 Okay, so welcome to this first course on algebraic geometry, so let me begin by trying to tell you what algebraic geometry is all about in the in its greatest generality it is trying to study the geometry of the set of common zeros of a bunch of polynomials okay, so let me write that down, so that is in one line what algebraic geometry is okay, so beginning with this let us try to take the discussion further, so when I say a set of polynomials of course I should tell you where these polynomials are from okay and then I should also tell you by that I mean I should tell you these are polynomials in how many variables okay, so you must have a fixed number of variables and then I should also tell you about the coefficients of these polynomials because of course for example we are used to writing polynomials over real numbers which means polynomials with real coefficients and for example also with complex coefficients okay or sometimes we also look at polynomials with just integer coefficients okay, so the coefficients usually come from a ring okay, so what is happening is that you see in this way the ring of polynomials in several variables over a given ring comes into the picture okay, so you see, so what you do is so we pick or rather we are given, I should not say we pick, so we are given a collection of polynomials, let me say a subset of polynomials, so let me write them as f sub alpha of x1 etc xn, s is the set of all polynomials f sub alpha x1 etc xn and this is each one of these f alphas is a polynomial in a polynomial ring okay, so this ring is the ring the commutative ring that consists of polynomials in the variables x1 through xn and with coefficients in the commutative ring r okay, so of course I will not repeat this often we are always going to be worried only about commutative rings and commutative rings are assumed to be with unit element that is with one and we will always assume that homomorphisms of commutative rings carry one to one okay, so r is a commutative ring with one okay and x1 through xn are variables n variables and this is the polynomial ring over r in n variables and each f sub alpha is a polynomial in this ring and you take some of these subset of these polynomials, so this is alpha is some indexing set let me put it as lambda if you want okay in which case I can rather change the index to small lambda for more coherence, so this S is a subset of the ring of polynomials in n variables over r okay and we want to study the geometry of the subset of the subset E set of S, the set of common zeros of S and that is defined to be the set of all r1 etc rn n tuples of elements of r okay, so this is just r cross r n times this is just the Cartesian product of r taken with itself n times, so these are n tuples of elements of r each one each r i is an element of small r i is an element of capital R such that the if you plug in xi equal to r i okay that is if you substitute for xi the corresponding r i in this tuple okay or in other words if you substitute for the tuple x1 etc xn the values r1 etc rn in that order then you will get a and when you evaluate this polynomial okay then you will get an element of r and that has to be 0 and that should happen for every lambda okay then and only then is a point of rn in this set okay, so such that f lambda of r1 etc rn is 0 for every lambda belonging to capital R okay, so maybe this I will just write it as such that okay, so you see so basically what is happening is that you already have 2 objects here on the one hand you have rn which is r cross r cross r n times okay and on the other hand you also have the polynomial ring over r in n variables okay and what is happening is that I am given a subset S of the polynomial ring and I am associating to that subset the set of common zeros of that subset which is a subset of rn okay, so and the purpose of algebraic geometry is to study the geometry of this common set of zeros okay, so this is the general picture so you can see that already there are 2 sides to the picture there is one side which is the geometric side okay where you have the space where you are looking at the zeros okay and you have here the algebraic side which consists of essentially you are looking at the polynomial ring okay, so you can already see that there is a there is a commutative algebra side and there is an algebraic geometry side, so algebraic geometry is all about going from this to that and that to this based on properties of both sides for example you could have properties of this set which are geometric properties okay and they would translate into some properties connected on this side and the properties that are connected on this side ring theoretic properties or which may be ring theoretic properties or they therefore they could be ideal theoretic properties or more generally they could also be module theoretic properties okay because modern commutative algebra is not just the study of rings and ideals but it is also the study of modules because module the notion of a module generalizes the notion of an ideal and also that of a vector space at the same time and this is more is more versatile okay, so the properties on this side are geometric properties the properties on that side are commutative algebra properties and it is this dictionary it is setting up this dictionary which is the subject of algebraic geometry okay and but of course there are several things that I will have to explain to you first of all I have not so let me write that here, so this is the geometric side and this is the commutative algebraic side and in some sense therefore you can say that algebraic geometry and commutative algebra are kind of married to each other in that sense okay there are two sides of the same coin okay, so but then I will have to explain to you what do you mean by what do you mean by the geometry of a subset okay because that involves several things but for example it is at the base level it will involve some topology and then on top of that it will involve further properties topological properties and on top of that it will involve some properties connected with manifold theoretic properties okay and so on and so forth which we will try to explain but let me at