 Okay, so this is a continuation of the previous lecture, so we have just seen that projective space is covered by n plus 1 copies of open sets which are isomorphic to affine n dimensional space. Now what I wanted to say is that from this the first thing that we should notice is that you know if you take any projective variety, a projective variety is an irreducible close subset of projective space then that projective variety if I intersect with ui I will get a projective variety and take its image I will get a projective variety I will get an affine variety in n okay and therefore from this you will get that any projective variety is a finite union of affine varieties okay, so that is so these are all corollaries of this of this construction. So these are all observations observations which are very important for us number one the projective space itself is a union of all the ui's I equal to 0 to n each ui isomorphic via phi i to an, so you have this to if x is a subset of pn then x is a union of from i equal to 0 to n x intersection ui. So any projective respectively quasi projective variety is covered by finitely many respectively quasi affine varieties okay and the most important thing is that any variety as therefore is a finite union of open subsets each isomorphic to an affine variety okay. So for that if you take pn or if you take a projective variety it is very clear that by the first two remarks that it is certainly a finite union of affine varieties the only case that one has to cover is the case of quasi projective or quasi affine varieties okay. So what I want to say is that if you take first of all I want to say that if you take a quasi affine variety that also can be covered by a finite union of affine varieties okay. So any quasi affine variety is a finite union of affine varieties why is this so because you see you take y inside some am this is irreducible closed which means that y is a close sub variety of affine space so it is a y is an affine variety and then in that you take a u non-empty open then this u will be a quasi affine variety by definition of quasi affine variety is a non-empty open subset of an affine variety and now y is z of p where p is a prime ideal in the p inside the affine coordinate ring of this affine variety am prime ideal so it is 0 set of a prime ideal because it is a variety okay irreducible closed subset so it corresponds to a prime ideal this is 0 set of that prime ideal and this of course any way this ideal is in this which is a polynomial ring in m variables and it is by Hilbert's basis theorem it is noetherian so the prime ideal p is also finitely generated any ideal is finitely generated so you can write if you write p as the ideal generated by g1 through gl by the Hilbert's basis theorem then you know that no this is not what I want sorry it is no what I want so I have so u is an open subset here alright what I want is a following I want to show that this u which is a quasi affine variety is a finite union of affine varieties I want to show that and the way to do that is to realize that the fact that y that u is a so the compliment of u inside y is a closed subset of y alright and it therefore corresponds to a an ideal okay so y-u is z of j bar where j where j in a of an is an ideal containing containing p okay this is what will happen so you see I have I have this quotient map quotient homomorphism from this to the affine coordinating of affine space affine coordinating of the affine variety y and this is just given by mod p okay and what is a what is y what is y-u it is a closed subset of y okay and y is already a closed subset of am so it is a closed subset of am also a closed subset of a closed subset is also a closed subset therefore it is given by an ideal in the in the am in the affine coordinating of the big affine space and in fact if you want to look at it it will be given by 0 set so in fact maybe I should not put j bar I should simply put j okay just put j it will be the 0 set it will be the 0 set of j and if you consider it as a 0 set in the full affine space okay and of course j will be an ideal which will contain p because zj is contained inside zp okay and now now j is generated by finitely many elements so g1 through gl okay and what will happen is and this is of course because any ideal is finitely generated because this is a noetherian ring this is a polynomial ring in m variables that is a noetherian ring by Hilbert's basis theorem or Amino Aether's theorem so this is finitely generated so if you take this as generators then of course if you take the compliment of j in the full affine space that will be a union of the basic open sets given by all these gis okay so you know Amk-zj will be union dgi i equal to 1 to l okay and therefore if you take if you take y-zj so y-zj which is u okay this will be Am-zj intersected with y okay and so it will be union i equal to 1 to l dgi intersection y alright and therefore but the point I want to make is that each dgi inside Am you know is an affine variety itself though it is an open subset of Am it is isomorphic to an irreducible closed subset of affine space of what dimension one more by the Rabinovich trick so this is so you know from Am so you have your Am plus 1k and you take the 0 set of gi into some extra variable t minus 1 okay so tm plus 1 if you want and you know that there is this there is a there is an isomorphism varieties like this okay each dgi basic open set in affine space is itself an affine variety okay which means it is isomorphic to an affine variety what is affine variety it is the affine variety given by g times gi times a new variable an extra variable minus 1 0 