 Okay, so what I am going to do today is uhhh uhhh so the first thing I want to do is to give the proof that there are no global regular functions that are non-constant on a projective variety okay. In other words on a projective variety the only global regular functions are constants okay. So uhhh you know uhhh so let me tell you couple of things. First thing is that it is uhhh this is this result should be thought of as uhhh an analog uhhh or you know part of a general philosophy of results which say that on compact objects you do not have any uhhh the only global nice functions on compact objects are constants okay. So, for example you know uhhh one such uhhh example is for example you know if you take uhhh suppose you you take the Riemann sphere okay namely the ordinary real sphere S2 and make it into a Riemann surface which means what you do is uhhh to make it into a Riemann surface you just have to be able to talk about uhhh holomorphic functions at a point or analytic functions at a point and for that you use a stereographic projection to identify the complement of uhhh a point with the plane. And by taking two different points you can cover the uhhh uhhh sphere by two planes okay by the stereographic projection and you can check that you can use this to uhhh define what is called a Riemann surface structure on the sphere and then you can define what is then then then you can look at global holomorphic functions on the Riemann sphere. And then uhhh it will turn out that the only global holomorphic functions on the Riemann sphere are constant. The reason being that the Riemann sphere is compact because it is after all topologically it is compact it is a sphere it is uhhh if for if for example if you want it is both closed and bounded so it is compact. And uhhh uhhh any global holomorphic function on the Riemann sphere if you uhhh restricted to the complement of a point then you are then you will get a holomorphic function on the plane okay. And it is image will be bounded because it is the image is compact okay. So you get a holomorphic function an entire function which is bounded and Liewel's theorem will tell you that it is constant. So what will happen is that your your holomorphic function on the Riemann sphere if you if you throw outside of if you throw out a point uhhh in the complement of a point it will be a constant and therefore by continuity it will also be a constant at that point. So it will globally it reduce to a constant okay. So this is for example uhhh in tune with the general philosophy of results that says that whenever objects are compact then uhhh the only global good functions on such objects are constants okay. You cannot expect non constant good global functions on objects that are compact. And the fact is that in algebraic geometry uhhh the correct analog of compactness is projectivity uhhh in fact more generally the correct uhhh analog is called completeness okay. And I want uhhh uhhh be explaining about that probably in this course I I do not know whether I will do that but you will come to know if you take uhhh second further courses in algebraic geometry. But I want to tell you that you must always think of projective varieties as compact as the correct analog analogs of compact objects okay. And the usual compactness does not make sense I mean that does not give you anything uhhh special in Zariski topology because you know Zariski topology is already compact I mean that is the reason why we use the word quasi compact. In fact compactness in the uhhh compactness in the sense that every open cover has a finite sub cover that compactness in algebraic geometry for the Zariski topology is renamed as quasi compactness okay. And it just comes free of charge I mean it just comes free okay. So there is nothing special if you define compactness to be just uhhh you know uhhh every open cover uhhh admitting a finite sub cover. So you do not you do not get anything uhhh I mean there is nothing to uhhh that is not a condition because it is always true alright. That is the reason why the usual definition of compactness is of no use in algebraic geometry and it therefore it is re it is re called uhhh it is re designated as quasi compactness. And therefore you can ask what is the correct analog of compactness the correct analog is being projective okay at least when you study varieties but uhhh the more perfect answer would be that the correct analog is that you should look at what are called as complete objects okay and projective varieties are examples of complete objects. So this is one fact I am I am going to prove that I mean we are going to show that on a projective variety the only global regular functions are constant then the other thing that we are going to show is uhhh the other the other thing that one has to be worried about is uhhh uhhh the following. See if global regular functions are constant then how do you study the object okay there are no global regular functions of object and how do you study the object. Now uhhh it presents a problem because you see in the case of affine variety is a global regular functions uhhh they give you the affine coordinate ring and you know the affine coordinate ring is an invariant okay it describes the affine variety completely okay. In fact if you give me uhhh the affine from the affine coordinate ring you can get back the affine variety by just taking the maximal