 Okay, so let us continue the discussion, so I will go back to this diagram on this side okay and what I am going to do is I am going to just rub off several things so let me rub this off and also rub this side and let me just write down the implication of this lemma here okay, so basically on this side the biggest ideal you can think of is the whole ring course and well the z of that is of course in null set which is the kind of the smallest set here okay and maybe I should because there is an inclusion reversal I should write it here, you have the null set here which is the 0 set of k x1 xn and you know well then of course if I take a maximal ideal okay m is a maximal ideal so more generally I could have started with an ideal I and rather let me write it okay so let me write it here does not matter so I have an ideal I here I would get the 0 set of the ideal okay and of course you know if I start with the 0 ideal okay then the that would give me the whole space this will be just be 0 set of 0 is the whole space okay and if I start with an ideal I1 I end up with well the 0 set of I1 which is the sum of the whole space right and you know if I take a bigger ideal I end up with a smaller set of zeros so it goes like this if I go all the way to a maximal ideal then you see the corresponding 0 set of a maximal ideal is just a point it is a single point okay and well if the ideal if the ideal is contained in the maximal ideal then this point belongs to the 0 set okay so it is like this and of course the null set is contained everywhere so you have on this side the subsets increasing from the null set which is smallest possible to the whole space and on the other side you have the ideals decreasing from the largest possible ideal which is a whole ring to the smallest possible ideal which is the 0 ideal okay and there is this order as you can see these are an order reversing correspondence so the point I want to make is I want to explain first of all that if you take a maximal ideal you will end up with a point okay and why is that so that is again basically if you want due to the null set okay. So let me tell you few more lines about that so here is a lemma this is a lemma from probably from field theory which you could have come across in a first course in commutative algebra but then it is even otherwise it is easy to prove if capital K is a field so capital K is a field and mind you this is any field it is not necessarily algebraically closed then an ideal of the form x1-lambda x1-lambda 1 x2-lambda 2 and so on xn-lambda n in K of x1 etc to xn with all the lambda i is in K is a maximal ideal okay so the point is that so if you take an n triple of points of the field then there is an automatically a maximal ideal associated to that so what this tells you is that there is a so let me write that this gives a map from KN to let me write max spec capital K x1 etc xn which sense the point lambda 1 etc lambda n to the maximal ideal given by so there is a there is a map like this and you know let me quickly tell you how to prove this so what you do is you assume that this point is origin namely you assume all the lambda i is at 0 then this becomes ideal generated by all the variables and you can see that this polynomial ring in this polynomial ring the ideal generated by all the variables is a maximal ideal and the reason for that is how do you check is the following how do you check an ideal in a ring is a maximal ideal you just if you go mod the ideal you should get a field you can show that if you take this polynomial ring and go model of the ideal generated by the variables you end up with the field K okay and therefore you will get that the ideal generated by the variables is a maximal ideal and then all coordinates being zeros not something special this also holds for other points okay because you can always find an automorphism of the ring which maps which translates any given point to the origin and an automorphism of the ring is a self isomorphism of the ring so it will carry maximal ideals to maximal ideals so the fact that the all the variable generate the maximal ideal will also tell you that ideals like this are also maximal okay so that is the this is a very simple that you can work out okay so the model of the story is well you know that if I take a maximal ideal of this form then you know what is the point you are going to get if this maximal ideal m is going to be x1-lambda1 etc xn-lambda n then the point you get is going to be just the point lambda1 it coordinates lambda1 etc lambdan this is the single point you are going to get because what is the point of kn which is a common zero of all these polynomials for such a point the first equation is a zero such a point is zero of the first equation means that the first coordinate has to be lambda1 the second it is also zero the second equation means the second coordinate has to be lambda2 and so on that will tell you that the point has to be just lambda1 lambda2 lambdan okay the ith coordinate has to be lambdai so it is clear that if you take a maximal ideal like this the corresponding point you get is this okay and what is more serious is that every maximal ideal is of this form that is also another avatar of the Hilbert's null sense okay so fact another avatar of the Hilbert's null sense is that the map above is surjective if capital K is algebraically closed so this is so in other words you take any maximal ideal in the polynomial variables it is of this form it arises from a point in this way and of course I also want to remark that it is easy to see that this map is injective okay because if you take two different points they will go to different maximal ideals or you can show that if you have lambdas here as coordinates of one point and lambda primes here which are coordinates of one of the points such that the maximal ideals coincide then the lambdas have to be equal to the corresponding lambda primes okay so this map is injective is trivial okay it is a surjectivity which is more serious and that surjectivity is also another avatar of the Hilbert's null sense okay so what this really tells you is that in our case since you are working with an algebraically closed field the points of the affine space are precise they correspond the points of the affine space on this side as closed subsets they first of all they are closed subsets and they correspond to maximal ideals on this side okay and the