set of that in an affine variety of dimension one more we have seen this earlier so now what I wanted to understand is that you know y y is a closed subset of Am therefore y is also closed subset y y intersection dgi is a closed subset of dgi okay y is a closed subset of Am of the affine space so y intersection dgi is a closed subset of dgi for the induced topology therefore it will be and y intersection dgi is also irreducible that is because of the following fact dgi intersection y is a non-empty open subset of y and y is irreducible therefore dgi intersection y is also irreducible so the model of the story is that dgi intersection y is an irreducible closed subset of dgi and via this isomorphism it will give you an irreducible closed subset of this and therefore it will become an affine variety so I am just trying to say why each dgi intersection y is itself isomorphic to an affine variety okay and the finite union of all such is your u so what this argument shows is that any quasi affine variety is a finite union of affine variety and if you combine that if I if you combine that with with 1 and 2 okay you will get that any even a quasi projective variety will also be a finite union of affine varieties so if you put everything together you will get that any variety is a finite union of affine varieties okay any variety has a finite cover by open subsets each of which is isomorphic to an affine variety and thus we say that affine varieties are the building blocks of varieties okay so let me write that down here that is very important so thus dgi intersection y is a closed subset of dgi dgi intersection y is an open subset as long as it is non-empty it is an open subset and as long as it is non-empty is irreducible thus dgi intersection y is itself an affine variety because it is irreducible closed subset of an affine variety or something that is isomorphic to an affine variety thus u is a finite union of open affine finite union of open subsets each isomorphic to an affine variety so this is point number this is observation 3 that any quasi affine variety is a finite union of affine varieties and then 4 by 2 and 3 any quasi projective variety is a finite union of open subsets each isomorphic to an affine variety and the upshot of all this is the theorem that any variety whether it is projective or quasi projective or affine or quasi affine is a finite union of open subsets each isomorphic to an affine variety so sometimes we say this also in the following way we say that any variety admits a finite cover by open affine varieties okay that is one way of saying it the other way of saying it is that affine varieties are the building blocks of all varieties okay so let me write that down we may rephrase above any variety by open affine varieties the other way of saying it is that affine varieties are the building blocks arbitrary varieties so I mean the importance of this is you will see the importance of this in the following sense so the first immediate thing is if you are going to study if you are going to focus attention in a neighborhood of a point of a variety then you could as well look at an affine neighborhood of the point and reduce your question to studying a point on an affine variety that is an advantage okay so let me repeat that if you have a variety and you have want to study in a neighborhood of the point the variety then what you do is you choose a neighborhood which is an open set which itself is an affine variety okay the variety you started with may not be affine it could be projective could be quasi projective okay it could even be quasi affine but then if you want to restrict attention to a point you can find an open neighborhood which is itself isomorphic to an affine variety and then thereafter it reduces to studying a point on an affine variety okay so the advantages that if you are trying to restrict attention to a point or a neighborhood of a point on a variety is just enough to study points on affine varieties okay that is one advantage the other advantage is rather philosophical it is that when you it is it concerns the more general definition of what is called a scheme which you will see in a second course in algebra geometry so the roughly a scheme is something that locally looks that is locally isomorphic to something which is affine okay so the affines they build up what is called a scheme alright so you can so roughly speaking here you know if you take an affine variety you have its coordinate ring okay and then from the coordinate ring how can you get back the affine variety by applying max spec okay so you can say any variety is a is got by topologically at least as topological space it is got by gluing max specs okay alright if you take a variety which is which is covered by finitely many affine varieties okay then it is the same as saying that you are gluing all these affine varieties together to get that big variety okay but then each affine variety is a max spec of its coordinate ring so you can say that this big variety is gotten by gluing the maximums maximal spectra of these various affine coordinate rings okay and more generally the more more general thing is what is called a scheme and that is an object that you get by gluing just not maximal spectra but just prime spectra of rings okay so this is helpful this will be helpful later when you take a second course and as an example but I am saying that the philosophy is already here that affines are the building blocks okay alright so that is a very important thing now you know there is one fact that I need to prove I need to tell you that the only regular functions on a on a projective variety are the