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.  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.  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.   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.  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.  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.


Hey, welcome back to the channel. So they say the average millionaire has seven income streams, and I'm not a millionaire yet, but I have been an entrepreneur full time since 2016. I have seven income streams now that do right around 35k per month in profit. And I want to share with you what those income streams are, how you can get started creating these income streams for yourself, and then also how these do require some work, right? So these are if you're looking for fully passive income streams where you're kicking your feet up, you're drinking a Mai Tai, you're not doing any work, this is probably not going to be for you because only one of these income streams are actually fully passive. But what I do want to help you do through this video is maybe give you some inspiration on things that you can work on to start bringing in more income so that you can invest into more fully passive income streams. So with that said, let's jump into the first one right now. So income stream number one are services. And all of my services that I offer right now are mainly around email and helping people and businesses sell more with email. So my two core services, I have one that is more of a product type service, a product type service means it's it's in between a product and a service and that you're delivering the same service over and over and over as you would with a product to every client. So I have a service that includes cold email and LinkedIn and we do there is we help companies mainly in the tech space, we help them book more phone calls with the types of clients that they're looking for. And on those phone calls, they usually close a deal from there. So that is the product type service that I offer. And I also have freelancing services where I'm mostly doing the work here. So I go to companies that have lists of customers and buyers that they're not fully leveraging and they're not fully emailing. So I go to these companies that have lists of customers and help them make more sales out of their customer list. So I send emails and I write emails for their list and help them get more of their sales. And then I take a cut off of that. So that is income stream. Number one is services. And it's mainly around email marketing and helping companies make more money with email income stream. Number two is coaching and consulting. So I do a lot of coaching one on one and in groups. And I help people learn more about email marketing grow their business with email marketing and also digital and info products. So there is different options that I have. I have one where if you want to book one on one time with me, you can book a 60 minute session on zoom where I just answer any of your questions. It's $300 an hour. And I get a handful of those every month anywhere from four to eight of those one on one calls every month. And then I also have a group coaching programs where we're in small groups and we're working towards a goal together. And I'm helping people and coaching them along through the process. So one that I have right now is a new program that opened up called the email side hustle program. And the goal of that coaching program is to help people get one K to five K per month clients by writing simple emails for them. So they go out to businesses, they learn how to get these businesses to pay them money to write them emails to make more sales. Pretty much how I've done it with with growth response, which is my my service business. And I coach people and I teach people how to do this in small group settings. So I love coaching, especially in the small group coaching environment, because I'm able to work closely with people with clients and students, I'm able to help them get a better result in their life and in their business. And I'm able to impact more people because one on one is great, but I can only impact one person through that 60 minute session. When I'm doing group calls and group coaching, I'm able to talk to 20, 30, 40 people at one time. And then they go out and get results all at once. So I love this group coaching setting. I love working both one on one and in group settings. But the group settings the upside is I'm able to work with more people at once and help more people and impact more people all at once. So that's Income Stream number two. Income Stream number three is digital products. So I sell things like ebooks and mini courses and template packs that help people with different things to do with their business. And again, this is mainly around email marketing and copywriting and creating digital products. Most of the things that I roll out are around those things. So I have these lower ticket digital products. They're usually between $5 for an ebook all the way up to $100 or $200 for like a mini course or a template pack. And again, these help people get very specific results depending on what they're looking for. So I have something called the super fast sales email challenge. And that helps people for $29 right there first email that sells. So it's developing the skill of learning how to write a sales email for $29. And that's an example of a digital product that I sell. That's probably my most popular digital product right now under $100. And someone buys that product. If they're more interested in taking the next step, I have my group