other thing that I want to tell you is that if you take closed sets here they will correspond to radical ideals there okay so let me make that statement so again lemma or let me just put it as fact closed subsets of kn they correspond to radical ideals of k x1 etc or xn okay. So you see I am just trying to concentrate on this dictionary the first thing is that it is order reversing the second thing is that you if you want an exact equivalence bijection then on this side you have to take radical ideals and on this side you have to take close sets okay and in fact you can have closed sets along with inclusion that as a partial order and you can have radical ideals along with inclusion here as a partial order and then this bijective correspondence will be bijective and it will be inclusion and reversing okay and I think with what we have seen so far we can more or less you can more or less see why this is true of course I should tell you the map on this direction is z a map in this direction is script I okay but you can check why both these maps are inverses of each other it might require a couple of results so let me do that if you start with let f in an be closed then if I go here and come back I will get I of z of I of f this is what I get when I go when I go I apply script I when I come back I apply z so I get z of I of f but what is this supposed to be f bar you have already seen that and but then f is already closed if a set is closed then its closure is equal to itself okay so taking the closure of the set is essentially adding the boundary okay the limits in a very naive sense okay. So I of z of I of f is f bar but that is equal to f since f is closed so what it means is that if you go in this direction and come back you get the identity map on the set of closed subsets of an okay and let us go from this direction so from this direction if I start with let I in k of x1 xn be an ideal then if I take z of I and then take script I of that you know that this is because of the Nulster and this is rad I the radical of I this is the enlarged ideal which consists of all those elements some integral power of which is in the given original ideal okay but then if I is already radical it means that the radical of the ideal is same as the ideal itself an ideal if the ideal is we say an ideal is a radical ideal if you take the radical of the ideal you do not get anything bigger okay what it means is that if some power of an element is in that ideal then that element is already in the ideal it means that you do not have to expand the ideal further by taking all possible n th roots for all possible n's okay that is what it means. So if I is radical rad I is equal to I I of z of script I of z of I is just I what it means is that if I start with the radical ideal here I go back go here and come back I end up with this now therefore these two statements should tell you that these two are inverse maps of each other from this set to that set okay and that gives you the bijective correspondence here okay and this well there is one more there is one more point that needs to be noted you can ask when suppose you are not suppose you are not worried about radical ideals suppose you are just worried about any two ideals you know that already any two ideals can still have the same zero set here for example an ideal which is not radical and it is radical they can have the same zero set. So you can ask more generally if what is the condition when two ideals here have the same zero set and the answer is that they should have the same radical in other words they are radicals are the same okay so here is one more fact so this is another fact that you can try out as a simple exercise I1 z of I1 is equal to z of I2 if and only if radical of I1 is the same as radical of I2 two ideals will have the same zero locus common locus of zeros if and only if they are radicals coincide okay. So this is something that you can easily check so that clarifies the bijective correspondence okay now there is one question that one can ask of course coming to the commutative algebraic part the question that you can ask is well maximal ideals have come then of course you know the other important ideals you are worried about are the prime ideals so you can ask well if you have a prime ideal here on this side what is so special about the zero set here okay so you can ask so you know now once you start building this dictionary you can take properties here and ask what they correspond to on this side so for example you can take z of I where I is radical ideal and suppose z of I has some geometric property you can ask what does it mean for the ideal I that is trying to come from the geometric side to the commutative algebraic side on the other hand you could do the other thing you could start with an ideal here which has a certain property okay for example maximality of an ideal is a property okay and of course primeness of an ideal is also a property and you can ask what does that property correspond to on this side when you take the zero set of the ideal so that is the question I am asking if you start with the prime ideal here what do you get here what is so special about what you get here the answer to that is what you get on that side is a strong form of connectedness of the corresponding zero set and this strong form of connectedness is called irreducibility okay so the answer is that the prime ideals here they correspond to sets on the other side close they are of course close sets on the other side but these close sets are topologically going to be what are called as irreducible sets and these irreducible and irreducibility is a very strong form of connectivity connectedness okay so I will explain that next so let us do that so this again goes so let me so definition a subset y of a topological space is called irreducible if we cannot write y is equal to y1 union y2 where y1 y2 are closed subsets of y. And of course I should say a proper close subsets of y and maybe I should also just to make sure that some silly contradictions do not come I should say proper close and non-empty subsets okay so you see I want you to reflect about this with respect to the notion of connectedness see when do you say a topological space is connected okay you say a topological space is connected if it cannot be disconnected and what is a disconnection a disconnection is breaking the topological space into two disjoint pieces