spectrum and putting the Zariski topology on it okay. Therefore the for an affine variety the affine coordinate ring is the same as global regular functions and that completely captures the affine variety but it is not true for projective varieties for projective varieties first of all global regular functions are uhhh constants okay. So the there are no non constant global regular functions the second thing is if you take the analog of the affine ring you will get the homogeneous coordinate ring the homogeneous coordinate ring is also uhhh not uhhh will not characterize your projective variety. So in fact the homogeneous coordinate ring will depend on the embedding okay. So if you change the embedding into into a projective space if you change the if you take the same projective variety okay and put it in a different uhhh projective space then the homogeneous coordinate ring will change it will not be this uhhh same up to isomorphism. So you cannot keep track of the projective variety so usually just either using regular functions because there are not any non constant regular functions and you cannot use the homogeneous coordinate ring to track your projective variety okay. Therefore the only way of studying projective varieties is to study embeddings into projective space okay and uhhh this leads to the study of morphisms into projective space the so called uhhh classically it was done by studying what are called as uhhh uhhh linear systems and then uhhh uhhh uhhh uhhh and things like that and uhhh in modern language we study line bundles and uhhh things like that okay. So but these are all things that you will come across probably in a second course in algebraic geometry but the so what I want to say is that the fact that uhhh there are no non non constant regular functions on a projective variety and the fact that the homogeneous coordinate ring of a projective variety does not characterize a projective variety leads you to study embeddings of projective varieties into different projective spaces and the embeddings are uhhh they need to be studied carefully okay. So uhhh anyway with that preamble let me start with uhhh let me start with uhhh the theorem that we want to prove. So here is the theorem if y is a projective variety O of y is equal to k. So uhhh in other words every global regular function on y is a constant okay where when I write k I mean every element of k is a constant and it define it is it is thought of as the constant regular function uhhh constants are of course regular functions okay. So what is the proof? So the proof is uhhh you know uhhh proof is a few ideas from the commutative algebra and module theory but I will explain I explain that. So you know your situation is like this you have you have y uhhh which is sitting as an irreducibly closed subset of some projective space. Of course we are always as usual working over small k which is an algebraically closed field okay and you know what uhhh then you also have. So this diagram if you go to homogeneous coordinate rings it translates to a quotient. So closed subsets always correspond to quotients in algebraic geometry uhhh and of course going to uhhh open subsets uhhh corresponds to going to a union of localizations if you want okay. So uhhh so this is the polynomial ring in uhhh n plus 1 variables you know this is the affine coordinate ring of the affine space uhhh which whose for for which the corresponding punctured affine space uhhh sits above this projective space. This is the quotient of the punctured affine n plus 1 space okay and that affine n plus 1 space has affine coordinate ring equal to this and that affine coordinate ring is defined to be the projective homogeneous coordinate ring of the projective space. And then this closed why being a closed subset of that corresponds to a quotient you go modulo the ideal of y as well I put a subscript h this is the homogeneous ideal of y mainly it is all those homogeneous polynomials which vanish on y okay uhhh and uhhh then you get S y. So S y is just S p and mod uhhh I y and of course the fact with uhhh uhhh the fact with all these things is that everything is graded okay. So there is a gradation here you know uhhh the gradation here just corresponds to uhhh uhhh every polynomial being broken down into its homogeneous parts okay and uhhh uhhh uhhh so uhhh uhhh and therefore you also graded gradation here and the gradation here is just gradation uhhh when you read it mod I y okay. So uhhh well well now what one does is that we have already seen that you know if you take uhhh if you take the uhhh so so you know you have so we have a we have a picture like this uhhh we have the we have o y this is the uhhh ring of regular functions on y global regular functions on y and of course case it is inside this as constant as the constant functions okay because every uhhh scalar is being thought of as the as a regular function uhhh which is equal to that constant scalar right. And then o y goes into k y k y is the is the function field of y okay that is what we discussed in study in the last lecture and what is this function field of y this is actually we calculated it for a projective variety and we showed that this it you take s y okay you take homogeneous localization and 0 okay 0 is a prime ideal okay the 0 ideal is a prime ideal because it is an integral domain uhhh even here you take the 0 ideal here it is a