this point go to something else okay. So first of all you know if you give me an equation like this forget even a set of equations suppose I give you a single equation over a ring okay it might turn out that this set may be empty okay that is the problem this set if it is going to be empty it does not there is nothing interesting to study because there is nothing to study first of all. So that can happen very easily for example you know if you take the ring R to be real numbers and you know if I take something like just one variable okay and if I take the and if I call that variable so I just take real numbers with one variable X1 and if I take the equation X1 square plus 1 then you know that it has no zeros in real numbers okay because obviously you know because the zeros are non real they are complex okay. So it is very possible that if you work over a general ring okay this particular set the zero set can turn out to be empty and then you are there is nothing to study okay. So at this point this point there is a I should say a dichotomy the subject actually breaks into or rather can be divided into two parts. So there is one question that tells you that if you are going to work over rings such that this is never going to be empty okay then of course you have these are good rings over which you can do this kind of study okay. The other thing is the question that what do you do if this ring does not have those properties so the answer to that is there is an answer to both. So if you so the first question the first part is you restrict to rings which are algebraically closed fields okay if you restrict to rings which are algebraically closed fields then this set is never empty for decent collection of polynomials okay I will explain what decent means later and you can do geometry okay. So that is called variety theory and that is what is usually done in a first course in algebraic geometry then the other thing is what do you do with a general ring okay for example I would like to have I would like to work over the integers for example you know questions like Fermat's last theorem it also involves an equation in three variables it is an equation with coefficients in integers and then you are trying to ask whether there are non-trivial integer solutions apart from the easy solutions okay then you also need to solve questions like that and to solve questions like that of course there are equations over integers which do not have solutions for example x square plus 1 equal to 0 is very well an equation with integer questions and it has no solutions in integers it does not have any solutions even in real numbers so how can it have solutions even in integers. So the question is how do you deal with such things so there is a part of algebraic geometry slightly more sophisticated area of algebraic geometry which deals with such things and that is called scheme theory and the scheme theory is the it is a modern language of algebraic geometry and that is usually what is covered in a second course in algebraic geometry because it involves far more machinery okay but what we will be doing this course is that we will be safely restricting ourselves to the cases when this set is non-empty okay so what I am going to do is I am going to tell you something about I am going to tell you something about solving equations or rather okay zeros of equations so well so the first thing is you know as you would have come across in any first course in algebra normally the usually we start with integers and then you know you extend them to rational numbers well of course I could also take natural numbers before this okay you can well maybe I can even do that it is not a big deal I can take natural numbers then I extend them to integers then there are there is a field of rational numbers then there is a field of real numbers there is a field of complex numbers this is how it goes and every time you have a bigger number system and that is because essentially you want to solve equations so number systems become bigger and bigger because you have some equations for which you do not have solutions so you have to make the make the make your set of numbers bigger okay so you know for example natural numbers the counting numbers 1, 2, 3, 4 does not have 0 so an equation like x plus 1 equal to 1 which has a solution x equal to 0 will not have a solution here and an equation like x plus 1 equal to 2 or I mean x plus 1 equal to or rather x plus 2 equal to 1 which is a solution x equal to minus 1 also does not have a solution here so you are forced to go to integers and then you can have an equation of integers which does not have a solution integers which have a solution only in rational numbers for example things like 2x equal to 3 it is very well in equation over integers okay but the solution is x equal to 2x equal to 3 the solution is x equal to 3 by 2 which is a rational number so you have to extend so finally what happens is that you see that you come to fields and you come to field extensions okay and the point is that every time you ask the question when you get a bigger number system you ask the question well if I write a polynomial in that with that coefficients will the zeros of the polynomial always lie there and if the answer is yes at some point then that is called an algebraically closed field okay so the fact is C is an algebraically closed field okay C is an algebraically closed field. So and what does this mean that is so that is any polynomial in one variable with C equations has a 0 in C okay and in fact you know if a polynomial has a 0 if you call the polynomial as f of x of course it is one variable so I should just call it as f of x where x is a variable and if it has a 0 which means it has a value lambda such that f of lambda equal to 0 then you know x minus lambda is a factor of the polynomial and by the division algorithm you know that the polynomial becomes x minus lambda times another polynomial of lower degree of 1 degree less okay and in this way you can continue and well the polynomial of lower degree that you get that also will have a 0 in the complex numbers and you can factor that linear factor out and if you do like this what it will tell you is that