 Okay so what I am going to what I am going to do now is you know in continuation with our discussion on the importance of local rings I am going to explain the notion of non-singularity okay which is in you know analytical language classical analytical language you are trying to say when something is a manifold okay so or when something is smooth okay so the so in other words you know the idea is that an object is smooth at a given point if the dimension of the object is the same as the dimension of the tangent space at that point usually what happens is that you know the dimension of the tangent space will be more and if the dimension of the tangent space is more at a given point then the dimension of the object that you are studying then that point is a singular point it is not a smooth point. So you know you know for example its smoothness in terms of in the language of classical or you know in the language of analysis so you know if you take a so you know if you take a if you take a surface and you know you take a point on surface then the surface is so suppose you are looking at you know the usual topology okay and you are in nuclear in space and you look at a surface for example surface of sphere or surface of cylinder or whatever it is okay then you know if you look at the so there is a dimension of the surface which is say M okay so in so if you are I mean if you are looking at a surface in if you want in so if you are in R3 okay real nuclear in space and you are looking at a surface it will have dimension 2 okay a curve will have dimension 1 alright. So a surface will be dimension 2 a curve in 3 space will be dimension 1 and if you take a point on the surface alright and draw you try to draw all possible tangents to the surface at that point you get the tangent space at that point and if the surface is smooth at the given point P okay then what happens is that the tangent space will also be 2 dimensional okay so what will happen is that you will get a unique tangent plane you will get a unique tangent plane at the point P okay. And if the surface is not smooth at a given point what will happen is that if you look at the tangents the tangents space could have higher dimension so for example you know if you take something like a cone you take something like a cone and if you consider the vertex to be the point okay then what happens is that if you how do you calculate the tangent space it is a space spanned by all tangent vectors and what are tangent vectors I mean these are I mean they lie on tangent lines passing through the point okay. So you know if I try to draw tangent lines passing through this point I can easily draw three you know I can easily draw three linearly independent vectors I can draw three different lines which are given which lie in the directions of three linearly independent vectors okay. So you know if the cone is like this right I can draw one like this and then I can draw one like this and then I can draw one like this okay so I can easily draw three of them they I get three linearly independent vectors and therefore you know if you take the take all such vectors the space that they span will be R3 so you know tangent space here this is the vertex of the cone has tangent space isomorphic to R3 as a vector space because you are easily able to find three linearly independent vectors and you know and these are vectors in three space alright and therefore the subspace that they span will be all of three space. So what happens is that the so here the dimension of tangent space at the point P is three which is greater than or equal to two which is a dimension of the surface the cone is two dimensional but you take that point which is a vertex of the cone there you look at the tangent space namely the space the vector space span by all tangent vectors the tangent space is three dimensional so the tangent space dimension is more than the dimension of the cone and so this is this tells you that P is a singular point P is a singularity or singular point or a non smooth point sometimes in classical I mean in analysis in the language of usual analysis you also say it is a non-manifold point it is also called as a non-manifold point alright so the idea is that at any point if you want to check whether it is a smooth point or not okay what you do is that you try to look measure the dimension of the tangent space at that point if the dimension of the tangent space is equal to the dimension of the object the dimension on of the space on which are considering the point then the point is a smooth point otherwise the dimension could very well be more if it is more then the point is not a smooth point it is a singular point so it is also the case in the case with the line I mean with the curve see if you take a point like this on the curve then you know if it is a smooth point I will get a unique tangent direction to the curve at that point and the tangent space will just be a single line it will just be the line span it will just be the space span by a single vector so you see the tangent space you will get a unique tangent line you will get a unique tangent line at the point P the tangent space at the point P has dimension 1 and that is equal to the dimension of the curve on which the point P is lying that tells you that the point P is a smooth point okay but however you know if I take a curve which is not a smooth curve then things can be different for example you know if I take something like you know on the plane I can easily draw a curve which is not smooth so you know for example if I purposely draw something like this with the kink here if I draw a curve like this and take this point then at this point if you calculate the tangent space you will easily see that you can draw two tangents you know from the if I approach from the left okay the I will get a tangent like this if I approach from the right I will get a tangent like this at this point okay and these two are two linearly independent directions therefore the dimension of tangent space at this at this point is two time is two whereas the point is lying on a curve which is one dimensional so the dimension of tangent space is more than the