 Ok, so let us continue with our discussion on algebraic geometry, so you know so let me recall what we have seen until last until the last time, so broadly algebraic geometry is study of the geometry of the set of common zeros of a bunch of polynomials, so what we do is we take k to be an algebraically closed field and of course what is the significance of this is well the significance is that you have the Null-Stokes arcs which tells you that zeros of common zeros of a bunch of polynomials is this set of common zeros of a bunch of polynomials is not going to be an empty set provided the set of those bunch of that bunch of polynomials does not generate the whole ring ok. So that is the reason why you choose an algebraically closed field and what we have is basically we have this you know there is a dictionary or we rather try to set up a dictionary on one side which is the geometric side on the other side being the commutative algebra side. So what we have in the on the geometric side is the affine space over k which is actually kn namely k cross k cross k n copies this is the Cartesian product of k taken with itself in n times of course n is greater than or equal to 1 and it is kn not just kn but it is with the so called Sariski topology, so well if you just look at kn you would normally think of it as n dimensional vector space over k ok but the point is and in a vector space 0 is a very special vector ok but we want we do not want to think of it as a vector space we want to think of it as affine space. So 0 does not have the point 0 does not have any special meaning in the case of vectors it does have a special meaning because vector always needs an initial point and a terminal point and we always take the initial point with the origin ok but here in affine space any two points are literally the same except that only the coordinates have changed ok. So but more importantly what we have is not just a set it is not just the Cartesian product this Cartesian product is consist of n tuples of elements of k ok it is not just the Cartesian product as I said but there is a so called what is that Sariski topology it comes from the it rather comes from the commutative algebra side. So let me put that on this side this is the commutative algebra side on this side we have the polynomial ring in n variables over k so x1 through xn are n indeterminates or n variables and of course this is the polynomial ring in n variables ok. So these are polynomials in these n variables with coefficients coming from the field k ok and how do you get the Sariski topology you get the Sariski topology by prescribing of course any topology is given by either prescribing a collection of closed sets or a collection of open sets and these two are complementary to each other because a closed set is a complement of an open set and vice versa. So the approach to the Sariski topology is by specifying a collection of closed sets and how do you specify this collection of closed sets what you do is well you take any subset of this polynomial ring which means you are just taking a bunch of polynomials in n variables with k coefficients and what you do is associate to that subset the set of zeros of the set of common zeros of that subset. So this subset here will be all those n triples which satisfy each and every polynomial in this collection in this subset ok and we declare sets like this to be the closed sets. So they are given a name they are called algebraic sets because they are the common zero locus of a bunch of algebraic equations ok the solutions to a bunch of algebraic equations you can think of the equation corresponding to each polynomial as a polynomial being equated to zero and then the solutions are nothing but the zeros of the polynomial ok. So the algebraic sets are sets like this and they are if you declare them as closed sets then you see that they satisfy the axioms for closed sets and therefore this sets of subsets of this kind taken as closed sets does define a topology on these on the set kn and along with this topology we call kn as the affine space n dimensional affine space over k we give the special symbol and we call this topology as Zariski topology ok in order of Oskar Zariski whose whom you could say is a founding father of algebraic geometry from the view point of combative algebra ok. So and then I told you that the so yeah so there was a so on this side we have a collection of nice sets namely the algebraic sets which are the closed sets but on this side you seem to only have subsets of the polynomial ring but then a subset does not make sense as a sub-object of the polynomial ring and the right sub-objects are ideals ok and of course these everything here is commutative the k is of course always a commutative field we are in this course we are only worried about commutative rings so and therefore you know ideals are always two sided ok. So the point is you if you want the right objects on this side more interesting than the subsets are the ideals and how do you pass from subset to one ideal this is a very general yoga it is a general philosophy that you can always take the smallest sub-object which contains your subset in any mathematical structure if you have a subset which is not a sub-object then how do you get a sub-object you just look at the smallest sub-object which contains that subset it is called the sub-object generated by the subset ok. So in this case you can take the ideal the