finally any polynomial can be completely split into linear factors okay that is what it says and this is the property of the field of complex numbers which makes it algebraically closed and in general this is the definition of what an algebraically closed field is an algebraically closed field is a field such that you take any single polynomial in one variable over that field then all its zeros are in that field that it means that you can find all the zeros in the field itself you do not have to extend the field you do not have to extend your number system to something bigger where you will get the zeros okay. So basically what it tells you is that you know if you are so what it tells you if you go back to our setup what it tells you is that if I take r equal to c if I take my ring to be c which is in fact field okay and if I take n equal to 1 then I have c here and here I will have c of x okay polynomial ring in one variable and then it tells you that you would give me if you give me a single polynomial if this s is set s is a single polynomial then the zero set of that is non-empty in fact it will have the number of zeros will be equal to the degree of the polynomial but of course you have to count the zeros with multiplicity okay you have to count the zeros with multiplicity after all the polynomial may be x minus 1 to the power of 5 okay and then well if you want to if you want to count the zeros with multiplicity then you can think of them as 5 zeros but well if you think of it as a subset here you get only one point namely the point 1 okay. So what this tells you is if you are working on an algebraically closed field and there is only one variable involved and you are looking at only one polynomial then you end up in a situation where this set is non empty okay now the question is mind you our original question is we are not looking at a single polynomial we are looking at a bunch of polynomials okay and this need not even be finite this collection of polynomials need not even be finite and not only that we are not looking at a polynomial in one variable we are looking at a polynomial in several variables okay what the algebraic closure property tells you is that if you are looking at a polynomial in one variable and if you are coefficient ring is an algebraically closed field then if you look at the zeros of a single polynomial then it is not going to be empty okay but we want something very very gentle to happen okay the answer to that is so you might expect that should you put something more some more conditions than just the field being algebraically closed for such a thing to happen and the answer is no the amazing answer is yes the amazing answer is this itself will ensure that so long as the set S is good in a decent way if R is an algebraically closed field then Z of S can never be empty okay and this is a very deep fact and this is called the this is one form of the Hilbert Nullstahl and Salz okay so let me write that down so fact so if we take for R an algebraically closed field then for F in Rx Z of F is non-empty of course provided F is non-constant of course you know when you are looking at zeros of polynomial suddenly you are not looking at a constant polynomial what that means is if you are looking at a constant polynomial if that constant is non-zero then there are no zeros and if that constant is zero then the whole space satisfies the condition okay so if you take a non-zero constant and consider that as a polynomial then it has no zeros so it is zero set is empty if you take zero as a constant polynomial then it is zero set is a whole space okay so what it says is that if you take for R an algebraically closed field and this set S to be a singleton consisting of only one polynomial which is of course non-constant then the zero set is non-empty so you ensure that this is non-empty so you can do some geometry okay but then here is the important thing so what we want is not for one variable X but we want it for several variables and we do not want it for a single polynomial in several variables but we want it for a whole collection of polynomials in several variables okay so that is the that is our deep requirement and that is what the Nulstel and Schatz as it is called Hilbert's Nulstel and Schatz that is what it promises so let me write the following down so first let me say the following thing suppose so here is a very simple let me call it a lemma I will write a very simple lemma if the set S is such that that does not that does not exist a finite subset let me write f lambda 1 f sub lambda m and polynomials g lambda 1 g lambda m in the polynomial ring such that sigma f lambda i g lambda i is equal to 1 i equal to 1 to m then so I should say and only then so I should say the following thing so there is one part of the lemma which is very trivial there is the other part of the lemma which is actually the Nulstel and Schatz when R is an algebraical course field so let me say that correctly I have to modify this statement a little bit so let me make a small modification if the set S is such that z of S is non-empty then that cannot exist a finite subset such that this is equal to 1 so this is a statement okay so I am saying that suppose so you see the whole point is the following the whole point is we are trying to look we are trying to we are trying to study this set of common zeros okay we do not want that to be empty you do not want that to be empty and the statement I am making is that if it is non-empty then no finitely no finite subset of S can generate 1 okay no finite subset of S can generate 1 that means you cannot get finitely many elements of S and finitely many polynomials from the ring of which this capital S is subset of such that you take multiply them and then add them you get 1 that cannot happen that is obvious because you see if the 0 set is non-empty that means there is a value small r1 etc small rn in the 0 set and this value small r1 etc small rn when you substitute it in each of these f lambdas it is going to vanish okay so in particular if I substitute it in this relation on the left side okay the left side is going to vanish and I will get 0 equal to 1 and in a ring if 0 equal to 1 then the ring