prime ideal because this is an integral domain you have only gone modulo I y and I y mind you is a homogeneous prime ideal okay and the reason why I y is a homogeneous prime ideal is because y is irreducible alright. So since you have gone modulo prime this is still a domain the 0 ideal is uhhh prime ideal so you take the homogeneous localization and 0. So I wanted to uhhh I wanted to distinguish between this and the homogeneous and and the non-homogeneous localization. So this this s y sub 0 this 0 here I do not put a round bracket around that 0 see this 0 is you invert everything outside 0 okay this is actually the quotient field of s y this is the quotient field of s y okay. So this consists of literally taking quotients of 2 polynomials in s y of course by a polynomial in s y I mean every polynomial in s y is some polynomial here read mod I y okay of course here I y same as I y sub h okay I keep putting this h sometimes to just remind you that it is a homogeneous ideal right it is generated by homogeneous elements. So this is the quotient field of s y but this is different see this is this is homogeneous localization. So what you do is you do not here you uhhh invert everything that is not 0 but here you invert only those things that are not 0 and which are homogeneous okay and the reason why you do that is when you do it like that then you know this consists of uhhh elements of the form numerator by denominator the numerator coming from s y and the denominator being uhhh non-zero homogeneous element okay and therefore if the numerator is also homogeneous since the denominator is also homogeneous the difference in the degree of homogeneity will give you a degree. So this will become graded this will be graded and for this graded you take degree 0 part that is precisely the uhhh function field of k y that is what we proved in the last lecture right. So I am I am I am I am just ask that telling you I am just recalling that this is uhhh k y the function field of of y is actually this homogeneous degree 0 part of the homogeneous localization at 0 alright which it is inside this huge field this is a huge field this is the full quotient field of the of s y okay and uhhh uhhh of course you must remember that this is so you know you see s y is sitting inside uhhh uhhh of course this is the after all this is the quotient field of this integral domain this sitting inside that and you know what I am going to I am going to do the following thing I am going to start with an f here I am going to start with a global regular function on y. So you know f is f is global regular function f is f is from y to a 1 so morphism a global regular function is uhhh regular uhhh regular function is just a morphism into a 1 okay you have seen that before. So this f is a global regular function it is defined on all of y and it takes values in a 1 it is a morphism into a 1 and I am going to show that f is a constant okay I am going to show f is a constant and how am I going to do that I am going to do the following thing I am going to show that f satisfies uhhh I am going to show f satisfies a monic polynomial with coefficients in K okay I am going to show f satisfies a monic polynomial with coefficients in K okay. That means if you consider everything uhhh as being uhhh inside this field this is a field extension of k, this is k is a field, this is also a bigger field and this is a field extension of k and this is an element there and beautiful thing is that this is algebraic over k, if I prove that f satisfies a monic polynomial with coefficients in k I am just saying that f is algebraic over k but k is algebraically closed therefore f has to belong to k and that is how I prove f is equal to some constant okay, so this is how I am going to do it and how am I going to get a monic polynomial that f satisfies with coefficients in k what I am going to do is I am going to look at this ring, so I am going to look at sy, I am going to look at polynomials in f okay which is also sitting inside this, I am going to look at this ring this is the polynomial I am writing polynomials in f with sy coefficients okay I am going to study this ring okay. So this is the broad idea of the proof alright, so now let us get to the details, so the first thing I want to say is well you see y is you know y is of course inside pn and you know pn is the union of all these ui's i equal to 0 to n these where each ui you know inside pn this corresponds to the coordinate xi not vanishing okay, so this is the affine piece this is isomorphic to an, so in fact you know that there is this isomorphism phi i of this with an you have this and then I have y intersection ui inside this, so well this is open this is closed this is closed this diagram commutes well this is irreducible well this is also irreducible okay of course and here also this is also irreducible that is also irreducible okay, so this is your diagram and the whole point is that you know we know what we know what see you know you see this isomorphism is how we showed that ui is actually affine we proved that ui is actually isomorphic to affine space therefore ui is affine and this is an irreducible closed subset of ui therefore this is also an affine variety okay and therefore you we have that O of y intersection ui is just the same as A of y intersection ui because y intersection ui is affine and what is this this guy this is just we