sub-objects which are interested in our ideals of course you know you can think of sub rings also as sub-objects ok but sub rings are not going to help because the point is the moment this set contains the unit then the zero set will become empty ok because the unit has to be just a constant polynomial which is non-zero ok and that constant polynomial is never going to vanish anywhere so the zero set will become empty. So certainly you are not interested in sets like this which contain one or which can generate one so basically you are only interested in so that tells you that you are not interested in sub rings and what other objects sub-objects can you think of you can think of ideals and why ideals because ideals are really nice sub-objects because the nice sub-objects are the ones which can give you quotient objects ok. So you know if you have ring and you have an ideal then you can get the quotient ring ok whereas of course if you have ring and a sub-ring I am not going to get any quotient ok. So from the point of view of sub-object being the right one if it gives a decent quotient object you see that ideals are preferred ok ideals are the right choice and sub-rings are not but anyway we do not want sub-rings because this is going to end up as a null set if you are going to take a sub-ring ok. So the up short of all this is that you look at the ideal generated by this subset which I put as bracket S and then you see that if you take the 0 set of the ideal generated by S ok that turns out to be the same as 0 set generated by S the 0 set of S the 0 set of the ideal generated by S is the same as 0 set of S. So what it tells you is that if you replace a set by the ideal it generates you are not going to change anything here ok. So the advantage of all this is on this side on this side you have close subsets on this side you have ideals ok and not just subsets and what is a prescription though we started out with subsets we always take the ideal generated by the subset and work with that ok and that does not change anything on this side ok because of this equality. Now so as I told you the you know the whole purpose of algebraic geometry is to look at this these two sides of the picture ok and go from one side to the other and keep translating what properties on one side mean on the other side. So geometric properties on this side should mean they should give rise to some ring theoretic properties which means they give rise to some ideal theoretic properties more generally they might give rise to module theoretic properties on this side and conversely some module theoretic or ideal theoretic properties or ring theoretic properties on this side will correspond to geometric properties on this side and our and the aim of algebraic geometry is to is to discover this relationship okay, so the first thing I wanted to say is you know well but there are two points I have to mention. So the first point is so let me again recall you know an algebraic closed field is a field in which if you take a polynomial of one variable in one variable over that field and if it is non-constant then all its zeros are there in that field okay. So normally field theory tells you that if you have a polynomial over a field then you may have to go to an extension field to find the zeros of the polynomial okay and in fact a field theory it gives you what is called as a splitting field for a given polynomial which is a kind of a smallest field extension over which the polynomial completely splits into linear factors but you do not have to worry about such things I mean if you are working with an algebraic closed field because the definition of algebraic closed field tells you that if you have a polynomial already split into linear factors that means all the which is equivalent to saying that all the zeros of the polynomial are already elements of this field okay but the story does not end there the important thing is the when you define algebraic closed field it is only for one variable okay but when we do algebraic geometry in this general sense we are worried about polynomials not in just one variable we have worried about polynomials in several variables and then the question that arises is if you give me a you know subset of polynomials or for that matter the ideal that it generates and look at the zero set what is the condition that the zero set is non-empty okay and Hilbert, Null, Stel and Schatz tells you that this will be non-empty so long as this ideal is really a proper ideal so long as this ideal is not the whole ring or the unit ideal okay. So in other words this should not contain a unit we all know that if it contains a unit if S contains a unit it is very clear that this is empty okay and the Hilbert, Null, Stel and Schatz tells you that that is the only case when it is empty. So long as this does not contain a unit and of course it is very important that you know well I should correct my statement a little this may not contain a unit but this might generate a unit okay so to be very strict this ideal should not be this ideal should not contain a unit which is the same as saying this ideal should not be the whole ring okay in that case and then and only then will the zero set be non-empty okay and that assurance is given to you by the Hilbert, Null, Stel and Schatz okay that is one important thing. Then the other thing is the other important theorem that comes is the Hilbert's basis theorem or Emi's no