into two pieces which do not intersect such that each piece is open okay but then since each piece is a complement of the other that is because they are disjoint and the union is the whole space it is also same as saying that each piece is closed okay so saying that topological space can be disconnected means that you can write it as a in two pieces which are closed okay and mind you it can happen that you may not be able to write the set as two pieces two disjoint pieces which are closed but you might still be able to write it as a union of two pieces which intersect and which are closed okay so you know for example you know if you take an interval on the real line an interval on the real line with usual topology is connected and which means you cannot disconnect it if you try to write it in two pieces okay then you will see that in the simplest case one of them will be a half open interval the other will be half closed and therefore you can never break it into two pieces with both pieces being disjoint and closed okay but if you remove the disjoint readiness then you can do it okay so you know if I have an interval from 0 to 1 I can write it as the union of let us say 0 to 0.8 union say 0.2 to 1 these are two closed subsets take the closed interval 0 to 0.8 take the closed interval 0.2 to 1 these are two proper closed subsets because they are closed intervals and they are non-empty the union is again 0,1 okay so you see that you can always do but what irreducibility says that even that is not allowed irreducibility is very very strong so irreducibility says you cannot write it even as a union forget disjoint union so when I say that you cannot write it as union of two proper closed non-empty sets it follows that you cannot write it as a union disjoint union of two proper non-empty closed subsets so what you must understand is that by the very definition irreducibility is very strong condition it is a very very strong form of connectedness and for example our interval on the real line any closed interval on the real line or for that matter any interval on the real line it is not irreducible but it is connected okay so I wanted to understand I wanted to think of irreducibility as a very strong form of connectedness okay and the definition of irreducible reduces to connectedness when you make this union disjoint okay. So here is a remark the remark is irreducible implies connected but not conversely so the conversely is not whole okay so irreducibility is a very strong form of connectedness then the so one wants to understand what are the properties of irreducibility so the there is one thing you can expect some properties that are true of connected sets will also hold for irreducibility. So one of the properties for connected sets is well you know if a set is connected then its closure is also connected okay this is a very simple exercise in term policy if a set is connected then its closure is also automatically connected and the analog of that result also holds for irreducibility if a subset is irreducible then its closure is also irreducible so let me write that properties if y is irreducible so is its closure y bar this is something I mean this is something that you can you should expect okay you know the point is when you take the closure of a set you are only adding the boundary so when you take the closure you are not removing anything and you should always think that trying to remove things might disconnect so after you are adding the boundary so it should not disconnect. So that is the reason you take a topological connected subset when you take its closure it will be connected but of course you know just adding something will not help if you add something that is away then you already made it into two pieces so you must add something only in the boundary okay that is so just like adding something in the boundary does not affect connectedness the same way adding something in the boundary is not going to affect irreducibility okay that is exactly what this statement is that is one thing. The second thing is there is something even more serious it is about open subsets of an irreducible set okay so but before I go to that let me again remind you what do you mean by closure the closure of a subset is the smallest closure set which contains that subset so it is the intersection of all the closed sets which contain that subset that is how it is defined okay fine so let me continue with my earlier statement irreducibility means a lot for open subsets and what does it mean it means means the following thing you take an you take a subset which is irreducible okay then every open subset is not only irreducible again but it is also dense okay so that is a way I mean that tells you how strong irreducibility is okay so but 2 if y is irreducible every open subset of y is irreducible and dense. So let me explain so I have to there is something that I wanted to say that if I got to say let me say it now see when I say y is a subset of a topological space and it is called irreducible if you cannot write it as a union of 2 proper closed non-empty subsets what do I mean by closed subsets okay so what do I mean by closed is with respect to the induced topology so if you have a topological space and you have a subset then the subset itself becomes a topological space in what is called the induced topology and what is this induced topology it is very easy you simply call you know to define a topology on a subset on a space you have to just give me a class of subsets which you might call as open or closed if you call them as open they should satisfy the axioms for open subsets and if you call them as closed they should satisfy the axioms for closed sets. So for example when I say y1 is closed in y what it means is that y1 is gotten by intersecting y with a closed set in the bigger topological space for which I have not given a name here if you think of the if you think of y as sitting inside topological space capital X then when do I say a subset y1 of y is closed it is said to be closed if it is gotten by intersecting y with a closed subset of x the ambient the bigger space ok. So whenever you say closed or open with respect to a subset it means it is gotten by taking a closed or open with respect to the big topological space after intersecting with a subset ok that is what closed or open in a subset means this is called the this is the language of induced topology