equal to 0 okay where S D corresponds to homogeneous polynomials of degree D S0 is of course is going to be just k the constants which are homogeneous polynomials of degree 0 and it is this so called graded structure which is very very important and in fact what we do is that the Zariski topology is defined like this on the projective space you start with I in S homogeneous ideal a homogeneous ideal namely I the condition for homogeneity is that if you take the ideal I should be the same as it should be the same as direct sum of all its pieces intersected with S D so you take the ideal I and intersect with S D what you will get is all those elements in the ideal which are homogenous of degree D and of course if you take the sum it will of course be a direct sum because it is already a direct sum okay and this will certainly be a subset of this always for any ideal but then the condition that ideal should be homogenous is that this is exactly equal to this okay which is the same as saying that given any polynomial we given any element here each of its homogenous components is also back in this ideal okay that is what it means and you know we saw this as a geometric condition for a polynomial to vanish on a line it is necessary a line passing the origin it is necessary that the polynomial has no degrees has no constant term that is its constant term is 0 and every degree D piece every homogenous piece of the polynomial also vanish on that line okay and that is exactly the this homogeneity condition okay. So once you have a homogenous ideal then you can define the 0 set of this ideal in the projective space which is the set of all points here which are common zeros of all the polynomials here okay and of course it is the homogeneity of the polynomial which allows you to decide for sure that the polynomial vanishes at a point on the projective space it is a because it is a homogeneity of a polynomial it tells you that if it vanishes on a line passing through the origin then it will vanish at every point on that line okay and so we get this and these are the so called closed these are the closed or algebraic subsets the in projective space and this gives the projective space is a risky topology topology the proof that this is a topology is very similar to the affine case okay you can check it and of course you have to remember that the property of an ideal the property of homogeneity of an ideal behaves well under some product intersection and taking radicals okay namely a sum of homogenous ideals is homogenous a product of homogenous ideals is homogenous an intersection of homogenous ideals is also homogenous the radical of homogenous ideal is also homogenous okay these are all simple facts that you can check and moreover you can also check that to check that homogenous ideal is prime you can check the primeness condition only for homogenous products okay. So in general if you want to check up ideal is prime you take a product in the ideal and show that one of the factors of the product is also in the ideal but then you can restrict this checking only to homogenous elements if you want to check homogenous ideal is prime okay. So the fact is that as in the affine case you get a very nice picture you get you have an arrow going in this direction and well there is also an arrow that is going in this direction. So this is the z this is the I and this is give me any subset Y of projective space then I have I of Y this is the set of all so you take the set all those homogenous polynomials which vanish on Y and then you take the ideal generated by that. So this is the ideal so this is well so this is the ideal generated by all homogenous polynomials vanishing on Y okay. So and I also told you if you recall in the last lecture that yet another way of saying that an ideal is homogenous is by saying that it is generated by homogenous elements okay. So since this ideal is generated by homogenous polynomials which are of course homogenous elements it is obvious that this ideal is a homogenous ideal. So you get a kind of you know mappings back and forth in this side on this side you can have you know close subsets or algebraic subsets of projective space and on this side you can have homogenous ideals and you have mappings going in this direction and the reverse direction. But then if you want to make this into bijective correspondence you will have to restrict of course well in fact on this side I can take all subsets okay but here I can take all subsets and here I can take all ideals but of course the point is I cannot take all ideals here I have to take homogenous ideals okay but then this map always gives me something closed here okay because that is how the risk it of policy is defined whereas if you give me any subset this always gives me homogenous ideal okay. So if you want a bijective correspondence what you will have to do is that just like the affine case you will have to restrict here to you know you have to restrict here to radical ideals okay and on this side you will have to restrict to close subsets and then you have bijective correspondence okay and as before this is as in the affine case this is an inclusion reversing bijective correspondence between radical homogenous ideals on this side and close subsets here the only thing you have to remember is that you should take homogenous radical ideals that is the first point the second point is you will have to leave out one particular ideal and that is the so called irrelevant maximal ideal that is actually the maximal ideal that corresponds to the zero in the affine space above of which has been thrown out when we considered the projective space okay. So this is the fact that I told you last time and of course we have nice things like we have the statements like so let me say that just repeat it here you have closed subsets on this side and you have a bijective correspondence on this side you take homogenous radical ideals different from the irrelevant maximal ideal which is usually written as S plus it is written as S plus because it is the sum of all the it is the direct sum of all the SD's for D positive okay if you take S1 plus S1 direct sum S2 direct sum and so on what you will get is exactly the ideal generated by all the variables so it is written as S plus okay and this called irrelevant maximal ideal okay. So you have this bijective correspondence and then of course as you we have these facts like I of z of I is rad