Emi Noether theorem okay so you see you already see the impact of results on this side which means something on this side you see the fact that the Hilbert's Null, Stel and Schatz is basically a result which comes from this side of the diagram okay if you want to think of commutative algebra also as involving field theory for that matter because you know fields come because you know you start with a ring basically you start with rings which are integral domains and then if you go to the quotient field or the field of fractions then you know studying things over that already leads you into field theory okay and so you know Null, Stel and Schatz is the kind of result that you know comes on this side of the picture but geometrically it means that you know the zero set you are working with gives you conditions when the zero set you are working with is non-empty okay that is already a transmission from a result on this side to this side okay and but the way I have given it I have already given it on this side okay I have not I have not told you the commutative algebraic version of the Null, Stel and Schatz which I will do okay and the other statement that I am worried about is that I want to talk about is about the Hilbert basis theorem or the immuno ethyl theorem what does it say? It says well it says that if you start with a ring R a commutative ring of course you know you must always remember that we always work only with commutative rings with one and we always assume that all ring homomorphisms take one to one okay. So if you start with the Noetherian commutative ring okay which means ring is which means that the ring satisfies a property that every ideal is finitely generated then if you take a polynomial ring in finitely many variables over that ring the polynomial ring also becomes Noetherian okay. So if you take for the ring a field you know a field is always Noetherian because it has only two ideals namely the zero ideal and the full field which is unit ideal so it is Noetherian and now if you apply Hilbert's basis theorem or immuno ethyl theorem you get that this ring is Noetherian what it means is therefore that every ideal is finitely generated okay. Now what is the importance of saying that an ideal is finitely generated the importance is that every element in this ideal can be written as a finite linear combination of a fixed number of elements with ring coefficients that is what it means but what it really means is that you can take for this generating set only a finite set of polynomials and what it means therefore is even though you start with a set which is probably infinite okay or you start with an ideal which is infinite okay in fact an ideal will be infinite because even if it has one elements one element then it will contain all multiples of that element by the ring elements and this is going to this is an infinitely many elements here because k is any algebraically closed field is infinite that is the result from field theory okay and this polynomial ring is also infinite so all these ideals are all going to have infinitely many elements and when you look at the common zeros of elements in the ideal you seem to be looking for common zeros for infinitely many polynomials but then what the Hilbert's basis theorem tells you is that that is not what is happening really what is really happening is you are just looking at the zero set of finitely many polynomials so what it tells you is that even if S is infinite in any case any ideal like this will always be infinite that ideal is the same as the ideal generated by finitely many polynomials okay and therefore the zero set is just the common zeros of this finitely many polynomials and in fact this is a set this is the same as the intersection of the zero sets of the individual polynomials okay z of f i is just the zero the zeros of f i okay and then if you take the intersection you will get the points which are zeros of all the f i's which is exactly what this means this is the set of common zeros of all the f i's and of course this tells you that you are always only going to solve finitely many equations okay and why this is important it is also important for computation okay because once you have finitely many things to deal with you can have inductive procedures you based on some ordering for example you can do lot of computations in commutative algebra using software and all this is possible just because of this result that you are only dealing with finitely many polynomials at a time okay. So that is the importance of these two very basic but very important theorems one is the Hilbert Null's theorem and the other is the Hilbert Bezos theorem okay fine so that is the set up now. So what I want to do is I want to give you something that goes in both directions okay so already I have this z which associates to every subset or every ideal on this side the common zero locus on this side okay I also want to give something on this in this direction okay I want to give something in this direction so when I go from this direction so from how do I go from here to there so you know if so let T be a subset of An okay and mind you this is just a subset is just some collection of points okay I am not requiring that it is closed or open or something like that I am just taking any subset and what I do is from here I associate to T I of T which is called the ideal of T okay this is the ideal of T okay and what is the ideal of T this