ok. So that is what I mean when I say an open subset of y an open subset of y is nothing but an open subset of the big space in which y sits intersected with y ok and of course the more important thing is so the point is that any open subset is irreducible ok which means that irreducibility is a property that passes on to open subsets ok and you see this is not true for connectedness connect an open subset of a connected set need not be connected for example if you take the real line ok if I take union of if you take the whole real line it is connected and take the open subset to be a union of two disjoint intervals that is an open set because any open set on the real line looks like a union of intervals. So you take two disjoint in intervals open intervals that is a subset but is that connected it is not ok so you see but irreducibility is something that passes on to an open subset of course I should again say that whenever I say subset I should always keep worrying about the non-emptiness so I should add that every non-empty ok and you might ask what about the empty set the answer is the empty set is not considered to be an irreducible subset ok. So the reason is it is a matter of logic so the rule in logic is you team a statement to be true if you can test it and prove its truth or if there is nothing to test then also the statement is true so you know if my y is already empty ok then I have nothing to test so it will you know probably in that sense it is fair to think of the empty set as not irreducible I think that is the standard but anyway let me check once more yeah the empty set is not considered to be irreducible so I am so the book that I am following which is also given as a reference for this course is with the standard book by Robin Hartschoen titled algebraic geometry it is a graduate text in mathematics series GTM52 by Springer Verlag and it is more or less the first chapter that I am trying to cover in this course ok fine so as I told you the empty set is not considered to be irreducible right so whenever I say open subset of course I am taking a non-empty open subset so you see irreducibility passes on to a non-empty open subset which is not true of connectedness ok and more importantly this is the more important condition it is dense in other words you take an irreducible space take an open subset not only is that open subset irreducible but it is dense what does it mean to say it is dense it means that the closure of that will be again the whole set so let me write that down that is the closure of the open set in the in y is y itself. So you know why this is so important for algebraic geometry is because you see it is like you are saying every point of y is in the boundary of every open subset non-empty open subset of y that is what you are saying. So what this means is that you know if you want to test things you want to test properties which are going to be preserved under limits ok for example but when I say limits take it in the naive way ok because I cannot really talk of limits unless I have a metric and I have notion of convergence and so on and so forth I do not have all that here but I am just thinking of limits as trying to add the boundary ok. So when I say any non-empty open subset is dense what I mean is that it is closure it is boundary when you add the boundary to it you get the whole set it means that every point of the set is either a boundary point that occurs in the closure or it is in that open set and so any properties that are preserved when you go to the boundary they can be tested on an open set ok because testing if there is a property that is true if there is a property which is such that it is true on a subset then it is also true on its closure when such a property can just be tested on any non-empty open set then it will automatically be true on its closure which will be in the whole space. So that is the importance of that is one of the importance of important outcomes of it being dense then the other thing is that you know if you take any two open sets they will always intersect that is again because of this denseness ok. So what this so that is another thing you cannot you take any two non-empty open subsets they will intersect ok. So what this tells you is that in an irreducible space the open sets are huge see in the if you are for example thinking of an open set on the real line ok I can make the open set smaller and smaller ok or if I take the open interval 0, 1 ok I can find two small open sets two small open subintervals of that which are disjoint from each other which do not intersect and then I can make them as small as I want but I cannot do that here in the case of an irreducible topological space because if you take an irreducible topological space any two open subsets will intersect you cannot make them very small. So this is one seeming disadvantage with irreducible subsets namely that you cannot get very small open sets but then the fact is the amazing fact about algebraic geometry is even with this much of information you are able to do all the geometry that you want ok. So let me write that down here in particular any two open sets any two non-empty so I have to keep one has to keep remembering that one is working with non-empty open sets intersect and irreducible I mean this is the fact that open sets are huge ok you cannot find a very given a irreducible space and given a point in the irreducible space you should not think of you should not think of being able to find very small open neighbourhoods of that point that would not happen. Any non-empty open any neighbourhood of the point will be non-empty because it contains a point the moment it is a non-empty open sets it will be dense you cannot expect to have very small neighbourhoods you see this gives you the feeling again let me repeat this gives you the feeling that you cannot take you cannot make an infinitesimal study around a point like you do in the usual analysis you cannot take smaller and smaller epsilon neighbourhoods and two analysis it gives you that feeling but that is not true the fact is that the analysis is not done on the geometry side the analysis is done on the commutative algebra side ok by studying the so called local rings of the commutative ring which are obtained by localizing the commutative ring with respect to various prime ideals or maximal ideals. So the limit process and it is and what you get from it