know what it is it is just sy value you localize at xi and then take the degree 0 part this is what it is okay, so this is affine coordinate ring of yi okay, so you take sy you localize at xi okay when you because your localize at xi means you invert powers of xi okay and xi is of course homogeneous of degree 1 therefore when you invert powers of xi there is a natural there is a natural gradation on this then I am saying take degree 0 part which means that you are just looking at some homogeneous polynomial in sy okay mod xi to the power of degree of that polynomial that is all that is what you are looking at that is what this thing is okay, so now what you should realize is that you see thus f if you see if you take f and restrict it to y intersection ui this will belong to O of y intersection ui because you know you take a regular function and restrict it to an open set you will get a regular function alright but O of y intersection ui is this, so this implies is of the form well it is an element here, so it will be it will look like gi divided by xi to the power of ni where ni is equal to degree gi, gi belonging to sy sub ni the this is the homogeneous part of degree ni in sy okay mind you see I wanted to let me again recall see this is a homogeneous this is a graded ring, so this is a direct sum of j greater than or equal to 0 stnkj where sj consists of homogeneous polynomials of degree j and s0 is going to be k homogeneous polynomial of degree 0 alright and this is just the standard fact that any polynomial can be broken down into its homogeneous components and this homogeneous components are unique and each homogeneous component has a homogeneous degree okay. So you know the polynomial has first degree 0 homogeneous component which is the constant term then it is a degree 1 homogeneous component which is the linear term then you have degree 2 homogeneous component which is the quadratic term and then the cubic term and so on and that is this decomposition alright and this decomposition also gives you a decomposition here okay you are also going to get a decomposition of this ring okay the only thing is that you read you read everything modulo this ideal it is a homogeneous ideal that is the whole point. So in particular you know if you take a polynomial here if you take a polynomial in Iy okay that will go to 0 here, so if you take a polynomial in Iy which is homogenous of some positive degree it will suppose it has degree j it will be in sj but if you go to the quotient it will become degree 0 because you have read it you have to read whatever you get you have to read it mod Iy okay. So you have this induced gradation here alright and so I think by notation I should use not s so I should use s sub ni okay so every element here every element here is going to look see without this 0 it is going to look like some element of sy modulo some power of xi and of course this gi is actually gi is actually being read mod Iy mind you okay I am if you want actually I should put gi bar but I would not do it okay gi is just the image of gi here and I have to read it mod Iy alright so it is going to look like this and the point is since I am taking degree 0 part the gi is homogenous of degree ni and the denominator is also may written so that you know the power of xi becomes ni so that the denominator is also homogenous of degree ni so the induced degree is going to be 0 that is how it is adjusted alright. Now this is how f restricted u y intersection ui looks like alright in fact you know if you I am just trying to say think of gi as a think of gi as a degree ni homogenous polynomial okay then if you take this quotient I have a degree ni homogenous polynomial divided by another degree ni homogenous polynomial it is a quotient of two homogenous polynomials so this is certainly a regular function and this regular function will live on ui in fact okay this is a regular function that will live on ui because I have I should not say the whole projective space it will live on ui alright so I want to make the following statement you see this o y so f is in o y but you know so let me try do something here so you know this is contained in o of y intersection ui okay because you know the this is something that we have already seen if you have regular functions on an open set then you can restrict them to a further smaller open set okay and the restriction map from a regular function on a larger open set to a regular function on a smaller open set is an injective map because the injectivity is because if two regular functions coincide on some open set they coincide everywhere okay therefore this is contained inside this and you know I started with an f here okay and what is the image of this f this the image of f under this inclusion is f is precisely f restricted to y intersection ui okay so I need some more space here so let me write it correctly okay so f goes to f restricted to y intersection ui but the point I want I am going to do something now what I am going to say is I am going to say I am going to identify these as these two together I am going to identify these two thinking that everything is be happening here so this is my big field where everything is happening all things are happening here okay so everything is happening in this big field which is a quotient field of sy okay so there is no difference between this and this after all because