coaching. So you can see that digital products are a great standalone income stream, but it also feeds into my other offers and it feeds into my consulting and coaching and other things that I have going on, maybe even my services there. So that's income stream number three, digital products. Income stream number four is an online course. So I'm making this its own income stream because some of the online courses I have are standalone, but a lot of them are grouped in with coaching. So in online course, I use a platform called Kajabi where I upload different videos, different workbooks and different tools and resources to help people get some kind of results in their life. So the email side hustle coaching program that I talked about, there is a video course component to that over five weeks that helps people take the steps needed to get clients and grow to 1k to 5k per month plus in their own email side hustle business. So the online course, there's videos. It's great because anybody can view those videos at any time. We don't have to be in a live group coaching call or a zoom call for them to go through it. So they can log into this area. I use Kajabi. There's other platforms like Teachable or Pandaya. There's tons of different course platforms out there, but I have videos in there and training and tools and resources that helps people move along and get a result in their life depending on what the course is about. So that's income stream number four is an online course. I love these because I'm able to impact a lot of people at once and I don't have to be present and live for them to go through the online course. Income stream number five is affiliate marketing. So affiliate marketing, a lot of people look at it like the first business model to go into. I don't fully agree with that. I only started doing affiliate marketing pretty recently, like in the last year, and affiliate marketing is mainly when you have an audience built up, people who want to learn what you're doing and who want to follow after you, and you're basically recommending products and you're getting a cut of whatever sales come in through your recommendation. So let's say I'm selling, I'm using a product called Active Campaign to send my emails and I recommend that to my audience. I say, hey, you guys should check out this tool. It's awesome for sending emails. I get maybe 30, 40% of whoever signs up using my link. Now the reason I say it's not a good beginner business model is because I needed an audience before I had people wanting to sign up through my link, right? They respected my opinion. They wanted to learn what I was doing and because of that, they were willing to sign up for whatever tool I was recommending. If you don't have an audience, if you don't have people who are listening to you through your content or wanting to learn more about what you're doing, then it's, but you're better off focusing on something like services or coaching or consulting first and building up an audience before you start diving into affiliate marketing. At least that's my opinion. So now what I do is I just take all the tools that I'm using to run my business and I just recommend it to people. So now it's like a semi passive income stream because all I'm doing is just recommending things that I'm already using. And that also brings up an important point. If you get into affiliate marketing, never recommend something that you're not already using. I see a lot of people teaching this like just find products that are selling well and recommend them. I don't agree with that. And it's another reason why I think that you need to be running, you need to have an audience at some point because you need to be using these tools whether it's for your business or for a hobby, whatever it is, using these things before you recommend them. That's my opinion there. So that is income stream number five is affiliate marketing. Income stream number six is YouTube ad revenue. So everyone always talks about YouTube ad revenue, right? There's tons of videos of this is how much I made in 2020 with YouTube. I make anywhere from $600 to $800 per month from YouTube AdSense ad revenue. And all that is is whenever you watch one of my videos, you might see like an ad before the video or in the middle of the video or near the end. And I'm getting paid for how many views of those ads there are on my videos. So I'm in a pretty good niche right now. I'm in like the business and money and finance niche. So my CPMs are pretty high. I think I'll make a separate video on this. I think they're around $52 or so. And a CPM is a cost per 1000 impressions. So it's basically how much you get paid for 1000 views of your video. So mine is pretty high. It depends what niche you're in. I'm in the personal finance and the money and the entrepreneurship niche, which pays pretty high. And I just get paid AdSense revenue for however many views of an AdR on my video. And finally, income stream number seven. This is probably the one you've been waiting for, the one that is fully passive income at this point. That is dividend income. So what are dividends? It's basically you invest in a stock or a mutual fund or an ETF or an index fund. And depending on what you're investing in, if you pick the ones that pay dividends, the companies that pay this, then what happens is every quarter or every year, depending on the company, they pay out a percentage of their profits to their shareholders as a dividend. And this is fully automatic. It's fully an autopilot. You don't really have to do anything other than pick the stocks and funds to invest in. So I get paid for my business every