dimension of the curve the dimension of the object at the on which the point lies and that tells you that that point is not a smooth point okay. So here what happens again P is a singular point as the dimension of the tangent space of the tangent space at the point P is 2 which is strictly greater than so here also I should not put greater than or equal to I should put strictly greater than 3 is strictly greater than 2 and here this 2 is strictly greater than 1 which is the dimension of the curve this is the so this is the curve on which the point is lying the curve is one dimensional okay but the tangent space at that point is two dimensional where the tangent space dimension exceeds the dimension these are the singular points okay. So this is what happens in so I have of course looked at dimension two dimension one you can therefore say if you are looking in if you are looking at an n dimensional hypersurface okay in say some which has to be thought of in some Euclidean space of dimension greater than n okay then at a point how do you save the point is smooth or not what you do is that you check the dimension of tangent space at that point if it is strictly greater than the dimension of the space on which the point lies then it is not a smooth point if it is equal then it is a smooth point okay. So this is the idea from at least from calculus and geometry usual analysis okay now the analog for this in algebraic geometry is of course there and everything is I mean this business of estimating the this business of calculating the tangent space and its dimension at the point P is done by looking at things connected with the local ring at the point okay. So here is a definition so now I am switching from you know some classical or analytic analysis based situation I am going from there to algebraic geometry so you know so I am going to take the following thing I am going to define when a point of an affine variety is smooth I mean when it is non-singular okay and when it is singular right so here is so in algebraic geometry so how do you do it so what you do is that you take x to be an affine variety okay it x to let x be an affine variety P a point of x how do you define that P is a smooth point or a non-singular point okay. So for that what you do is you do the following thing so let x sit inside some an affine space over k k is of course an algebraically closed field x is an affine variety so it is an irreducible subset it is isomorphic to some irreducible closed subset of some affine space so this by definition x is isomorphic to an irreducible closed subset of affine space okay and then you know you have the you have the ideal of x you have the ideal of x which is which is which is a which is the ideal inside the affine co-ordinate ring of an okay which is actually you know well it can be identified with the polynomial ring in n variables if you want okay if you take capital X1 through capital Xn to be the coordinates coordinate functions then you take the polynomial ring in the in those coordinate functions that is the affine co-ordinate ring of affine space okay and x is an irreducible closed subset so it corresponds to a prime ideal so if x is its prime ideal and of course you know the affine co-ordinate ring of x is given by the affine co-ordinate ring of affine space namely the polynomial ring mod the ideal of x there is a finitely generated k algebra which is an integral to main okay but the point is more importantly the point is about this ideal see the ideal of x this is an ideal in this polynomial ring which is noetherian ring so it is finitely generated okay so let us look at a set of generators so f1 say g1 etc up to gm the ideal generated by finitely many polynomials okay this is true because in a noetherian ring any ideal is finitely generated and the polynomial ring is noetherian because that is as you know Hilbert Hilbert's basis theorem or I mean Oithers theorem so you choose a set of generators alright now what you do is you do the following thing you compute calculate the Jacobian of this m tuple of functions with respect to these n variables okay you get an m by n matrix of polynomials matrix with polynomial entries and that you evaluate at the point p okay and then you get a numerical matrix a matrix with entries in the field and calculate its rank okay so this is the this is the thing that you will have to do so you calculate rank of Jacobian of g1 gm with respect to Jacobian with so it is with respect to these variables so let me just write it like this calculate rank of the Jacobian of all this at the point p calculate this number okay so what you are doing is well basically what you are doing is you are taking g1 partially differentiating it with respect to x1 and then you know evaluating it at p okay and then you do it so on with g1 with respect to well all the variables xn and then now you repeat it with g2 with respect to x1 at p and this is do g2 at xn okay and then you do it like this you calculate this you have this matrix okay now mind you when I say partial derivative you do not have to think of derivative in the sense of calculus because derivative in the sense of calculus will require a limiting process but do not think of it as derivative in sense of calculus but think of it as formal derivative because you know you can always take any polynomial in so many variables and you know how to define the derivative okay using the usual rules of formal rules of differentiation so in calculating these derivatives there is no need for I mean you are not going you are not actually computing derivative in the calculus sense okay but you are you are directly using the formula for the derivative which so you know formulas for differentiation of polynomials can I mean these are the same formulas that you get in calculus but then they make sense even without calculus you take those differentiation formulas as the definition rather than getting them by using a limiting process okay. So you have this matrix okay these are matrix you calculate this rank okay and what you do is you now you do the following thing we say P is a non-singular point of x if so this is the definition the rank of the Jacobian