is a 0 ring okay it has only 1 element which is 0 ring and certainly we are not interested in working with a 0 ring okay so if the ring is not the 0 ring then what this tells you is that if the 0 set is non-empty this can never happen okay it is obvious this is a very very simple thing okay but the converse to this that the converse to this holds when the ring is an algebraically closed field is the is one form of the Hilbert-Nultz transfer so let me write that the converse is true if r is an algebraically closed field and that is that is the that is one form of the Hilbert-Nultz so maybe so let me keep it like this let me keep it like this let me not state it the converse will be that if the subset S if r is an algebraically closed field and if the subset S of polynomials in the polynomial ring over r in invariables is such that no finite subset of that can generate 1 then the 0 set defined by that subset is non-empty okay that is one form of the Nulstra Nsats which is usually called the weak form of the Nulstra Nsats okay so of the Nulstra Nsats so maybe I will write it down so let so what is that so if k is an algebraically closed field and S is a subset of polynomials over k in N variables of course small n is greater than or equal to 1 okay then such that such that that does not exist a finite subset subset let me write f1 etc fm in S and polynomials so let me continue here g1 etc gm so this is m again in the same ring such that sigma fi gi is equal to 1 i equal to 1 to m then the 0 set defined by S in kn is non-empty you can find solutions. See what you must understand is that the Hilbert Nulstra Nsats is a grand generalization of the property of being algebraically closed you know if I put in the Hilbert Nulstra Nsats if I put n equal to 1 then I am looking at polynomials only in 1 variable and if I take the subset S capital S to be a singleton set okay then this statement is obviously true if for an algebraically closed field see what you must understand is Hilbert Nulstra Nsats when you put n equal to 1 is true just by the definition of algebraically closed field because what happens when you put n equal to 1 and when you take the set S to be a singleton consisting of only one polynomial and then if you have this condition that then this condition will become that polynomial multiplied by no other polynomial gives you 1 which is the same as saying that that polynomial is non-constant okay then you are saying that there is 0 so that polynomial in k1 which is just k exist so you are just saying that every polynomial has 0 which is the definition of what an algebraically closed field is. So what you must understand is that Hilbert Nulstra Nsats is a grand generalization of the definition of algebraically closed and it is a very very important theorem and this is the theorem that guarantees that if you are working over algebraically closed fields then you can really do geometry okay. So and this is called the weak form we will come to the stronger form later on okay and I will see later on if I can hint at proof of the Hilbert Nulstra Nsats usually a proof of the Nulstra Nsats is given in a course in commutative algebra but then there is also a way of looking at that proof completely in algebraic geometric terms. So this is something that keeps happening that you must always keep at the back of your mind there are many things that can be said in this on the language on this side this is the algebraic geometry language in this there are the same things can be said with the language on this side this is the language of commutative wings and modules and ideals and things like that okay. So proof here will involve you know ring theoretic arguments ideals and homomorphisms and modules and things like that whereas the same proof when you translate it here it will have geometric meaning and it is this it is trying to go from one to the other that really enriches both the sides okay. So okay so incidentally I should tell you what this Nulstra Nsats is so Nul this is German you know Hilbert was a German mathematician of course one of the greatest of all time and Nul stands for 0 and Steyler stands for position or point and Sarts means theorem or statement so it is actually so if you translate it properly it means Hilbert's theorem on zeros of polynomials of a bunch of polynomials okay fine. So okay so what we are going to do is so from now on now on we will always work over an algebraically closed field K that is that is that is we take R equal to K so if you want for convenience you can even think of the algebraically closed field as complex samples so that you know if it makes it easier for you to think about things and visualize things okay. So what we are going to do so our picture becomes so our picture goes from that generality into something more concrete so we have the following picture on the commutative algebra side we have K of x1 etc xn this is the polynomial ring in n variables and on the geometric side we have Kn which is k x k x k n times okay. Now well how do you think of Kn you are used to Kn from linear algebra I mean the always one would the simplest way one would look at it is a vector space of dimension n okay for people who have done module theory Rn can be thought of as the n dimensional free module over R okay so in fact we do not use the word dimension for modules so I should amend my statement to Rn is the free module of rank n over R okay. So dimension is usually reserved for fields so Kn is an n dimensional vector space over K a module over a field is a vector space okay so but it is not the vector space properties we are interested in so you see the vector space the properties of vector space are you have the 0 vector and then you know you have addition of vectors and so on and so forth. So in that sense you know the vector space studying the vector space properties is literally studying linear algebra but that is not what we are interested in is actually trying to study the points of the space okay without any regard to the vector properties okay so you so it means that all points of the space are alike I mean if you take the plane and throw away the vector space structure that means you take the plane and how do you