month. It goes from my business to my personal account. And then automatically without me having to lift a finger, it takes a percentage of that paycheck, invest it into certain stocks or funds that I have picked. And I like to pick things that have dividend because it's income. If you invest in a company like Tesla, yeah, the company stock can grow a lot, but you're not getting really paid for it over time. So I have dividend stocks that my grandparents have invested in for me since I was a kid that have grown to a decent amount now off of just a very small amount over time because of the dividends. So I like to pick dividend paying stocks and funds because I get paid income. And my goal is by the age of 35, I'm 30 right now, I'd like to have $5 million invested into dividend paying stocks and funds and just live off of that passively where I would never have to work or worry about money again. If I'm getting paid anywhere from 2% to 4% in dividends off of $5 million, that's a pretty solid life that covers all the essentials there. So I like dividend paying stocks and funds because it's fully passive, but I want you to remember you can't just jump into you're not going to get rich off of dividend stocks and index funds and ETFs if you don't already have active income that's fueling your dividend stocks and your investments. So I need all these other six income streams that are making me money in order to invest into dividend stocks that makes me more money and multiplies what I already have. So really important to remember, right? A lot of people want to start with passive income, but passive income really only exists once you have truly passive income at least, only exists once you have enough money to make it worth it, right? Because the percentages 2% to 5% that you're making passively, you need a lot of money to make that work for you where you can just live off of that with a Mai Tai on the beach, okay? So those are my seven income streams right now. I'm not a millionaire, but it does make around 35 to 40k per month profits, sometimes higher, sometimes lower from those seven income streams. And I wanted to share that with you so you can get some inspiration on the different income streams that you can work on. And I share how to get started with all these income streams on the videos on my channel. So whether you want to start in services or digital products or online courses or coaching, I want to be talking about all those different ways to create internet income on this channel through my videos, through my past videos and the future ones coming up. So hopefully you guys liked that. Give me a like if you did comment below if you're already working on these income streams or which ones that are interesting to you or you want to learn more about. If you have any questions, let me know. Happy to answer them. And yeah, that's it for this video. Hope you guys enjoyed it and I'll see you in the next one. Take care.


Good day, ladies and gentlemen. I'm going to give you a briefing on the development of Chinese military bio weapons. Good day, ladies and gentlemen. I'm going to give you a briefing on the development of Chinese military bio weapons. As I mentioned my opening soliloquy, this will be a reflection of the Chinese Communist Party, as well as the Chinese military. If you have any questions, please ask those. You may not like the answer, but I'll give it to you anyway. Well, this is being recorded and presented under traffic light protocol red. And of course, Chatham House rules. If you have any concerns or questions about this, please stop watching. If you're unfamiliar with Chatham House rules, please Google that before you continue. So today we're going to talk about China, specifically the military use consideration planning and development of bio weapons as a prescription for enduring conflict capacity and defeating adversaries on the battlefield. Now as we go to China, you'll note that the middle of this map represents their view of the outside world. It's a person dominant and less than equal to the Chinese people. This is just culturally the way it is if you're offended by that, you know, too bad. What the Chinese are calling themselves don't go, which means literally middle kingdom, John is middle. Go means the king or Queens riches surrounded by the four borders of their kingdom. So today's dream and the manifestation of their culture is one of dominating others and unfortunately you are if you're not watching this in our Chinese you're already perhaps curious about where this is going. If you're a foreigner like me you are subservient to the Chinese domination and that's just the way it is. So this is a third part in the series that we'll talk about how unrestricted weapons of war will shape future conflicts specifically bio weapons as developed by the Chinese military. Now the abstract of course is just to let you know that this is third in a three part series examining the Chinese military use of violet logical reagents in a kinetic capacity that's a mouthful isn't it. There's one in two presented at the biohacking village I suggest that you go back and look at those on YouTube. Year one in 2019 essentially predicted what was going to happen with coven 19 year to last year, essentially demonstrated that indeed the Wuhan virus or virus called coven here in the western world we will not say what it is. Some failed leaders have talked about what the cultural attribution of this is really in year two I've demonstrated that it did in fact come out of Wuhan. I'll go back and look at that if you'd like, but today we're going to talk continue to discuss the unrestricted