of the generators okay of the ideal of x at the point P should be equal to the co-dimension of x in An and what is co-dimension it is co-dimension is just dimension of An-dimension of x so it is just n-dimension x so co-dimension of a subspace is just the difference of the it is you take away the dimension of the subspace from the dimension of the ambient space okay the ambient space here is affine space An the subspace is x okay which is embedded sitting inside An and you take the dimension of the ambient space minus the dimension of x that is called the dimension of the bigger space minus the dimension of the smaller space is called the co-dimension of the smaller space in the bigger space okay. So the condition for P to be non-singular point of x is that the rank is equal to the co-dimension alright so this is the condition and by the way rank of the Jacobian at the point P is actually rank of this matrix okay so this is equal to n-dimension of x so this is the condition for P in x to be non-singular or for it to be a smooth point okay. Now the beautiful thing about varieties is that you know they are not always smooth okay they will involve singularities but the point is that where the set of points which are singular will form a very small subset where if you take the set of points which are non-singular that will be a huge open set it will be a dense open set okay. So you know if you want to compare if you want to compare a variety with classical smooth object in a smooth object in analysis okay the comparisons should be the if you want to think like that what you should think of is that a variety is something like a smooth object on an open set plus your boundary which is the complement of the open set which will have singular points. So you know something like a cone okay if you throw away the point which is the vertex of the cone the rest of it is all smooth okay and that is a dense open set and the boundary is this point which is a singular point. So a variety also looks like that there is a big open set which is full of smooth points okay where it is like a smooth where it is the analog of a smooth object in analysis okay these are all the points where the dimension of the variety is the same as the dimension of the tangent space okay and that is what is actually being said in this definition but then you will have to unravel these definitions and try to literally see that this is the same as that okay but there is some translation that one has to do okay which we will do okay. So when you think of a variety what one needs to remember is that there is an open set dense open set where it is smooth okay where it is like a manifold a smooth object in analysis and then there the complement of the open set is a closed set it is a boundary and that closed set will consist of singular points okay. Of course there could be varieties which are totally smooth that also can happen and such varieties which are totally smooth are called non-singular varieties right. So now let me so this definition looks a little involved okay but the advantage of this definition is that you can make some calculations okay so for example you know so if you want to apply it so let me take the example of hypersurface in an okay what is a hypersurface it is a co-dimension one sub variety of an it is a co-dimension one sub variety and we have seen this a co-dimension one sub variety means that it is a it is an invisible co-sub variety of dimension one less so it is dimension n-1 and we have seen that this will happen if and only if the ideal of the variety is generated by a single non-constant irreducible polynomial okay. So you know so if I call that as x if we call this as x is equal to hypersurface in an so ideal of x is generated by f where r let me use g itself g an irreducible non-constant polynomial so we have seen this and now what is if you take a point p on this hypersurface when is the point p smooth so if I apply this condition so I will get I will have to look at rank of the Jacobian of g which will be just at the point p and that is just going to be that is equal to rank of 2g by du x1 2g by du xn okay. So you are just looking at all the first partial derivatives of that polynomial and then you are evaluating in them at a point p and this is equal to the co-dimension of x and what is the co-dimension of x is 1 okay because x is a hypersurface so the co-dimension is 1 if and only if p is a smooth is a non-singular point of x. So you know if you want the hypersurface to be non-singular that means you want all the points on the hypersurface to be smooth non-singular points then the condition is that all the first partial derivatives of g should not simultaneously vanish at any given point okay and such a polynomial is called a non-singular polynomial okay it is called a smooth polynomial. So x is non-singular if so x is non-singular means x is every point at every point of x is non-singular that is what it means okay. So x is non-singular if and only if the rank of this is always 1 okay that means that given any point at least one of the partial derivatives should not vanish okay at least one of the partial derivatives dou g by dou xi never vanishes does not vanish at each point of x. So you know it is very easy to check that a hypersurface is you know is non-singular and is smooth right. So examples of these things are well examples of smooth I keep using the word smooth but in algebraic geometry the word smooth is reserved for something more general than this so the word that we use is actually non-singular. So examples of non-singular hypersurfaces are well hyperplanes which are given by you know f is linear homogenous f is a need not be homogenous well f is just a linear polynomial you take a linear polynomial right. So that is f of yeah so f of x1 through xn is just sigma alpha i xi minus some beta 0 i equal to 1 to n something like this and of course you know I am not looking at the case when all the alpha is and the beta and this beta are all 0 I mean so I really want a linear polynomial which is not 0 polynomial right or a constant polynomial so at least one of the alpha is survives is non-zero. So if you know if a certain