 Okay, so let us continue our discussion about you know rings of functions of varieties okay, so let me recall the picture as it was explained in the previous lecture. So on one side we have the that is the geometric side we have if you want affine varieties and on the on the commutative algebraic side we have the so called rings of functions on affine varieties. So let me recall you see an affine variety is supposed to be a closed irreducible close subset of some affine space okay and affine space is of dimension n is just kn k cross k cross kn times given this a risky topology alright and by the risky topology of course if you recall it is just the topology that declares closed sets to be common 0 loci of a bunch of polynomials okay in the appropriate number of variables. So here on this side you have close subsets irreducible close subsets of affine space so standard examples are affine space itself is irreducible of course we are always working with k an algebraically closed field k is an algebraically closed field okay if you give if you take the affine n space then the ring of functions for affine space is a of an which is the polynomial ring in n variables over k okay so there is this association that takes to every affine variety swing of functions okay and of course the functions we are interested in are polynomial functions right and more generally if you give me a closed subset of an affine space which is also irreducible this is what a general affine variety is defined to be an irreducible closed subset of some affine space and for such a irreducible closed subset what is the ring of functions the ring of functions is defined to be the ambient ring of functions ring of functions sorry the there is a ring of function of the ambient space the largest space in which x sits in the ring of functions on this divided by the ideal of that subset and the ideal of this subset mind you is the ideal of functions in the ideal of polynomials which vanish on all of x okay and you know that that is prime because x is irreducible alright and therefore this quotient is an integral domain okay because you are going you are taking the polynomial ring and you are going model of prime ideal okay basically you are going model of prime ideal so the result the quotient is an integral domain and you see this closed this irreducible closed subset here shows up as a quotient here this is a quotient and this is just the you are just going mod ideal of x okay and well see the point is that what is actually happening is whatever is happening here is on the geometric side is an exact image of mirror image of what is happening here on the commutative algebraic side and algebraic geometry derives from this translation basically okay so that problems on geometric problems can be translated into commutative algebraic problems and vice versa and you can and a problem that you cannot solve here may be you can solve it here and the other way around okay. So the first thing I want to tell you is of course on this side it is affine varieties on this side it is rings of functions on affine varieties and I am saying that there is some kind of a bijection so there is also a map that is going on this side about which I mentioned the existence of but I did not tell you what it was in the last lecture I would like to tell you what that is and that is the that is the so called max spec so I will come to that and these two put together will tell you that whatever is happening here and is exactly a mirror image of what is happening on this side okay and so let me recall see for example if you have that X is a geometric hyper surface that corresponds to on this side X being X being and a hyper surface in the in the commutative algebraic sense okay and in fact on this side I should not write X actually I should write something about its ring of functions and the fact that X is a hyper surface in the commutative algebraic sense is the fact is that X is a hyper surface in the commutative algebraic sense is something that is defined by a single equation it is a locus defined by a single equation and so I of X so the condition is I of X is generated by single polynomial F and so X is just the zero locus of single polynomial and since I of X is prime okay it will force that F has to be an irreducible polynomial and it has to be irreducible non constant polynomial okay so that is the condition for hyper surface in the commutative algebraic sense okay and X being a geometric hyper surface is what is the geometric meaning of being hyper surface the geometric meaning is via dimension so it means that dimension of X is actually one less than the dimension of the ambient space the bigger space so it is n-1 okay where n is the dimension of the largest space and of course here by dimension I mean the topological dimension alright and we saw last time that these two are equal and that was basically an application of both the Krull's how TDL such and also the important theorem that you know and that a noetherian integral domain is a U of D a unique factorization domain if and only if every ideal every prime ideal of height 1 is principle okay so you know for both these theorems we do need that the ring involved is noetherian okay Krull's principle ideal theorem says that if you take a noetherian ring and you take an element F in a noetherian ring which is not a unit and it is also not a zero device okay then every minimal prime ideal that contains this element has height 1 okay that is Krull's principle ideal theorem and so the geometric content is that you know while both these theorems I tell you that these two are actually equivalent okay and the other beautiful thing is that you can so you can ask the question of when the ring of functions are an affine