warfare strategy that goes back to the early 1990s. The Chinese military strategy and initiative, including things such as the supply chain program 863 and other supporting components that discuss the Chinese military bio weapons strategy. All of those are going to be revealed during this discussion remember. This is traffic light protocol read. On the agenda, we're going to take a look at three basic elements, everything from no hyperbole bioweapons programs, the Chinese Chinese Communist Party or CCP science and technology programs, and then an examination of future state and planning using the dime mnemonic, meaning diplomatic Intel information and economic will be the focus for this. Now, as we go into this you'll note that the biological weapons will have very much a military flavor. Last year it was research from a military civilian perspective. Year before that it was definitely a civilian with a taste for an order of military response to coven. Now, we're going to of course talk about nuclear biological and chemical if you've served in the US military military of any other country. When you heard the words NBC you remember your guests chamber training, many of us remember that with such rapid abandon it's yesterday. Well, we're going to talk about that from a non kinetic perspective, as we progress through this now the old version of China is important to remember that historically they have indeed had foreign troops station on their soil, and the response to that occupation in the 1900s impacts, not only the information warfare component which I'm known for primarily but also in this age, the bio weapons capability. I don't know why that's important because the Chinese fear above all, the fact that a foreign power will occupy them again. And as you know with some of my previous discussions the Communist Party the current regime. Historically knows that if the population is dissatisfied with the discourse that it is giving from a leadership and political party perspective, they will rise up and essentially be requesting that the CCP step down communist party will never allow that to happen as long as it does. Now when we talk about China you think of the dragon, you think of the red dragon that's my hacker alias, but more importantly what I would like you to do is know that China is a major superpower.  My foreign research, as you know from other presentations that I've given has been redacted removed and cannot no longer be accessed. In this particular presentation I have started to make a decision to not include those URLs if you want those URLs and you're in a position that will help you research. I asked that you reach out to me I'll be happy to provide that. This presentation will be seen by the Communist Party. It will be seen by my friends at the Chinese military think tank. It is not meant disrespectfully or any sort of dishonor towards the culture history of China that is not the point it is really to educate from a military perspective what is going on in China. Now, that military revolution that you probably heard about the revolution of military affairs coined by Thomas. Thomas down at 411 worth really also speaks to the some of them breakthroughs of biotechnology and how future war will be able to use this as a combat force multiply. And it's interesting because in the Chinese military press, they are really researching and trying to understand this especially in a post COVID world. And I'll tell you about that in terms of one of the conclusions is why it's important to listen to the end. It comes to Chinese military history and certainly the history from a cultural perspective of the way that the Chinese view bio weapons control it's important to revisit the fact that they are not part of the OPCW and hey, while they adhere to it they do not belong to it and the Chinese military as you can see in this URL, which has been redacted I believe firmly believe that while this is necessary for other countries it does not apply to the Chinese military. And as such they will prepare vigorously in order to prevent this from happening now why you may say are they so aggressive. Well it's important to understand that during World War two the Chinese were invaded by the Japanese here is yet another historical of what the Chinese fear, which is foreign domination, and perhaps colonization by a foreign force. Thus there is no love lost between the Chinese and the Japanese and in fact if you go to Beijing. As I have done recently you will note that there is the Japanese war of aggression museum outside of Beijing it's a fantastic historical view of what the Japanese military did to the Chinese civilians and military in the form of biological weapons testing some very nasty stuff was essentially leveled against men women and children. The Japanese were very brutal and the Chinese remember this as a result they have developed their own weapons grade chemical weapons and toxin agents and I'll give you this as a review in terms of what the Chinese military is doing in the present day in response to what the Japanese did to them historically. Now you may have seen this during last year but I want you to remember that weaponization of biological agents. Really is something that they take very serious and they look at it from a intelligence perspective. There's no secret here believe me they want the US main protagonist to know what they're doing in terms of developing chemical weapons because they want it to be a fair and equal ground when it comes to war. They will basically state from a military perspective we told you so you need to be prepared for this. Now what's interesting is I will