this is this element f of f restricted to y intersection ui is just coming from this f okay which is a subset of this and when I consider everything here mind you this is my universe this is the big field where everything lives so you know I am going to identify this with this it is correct alright now what I want you to understand is you know if f restricted to this is this then xi power ni multiplied by f will can be identified with gi which is here okay so I am going to write this xi power ni okay times f belongs to s ni y okay this is true for every i alright and in fact you know what I actually what I should write is if you want xi to the ni f restricted to y intersection ui okay that is what I should write but I am identifying f with f restricted to y intersection ui because everything is sitting inside this huge field and this element goes to that okay alright now so you know but what you should understand is that you know you see if I instead of ui if I take uj okay then I will get to y intersection uj alright and well the f will also go to f restricted to y intersection uj but then the fact is both of them this f restricted to y intersection ui and f restricted to y intersection uj will define they will correspond to the same element here that is something that you should not forget okay so even as the i changes these things change but they all come from the same f so they all all as i changes the various f restricted to y intersection ui are all one in the same element here I am calling them just as f okay so this is a small thing that you have to set the analytic thing that subtlety you have to notice okay now okay so I have this now I am going to play with this and I am going to say the following thing I am going to say that you see you know basically the idea is f is a regular function see a regular function on a projective variety or a quasi projective variety is just a quotient of two homogenous polynomials of the same degree so essentially it is degree 0 you see you must understand that it is degree 0 if you think of it as locally as a quotient of two homogenous polynomials of the same degree which is what it is it is a degree 0 object and this is so you know this is correct to expect you take a degree 0 thing you multiply it with degree ni the resulting thing is of degree ni it lands in the piece with which is of homogenous elements of degree ni so the statement is correct alright I mean it is believable now what I want to do is put n is equal to sigma ni take the sum of all those okay nice okay consider take any m greater than n greater than or equal to n okay take any monomial in x0 dot dot dot x0 and so on up to xn of degree of degree m okay so this monomial will look like x0 power m1 into x0 power m2 and so on x0 power mn with sum of all the mis equal to m this is how a monomial in all the xis of degree m will look like alright oops something wrong okay I am oops this should be 1 should be n maybe this should be 0 my numbering is bad alright so this is how a monomial of degree m looks like and the point is you multiply a monomial of degree m with m greater than n with f and you will again land inside degree m piece because of this observation now you see x0 to the power of m0 xn to the power of mn times f if I calculate okay see this is you must understand that there since m is greater than or equal to n which is sigma ni okay see there exists j such that you know mj is greater than or equal to the corresponding nj this has to happen because if every mj small mj is less than strictly less than capital nj then the sum of all the small mj which is m has to be strictly less than the sum of all the ns nj which is n whereas I assumed m is greater than or equal to n so this is an obvious thing that has to happen so then you know I can write this x0 power m0 blah blah blah then you know when xj comes I will put mj minus nj then I will write xj plus 1 blah blah blah power mj plus 1 and go on up to xn mn I will take out this xj power nj times f and I know this xj power nj times f is in s sub nj y this is in s sub nj okay so this is a degree nj homogeneous object okay it is a degree nj homogeneous polynomial and what is left out is a homogeneous polynomial of degree m minus nj so the moral of the story is that this whole thing is going to lie in s m y okay this is what happens because this fellow lies in this belongs to s nj y that is because of this observation okay so this part is homogeneous of degree nj the remaining part is a monomial of which is homogeneous of degree m minus nj okay therefore when you multiply it you will get the total degree will be m minus nj plus nj so you will land in m okay now this happens for every monomial of degree m okay but all these monomials of degree m what do they span they span precisely s m okay after all s m is the space of all homogeneous polynomials of degree m of course red mod I y red mod I y because you are in s y you are not in s of p n okay you are in s of I y therefore the moral of the story is that since such monomials span s m y we have s m y dot f goes into s m y okay you have this you take an element of s m y multiplied by f you end up inside s m y okay because any element of s m y is just a linear combination of k linear combination of such one finite linear combination of such monomials and each monomial is going to push f into s m y alright so you are going to get this