variety is a unique factorization domain and the answer to that is that you know you take any affine variety and look at its ring of functions the condition that it is a unique factorization domain is equivalent to saying that every geometric hyper surface in that variety namely every co-dimension 1 irreducible subset is defined by a single equation okay that is the geometric translation of the fact that every prime ideal of height 1 is principle okay the prime ideal of height 1 condition tells you that the dimension of your locus goes down by 1 okay that is it has co-dimension 1 it goes down by 1 from the dimension of the bigger space in which you are considering it okay and so you know therefore the geometric content is that if you have a unique factorization domain I mean when are rings of for which affine varieties are the rings of functions unique factorization domains they are precisely the affine varieties for which every co-dimension 1 irreducible subset is defined by the locus of a single equation that is what that is a geometric content okay so this is one particular example so you know I still I want to say a little bit more going from this side to that side. So you see let me give you two examples one is the let me take the two plane okay so the two plane there are many avatars of the two plane so for example I can take well I can look at a2k this is a two plane okay and if I look at the corresponding you know ring of functions I will get the afa2k which is just k if you want x1, x2 okay it is a polynomial putting in two variables then I can also consider you know two plane as if you want the xy plane in three space that is also after all the two plane only that you are considering it in a bigger space. So you know for example you can take xy plane in a3k that means you know you are looking at three dimensional space k cross k cross k you are calling the coordinates as xyz if you want and then you are looking at the xy plane that is also the plane two plane anyway and if you look at the ring of functions on this you will get a of you know well let me let me write that a of xy plane in three space what am I going to get I am going to get the ring of functions on a reducible closed subset is defined to be the ring of functions of the ambient space which in this case is the affine three space so it is a of a3 okay modulo the ideal of that irreducible subset okay that is the definition. So it goes by modulo you have to go modulo the ideal of the xy plane and if you see you see what this will give you is you will get a of a3 is of course k xyz because we agreed to call the coordinates on this three plane as xyz okay and it is xy plane you are interested in and what are the functions that vanish on the xy plane it is the z coordinate where xy plane is simply written as z equal to 0 and so it is actually the ideal of xy plane the xy plane is just the ideal generated by z okay and so it is the polynomial ring k xyz modulo z which is actually isomorphic to k xy so and so you see if you watch these two are one and the same geometrically they are it is just the two plane here it is a two plane considered as the whole space and here is a two plane considered as the xy plane okay and see what is happening on this side what you get is these two are isomorphic you get that the ring of functions are isomorphic this is also polynomial ring in two variables that is also polynomial ring in two variables so they are isomorphic okay. So let me give you one more example in fact let me take more generally you can take some plane a more general plane in n space okay if you take a two plane in an okay and if you associate to it the ring of functions a of the two plane in n space okay what will happen is you will still find that it is still isomorphic to polynomial ring in two variables it will happen so what is it that I am trying to say I am trying to say that geometrically you are looking only at the two plane whether you are looking at the two plane completely the two plane itself or whether it is embedded as a as the as a plane in three space or whether it is embedded as a plane in n space no matter how you look at it this definition of the affine coordinate ring the ring of functions on an affine variety this definition gives you isomorphic rings it does not give you something different and the fact that you all the three cases give you up to isomorphism the same ring is an indication of the fact that you are looking at the same geometric object you are looking at only the plane okay. So the moral of the story is that the ring of functions okay is go easy it really captures the object that you are looking at it is not going to change just because it is not going to distinguish between a two plane in three space or a two plane in n space because the two plane is always a two plane there is nothing different geometrically about it the only thing is the way it is embedded right. So the moral of the story is this gives you one rational as to why you should study rings of functions the ring of functions that is a geometrically it is an intrinsic object when you take an affine variety okay intrinsic means up to isomorphism it depends only on the affine variety not on where you see it okay. So that is one of the justifications of studying rings of functions and affine varieties. So I should now tell you that you know there is another common name in the literature people also use the word co-ordinate affine co-ordinate ring okay so people instead of saying rings of functions and affine varieties people just use the word affine co-ordinate ring so this is the word that you will often see in the literature okay. So the moral of the story is that so you see so you can expect what does this isomorphism here correspond to here