constants okay but for that I need I need to understand what is happening at a point okay in a neighborhood of a point and I also need to keep track of information of what is happening on an open set okay so I have to study regular functions in the neighborhood of a point and also regular functions on open sets and the devices that handle these things are respectively called the local ring at a point and the function field or field of meromorphic or rational functions okay which is what I am going to do next okay so so what I am going to do next is focusing attention at a point this focusing attention at a point so well geometrically study functions close to a point and algebraically commutative algebraically this is the same as studying local rings okay so so you know one important function between algebra and analysis is that in analysis you have the limit concept okay so you have a limit and therefore it helps you for example to take small neighborhood about a point and you can take smaller and smaller neighborhoods and using that you can define various things existence of a limit and then you can define continuity differentiability and you can do analysis but in algebra handicap is that you do not have any such notion of limit a priori so it appears that you cannot do for example study smaller and smaller neighborhoods of a point so it appears like that because to study smaller and smaller neighborhoods of a point I need smaller and smaller neighborhoods first but then the neighborhoods if they are open neighborhoods they are Zarisky open neighborhoods and Zarisky open sets are you know irreducible and dense so they are huge sets okay so you cannot think of epsilon neighborhoods as you would think of in usual analysis so it appears that it is not possible to do that kind of analytic study at a point but the answer is wrong that is not true the truth is that you can even though you do not have small enough neighborhoods the information is still available okay and that is that comes by studying so called local rings okay so if you recall a local ring in community algebra is a ring which has a unique maximal ideal okay and so in other words if you take its maximal spectrum it is just a single term okay and every element outside that maximal ideal will be a unit okay and you know conversely any ring with an ideal with the property that every element outside that ideal is a unit has to be a local ring with that ideal as a unique maximal ideal okay so these local rings are what we need to study infinitesimal neighborhoods around a point okay and we do it even though we do not have small neighborhoods Zarisky neighborhoods even though we do not have the notion of limit okay so let me explain so what we will do is so the idea is that in algebraic geometry whenever you want to study something close to a point you have to always think in terms of local rings okay so what I am going to do now is I am going to define the local ring at a point of a variety okay so the local ring OXP is the notation of a point at a point so I maybe I will use okay at a point P in the variety so here of course X is a variety so we are going to define this and how does one define it it is defined in a it is defined in a way in which which extends to defining local ring of functions of any type you can use this to define local rings of continuous functions local rings of differentiable functions local ring of if you want holomorphic functions and so on and so forth so the philosophy is very very general so what you do is that so you so you do the following thing define OXP tilde to be the set of all pairs U, F such that U inside X is open X is a point of U and F is a regular function of U okay so you would you look at pairs like this okay so what I am doing is I am just looking at regular functions defined on open neighborhoods containing oh yeah okay my point was not small x it was capital P so I will change it thank you for pointing that out I have taken my point as capital P right so you are just looking at pairs consisting of regular function defined on an open neighborhood of the point P alright and what we will do with this is that we will introduce an equivalence relation okay so define U, F equivalent to VG if there exist W open in U intersection V such that F recited to W is equal to G recited to W okay so you put this condition you define two such pairs to be equivalent if they define the same function in a smaller neighborhood of P okay of course you know this implies that this is true this implies that F and F recited to U intersection V itself will be equal to G recited to U intersection okay this will happen because you know F and G will now be regular function F recited to U, F recited to U intersection V and G recited to U intersection V will be two regular functions on U intersection V which is our ITU okay and if they are equal on they are equal on if they are equal on nonempty open set then they have to be globally equal I only have to ensure that U intersection V itself is reducible okay and I think that is alright because U intersection V is an open subset of U and U is reducible in therefore U intersection V is also reducible okay so these are two regular functions on U intersection V which coincide on an open subset of U intersection V therefore they coincide everywhere. So well though the equivalence only requires F and G to coincide on a small neighborhood containing of course I want P to be point of W that is very important okay probably I do not even need that I mean if it should not create any problems yeah probably I do not need this I do not need this but anyway I will add it okay yeah so this is