and now from this what you can get is that s m y of f squared will go into s m y of f dot f which will go into s m y dot f which will go into s m y and if you continue by induction you will get s m y will take f power r into s m y okay so multiplication on the right by powers of f plus non-negative powers of f is going to push s m y into itself okay so you have this so you know so in particular you know in particular what I want you to notice is that you know if you take x not power so I want to so here is a here is the observation that is very important for us the observation is that s y f okay I want to say s y f is contained in x not to the power of minus n times s y so here is the important observation I mean it is a it is a that is a result of this actually okay see and why is all this happening this is all mind you all this is happening in the quotient field of s y the big huge field where everything is contained it is happening there see x not to the minus n makes sense there in the quotient field of s y okay so everything is working there everything is living there right why is this true that is this is because you see you take you take any you take any homogeneous piece here I mean you take any element here and multiply by f and if you further multiply by x not power n the result is going to go into s y okay see any element here is going to see any element here is going to look like sigma hi f power i i equal to 0 to some l this is how something is going to look like and you see this this each hi is in s y okay but now if I take if I multiply outside by x not power n okay if I multiply outside by x not power n then this thing if this x not power n into hi will push each homogeneous component of hi into a degree greater than or equal to n okay see each hi is in s y alright each hi is s y so each hi breaks down into homogeneous components and it components will it could have components from degree 0 onwards up to some value finite value but multiplying by x not power n hikes all these degrees homogeneous degrees of these homogeneous components to make even the minimum to be greater than or equal to n okay therefore when I multiply x not this goes into s y because of this observation whenever you take any homogeneous degree greater than or equal to small n degree greater than or equal to small n I mean degree greater than or equal to capital n polynomial and you multiply by any power of f you again get degree greater than or equal to I mean you again get a homogeneous polynomial mod i well. So this goes into this therefore this is in x not power minus n that means it implies that any polynomial in the f f with coefficients in s y this belongs to x not to the minus n s y so that is how you get this in here okay now the nice thing is so the moral of the story is you see this fellow here this fellow here is actually contained so what I have got is that this fellow is actually contained in x not to the minus n s y which is of course also contained here okay. So this see this object this polynomial ring in f f with coefficients in s y this is caught inside this and we are more or less done you know why the fact is because you see this fellow here this is as a now think of everything as a s y module this is a finite this is a s y module generated by the single element x not to the minus n x not to the minus n s y is the s y module is the s y sub module of the quotient field of s y generated by x not to the minus n so it is a module is generated by single element so it is a finitely generated module okay and s y is what s y is noetherian ring this is a finitely generated module over a noetherian ring and this is a sub module of that therefore this also finitely generated okay. So you see s y since it is generated by one element it is finitely generated over s y module and since s y is noetherian so this is where we are using the fact the polynomial ring is noetherian see the polynomial ring is noetherian and s y is just a quotient of a noetherian ring a quotient of a noetherian ring is also noetherian therefore s y is a noetherian ring and you have a finitely generated any finitely generated module over a noetherian ring is also noetherian and every sub module of a finitely generated module over a noetherian ring is also noetherian is also finitely generated okay. So uhhh the final condition is that sy of f is finitely generated as a module over sy this is what I want okay. Now you know now we uhhh we go into a little bit of combative algebra. You see the fact that sy f is finitely generated as a modulo sy is equivalent to saying that f is integral over sy okay that is that f satisfies a monic polynomial with coefficients in sy okay by by the combative algebra of integral extensions sy f sy f is an integral extension of sy f is actually integral over sy okay and f is integral over sy. So this is some combative algebra which you can uhhh I mean you can easily look up in a book in in a book on combative algebra for example standard references Atia and McDonald's introduction to combative algebra you look at the chapter on uhhh integral extensions and you will find and it is very easy to prove it is a it is it is just an argument that involves some determinants okay. So by the combative algebra of integral extensions f is integral over sy that is uhhh f satisfies a monic polynomial with sy coefficients. So you know you will get something