what it is telling you is you know you should expect that this should be isomorphic to this and this should be isomorphic to this and this isomorphism as varieties as affine varieties is precisely what is getting translated on that side as isomorphism of their affine co-ordinate rings that is what you should expect and that is exactly what happens in fact this association is actually an equance okay I will tell you how it is an equance how strong it is on this side if you put if you take only isomorphism classes of affine varieties okay then on this side you should put isomorphism classes of such rings affine co-ordinate rings okay then this is a bijective association because if you change the affine variety up to isomorphism then its affine co-ordinate ring will also change only up to isomorphism okay that is one point then the other point is if anything that you that happens on this side has a meaning on that side so a close sub variety corresponds to a quotient okay suppose I do not take the set of isomorphism classes of affine varieties on this side and there also I do not take the set of isomorphism classes of affine co-ordinate rings okay then if you give me a closed sub variety like this this closed sub variety will automatically correspond to a quotient so you see this closed something being a closed subset of the irreducible closed subset of another thing that will this is called a closed immersion okay the word use is a closed immersion some affine variety being thought of as a closed irreducible subset of some other affine variety okay this closed immersion reflects on this side as a quotient okay so quotients here correspond to closed immersion okay it does not stop there isomorphism here correspond to isomorphism there that is exactly what we saw here okay more what do open subsets correspond to okay the beautiful answer is open subsets here will also correspond to ring homomorphism here in fact there will be K-algebra homomorphism and you know what they will be localizations so beautiful thing is closed immersed will correspond to quotients open immersion namely the inclusion of an open inclusion of a variety a variety being sitting inside another bigger variety is an open subset okay and this is something that I will explain soon okay that will show up there as a localization of rings okay. So and this is a very beautiful equivalence of categories so I should only tell you one thing what is what are the objects here so what are so if I say they are just coordinate rings of affine affine coordinate rings of affine varieties that means I am giving a definition here which actually depends on this side but if I want to give a definition only on this side then the correct definition is these what you should put on this side is finitely generated K-algebras which are integral domains okay. So what you should put on this side okay so on this side so let me write that down somewhere here I am a little short of space but anyway let me write it so let me write it here finitely generated K-algebras which are integral domains so what does this mean this means they are algebras of this form K of some polynomial ring K of y1 etc ym modulo p where p is a prime ideal let me write it below so that I can modulo p where p is a prime and you know how it is going to go from here to here okay do not worry about this max spec for the moment but you know what you know the object on this side for which this is the affine coordinate ring you know what it is it is actually am okay whenever you write a quotient I told you it is the affine coordinate ring of the 0 locus of the denominator which has to be a prime ideal considered in the ambient space for which the functions are the numerator so the numerator here is a polynomial ring in m variables that is a ring of functions on am okay and p is a prime ideal there therefore it is 0 locus will be an irreducible closed subset of am and that therefore defines an closed sub variety of am okay m dimensional affine space over K and each ring of functions will be precisely this so you know this will correspond you know if I draw a arrow like this this will correspond to well let me write it here it will be some this will correspond to z of p as an irreducible closed subset of am that is what this corresponds to okay. So on this side what I am putting is so called finitely generated K algebra which are integral domains and the definition of that is actually it is simply a polynomial ring in finitely many variables over K modulo a prime ideal I mean this prime ideal comes because I want the quotient to be an integral domain so you have to go modulo a prime and finitely generated is because it should be quotient of a polynomial ring in finitely many variables because the yi's they generate the polynomial ring in m variables therefore their image is in the quotient namely the yi bars okay the cosets yi plus p which are the yi bars in the quotient ring they will generate the quotient ring as an algebra over K okay so that is why this is a finitely generated algebra over K which is an integral domain and conversely every such algebra is defined to be something like this okay. So what I am trying to say is we have an equivalence of actually categories okay we have an equivalence of categories on this side you have affine varieties and morphisms between affine varieties on that side you have these finitely generated K algebra which are integral domains and the morphisms are K algebra homomorphisms and it is an equivalence of categories okay I will explain all these things soon but there is one thing I should say at the outset I have been saying morphisms I said isomorphisms of varieties I said morphisms of varieties so that is something I have to define I have to define what is a