get the vector space structure you have to first have an origin because for a vector you need an initial point and then you need a terminal point so normally what we do is we have an origin and every other point to every other point we associate the vector which starts at the origin and goes to that point we call it the position vector of that point and then we study all these position vectors and add them as usual with parallelogram law and then so on and so forth. But what we are going to do now is not call any not worry about doing all this we are going to think of all points as equal as one and the same so think of Kn but do not think of the axis do not think of the origin as the origin of the vector so you just think of it for example when is equal to 2 think of the plane but without the 2 axis there is a point called the origin but that point is not you do not try to for every other point you do not try to draw position vector and think of it as a vector. So you know this process of trying to think of Kn as a space not of points but not as space of vectors is what makes it into what is called an affine space okay so the affine space Kn is different from the vector space Kn in the affine space you are only worried about the points not about the vectors okay so you will see often in algebraic geometry people use the word affine and that is the whole point okay. So the first important thing is do not ever think of these vectors okay though it is a vector space of dimension Kn but that is not what we want right so well so this is the affine space literally and given a subset S to get the 0 set of oops yeah that is right here so here is our this is the picture we are going to look at and we are going to see what is there on both sides okay. So the first thing that I am going to do is so that brings us to the following question that brings us to the following question what are we going to do with this set we have thrown away the vector space structure is just a set what are you going to do with it so the first step is to give it a topology okay and that topology is called the Zariski topology okay so I will explain what that topology is so well so let me write that so let me write the following thing Kn along with the Zariski topology is called affine n space over k and denoted Kn okay so the first thing that we do to get some geometry on the left side is first of all have at least a topological space okay so we make Kn into a topological space what does that mean it means that you have to specify a topology on it and specifying a topology on it means what you try to give a collection of subsets which you make all as open subsets and this collection has to satisfy the axioms for a topology and what are the axioms for a topology the whole space should be an open set the null set which is the empty set should be an open set an arbitrary union of open sets should be open a finite intersection of open sets must be open so these are the four conditions for a collection of subsets of a topology of a space or of a set to make it into a topological space okay so so what so I saw that explain what the open sets are okay but then you know in when you go to topology there are there are two ways to approaching a topology one way is by open sets the other way is by using closed sets okay because closed sets are just complements of open sets so I can also give a topology on a set by giving a collection of subsets called closed sets but then the conditions will be complementary okay because of D Morgan's laws so the conditions will be that the whole space is a closed set then the empty set that is the null set is a closed set any if you take a finite collection of closed sets their union is again a closed set okay and if you take an arbitrary collection of closed sets their intersection is again a closed set these are just translations using D Morgan's laws for the corresponding axioms for open sets so basically what I have to do is that I have to tell you either I should either give you a collection of open sets or I should give you a collection of closed sets which satisfy the corresponding axioms and what happens in geometry is that is easier to begin with by looking at collections of closed sets and guess what what are going to be the closed sets the closed sets are going to be just common zeros of functions of this type so you see the so the idea is the following you look at this space the look at kn think of these as functions on kn of course you take any polynomial here and you take a point here if you evaluate the polynomial at that point you get a value so certainly a polynomial the elements here which are polynomials are certainly functions on this space and what you do is you call a subset here to be closed if it is the in the zero locus common zero locus of a bunch of polynomial functions which is exactly what this is okay so the moral story is that if you declare subsets of kn of this form zero loci of a bunch of polynomials common zero loci of a bunch of polynomials call them as closed sets then you get the Zariski topology that makes this kn into Zariski topology so what that will tell you is that what are the open sets open sets will be loci where certain where functions do not vanish the loci where the functions vanish for example the loci where locus where a single function vanishes is a closed set and the complement of that locus where the single function does not vanish is an open set okay and this is in tune with our common sense okay because if you take a real valued function on some subset of Rn or you take a complex valued function on a subset of n dimensional complex space Cn then the set of points where the function takes the value zero is going to be a closed set provided the function is a continuous function because it is a point is always a closed subset and if a function is continuous the inverse image of a closed subset is closed therefore the set of points where the function is zero is exactly the inverse image of the point which is zero and that has to be closed okay. So it agrees with our usual intuition so the idea is to give the Zariski topology in that way okay so we will look at that in more detail in the next lecture okay.