I and you have z of I of y is y bar okay these are all facts that we have in the affine case we have the and of course in one direction this is trivial the other direction is the so called projective Null-Stellen-Satz okay and so you have projective version of the Null-Stellen-Satz and you have these two facts and you also have as in the projective as in affine case you also have this fact that if you take prime ideals homogeneous prime ideals that is a subset of this because the prime ideals always radical so if you take the subset of homogeneous prime ideals that will that under this correspondence will go to what are called as projective varieties these will be irreducible algebraic sets in projective space. So on this side I will get projective sub variety here and by projective sub varieties I mean closed or algebraic subsets of projective space which are irreducible. So in other words what I am saying is just like in affine case a subset here is a closed subset here is irreducible if and only if it is idealist prime okay and so these are all things that we have just as in the affine case and now what we do is that so we define so what we do is that we enlarge the notion of variety to include so far our varieties were either affine or quasi-affine so affine meant that you are looking at irreducible closed subset of some affine space and quasi-affine means you are looking at an open subset non-empty open subset of an irreducible closed subset of affine space. So this is X irreducible closed in some affine space and quasi-affine meant something that is an open subset of such an X that is an open subset of an affine so you can think of it as U sitting inside X this is non-empty open subset of X which is an irreducible closed subset of some affine space okay. So this is what is meant by a quasi-affine variety so we have already dealt with these two now we extend the definition to include projective varieties and quasi-projective varieties. So what are so projective varieties are similarly irreducible closed subsets in projective space so it is some X which is irreducible closed inside projective space and of course you can again define quasi-projective varieties and quasi-projective varieties are open subsets of projective varieties. So quasi-projective these are open subsets of projective varieties so they will look like an open subset open non-empty inside X which is irreducible closed inside some projective space. So now variety means any one of the following four possibilities so it is either an irreducible closed subset or an open subset of that in affine space or in projective space okay. Now so you have to be enlarge the notion of what a variety is and then you have to note that talking about irreducibility you have to note that projective space is also noetherian so I just wanted to remind you that projective space is just the quotient of the punctured affine space and of course you know if you take and the punctured affine space is noetherian so the projective space is noetherian. For example how do you verify that the space is noetherian you show that it satisfies the DCC descending chain condition for closed sets so if you give me a descending chain of closed subsets in projective space you simply pull it up by the projection map to the affine the punctured affine space above and then you add the origin okay so that you will get a descending closed sequence of subsets in affine space but then you know that the affine space is noetherian therefore that sequence stabilizes and therefore its image below will also stabilize okay. Of course you will have to remove when you take the image below you will have to remove the origin and then take the image under the projection from the punctured affine space to the projective space so it is obvious that the projective space is going to be noetherian okay and then you know the moment you have noetherian topological space then every closed subset has noetherian decomposition namely decomposition into unique decomposition into irreducible affinitely many irreducible closed subsets the decomposition be unique except for permutation of the elements occurring in the decomposition except provided you assume that there is no redundancy in your decomposition namely no irreducible closed subset in the decomposition is a subset of some other irreducible closed set in that decomposition and therefore and such sets irreducible closed subsets though finitely many irreducible closed subsets the union of which in a unique sense is the given closed subset of projective space they are called irreducible components okay. So this is just so you have noetherianness projective space you have irreducible decompression you have the noetherian decomposition for any closed subset okay that is because of noetherian property and then you will also have this fact that topologically you know that any noetherian space is quasi-compact therefore you will get that projective space any projective space of course quasi-compact and in fact direct demonstration of that is that we have seen that a projective space is actually a union of n plus 1 affine spaces okay. So that is already a finite cover by affine spaces. So Pn has a union Pn is the union of finitely many an's n plus 1 an's okay so but in fact any close any subset of Pn being a subset of a noetherian topological space will be noetherian and you know it and since the noetherian topological space is always quasi-compact any subset will be quasi-compact okay. So well now so these are all nice things that are going on here so in particular you must remember that if you take an open subset if you take a quasi-projective variety then that is both irreducible and dense in its closure which will be a projective variety just like if you take a quasi-affine variety it will be both irreducible and dense in its closure which will be an affine variety okay. So this is the situation now what I wanted to do is I have enlarged the objects in the category of varieties like this okay I also want to enlarge sorry I am thinking of the category of varieties which means I am thinking of both I have to think of both objects and morphisms okay the objects of course I have enlarged because I have added projective and quasi-projective varieties but then I have to enlarge the definition of morphism and you know the definition of morphism the affine or quasi-affine case is that it is a continuous map that pulls back regular