morphism between one morphism from one variety to another okay so that is going to come very soon but I am saying that this is the general picture in which you get a complete equivalence of categories and in that sense everything that is that you see here is actually a mirror image of that and vice versa okay. So all this tells you that studying the affine coordinate ring the ring of function on affine variety is a very important thing because that captures everything okay whether you look at it as coordinate rings of affine varieties or whether you look at on this side the affine varieties it is one and the same because of the sequence. So this is why I am saying all this I am saying this is why you can say that studying rings of functions on affine varieties is important and it also kind of you know it also justifies in a way Felix Klein statement that you know the geometry of a space is controlled by the functions on the space okay. So you know somehow these functions they control the geometry mind you we had only affine n space which came from kn and all we normally know about kn is only that it is an n dimensional vector space over k but we forgot the vector space structure and the geometry came in by looking at polynomial functions and using them to define the Zariski topology. So you see the even the topology on this side was gotten by functions on that side you must understand and the beauty is the functions on this side completely capture the topology okay. In fact not only the topology the most deeper statement is that everything the functions on this side completely capture the geometry on this side. So geometry at the lowest level is the topology then the next level is more deeper properties like the manifold theoretic properties, smoothness, regularity and things like that okay. So everything on this side the geometric side is captured by the competitive algebra which is the ring of functions okay. So all this is to tell you why it is so important to study rings of functions okay. So I will elaborate on all this whatever I said very soon but now what we will do is I wanted to you know okay so let me tell you about this max spec thing so let me explain this max spec that is the arrow that goes from this side to that side alright. So you know so let me again repeat the following thing. I will show later on that if you have two affine varieties on this side and you have a morphism that will give rise to a morphism in the reverse direction of their coordinate rings and conversely. Conversely if you have two coordinate rings of two affine varieties and you have a k-algebra home morphism that will give rise to a morphism in the other direction okay that this will be a pukka equivalence and under this equivalence a closed subset will correspond to a quotient and open subset will correspond to a localization okay this is what will happen. You can see the morphisms are going in the this is an arrow reversing equivalence because you see from X to An if you have closed embedding okay then you have from the functions on An to functions on X this is just restriction of functions which is seen as a quotient okay. So this is what will happen and when if you have U affine variety U sitting as an open subset of An what will happen is from An to A of U you will get a home morphism which will be actually localization of this ring at a suitable multiplicative subset in the sense of commutative one. So everything on this side will translate to something on this side and conversely okay but all that needs to be proved we are going to prove it okay but I am just telling this to you in at the outset to tell you that why it is so important to study rings of functions on affine varieties okay which are of this form finitely generated K algebra which are integral domains. Now I will go one step further what I am trying to do is to go one step further and say that suppose I forget the picture on this side suppose I completely forget affine space I forget the points on affine space I completely forget everything. Suppose I have only the functions suppose I had then from the functions I can get back my space okay that is what this arrow from this direction to that direction is all about that is just from the functions I can cook up the space that means the whole space is already here in the functions okay and that is a much stronger proof of the veracity of the statement of Felix Klein that the geometry of the space is not only so you know what he said is geometry of a space is controlled by the functions on the space but more seriously if you give me the functions then I can give you the space I can cook up the space from the space of function from the if I know the functions on a space then I know the space I can consider the space from its functions okay it is a very very deep thing so that is what this max spec does and that is what I want to explain. So what I will do is I will start with something that you must have seen in a course in commutative algebra but nevertheless it is not very difficult to recall these things so you know you start with R a commutative ring with 1. So what you do is you look at spec R this is called the spectrum of R in fact it is called sometimes it is also called the prime more explicitly as the prime spectrum of R and this is supposed to be the set of all prime ideals in R okay so what you do is you look at the set of prime ideals of R okay and regard each prime ideal not as a prime ideal regard it as a point so this square bracket script P is supposed to tell you that you think of here you are thinking of the prime ideal P as a point in spec R when you think of it as a point in spec R you are not thinking of it as a prime ideal inside R you are thinking of it as a point on a space okay. So