like this you will get f power t plus At minus 1 f to the t minus 1 plus blah blah blah blah plus t not equal to sy okay where where the a where the ai are actually in sy and this is all happening in the quotient field of sy mind you this is all happening in this quotient field of sy is a huge uhhh extension field of k it is a huge extension field of k. In fact you know uhhh this itself is a huge extension of k because you know actually uhhh this is the quotient field of y it is transcendence degree over k is actually equal to the dimension of the variety y. So if y see y is inside tn okay so y can have at most dimension and if it has dimension it is the it is all of tn otherwise it will have lesser dimension but suppose y has dimension r then this quotient the transcend it means that the this this field extension the transcend the the quotient field of for the function field of y it is uhhh transcendence degree over k is equal to dimension of y. So this is a huge field extension okay this contains lot of transcendent elements. So this itself is a huge field extension it is a transcendental extension and this is a much more huge or one okay. So everything is happening that huge field okay uhhh now you see now each of this a sub i's they are in sy so they have homogenous components and what I want to say is that you can you can write out you can replace each of these with the degree 0 homogenous component okay and uhhh so what you do is now multiply this whole thing throughout by x0 to the n right. So you will get x0 to the n f plus x0 to the I mean a n minus 1 a t uhhh f to the t e plus a t minus 1 x0 to the power of n f t minus 1 plus dot dot dot uhhh x0 to the power of n times uhhh uhhh a0 is 0. Now this is this certainly makes sense inside uhhh so this makes sense inside sy okay that is because you know x0 multiplying f by uhhh any uhhh homogenous polynomial of degree uhhh greater than or equal to capital N pushes it into sy that is the that is the whole point here. You multiply any power of f by any homogenous polynomial or uhhh even any polynomial with every with the condition that every homogenous component has degree greater than capital N then the result will land inside sy. So this happens in sy okay now once this happens in sy sy has a gradation if something is 0 then every graded piece is 0. So if you take the degree 0 part so you know you take the degree x0 power n part of this which is the lowest degree. If you take the degree x0 power n part then what I will get is I will get this I will get x0 power n f t plus a t minus 1 and you know I am I am going to put 0 here to tell you that a t minus 1 0 is the degree 0 part of a t minus 1. See a each a t minus 1 is in sy and since sy has a gradation a t minus 1 breaks down into various parts of homogenous parts of various homogenous degrees and I am taking the degree 0 part then I will get x0 to the n f to the t minus 1 plus and so on and then finally the last time I will get is x0 to the power of n a0 upper 0 this a0 upper 0 is the degree 0 part of a0 I will get this is also equal to 0 ok and where and so and this will happen in sy this will happen in the degree 0 piece in the degree n piece ok this is because mind you whenever you have a graded ring an element in that ring is 0 if and only if every homogenous piece is 0. So this element on the left side is an element of sy which is a graded ring the fact that it is equal to 0 means that each of its homogenous pieces of degree 0 and what is the minimum degree homogenous piece the minimum degree homogenous piece is capital N ok. So I am taking the degree I should not say degree x0 to the power of n I should say degree n I am taking the degree n part ok the lowest degree part. So I end up with this alright now what I will do is I will cancel this x0 to the n and when I cancel into x0 to the n I have to go out of this but I will still lie I will still be in the quotient field of sy. So this implies that you know ft plus a0 t minus 1 f to the t minus 1 plus blah blah blah blah plus a0 0 is equal to 0 in the quotient field of sy. So this makes sense because I am just multiplying by I am simply multiplying by x0 power minus n which which throws me out of sy but certainly keeps me inside this quotient field of sy because x0 to the minus n lives there alright. But after all what are these guys what are what are all these at minus 1 0 what are all those they are all scalars they are all constants they are degree 0 polynomials see they are degree 0 a j0 belong to s0 y this is a degree 0 polynomials red mod i y and this is just k this is just k and so this implies that f is algebraic over k but k is algebraically closed so f belongs to k so this implies o of y it goes into k which means that o of y is equal to k and that is the end of the book ok. So finally I ended up showing that every fellow here is actually here I end up showing that every regular function small f in o of y is actually here so that means that this is actually equal to this ok. So the only thing that you will have to look up in commutative algebra is that is this integrality which is I which I think is very easy to understand it is a small exercise you can look it up very clearly in Atea McDonald book introduction to commutative algebra ok but anyway the moral of the story is that there are no non constant regular functions on a projective variety ok so I will stop here.