reveal here shortly. They do not only always test these on military troops in order to prepare for that next level of warfare unrestricted warfare using bio weapons, but they tested on civilian populations within the people's population of China. You may have heard of rest of Xinjiang province in the Muslim population there they are not without their mistreatment by the Chinese military and government this is not a political statement just as a fact and we need to come to grips with that. Some of those bio weapon bio weapons development terms as you can see here include a variety now I put this out there for the scholars that would like to research this. If you're unfamiliar with the terms in Chinese then certainly this gives you an opportunity to grab a screenshot again TLP read and research of these on your own. From a nuclear biological chemical weapon perspective we go into the military revolution. Understanding that bio technology is really the mainstay of a future battlefield if they need to go in the kinetic space in the way that they are researching this is to integrate not only academic institutions military scientific programs within China to develop some military oriented initiatives or programs which include active, aggressive collaboration and procurement of dual purpose commercial civilian technologies that can be applied in a civilian military purpose. And you may say bill yeah we've all heard about IP theft what's new. Well I was on a briefing yesterday with the FBI. They asked them about 863 plan and they said they would have to get back to me they had never heard about it. Well, let me tell you, with all due respect to the FBI. The Chinese 863 plan should have been top of the list of the discussion from that that agent yesterday. And he tells us about how that dual use technology goes back to the third month of 1986 in terms of the active cyber espionage targets and you may be saying yourself bill you've given us the URL. It's all in Chinese how are we to translate this well I'll tell you, you can easily do the Google translation as long as these URLs are up. And then these active targetings of all of the areas of critical infrastructure science technology and research are indeed biotechnology, and you may have said well, yeah this is nothing new and I would tell you that if you're in a counter intelligence role in the federal government of any country and you don't know what China's 863 plan is, you should look in the mirror and fire yourself because this is really the roadmap in terms of the companies within China that you should be targeting from an active intelligence and counter intelligence perspective. So part of that of course is the thousand talents plan and the success here's an example of a gentleman works for an interesting Institute of zoology as part of the Chinese Academy of Academic Sciences CAS. What's interesting is he is really working on developing not only biological weapons, but looking at different types of applications in a military perspective. And one of those is really genetic weapons. And what he talks about is that these new generations can be made by modifying the genetic code of design disease causing microorganisms think of Kobe for example, use gene editing technology to find the level of offensive capabilities from a military perspective that will affect the enemy at the genetic level on the battlefield meaning they can essentially defeat any nuclear biological or chemical weapons training and preparation, therefore giving the Chinese military that first mover advantage and competitive military success that they so are desiring to enjoy. And now of course, not to delve back into what happened last year certainly with Kobe but if you're familiar with a biosafety lab for that's the highest level. We have them here in the US, and certainly the one at the one Institute of Rology was a BSL for and you may ask yourself what that's such a tightly controlled environment for bio weapons development and bio research how did the COVID-19 escape that well that's another lecture from last year. What I mean to tell you is that they're talking about from these BSL for these genetic weapons are going to essentially impact positively the development of future warfare planning capabilities and future acquisition of technology to help the Chinese military be successful on the battlefield including a development of a biological atomic bomb. Achieving military goals, much like in the information warfare information space where you don't need to have soldier on soldier or Marine on Marine, and that they do believe that from a strategic deterrent role genetic weapons will serve that purpose. Now what's interesting about this is their national security does face some of those biotech challenges, meaning that they know that there are specific biological weapons and terrorism. That can take place through research methods that may go awry vis-à-vis maybe the one Institute of Rology. But of course our intelligence here in the United States is working on that according to President Biden's edict so we don't need to go into that. Suffice it to say, man made biological disasters will occur, and they can occur and they have occurred due to careless use this is one of those national security challenges that really sets the basis for the development of military grade bio weapons by the military. Now the application of this field of endeavor biotechnology has become more and more extensive meaning that they are integrating not only emerging bio weapons and equipment into the health of sustainable combat initiatives, but also looking at it to shape that future battlefield