the spectrum is just the points which correspond to prime ideals okay and then so you should not confuse this with equivalence class because normally you know you put a square bracket around an element to say that it is equivalence class of that element so you should not confuse it with that there is no equivalence class it is just to distinguish between the prime ideal as a subset of R and the prime ideal as a point of spec R they are different okay so my space is spec R now the spec R will become a topological space okay how so there is a Zariski topology on this just like you have a Zariski topology on affine space you have a Zariski topology on spec R and in fact you can say both ways this is inspired by that and that is also inspired by this okay it depends on where you start they are equally one the same so let me define this Zariski topology on spec R is given by taking sets of the form Z of I okay so define to be equal to a set of all P in spec R such that P contains I where I in R is an ideal to be close sets okay so you see the Zariski topology is defined like the Zariski topology is defined as the to be it is defined by specifying a collection of close sets what are the close sets the close sets correspond they come out of ideals ideals of R you take an ideal I of R immediately you can write down the close set Z of I okay which you should think of as a 0 set of I okay if you compare with the Zariski topology on the affine space and the 0 set of I is all those prime ideals it is all those points which correspond to the for which the corresponding prime ideals contain I okay this is the way it is defined and you can check that this satisfies the condition for a topology namely sets of this form satisfy the axioms for close sets in a topology and you get thus spec R becomes a topological space okay and then you see you can also check you can also check that you know if you have if you give me a given a ring homomorphism F from R1 to R2 okay so R1 and R2 are commutative rings with one and F is a ring homomorphism from R1 and R2 then the map spec F going from spec R2 to spec R1 which is given by give me a prime ideal of R2 and simply send it to its inverse image which is a prime ideal of R1 okay. If you give me a ring homomorphism from R1 to R2 then it induces a map in the reverse direction on the spectra on the prime spectra and that is simply given by pulling back prime ideals because you know the inverse image of prime ideal in a ring homomorphism is again a prime ideal it is not true for maximal ideals but it is true for prime ideals inverse image of maximal ideal need not be maximal but inverse image of prime ideals always is a prime ideal okay. So what will happen is well the fact is you get this map that is not the point the point is this is now a map of topological spaces see it is a map between two topological spaces the most obvious thing we can expect is that it should be continuous and it is okay so this map is continuous so this is another fact that you can that you should have come across in a course in commutative algebra if you have not you can it is very easy to check okay and in fact what happens is if you look at so let me maybe I will put a star here I will put on the star here then here is so it is like a bulleted list so but it is of bullets I am using stars so here the third thing I want to say is that what about what about a close subset like this okay what about a close subset like this so the so what I want to say is that z of i so if you take consider r to r mod i the canonical so I am writing c a n for the canonical quotient map from r to r mod i okay then if you look at speck of that which goes from speck r mod i to speck r okay because you know I have already told you if you have a ring homomorphism from one ring to the other you are going to get a map on the spectra in the reverse direction in particular you take the ring homomorphism to be the canonical ring homomorphism that you get from a ring to its quotient then the corresponding map on spectra what will be this will actually identify speck r mod i with the close subset z of i in speck r so what will happen so identifies speck r mod i with z of i so mind you z of i is a subset of speck r by definition it is a close subset of speck r because of the risky topology I have defined on speck r and speck r mod i the image of speck r mod i under this is identified with that okay so in other words z of i is also spectrum offering the model of the story is that the subset close subset z of i also is a spectrum so every we define a close subset inside a spectrum and it turns out that every close subset itself is a spectrum can be identified with the spectrum what is the spectrum it is spectrum of the quotient by that ideal okay and in some sense you know you must think of it like this you must think of it think like this if r is thought of as the ring of functions on speck r okay abstractly then r mod i is a ring of functions on z of i which is identified with speck r mod i okay so you see r is a ring of functions on speck r alright functions in what sense is something that has to be made formal more precise but do not worry about it r is a ring of functions on speck r then what is the ring of functions on z i it is r mod i okay because speck of r mod i can be identified with z of i okay and there is one more thing you see what is the situation with respect to localization so two particular kinds of ring homomorphisms are quotients and localizations okay and so what happens with localization consider r to S inverse r l which is a localization of r with respect to a multiplicative subset S inside r so if you remember in commutative algebra even a ring r we define what is meant by a multiplicative subset a multiplicative