an equivalence relation alright and now you define OXP to be so check this is an equivalence relation that is very trivial because obviously it is obviously reflexive symmetric and for transitivity as far as transitivity that is also quite obvious okay. So this is certainly an equivalence relation maybe I have to think for a moment about yeah so the point that when one thinks with respect to the usual topology one can be misled one should remember that in the Zariski topology any two non-empty open sets will intersect that is something that one should not forget. So you cannot have you cannot imagine a picture where you have three successive open disks the first and second intersecting the second and third intersecting but the first and third not intersecting that kind of a picture will never happen any two open sets will intersect any two non-empty open sets will intersect in the Zariski topology okay so that is something that you have to remember. So we take OXP to be the set of equivalence classes which is just OXP tilde modulo is equivalence set of equivalence classes okay and you have this so you know you have this natural quotient map OXP tilde to OXP and this is pi P if you want okay and this is this sense this sense an element u, f here to its equivalence class which I will put which I will signify by putting a square bracket this is the map okay. Now what I want to tell you is that I want to tell you that the two things I want to tell you first thing is that there is a name for these equivalence classes they are called germs of regular functions okay so equivalence classes are called germs of regular functions at P okay. So a germ of a regular function is represented by on some neighborhood of the point is represented by a regular function okay and it is also represented by another regular function on another neighborhood of the point if it is also represented by another regular function on another neighborhood then these two functions coincide on the intersection okay. So you know in some sense you must think of these equivalence class as defining a particular class of regular functions at that point so this makes sense as thinking of a function class of functions at a point right and two distinct classes means that they are they correspond to regular functions at that point which will not coincide on any neighborhood of the point okay because if you have two regular functions locally defined at the point and if they coincide in some neighborhood of the point then they will give rise to the same germ. So if you take two distinct germs they will correspond to regular functions which do not coincide in any neighborhood of that point that is what it means okay. So the moral of the story is that if you want to think of regular functions at a point and you want to distinguish them this is the way to do it okay so that is the first point the second point is that this is a local ring okay that is the point and more importantly you can do this whole construction not only in algebraic geometry you can do this in with respect to anything. So for example you could have started I mean you could be working on a topological space and you could just take you could fix a point P on the topological space and you could simply consider paths where you have a neighborhood of the point and you have a function which is continuous real valued or complex valued function okay and then you could add more conditions like you could make that function not just continuous you can make it differentiable if you want you can make it if it is complex valued you can make it holomorphic or analytic and you can put the same definition okay and you will see that again you will always land in a local ring. So this definition this whole process of identifying equivalence classes of functions which agree in a neighborhood of a point automatically produces a local ring so in a nice way you know this is how a local ring is completely an algebraic concept okay it comes out completely out of geometry okay if you restrict the if you want to you know restrict the study of functions good functions at a point you automatically end up with a local ring okay. So if I take my space to be an interval on the real line close interval on the real line and if I take my functions to be real valued continuous functions then I will get a I will get a local ring which corresponds to germs of real valued continuous functions okay or I could have taken the space to be the plane or some n dimensional euclidean space and I could have taken functions f which are defined on neighborhood of a point and which are c infinity which means which are infinity differentiable okay and then I will get the local ring of I will get the local ring of germs of differentiable functions c infinity functions and that is what you get when you look at a manifold okay and on the other hand I could also take an x to be a domain in complex plane or a domain in cn n dimensional complex space okay and I could have taken f to be I could take the functions to be holomorphic functions holomorphic in several variables okay which means which is equivalent to saying holomorphic in each variable separately okay. Then I will end up with again a local ring namely I will get the local ring of germs of holomorphic functions at that point so this method of producing local rings is completely geometric okay so this algebraic concept of local ring comes out through geometry in this way by restricting by trying to look at functions at a point okay. Now the fact I want to say is that this is actually a ring why this is a ring is because you can define addition and multiplication in fact it