making that strategic and tactical decision making will focus based on the success of these bio weapons. And of course, from that comes the consistent theme of combat power generation. If you get inside a adversary's observe oriented side and act or OODA loop, then you can essentially control the battlefield produce a strong army using science and technology and defeat them in space. Of course, the US military has had three decades plus of experience in Iraq and Afghanistan, and the Marine Corps my service has never seen the second place. So what that means is the Chinese are really trying to use biological weapons in this new military revolutionary affair, meaning that they are looking for an urgency and exigent development of bioweapons that will last them from a national defense perspective over the next three decades so think about that when you think about future planning. And of course there is a variety of relationships between both Chinese national defense and technical innovation of course this is in Chinese and I'm sure you can all read this but I've done some translation for you and suffice it to say that there is a subjective nature to this which means that they know as a Chinese military and scientific world that there is a significant ecosystem historically where a lot of these intelligence substances arrive and arise and originate from that they can take advantage of, but they do know that this complex ecosystem of virology is incomplete. What that means is that they can use big debt and information processing and artificial intelligence to enhance the biological bioweapons development for the military. What this means then again from a combat generation perspective is that they can get the full measure of combat power projection using biotechnology, but that what that means then is that they need to have security around this program, meaning that it is being somewhat secret, so that other nations adversaries that they may face on the battlefield cannot realize what is taking place. And of course some of those major factors as you can see from this narrative are quite concise and develop. Of course I don't intend for you to translate this other than to say that there are a variety of different factors now it should be no because as we apply the diplomatic information intelligence military and economic mnemonic dime. You can see that there are of course political and economic factors meaning they want to stay in powers the Communist Party. There are social factors meaning that that colonization and domination is the manifestation of the Chinese dream is supplemented by the fact that bioweapons and biotechnology must enhance national security. And then of course there are institutional and governments which means these factors are really tied to the national scientific national security focus of the people's Republic of China. Meaning while there are ethical issues on nonetheless those don't necessarily apply to the adversaries outside of China remember they are the middle kingdom and they are dominant and we are subverted and subversive. Meaning they intend to colonize all of us and bioweapons is a military projection of power is one of those institutional governance factors that they consider important. Now, another series of major factors is to essentially look at what other countries such as the United States are doing. Look back a few years in terms of their development it's all right here in plain Chinese for you to look at, however. I haven't taken the large s to translate this for you and of course there are competitive factors that the Chinese look at from a biological weapons perspective, taking a look at the defense advanced research project agency or DARPA in terms of what they have been doing. The Chinese military wants to dominate this field of endeavor, perhaps even the surpassing those programs of the US military, and what they consider to be DARPA is baby child. There are diplomatic factors they do realize that they cannot just lease or loose these bioweapons on other countries, however that they do realize that these major and significant changes cannot keep pace with the diplomatic world. When diplomacy fails, of course the projection of force and power is always necessary from a military perspective. As part of this indeed will be the fact that the Chinese military will be called to use bioweapons in order to colonize attack and making enemy secede to them. Now, of course there are right to speak factors in terms of what the US intelligence community has talked about in terms of gene editing and the genome editing project, for example, related to weapons of mass destruction. The fact that this is in the open or in clear text for the Chinese military to pick up is interesting because this gives them a signal that indeed they must remain competitive and keep pace with this. And then of course the internal factors within China include those scientific and technological hurdles that could impede or delay the progress of biotechnical innovation. But nonetheless, it is important to understand that this model for the future in terms of bioweapons is a necessary condition for success of the Chinese military. That will be the arm of diplomatic failure in the force of power projection that the Chinese military will undertake perhaps in the future against the US military other adversaries they may see on the air land and see. So please don't relegate this to just a land discussion. But of course this is really interesting because the evolution of this national tech biotechnology under the Chinese Defense Act really comes into the way that there are dual use