subset is a subset which does not contain 0 and which for which which is close under multiplication so if there are two elements in that set then the product is also in that set okay and we can in commutative algebra form the localization of r with respect to the multiplicative subsets which is just inverting the elements of the multiplicative subset so S inverse r which is the localization of multiplication localization by the multiplicative subset sometimes it is also written as r bracket S inverse okay so it can be thought of as a polynomial it can be thought of as polynomial ring in r in as many variables given by reciprocal of elements of S modulo the natural relations that come because of the elements of S being inside r okay and that is the reason why one writes it writes S inverse r as r square bracket S inverse but essentially it means that you are just inverting S so you know the simplest case is if you take r to be for example if you take r to be integers and you take S to be the complement of the complement of 0 okay then S inverse r will just be rational numbers more generally if you take an r as an integral domain and S is the complement of 0 which is a prime ideal okay the complement of a prime ideal is always a multiplicative subset and localization by that is always going to give you localization and in this case complement of 0 if you localize the integral domain at 0 at the complement of 0 then you are going to invert that means you are inverting everything outside which is different from 0 and that means you are forming the field of fractions so forming the field of fractions is a very special case of localization localization is a generalization of that you have seen this in multiplicative algebra but the fact is that if you take this localization map okay then what does it correspond to in the spectrum at the level of the spectrum then spec l will go from spec S inverse r to spec r okay you will get this map. Let me take a particular case put for example S to be just the multiplicative subset given by a single element r r squared where r is in r is not nilpotent okay I do not I want r to be the element r in r small r in capital R to be nilpotent because I do not want some power of it to be 0 if some power of it is 0 then the multiplicative set contains 0 and I do not want 0 on the multiplicative set there are some books which say that 0 is allowed in a multiplicative set but then the convention is that the localization becomes 0 ring and nobody wants to study the 0 ring for as far as geometry is concerned so we always never want 0 to come inside the multiplicative set that is the reason why I do not want small r to be nilpotent okay and then in that case we normally write S inverse r as r localize at small r okay we also write it as r bracket small r inverse some people write it as r of 1 by r okay these are various notations okay and the fact is spec and spec of if you calculate spec of rr what you will get is you will simply get all those prime ideals the points corresponding to prime ideals such that the prime ideal does not contain small r okay and this so you will get this identification okay and this is just compliment of z of r in spec r so you see spec r has a Zariski topology as I have explained and then there is the ideal generated by small r okay it is ideal generated by single element r and you take the 0 set of that ideal what is what is the 0 set of the ideal by definition it is all those prime ideals which contain the ideal r but a prime ideal contains the ideal generated by r if and only if it contains r itself so this is z of r precisely has for its points those prime ideals which contain small r and what is the compliment it is all those prime ideals which do not contain points corresponding to prime ideals which do not contain small r and that is precisely spec rr capital r of small r and how do you get this identification this identification is provided by this map the image of this map the image of spec l in this case identifies spec of S inverse r with this subset which is an open subset of spec r okay because it is the compliment of a closed set. So the moral of the story is you know the ring of functions on the compliment of the locus defined by a single element okay namely the ring of functions on the locus where a single element does not vanish is precisely localization by that element okay which is if you think of small r you know if you think of small r as an element of capital r and if you think of r as the ring of functions on spec r then small r is a particular function and what is z of r you should think of z of r as the locus where r small r vanishes okay and what is its compliment is compliment is should be the locus where small r does not vanish but where small r does not vanish if small r is a function that does not vanish 1 by small r should also be a valid function. So it in other words it tells you that in this locus where small r does not vanish which is an open set the functions are simply given by r of 1 by small r namely its functions are of the form some function of some element of r divided by some power of small r which is intuitively correct and in fact you get it here okay so this identification is via l in this case. So the moral of the story is the ring of functions corresponding to a closed subset is given by an ideal is given by the quotient ring and the ring of functions corresponding to an open subset a so called basic open subset where a function does not vanish is simply given by inverting that function that is what it says okay. So this is the background from commutative algebra okay and we can use this to go from this side of the diagram to the other side of the diagram which I will continue with this in the next lecture okay.