is set of all polynomials in the polynomial ring such that f of okay so let me write it like this f of T equal to zero for every T in small capital T okay this is called the ideal of a subset okay so there are two statements I want I want you to understand first thing is I am defining a set and I am calling it an ideal okay in general that is not correct I can define a set then I have to verify it is an ideal okay so the fact is that if that if you really look at this definitions obviously it is an ideal because you see if you take two such f's say f1 and f2 then if f1 and f2 both vanish at every point of T then so does the sum okay and of course zero vanishes to every point of T so zero is there so this closed under addition this has zero and of course for that matter if f vanishes at every point of T so does minus f okay and also if f vanishes at every point of capital T then multiplying f by a G will also vanish at every point of capital T so this is an ideal so it is in fact an ideal and this is called the ideal corresponding to the subset okay and so you see now we have so basically what is happening is that towards subset here we associate a set here which is actually closed okay it is an algebraic set and in fact for an ideal here also we can associate a subset here which is actually closed okay and in this direction given a subset you associate an ideal okay so you can see more or less that on this side you are worried about closed sets okay and on this side you are worried about ideals okay. So let us explore the properties of these two associations many of them are quite you know quite straight forward so the first thing is so let us look at the association in this direction okay that associates to every subset the zero set of that subset. So the fact I want to put is that this association is actually inclusion reversing okay on this see on both sides you have subsets only thing is here the special subsets that we are interested in are closed subsets and here the subsets that we are interested in are ideals okay but nevertheless inclusion makes sense on both sides as a partial order okay and what I want to say is that both of these associations they just invert they are not order preserving but they are order reversing okay so what it means is that you know if S1 so let me write that down as a lemma it is pretty easy to see if S1 is subset of S2 then well Z of S1 contains Z of S2 the second thing is the corresponding result on this side if T1 is contained in T2 then ideal of T2 contains ideal of T1 okay so it is if you look at it it is pretty easy to understand you see if you start with S1 inside S2 probably I am making a mistake somewhere okay probably I am making some obvious mistake so let me talk through this okay so you see S1 S2 has more equations than S1 okay so a solution a point of the affine space which satisfies every equation S2 will always satisfy every equation S1 so this is correct right I think there is not anything wrong there and look at this if T1 contains T2 okay then if you take a function which vanishes at every point of T2 then it will vanish at every point of T1 therefore probably this has to be the other way round okay so probably that was a mistake you were pointing out yeah so if T1 is contained in T2 then the ideal of T2 is contained in ideal of T1 because the way I first wrote it seems to be order preserving which is not correct okay fine so this is quite obvious but what is not directly obvious is the following thing you see what happens if you go and come back okay so if I start with the T on this side okay then I take the ideal of that and then I take the 0 of that what do I get so this is a this is something that one has to worry about and then the other one is the other way round if I start with well I said a subset here I take the 0 set of that then I take the ideal of that what do I get okay so this is what we want to investigate and the answer to that is well the answer to this is that you get the closure of T okay mind you I started the set T the set T need not be closed but when I take I of T it becomes an ideal and when I take Z of I of T it is a closed set because Z of anything is closed by definition therefore what I am going to get is a closed set which contains T, T is of course going to be here it is very clear any point of T will be a common 0 of all the functions which vanish on all of T and therefore it is going to be here so it is clear that this contains T and it is a closed set containing T but the fact is that it is a smallest closed set which contains T and therefore it is T closure okay and then as far as this is concerned this is literally the involves the commutative algebra version of the null silences so what it says is this is just this is just the radical of the ideal generated bias okay and so the so as for the proof I think 1 and 2 are obvious they quite straightforward okay the question is with 3 and 4 alright so if you look at 3 so let us prove 3 clearly T is contained in Z of I of T right that is very clear because every point of T is a common 0 of all those functions which vanish at every point of T that is what it says I of T is all those functions which vanish at every point of capital T and Z of I of T is all those points at which all these functions vanish okay and therefore T contained in Z of I of T is obvious but more importantly if T is contained in F and F is closed okay then what will happen is that