functions to regular functions therefore if I want to enlarge if I want to define morphisms which involve even projective or quasi-projective varieties I have to tell you what are meant by regular functions for projective or quasi-projective varieties okay and the answer is very very simple just like the affine case where regular function is just a quotient of polynomials locally in the projective case you only require that it is a quotient of homogenous polynomials of the same degree and you put the same homogenous degree so that you get a valid function okay. So let me say that so you know so suppose you are having a subset S in projective space okay and you take f and g in S super h this is the union of all the Sd's t greater than t greater than equal to 0 okay let me alright so let me take t greater than equal to 1 okay let me not take non-zero constants okay so you take 2 so I am taking 2 homogenous polynomials of course you know I cannot evaluate a polynomial on even if it is homogenous I cannot evaluate a polynomial at a point of projective space the homogeneity of the polynomial will only tell me that it is what I can uniquely always say is whether the polynomial vanish at that point of projective space or not but I cannot give it but if it does not vanish it cannot give you a particular value okay that is because if I plug in a point from projective space here then you know there is a common multiple which is floating around because the points in projective space are common ratios that is why they are called homogenous coordinates and that whatever constant multiple can always be pulled out of the evaluation and it will come out with a power which is equal to the degree of the homogeneity of the polynomial. So but the point is that if degree of the homogenous degree of f is equal to the homogenous degree of g then you know okay so maybe there is no homing including 0 also because anyway constant functions will make sense so if both of them have the same degree then you know f by g make sense as a function into k in S intersection the complement of z of g okay. So the point is that if I plug in problem is that if I plug in a point of projective space into homogenous polynomial then if I change the representation of the point then a scalar will come out and it will come out with the power which is equal to degree of the polynomial but if I take such quotients then these powers will cancel okay therefore you get a well defined function so all this is telling you is that you know homogenous polynomials are not enough to define functions but quotients of homogenous polynomials of the same degree certainly define functions on appropriate subsets of projective space of course appropriate subsets I should by that I mean the denominator polynomial should not vanish okay for me to be able to evaluate f by g at a point okay. So now this is the prototype of what a regular function is for a subset of projective space so we make this definition that so here is the definition of a function h from x to k where x is a is a variety is a is a quasi projective or projective variety is defined is said to be regular at x in x if it locally looks like a quotient of two homogenous polynomials and the number of variables is equal to one more than the dimension of projective space in which x is okay. So if x belongs to z of g where x is subset of p n and h restricted to u x is equal to f by g restricted to u x where u x is an open neighbourhood of x contained in z oops x should not be in z in fact I do not want I want to divide by g and I want to evaluate it at x so g should not vanish at x so this has to be corrected x should not be in the 0 set of g and this neighbourhood should be contained in the complement of this 0 set of g the complement. So this is the definition of what a function regular function at a point means of course here f and g are in the homogenous coordinate ring of the projective space okay which is well polynomials in the right number of variables so this is going to be k of x0 okay. So the idea is very simple and of course you know it is regular at a point automatically means regular in a neighbourhood of a point okay so because you are requiring this not only at that point you are requiring it in neighbourhood of the point so the definition of regular function as in the affine case already says it is regular at a point if and only if it is already regular in a neighbourhood of that point okay. So now what we do is again define the ring of regular functions on the projective variety or quasi projective variety we define O x as before and it becomes k algebra okay so regular functions O x is of course set of all global regular functions namely functions are regular on the whole of x which are that means functions are regular which are regular at every point okay and if you take the set of all such functions that is k algebra because sum of regular functions is regular product of regular functions is regular and when you multiply a constant function constant with regular function that is again regular because the constant is also sort of as a constant regular function okay. So well so we have this ring of regular functions on your quasi projective or projective variety and the point I want to make is that as before every regular function is always continuous okay regular functions elements of O x are always continuous and that is again something continuous of course for the Zariski topology. So elements of O x are going to give I have still not defined morphisms so let me come to that later regular functions of O x are always continuous okay and the continuity is obvious because of it is obvious if you look at the if you remember the fact that the Zariski topology on the projective space is a quotient topology of the topology above okay. So if you give me a regular function on a subset here okay then if you compose it with the projection okay you will get a regular function on the affine space okay above on a subset of the suitable subset of the affine space above and that is continuous and that will tell you that the inverse image of closed sets are closed because of the definition of the quotient topology and therefore what will happen is that regular functions are it is very trivial to see regular functions are continuous okay. Now that we have defined this what we can do next is now this paves the way to be able to define morphisms so now how do we define morphisms