weapons in the future being developed. And it's important to understand that when it comes to biological warfare. There are major things that are recurring thing remember I mentioned that there are arms controls and the international biological space OPCW that I talked about. The Chinese observe but don't belong to it. They also know that there are emerging technologies that will lead to new biological weapons that they can carry out. And of course there is always the fatal temptation of inception, meaning that one must not be reborn in the image that is one because it may flash back or blow back on oneself. And of course this leads to avoiding the nuclear trap the failure that the Chinese observe of the Russian loss of the Cold War, and the nuclear run up that the US face down Russia with they believe that they as the Chinese military and Chinese people will have no competitors in the bioweapons and defense biotechnology well they will control the space and there's nothing that we can do about it, according to this. Of course, 2035 not too far off in the distant future is going to be a tipping point for national defense biotechnology now I certainly hope that President Biden intelligence community is reading, watching and paying attention to this of course. I've been ridiculed on YouTube I've been ridiculed in space but I will tell you, I'm not going to predict this I'm just reporting on what I see. And I've taken the opportunity to let you know that when 2035 comes around that development from by the Chinese national defense biotechnology initiatives will be the core product outside of information warfare for their perfection. Of course they're going to look at organisms and controlling them micro ecosystems, and essentially looking at ways to operate a biological tech not taxonomy of military biological organisms from which they can pick and really direct at any adversary on the battlefield. They also understand that there are national treasure treasures of military technological innovation, meaning they can prevent early strikes by other adversaries including the United States by essentially creating a very highly maneuverable highly active responsive reactive reagent that can identify defeat delayed grade destroy an adversary's capability to control them. I hope you're paying attention out there US government and US intelligence community. And of course there is the calculation of biological bioinformatics, meaning using bio computing to essentially map these genes to the purpose which is intended for a military dual use capability. And what's interesting is, while there is information collection that may be needed for retention they believe that by examining new different forms of life in this bioeconomic economic system, preventing those from being shared by some of the adversaries or countries that have dominated them, they can maintain that first mover military advantage and destroy an adversary in the battlefield. And of course that gets into some of the more esoteric origin of life origin of consciousness, talking about cognitive memory dissonance which hits a little bit Jean Paul Sartre for me. But nonetheless it's something that they're talking about in terms of synthetic mind control. Certainly we've seen that in the US military, not meant to reflect anything that has or has not happened in the past, realistically. And of course then anticipated 15 years on beyond that in 2050 of course it's right here for you in Chinese, certainly you can read this, however I've taken the, the liberty of taking a look that a new world or, meaning a human machine is inevitable, meaning that man society and integration and the possibility of eliminating biological warfare through that competitive biotechnology and national security matrix is going to be necessary, meaning that a calculation of what will happen in predicting that is a necessary intelligent attribute of this capability that they will develop. The military within China will also combine bionics of weapons and equipment, meaning that these automated systems will essentially locate seek and destroy adversaries that was a problem to them in the form of third generation weapon system development. And then of course, there's environmental technology making sure that while they don't necessarily care about destroying the earth they're going to use it for their advantage now when I say earth I mean air land and sea. And they're going to make sure that if even if there is a incorrect interpretation of the United States Convention hope PCW, or the prohibition of biological weapons doesn't necessarily apply to the Chinese well they are part of the National Security Council of the United Nations. And that they believe that they can be a first mover and write the policy on that as they have tried to do with the internet control with Russia. And then of course cognitive revolution and human birth, meaning that there is a way to use nano technology in order to avoid what had happened during the nuclear trap and dilemmas of the late 60s early 70s into the 90s between Russia and China. And last but not least, here's a little bit of historical trip down some of the military origins of this. Of course they've been they being the People's Liberation Army or the PLA or June way have been really focused on the advances of biotechnology and the way that it can change future conflicts. In 2010, what y'all way worked on a significance of impact of biology on the future plans for Chinese military, meaning that over a decade ago they were already looking at this in terms of future warfare planning. Five years later, Major General for troops. We're going