alpha j survives then dou f by dou j if I take the partial derivative f with respect to xj I will get the alpha j which is non-zero so that will never vanish at any point on this hyperplane. So these are hyperplanes are non-singular and then you can you can you can you know take things like n equal to 2 and you can take f to be x squared f is y squared minus 1 okay then well this is the circle in A2 you know circle in A2 and you know if you calculate of course I am taking the variables as x and y okay well if I want I use standard notation I should take x1 and x2 alright. So let me do that let me write it as x1 and x2 then you see that you see if I calculate dou f by dou x1 dou f by dou x2 I get this I will get 2 x1 I will get 2 x2 okay and now you know now you want that any at any point of the circle okay one of these should not vanish so when will both vanish both will vanish at the origin okay both will vanish at x1 equal to 0 x2 equal to 0 which is the origin in A2 alright but then the origin is not a point on the circle so it does not give me so what it tells me is that this is not going to vanish at any point of the circle but there is a there is a catch the catch is that your is one issue this K could be characteristic 2 K could be an algebraically closed field of characteristic 2 in which case this will be identically 0 because in characteristic 2 2 is 0 so if K is an algebraically closed field of characteristic 2 then this x1 squared plus you know this will become you know this will vanish so you have to be careful about the characteristic of the field where you are doing this computations okay and so let me put characteristic is not equal to 2 for safety okay so whenever you get this some integer coefficients you have to really worry about whenever you are talking about something vanishing okay and you are in algebraic geometry you are working on algebraically closed field you should remember that it could be a any characteristic so if the characteristic divides one of these coefficients you are in bad shape because it will just vanish okay so if you take characteristic not equal to you know your circle is smooth and well in fact if characteristic is 2 something more serious is happening f is first of all not irreducible if you are in characteristic 2 this is the same as x1 plus x2 plus 1 the whole square because in characteristic p a plus b whole power p is a power p plus b power p okay so you know x1 plus x2 and minus 1 is the same as plus 1 in characteristic 2 so this is actually f becomes square of a linear polynomial in characteristic 2 it becomes x1 plus x2 plus 1 the whole square okay and so it is not it is not even irreducible alright. So you have to worry about characteristic alright of course if you are working over complex numbers one does not worry about these issues but then whatever algebraic geometry we are discussing about is over an algebraically closed field and you know you can have algebraically closed field of any characteristic right. So this is not equal to 0 for any p in the zero set of f which is the circle in a2 and well now you know you can start this is with one equation you can start looking at objects given by several equations okay and start checking which are which are the points that are smooth points and whether the smooth points that is a non-singular points or all the points are you get some points just singular some points which are non-singular so that way this definition is useful for computation alright but you know the problem with this definition is that there are two problems of this definition so the first problem is I have only defined it defined non-singularity for a point of an affine variety okay. I have not defined non-singularity at a point for any variety because any variety in general could be non-affine okay it could be quasi-affine it could be projective it could be quasi-projective. So well I but anyway I can get over this problem by saying that well any of I know that any variety is covered by finitely many open sets which are isomorphic to affine varieties therefore you give me a point on any variety I can find an open set surrounding that point which is isomorphic to an affine variety so that point is now lying on this open set which is an affine variety and then I can say it is non-singular or not based on this definition alright. So I can get over this problem of extending this definition of non-singularity to any variety just because of the fact that any variety admits a cover finite cover by open sets which are isomorphic to affine varieties that is an issue that is easily resolved but that is more something more serious the more serious thing is this numerical business here you see there is lot of ambiguity here you see the same affine variety could be embedded in different affine spaces okay I could embed the same affine variety in AM I could I could also embed it in some AM if I embed it in a different AM then the ideal will change okay so this ideal is this ideal depends on the embedding see this ideal of X is the ideal in the affine coordinate ring and that affine coordinate ring it is in the ideal of the affine coordinate ring it is an ideal of the affine coordinate ring of the affine space in which X is embedded but if I change this affine space where X is embedded then I am changing this ring therefore this ideal also changes this ideal is not an invariant see what is an invariant this is this is the only thing that is an invariant for an affine variety the affine coordinate ring is an invariant okay whether I embed X as an irreducible close of sub variety of an or am any affine space if I calculate this I calculate the affine coordinate ring of X then you know that is an invariant because that is also equal to O X there ring you know the regular functions on X but the ideal can change okay so that is the ambiguity of the embedding if you change the embedding the ideal will change alright that is the first ambiguity what is the second ambiguity second ambiguity is here when I write the ideal I write a set of generators for the ideal the same ideal can have different sets of generators the sets of generators are by no means unique okay so