between two varieties is just a this definition is the same as before it is a morphism between two varieties is just a continuous map that pulls back regular functions to regular functions okay. So definition remains the same only thing is now you have your objects are more you are not only considering affine or quasi affine varieties you are also considering projective or quasi projective varieties. So you can think of a morphism from an affine or a quasi affine or a projective or a quasi projective variety into another variety which is again one of the one of these four types okay. So the definition of a morphism is the same as before the definition of a morphism is the same as before and again what will happen is that we again get the following important theorem if x is any variety and y an affine variety then we have a natural bijection from the set of all morphisms of varieties from x to y to to the set of all homomorphisms of k algebras from Ay to O x we saw this theorem where we thought where we were thinking of x only as an affine or quasi affine variety but then the same theorem will the proof will go through now if you go back and look at this proof you will see that the same proof will work even if x is a projective or a quasi projective variety okay. So this theorem still holds the theorem still holds and if you remember I think I call this map as alpha and what was this map well if you give me morphism from x to y then it goes to alpha f which is just the pull back of regular functions it is a map from Ay to O x which will pull back a regular function phi to you give me a regular function on you give me a regular function phi on y then if you compose it with f you will get a regular function f on on x so first apply f then apply phi okay this is just the pull back this is just the pull back of regular functions and you must remember that Ay is the same as O y Ay and O y are the same because y is an affine variety okay. So the affine coordinate ring is the same as the global regular functions okay and so this is the map we defined and then you also have the inverse map which goes in this direction and what is inverse map if you start with phi k algebra homomorphism from Ay to O x then what you do is that you recall that y is an affine variety so y sits inside some a n. So it means the affine coordinate ring of y is just the affine coordinate ring of a n mod low the ideal of y this is how you define the affine coordinate ring of a affine variety and then and this is well this is this is going to be identified with k x1 etc up to x or let me put k y1 or maybe t1, t1 etc up to tn, tn mod low Iy and so you are you have the ti bars here which are regular functions they are just the globe they are just the coordinate functions on the affine space in which y sits okay and you are just a ti bar means just it can also be thought of just as ti restricted to y okay because after all taking this quotient amounts to restricting polynomial functions to close subset y okay. So now each ti bar will go to a certain regular function in x and use this bunch of n regular functions in x to define a morphism from x to affine space and show that that morphism actually that map is actually a morphism which factors through y and for which the alpha is phi. So you know so the diagram is that from y what you do is you get a map into an and this is given by so here is g and g is g of y is just phi of t1 bar of y dot dot dot phi of tn bar of y this is n tuple so and the fact is that this factors through so I rather call this map as g this is g tilde if you want and it factors through x oops my this should have been x so this should have been x so these all should have been small x's so this map is from x to an and it factors through y and through a morphism like this and phi is actually phi so let me write that below alpha the alpha of this g is actually phi okay. So this is the inverse map this is alpha inverse this is how we got this bijective correspondence you can check that the whole proof goes through if you allow x also to be a quasi projective or a projective variety there is no difference okay the proof does not I mean really the proof really did not depend on the fact that x was affine or quasi affine okay. So you can check this theorem so in particular you know if I take y equal to a1 it will tell you that the morphisms from x to a1 are the same as the regular functions on x okay. So just ask me affine case the regular functions are the same as morphisms into a1 there is no difference okay there is really no difference so the same proof works and the point I want to make is here is a very important theorem which is which I would like to say in this connection see we saw that if you put y to be a1 you will get regular functions okay but more importantly we saw that you know if you take any affine variety which is different from a point okay the fact that it is different from a point means that it is ideal is different from a maximal ideal and therefore when you take its affine coordinate ring it will have lots of polynomial functions okay so it is going to be polynomial ring modulo some ideal which is a prime ideal but it is not a maximal ideal okay this is a finitely generated k algebra which is an integral domain and this has lots of polynomial functions so if you give me a affine variety which is different from a point there are lot of global regular functions which are given by lot of polynomials okay whereas this is not the case for a projective variety okay so the theorem is that if x is a projective variety then O of x is isomorphic to k O of x is just k so maybe I will let me put isomorphic okay that by isomorphism I mean so what I mean by this is that every global regular function is the function that corresponds to a constant it is a constant function there are the only global regular functions are constant so you must think of this as an analog of the fact that you know if you have a compact complex manifold then the only global homomorphic functions on that will be constant and that is just because of Lieuville's theorem okay that a bounded entire function is a constant so it is somehow you must think of x as being compact and therefore it does not admit any global functions which are not constant okay but the proof of this will require some more definitions so I will differ that okay but what you must understand is that if your variety is a projective variety then