to talk about bio materials in terms of controlling an adversary to make them think that they have been defeated on the battlefield in order to fix them in place and destroy them. This sounds pretty far out but if you think about it. There are agents out there that can certainly render a adversary incapable and incapacity and he was certainly talking about this from an academic academic perspective. Major Retired General Jean Shibu, who talked about seven new domains of warfare, including modern biotechnology development. What's interesting meaning that he believed that the strong offensive capability could be included with specific genetic attacks that should be employed. Now think of the rest of Xinjiang province and there you have a living battlefield environment that from a laboratory perspective that they could use. Finally, signs of military strategy I've talked about this in the information warfare talks came out in 2017 you can actually get it on Amazon and English really talks about that domain of a military bio weapons capability, making note that this is indeed a force multiplier. Again with an emphasis on specific genetic attacks was suffice it to say they being the Chinese have been talking about this for some time. Certainly hopefully our military and our intelligence world here in the West is taking and keeping pace with this. Now, this gentleman I mentioned him last one it really was on the cutting edge for the biological frontier, and he talked about confrontation in new methods. When he met new methods he indeed meant genetic weapons in terms of how this can be done and I talked about this earlier. Vis-a-vis how these new genetic weapons will affect the Chinese use of in control of future war in really chemical weapons being the precursor to this, the bio atomic bomb with a huge lethality imagine a kinetic atomic bomb and what happened to the Chinese use these during World War two. Well the Chinese believe that this can be done from a bio weapons perspective bio atomic bomb, and meaning that really one soldier does not need to be placed enough in the face of war to achieve military use these genetic weapons will play that strategic role to preserve the tactical troops from being essentially pawns any larger adversarial role, they do not want foreign militaries to face them on the battlefield and have that first mover advantage and Chinese believe that indeed they will do that for you. And of course they as you saw last year do trained in the same standards that we do. It's interesting that this Kim biosuit and masks look similar to the US by digress this is not about intellectual property. You should be aware of the China national gene bank as part of their initiatives I mean it's right here in English it's right out on the web. Certainly you can all read Chinese and go to the URL yourself but it's interesting because they have been tasked with developing some of those genetic resources that will safeguard bioinformatics with a focus on national security. And it's really interesting because you've heard about seizing strategic commanding heights and the information or cyber warfare perspective, they believe that in the military domain of biotechnology that this company the national gene bank will do the same for them it's important to pay attention to organizations like this. And then of course, when we think of military use coming out of gas masks we've all done that in biological weapons as distasteful as it is, is something we should pay attention to. Last but not least as I conclude my presentation here are some of the conclusions. The dual Chinese military and civilian recollection of Japanese horrors of weapons of mass destruction during the war of aggression World War two really endures with them. Historically they distaste the Japanese. Economically they'll use them but from a military colonization perspective they hate that. They do study the horrors of this and prepare actively for the terrorist use of a bio weapon is a weapon of mass destruction. They do believe that dual use civilian military by weapons programs is necessary and they are pursuing us. And they also believe that bio weapons accent accents are inevitable in China, the as a view one institute of biology and the Western world as a result of that they need to prepare for. And then of course similar to what the foreign military militaries US and others are doing to prepare. They don't want a war but they know that vigilance is necessary. And really when you look above, they don't want to be subject to any sort of other country coming in and using these by weapons on the civilian or military population. Number six, of course, by weapons development and the procurement of that technology comes under the China's 1000 talents plan if you want more about that. I certainly come and see some of my presentations on that in the cyber warfare information dominance perspective and low. They are using human intelligence and active intelligence collection make no mistake about it. Number seven, of course, from a China manifest dream world domination culturally and economic is going to be their goal and weapons of biological mass destruction will be the military sworn that will take care of those that do not want to play. And of course they're actively preparing for weapons, by weapons deployment in the shadow of the world's failure to convey co contained cove it I apologize as Kobe, it's not a new acronym it should say Kobe. That concludes my presentation 32 minutes. If you have any comments or want to contact me. Please do so through the biohacking village. Thank you very much and have a great day. Thank you.