you will see that it will follow that Z of I of T is contained in F okay what this means is that Z of I of T because Z of I of T is already closed it means that Z of I of T is the smallest closed set which contains T it is a closed set which contains T and Z of I of T is a closed set which contains T and whenever some other closed set contains T that closed set also contains Z of I of T so this implies that Z of I of T is equal to T bar because by definition the closure of a subset is the smallest closed set which contains that subset and which you can obtain set theoretically as the intersection of all the closed sets which contain that subset okay. So I think it is I have written it will follow that it is something that you can very easily check okay and the as for the proof of 4 well so one way is kind of obvious and it is the other way which is not obvious and which involves the noose answers okay so you can see that I of Z of S is the same as I of Z of ideal generated by S because Z of S and Z of ideal generated by S are the same okay and I think it is very it should be very clear that this contains radical of this should be very clear because you see you what is an element of the radical of S radical of the ideal generated by S it means it is an element whose power is in the ideal generated by S okay and that but any element in the ideal generated by S will be in this ideal okay and therefore it will be essentially an element that will vanish at every point of Z of S okay and therefore the element whose you started with will also vanish at the points of Z of S so this will be obvious okay what is what is what is more difficult is the other way around namely if you started the function which vanishes at every point of Z of S then some power of the function is actually in the ideal generated by S that is the non-trivial part and that is precisely the commutative algebra version of the Hilbert noose answers okay so let me write that this is obvious since F belongs to root of I mean ideal of radical of the ideal generated by S means F power R belongs to F power M belongs to ideal generated by S this implies which implies that F power M belongs to I of Z of S okay and so in fact I should say more importantly not only this what it means is that F power M belongs to okay so F power M vanishes on Z of S which means F vanishes on Z of S okay so this means that I mean this means that if you start with an F in the radical of the ideal generated by S it has to be in the ideal of Z of S okay. So this is kind of this is kind of obvious what is not obvious is the following conversely if F is in I of Z of S that is F vanishes on Z of S then F power M belongs to S for so let me add for some M greater than 0 due to the commutative algebraic version of the Hilbert noose answers okay so they so what is the commutative algebraic version let me expand on that so here is Hilbert noose noose answers so this is the what is called the strong form strong form commutative algebraic if K is an algebraic noose field I in K X1 etc Xn is a proper ideal okay then the ideal of Z of I is contained in rad I this is the this is the commutative algebraic version in fact we often say it as I of Z of I equal to rad I okay but the fact that I of Z of I contains rad I is something that is obvious okay but what is not obvious is that I of Z of I is contained in rad I namely if there is a function which vanishes on Z of I then some power of the function is in I because radical of an ideal is just all those elements some power of which some positive integral power of which is in the ideal and you can check that that is a bigger ideal in fact okay and what this says is that if a function vanishes on the zero locus of an ideal then some power of the function has to be in the ideal which means that function has to be in there it may not be in the ideal but some power is in the ideal therefore the function is in the radical of the ideal okay so if you think of the radical as trying to expand the ideal by trying to take all possible nth roots of elements in the ideal okay. So this is the Hilbert's Nielsen sets this is the commutative algebraic form okay oh yeah so as one of the students rightly points out maybe okay so let me make the correction right from here let me call this let me use a script I okay for the for going from this direction to this direction so I will change this to script I okay so that things become far better so this changes to script I this changes to script I and this changes to script I so what does this yeah so it is it is a lot more helpful to have this kind of notation which does not confuse so I will change it to everywhere yeah of course the statement here that f power m is in the ideal generated by s for some m greater than 0 is another way of saying that f is in the radical ideal generated by s which is what we want for the other for the other inclusion okay and here let me let me put script I so that it becomes better to read okay and you know you see I gave a so called weak form of the Hilbert Nielsen sets okay what was the weak form the weak form was the assurance that so long as an ideal is not the unit ideal the 0 set defined by the ideal is going to be non-empty okay and that weak form can be deduced from this strong form as follows if an ideal is not the unit ideal then that translates to the fact that the radical of the ideal is not also a unit ideal okay and ideal is you can check that the