it has no global regular functions which are no non-constant global regular functions of course constant functions are always there but if you want non-constant regular functions there are none okay this makes life a little bad in the following sense because you know you can we have seen that if you have two affine varieties then they are isomorphic if and only if they are affine coordinate rings are isomorphic and you know for an affine variety the affine coordinate ring is the same as the ring of regular functions okay. So an affine variety can be kept track of by looking at its affine coordinate ring and the affine coordinate ring does not change no matter in which projective space you are embedding the affine variety as a closed subset of okay but this is not going to have so let me repeat that if you take an affine variety if you take its affine coordinate ring that is the same as its ring of regular functions that ring is independent of the embedding of this affine variety as a closed irreducible closed subset of some affine space if you change the affine space and you embed the same affine variety into some other affine space as an irreducible closed subset then if you compute the affine coordinate ring there you will still get an isomorphic ring okay. So you can keep track of an affine variety by looking at its ring of functions that is what it says the ring of functions completely controls and keeps track of the affine variety but this is not true for projective variety because for a projective variety you take two different projective varieties they unfortunately the ring of regular functions is just k it is just a constants so there is no way to it becomes hard for you to distinguish between two projective varieties okay then of course so this leads to other problems and in fact this is what leads you to study so in fact this should tell you should expect that if you take a projective variety and try to define the coordinate ring of a projective variety which is you know analogous to what you would do for affine variety namely take the homogeneous coordinate ring of the ambient projective space if you have a projective variety embedded in an ambient projective space what you do is you take the homogeneous coordinate ring of the ambient projective space and go model of the ideal of this projective variety and the result is again a graded ring because you are taking a graded ring okay and you are going model an ideal which is a homogeneous ideal, it is a homogeneous prime ideal so you again get a graded ring which is an integral domain which is a finitely generated K algebra but the problem is that if you change this embedding you take the same projective variety and put it into some other projective space and calculate again look at the coordinate ring, homogeneous coordinate ring it will change, it could change and it will so it is very so the way in which a projective variety is embedded in projective space is it does not have a uniformity about it okay and this tells you that you know it gives you the following fact I mean this gives you the following philosophy which is the basis of all higher study over projective space it is a fact that if you want to study all the functions if you want to study the geometry of a projective space you want to study geometry of a projective variety you must look at its embeddings in various projective spaces that should reveal its geometry okay. The way it is the way it is homogeneous coordinate ring changes as you embedded in various projective spaces okay that should give you some grasp about the geometry of the projective variety so it is but nevertheless this does not mean that there are not there are that you do not have properties of projective variety which are intrinsic to it as a variety okay. So what it tells you is that you can no longer work with global regular functions on it because there are not any non constant global regular functions okay so I will come to the proof of this later but then I want to tell you only one thing if you look up put these two together as a corollary you will get the only morphisms from the projective variety to an affine variety or the constants or the constant maps so this is something that you can see immediately because you know in this bijection suppose X is a projective variety if X is a projective variety then OX will become K okay and therefore I will get morphisms from X to Y in bijection with home morphisms of K algebras from AY to K okay but every K algebra home morphism from AY to K is projective because it is a K algebra home morphism the image has to contain K so every K algebra home morphism from AY to K will be projective which means its kernel will be a maximal ideal and therefore then the set of morphisms from X to Y okay will be the same will be in one to one correspondence with the maximal ideals of AY but the maximal ideals of AY correspond to points of Y and therefore what will happen is that what this will translate to if you look at it will be that the only morphism from a projective variety to an affine variety will be if the constant map that ends whose image is single point and how many points how many such morphisms will you have as many morphisms as there are points in the target variety okay and each point in the target variety which is an affine variety corresponds to a maximal ideal of AY mod which you get a home morphism from AY to K that is what this bijection says so as a corollary what you get is that the only morphisms from projective variety to an affine variety are the constant maps okay there are no non-constant morphisms there are no morphisms except constant maps okay and of course this also should tell you another corollary that you can get is that if a variety is both affine and projective and projective then it is a point okay this is also something that you can easily realize because you know if the variety is projective then its global regular functions are just constants and if it is an affine variety then you know the ring of regular functions is will now be constant just the constants and for what affine varieties will the ring of regular functions be constants only single terms which consist of points. So if you put the condition affine and projective on a variety then you are reducing it to a point okay so these are two easy corollaries of this theorem and these two theorems okay so I will stop here and continue in the next lecture.