in calculus is kind of is kind of already buried there in studying the local rings ok so you should not get the get a negative feeling that well I am not able to get very small open sets surrounding a point so I cannot do any analysis that is not true ok. So well so this is the other thing and ok so let me come back to the let me come back to this question here if I start with the prime ideal here then what do you get on that side if you look at the 0 set so the answer is if you start with the prime ideal here the 0 set of the prime ideal will be an irreducible closed subset and conversely if z of i is an irreducible closed subset then the radical of i has to be prime and if you already chosen i to be radical it means i has to be prime so what it tells you is that the prime ideals on this side they correspond to the irreducible closed sets on this side ok. So that gives you an answer as to what the commutative algebraic property of an ideal being prime means in terms of geometry the commutative algebraic property of an ideal being prime translates into geometry into the geometry that the 0 set of that ideal is an irreducible closed subset ok. So let me write that so let me keep this diagram here so here is a well if you want theorem if i in k x1 through xn is an ideal then z of i in an is irreducible is irreducible of course it is closed by definition but it is irreducible if and only if rad i is prime so what it means is that if you already have started with a radical ideal then the 0 set of the radical ideal is irreducible if and only if the ideal you started with was already prime and a prime ideal if you take the 0 set of a prime ideal that will also give you an irreducible closed subset ok and in fact the reason why we are interested in irreducible closed sets of course we got into this notion of irreducibility because it is the translation of primeness ok but then you might ask in what other ways it is useful so the answer is it is useful in several ways the first thing is that you know if you start with a prime ideal the advantage is if you go modulo of prime ideal you will get an integral domain ok you take any commutative ring and you go modulo of a prime ideal you will get an integral domain and in fact a commutative ring modulo of an ideal is integral domain if and only if that ideal is prime so prime ideals are good because when you take the quotient you get at least domain an integral domain and why are domains good because domains do not have 0 devices and you certainly do not want 0 devices because you see what are 0 devices that they are elements non-zero elements whose product becomes 0 but elements in our rings are thought to be thought of as functions ok so what it means is if you have a ring of functions which has 0 devices you are saying that there are 2 non-zero functions ok which when you take the product becomes 0 ok which is which never happens in for decent functions if a product of 2 functions is 0 at a point you expect one of them to at least vanish ok but so you know it is not good to begin with go modulo any ideal that might result in a quotient which is not an integral domain so in that sense if you go modulo only prime ideals then you get integral domains and why do you want rings to be integral domains that is because I told you you do not want this geometrically non-intuitive situation of having 2 functions which whose product vanishes at a point but neither function vanishes at a point which you do not expect to happen ok that is one thing. Then the other thing is I have already told you that you have for corresponding to prime ideals here you get irreducible closed subsets and what is so special about these irreducible closed subsets the answer to that is what is called as noetherian decomposition. So this noetherian decomposition tells you that you take any closed subsets you can always break it down into a union of finitely many irreducible closed subsets and this break up this decomposition as you might call it is unique provided you make sure that you are not repeating any of the subsets as being contained in one another. So if you so the important result is started the closed subset here the closed subset can be written as a union of irreducible closed subsets and this union is unique up to permutation of the factors of course if you assume that no component of the union is contained in some other component no piece of the union is contained in some other piece. So this is called the noetherian decomposition so what it tells you is that if you want to study any closed subset you can always break it down into irreducible closed subsets and therefore it is enough to study only irreducible closed subsets ok and of course since prime ideals since maximal ideals are also prime ideals this also tells you that a single point is irreducible and that is but that is anyway obvious ok. So the moral of the story is that we are somehow led to study just irreducible closed subsets ok and these are what are called as affine varieties ok. So the moral of the story is we study the irreducible closed subsets of affine space and we call them as affine varieties ok and open subsets of such irreducible closed subsets are called quasi affine varieties. So the word quasi is used whenever you take an open subset of whenever you go to an open subset use the word quasi ok. So let me make that statement so this theorem is quite easy to prove ok and probably the reason why I am putting this as a theorem is because it is also something that probably we will use the Null-Strahms-Arts ok. So this needs to be it is quite easy to prove but let me write down the following thing definition and irreducible closed subset An is called an affine variety and open subset of an affine variety is called a quasi affine variety. The whole object of the first step in algebra geometry is to study what is called as affine algebraic geometry and affine algebraic geometry is just the study of affine varieties and quasi affine varieties and I have already told you why affine varieties are important they are important because any closed subset any algebraic set any closed subset can be uniquely decomposed into a finite union of affine varieties. So you can analyze any closed subset in this way if you analyze affine varieties ok. So probably I will stop here and then I will indicate a proof of this theorem and proof of other statements that I made in the next lecture.