ideal is a unit ideal and ideal is a unit ideal namely the whole ring if and only if the radical of the ideal is also a unit ideal the reason is because if because of the fact that if a power of an element is a unit then that element itself has to be a unit okay. So I not being the unit ideal I being the ideal is the same as the radical of I being the unit ideal okay and then of course you see if Z of I is empty okay if Z of I is empty then the ideal of the empty set is a whole ring okay so if you assume that I is not a unit not a unit ideal then you will get a contradiction from this if Z of I is empty so you should read it in fact you should read it with equality okay because the other inclusion is obvious. So this strong form of the Hilbert's null set does ensure that you know so long as I is not the unit ideal the zero set is non-empty okay so that is how the strong form gives the so-called weak form okay. So what one needs to know is you know since you have two associations going in two directions you would like to make this into you would like to see this as a bijective you know equivalence and it is obvious that you will have to restrict the subsets here and there are subsets there and what we are going to do next is to go towards that okay. So the see the problem is on both sides the arrows are not injective the associations are not injective for example you know if I take an ideal I and if I take the radical of the ideal I they both go to the same thing here okay so if I take I here which is contained it is radical every ideal is of course contained it is radical okay then both of these things they go to the same thing the zero sets are the same the zero sets are the same okay. So you have two different things going to the same thing here okay so to avoid this what you will expect is that on this side you should replace ideals by radical ideals not just look at all ideals first of all not look at all subsets you pass from subsets to ideals and then do not just look at ideals look at radical ideals okay then you see that you can expect that this kind of a thing does not happen okay so on this side you put radical ideals alright and on this side you put closed sets then the then it is the fact that this is a bijective correspondence so radical ideals on this side and closed subsets on this side is the first layer of the bijective correspondence okay and the other important thing is you know this statement about inclusions being reversed what it tells you is that as the ideals become bigger the zero sets become smaller okay so in fact when I say ideals become bigger it is with respect to inclusion and you know the biggest ideals with respect to the inclusion non-trivial ideals with respect to the inclusion are the maximal ideals. So what you can imagine is that the biggest ones on this side are the maximal ideals and they will correspond to the smallest sets on this side and what you expect them to be they would be points okay so what will happen is that the biggest ideals here the maximal ideals they correspond to the smallest sets here which are the points and it is again corollary of the Hilbert Null seven sets and all this machinery that we have built up that the set of points in the fine space can be simply identified with the set of maximal ideals in the in this competitive ring okay and the beautiful thing is so therefore you know you are able to see the set of points here which is geometric as again a set of points there but these are not actually points here you have to form another space called the maximal spectrum the maximal spectrum of ring is a set which contains all the maximal ideals of the ring and the fact is if you take the maximal spectrum of this polynomial ring what you get back is a fine space okay but this is getting it back as a set the truth is it does not stop there the truth is that on this maximal spectrum here there is a Zaris ketopology and if you take that Zaris ketopology and give that topological structure to the maximal spectrum then that topological space becomes homeomorphic to this so it is not just a bijective correspondence but it is a topological homeomorphism so you see the in conclusion what is happening is that you have finally managed to completely rub off the geometric side and obtain it completely using competitive algebra so you see your affine space along with this Zaris ketopology can be completely forgotten and you can recover it only from the competitive algebra side by doing what by taking the so called maximal spectrum of this competitive ring which means take the set of all its maximal ideals and on that maximal spectrum impose the so called Zaris ketopology there is a Zaris ketopology on this side there is something called a Zaris ketopology on any ring you know on any competitive ring which would have come across in course in competitive algebra but anyway I will recall it and if you the fact is if you put that topology on this on the maximal spectrum amazingly you get back your affine space so the beauty you see the beauty of the dictionary is that I am able to see the affine space on this side without ever going to that side so you see this is the kind of you know translation that one is able to do and then there are there are there are many more things that happen on this side that that can be seen here and vice versa okay so this discussion will will proceed in that direction okay so I will continue in the next lecture.