if I change these generators then this you know this Jacobian matrix itself will change instead of well even the number of generators I have I do not know G1 through Gm may be one set of generators I may find these are M generators I may find some different number of generators and they may be all completely different polynomials and again I do this computation what is the guarantee that my definition is consistent okay for the same ideal if I keep the same embedding and therefore my ideal is fixed if I take a different set of generators what is the guarantee that if I compute this I will still get the same rank so well thanks to God that is the case okay and the point is why does that happen the one way to see it is using the language of local rings okay so this is where the power of local rings comes in to tell you that this definition is independent of the embedding it is independent of the generators for the ideal that you choose you will this definition is absolutely correct it is not going to fail you it is not going to become inconsistent okay so it is to that end that I am going to state something now so here is a this is a fact which was discovered and proved by Oskar Zariski who is who can very well be called the father of or even the grandfather of algebraic geometry father the grand man of algebraic geometry he was he was a person who initially wrote papers in the Italian style where he was where the papers were the proofs were based more on you know geometric ideas and there was there was no proper rigor but then being you know commutative algebraist and field theorist he developed the necessary commutative algebra and field theory to translate all that into the modern language and then he was able to rewrite everything gradually and show that everything is can be you know made rigorous using commutative algebra so this is so here is a theorem so here is a theorem let X be a variety then and P a point of X then P is non-singular if and only if dimension over k of mp mod mp squared is equal to dimension of k so this is the correct statement so what I want to tell you about this is you see you have you have the local ring OXP ok this is the this is the local ring X at the point P ok and it is a local ring so it has unique maximal ideal and that maximal ideal is given by this mp ok so with so this is with unique maximal ideal mp right and now what you must understand is you know the dimension of the local ring you know that this is the same as dimension of X there is something that we already know ok the dimension of the local ring is the same as dimension of X and so you know you can here I can add if I want it if I want to reflect the point P I can also write dimension of OOXP and so you know if I if I if I remove the if I do not look at the central term then I have condition which seems to have any only to do with the local ring I am just saying that the dimension of the local ring is the same as the dimension of m mod m squared so you know what I want to understand is the local ring modulo the maximal ideal will give you just k ok and what you must understand is that if you look at mp squared ok this is the square of the ideal mp so you know this is this is you know this is this is just consisting of elements of the form you know sigma ai bi i equal to 1 to some l where ai and bi are in the maximal ideal ok this is just the squared ideal alright sum of finite sum of products to taken at it taken to at a time from the ideal ok from this maximal ideal mp and you know this is if you if you if you look at it if you look at and of course this is contained in this is contained in mp ok this is certainly contained in mp because if you take two elements of this ideal and multiply them out the product is certainly the ideal this sums such finite sums are also here so this is contained inside this but the point is that if you look at mp mod mp squared ok this is a k vector space ok this is a k vector space and it is a k vector space is because of this fact because O mod m is k ok so this is a this is a k vector space it is a module over k well in fact you see you take O xp modulo mp you can you can define scalar multiplication like this by simply you know f so you know f bar, well g bar going to well f bar, g bar I think this should give you we should give you an obvious map which will make mp mod mp squared a module over O xp mod mp but O xp mod mp mp is just k therefore mp mod mp squared is a module over k so it is a vector space ok so you can well you know you can check that this is this map is well defined where this f bar is where f is an element of the local ring so it is a germ of a regular function at the point p and f bar is its image actually f bar is just that regular function evaluated at the point p ok it is just evaluation and here you are taking g bar is just the image of a g is just a regular function in a neighborhood of p germ of a regular function neighborhood of p which vanishes at the point p so g is in mp and its image in the quotient mp mod mp squared is g bar so you are reading g up to you going mod mp squared is just reading only the linear term when you go mod when you go mod m you are evaluating at the point see these are all actions that are going on at the with the elements here in the local ring what are the elements in the local ring the elements in the local ring are regular functions germs are regular functions how do you for how do you get this quotient isomorphic to k what you do is give me a regular function at that point you evaluate it at that point you give me a germ of a regular function here you evaluate it at that point that will give you a map from this to k its kernel will be exactly mp all those germs of those regular functions which vanish at the point p so the quotient will be k so this isomorphism is just evaluation arises just by evaluation of a germ of a regular function in a neighborhood of p at the point p okay and what is this what does m mod m squared stand for it is you take a function which vanishes at the point p germ of a regular function which vanishes at the point p that is what a function that belongs to mp means and reading it mod mp squared means that