is not just a ring in our case it is actually a k-algebra it is a k-algebra and so let me write out fact o xp is a local ring is a local k-algebra this is the fact. So first of all how do you make it into a ring how do you define addition multiplication and so on so you define so the operations so you know if you give me u,f and if I want to add it to v,g this is the germ of u,f this is the germ of v,g okay mind you u and v are neighbourhoods of the point p therefore u intersection v makes sense that is also a neighbourhood of the point p. So I will simply define this to be u intersection v and f restricted to u intersection v which is the same as plus g restricted to u intersection so this is how I define addition then I can define multiplication in the same way I simply restrict and multiply on the intersection okay. So you can check that since I am defining these operations on equivalence classes you can check that they are well defined okay so this is my addition this is my multiplication and of course the make this into a ring into a ring which is competitive it is of course a competitive ring because the way the product of functions is competitive okay the function defined by f times g is the same as function defined by g times f okay because functions are defined product functions and some functions are defined point wise and therefore everything is point wise being done in the base field and the base field is of course competitive and so this makes it into a ring which is a competitive ring with unit element with 0 element the 0 element is very simple it is just the class of x,0 so you have the constant functions are always there so you take the function 0 on the whole of x that is a regular function and you take its germ that is a 0 element and unit element is given by x,1 you take the constant function 1 constant functions are of course regular functions always so that is the reason why the constants k the field base field sits in as constant functions in the ring of regular functions on any open set. So every if you take u any open set then o u is of course a k algebra because k sits inside o u as a constant functions okay so this makes it into a competitive ring with this is 0 element and this is unit element and further it is a local ring why because the subset mp m of p so let me write mxp to be the set of all germs u,f such that f vanishes at p okay. So you whenever you take a germ it is defined it is represented by a regular function on a neighborhood and you take all those take the subset which corresponds to functions vanish at that point okay. Then this is a maximal this is an ideal in o xp that is very clear because if a function vanish at a point then a function multiplied by any other function will also vanish at that point and some of two functions vanishing at a point will also vanish at a point so it is an ideal alright and every element which is outside that you take a function you take a germ of germ which corresponds to a function it does not vanish at a point it will have a it will have an inverse because the moment f does not vanish at p then by continuity f will not vanish in a neighborhood of p okay on that neighborhood 1 by f will be a regular function and f into 1 by f will be 1 on that neighborhood therefore every element outside this ideal is going to be unit and that will tell you that this has to be the unique maximal ideal for that local ring and that ring will be local with this unique maximal ideal. that any element outside mxp is of the form p, g with g of p not equal to 0 since g is continuous g does not vanish in an open neighborhood w of p and clearly 1 by g belongs to ow 1 by g germ of this into v, g is equal to x, 1 so every element outside mxp is a unit in that is an invertible element in this ring which implies that ow xp is a local ring with unique maximal ideal mxp okay so this is a local ring at a point of a variety okay and so there are so immediately there are couple of facts about this so the questions that you will ask is we have defined this local ring just using regular functions in a very abstract way can you really write it down commutative algebraically and the answer is yes you can write it down if you are looking at a point on an affine variety okay you can write it down as the localization of the affine coordinate ring at the maximal ideal that corresponds to that point okay and that is one fact the other fact is that the local ring will not change if instead of x you take an open subset which contains the point p inside x so instead of computing the local ring of x at p if I compute the local ring of u at p where u is an open subset of x which contains the point p I will still get the same local ring which means that you know u the local ring depends only on the neighborhood of p on a neighborhood of p it does not depend on the ambient variety okay so the moral of the story is that if you want to really work with the local ring at a point of a variety you know already that that point can be surrounded by an affine open because I have told you that any variety can be covered by finitely many affine opens and therefore the local ring of the variety at that point is the same as the local ring of the open affine sub variety at that point but then when you look at the local ring of an affine variety at a point there is a precise community algebraic expression namely it is the localization of the affine coordinate ring at the maximal ideal corresponding to that point. So this all this helps us to reduce study of local properties to study of local rings of affine coordinate rings okay so that is the importance so with that I will stop now.