you take literally you know it is derivative because you know you are cutting if you read mod m squared that means you are not you are only taking the linear term you are not taking the degree 2 term onwards so in a sense this corresponds to taking only reading only the linear term alright and that is correspond and you know the first order term always is the derivative so going m mod m squared is reading of the derivative in a certain sense okay and therefore this is how m mod mp m mod m squared becomes a k vector space okay and it is it is certainly a finite dimensional vector space because you know after all o xp is a noetherian ring you know this local ring is a noetherian ring and this ideal mp in this noetherian ring is finitely generated you take a set of generators and take their images here they will give you generators for this quotient so it is a vector space which has finitely many generators and therefore it is a finite dimensional vector space it is a vector space which has a finite spanning set so it is a finite dimensional vector space okay. Therefore this is a finite dimensional vector space you calculate its dimension okay and this dimension if it is equal to the dimension of x then and only then is the point p p a non singular point. So you know what this quantity is you know this quantity is actually the dimension of the tangent space at the point p this quantity dimension of m mod m squared over k actually measures the dimension of the tangent space to the variety x at the point p and what normally will happen is that this will be more than this as we saw in those examples of a cone and a line and a curve with a kink at a point. What happened in the singular point was that the dimension of the tangent space shot up the dimension of the tangent space became more than the dimension of the object and for the vertex of the cone the dimension of the tangent space is 3 whereas the cone is only 2 dimension so the vertex is not a smooth point it is a singular point similarly if you take a line with a kink if you take a curve with a kink at the point where you have the kink you know the tangent space becomes 2 dimensional and the dimension is 2 which is greater than dimension of the curve which is 1. So that point which is a kink is not a smooth point it is a singular point okay so that is exactly what is happening here so it is a matter of a little bit of commutative algebra to check that you know if you have a noetherian local ring with maximal ideal m then the dimension of this vector space will always be greater than or equal to the dimension of the local ring okay and if the dimension is greater than the dimension of the local ring then that point p is not a smooth point it is not a it is not a non singular point it is a singular point it is a singularity if the dimensions are equal then it is a smooth point. So the point I wanted to understand is that what this see what this theorem is saying is exactly the analog of what we saw in the analysis calculus situation that a point of a variety is smooth if and only if dimension of the tangent space at that point is exactly equal to the dimension of the variety and what will happen if it is a non singular if it is a singular point this will be bigger than this you will have more tangent space dimension will be more than the dimension of your of the space on which the point lies okay but the nice thing that I wanted to notice is that that whole thing that is done using calculus all that has been captured just using local rings that is what I wanted to appreciate okay. So when you do usual calculus how do you define a tangent space at a point you take a point you draw a curve through the point and then you draw the tangent to the curve at that point okay and then like this you try to fill the enabled of the point by curves draw tangents and then I would take this all the space of all these tangents that is the tangent space. So it involves it involves usual facts from calculus thinking of curves passing through the point and drawing tangents and all that okay and of course even to find the tangent to a curve at a point it is a limiting process right because you take that point and you take a sufficiently close point and then you draw a chord and then you take the limit as the sufficiently close point tends to the given point. So the chord becomes a tangent at that point so in usual calculus even the process of getting hold of a tangent is a limiting process and then you in this way you build the tangent space you check out what the tangent space is you calculate its dimension and then you check whether the tangent space dimension is more or whether it is equal to the dimension of the object and then that is how you get smoothness or not and it involves lot of calculus but you see in algebraic geometry all that limit process is not there but still you are able to capture the smoothness the non-singularity the key is local rings okay that is one fact then the other fact is what this tells you is that you know this based on this definition tells you that this definition is independent of the embedding of X in affine space if X is an affine variety then a point of X is smooth okay that is a condition which is intrinsic to the point because that condition only depends on the local ring at that point and the local ring at a given point is invariant it will not change if you no matter how you embed your variety the local ring is an invariant and therefore this theorem tells you that the non-singularity that you have defined here actually really does not depend on this embedding or this ideal I mean this embedding which dictates this ideal and then the ambiguity of what generators you have chosen for this ideal okay. So it is a very intrinsic statement okay and that tells you that you do not have to worry about this definition but it is useful to make calculations and to check that a given you know variety defined by a bunch of equations is smooth or not at a point so in that way this definition is useful but that theorem tells you that you are not going to go wrong if you use this okay. So I will give you a proof of this in my next lecture okay so I will stop here.