 Alright, so you know we are trying to discuss uhhh uhhh the notion of focusing attention at a point which means you are trying to understand uhhh geometrically how to study functions at a point and the device for that is commutative algebraically the device is called the local ring. So, uhhh I told you last time that uhhh if X is a variety of course which means that X could be either an affine variety or a quasi affine variety or a projective variety or a quasi projective variety and you have P a capital P a point of X we have the local ring of X at P. So, uhhh well in fact to expand it is actually called local ring of uhhh germs of regular functions of X uhhh the local ring of germs of regular functions at P okay. So, uhhh sorry. So, in fact let me write this as local ring of germs of regular functions at P and how was this defined? This was defined the following way we uhhh we took uhhh uhhh OX P tilde to be the set of all uhhh pairs of the of the form U, F such that uhhh U contains P uhhh U is an open a open subset of uhhh X and F is a regular function on U. So, you take the set of all such pairs and then what you do is you go modulo uhhh an equivalence relation okay to get uhhh a surjection from this into the local ring. And what is this equivalence relation? The equivalence relation is well uhhh U F is equivalent to V G if and only if uhhh by definition uhhh F restricted to U intersection V is equal to G restricted to U intersection. So, uhhh so this is an equivalence relation you are just identifying functions uhhh uhhh you are just identifying functions on the intersection and of course you know uhhh for this uhhh this is also this is also equivalent to uhhh this is also equivalent to just assume requiring the requiring that F restricted to W is equal to G restricted to W for an open W inside U intersection because if a regular function uhhh if 2 regular functions uhhh coincide on an open subs non-empty open subset uhhh then uhhh they of course uhhh coincide everywhere right. So, uhhh so I need not uhhh require that the point P is in W uhhh so this uhhh is a local ring and uhhh if you if you start with an element here U, F it is image here is written as the square bracket and it is called the germ of the function F okay and the germ of the function F uhhh whenever a pair U, F is equivalent of pair V, G then in the local ring they will give the same they will give rise to the same germ okay. The germ of F and the germ of G will be one and the same right. So, and I told you that uhhh OXP is a local ring is a is a local ring with uhhh uhhh uhhh uhhh uhhh unique maximal ideal ideal uhhh MXP and MXP what is the maximal ideal it is germs of all those functions which vanish at the point P. So, MXP is all those germs U, F such that F of P is 0. So, germs of those regular functions defined in a neighbourhood of the point P which vanish at P okay and uhhh so we saw that uhhh uhhh how did we show that this is a maximal ideal here we showed this is a maximal ideal here by uhhh showing that if there is a germ of which does not vanish at P then it will by continuity not vanish in open neighbourhood containing P and on that open neighbourhood it will become invertible okay. So, uhhh so it will become a unit. So, in other words a germ of a function that does not vanish at P namely an element here which is outside here is a unit. So, you have a characterization uhhh of local rings in commutative algebra which is a simple lemma you can prove it for yourself if you have a commutative ring with 1 and if there is an ideal such that everything outside that ideal is a unit then that ideal has to be the maximal ideal unique maximal ideal and therefore the ring will become a local ring okay. So, this is what is happening here. So, you have a commutative ring with 1 and you have this this is obviously an ideal and everything outside it is a unit. So, this is unique maximal ideal for this local ring and this ring is local okay. Of course all these are even k algebras okay O O X P is actually k algebra. So, uhhh so the point is uhhh uhhh uhhh so focusing attention at a point corresponds uhhh geometrically uhhh focusing attention at a point uhhh which means trying to study regular functions at a point that commutative algebraically uhhh boils down to studying local rings okay. Now the point I want to make is that uhhh why is this important? So, here is a lemma. So, the lemma is uhhh uhhh uhhh by the way I forgot to tell you that there is an obvious uhhh way in which addition and multiplication is is is defined here okay. Maybe I will just recall it uhhh you have u, f uhhh you have a germ you uhhh you you add it with v, g uhhh by simply adding the functions on the intersection. So, you just take u intersection v and uhhh take f restricted to u intersection v uhhh plus g restricted to u intersection okay. So, you take the germ of this I asked you to check that this uhhh addition which is defined like this using representatives of equivalence classes is well defined it is pretty easy to see uhhh. So, of course instead of addition I could also put multiplication and if it is multiplication I will have to put multiplication here of course and uhhh the 0 element is uhhh the 0 element is given by the function 0 the germ of the function 0 and uhhh uhhh unit element uhhh the multiplicative identity element is given by the uhhh constant function 1 which is also of course constant functions are all regular functions okay. So, 0 and 1 are just given by the germ of the 0 function the constant function 0 and 1 is just the germ of the constant function 1 which takes the value 1 okay. So, uhhh now the the lemma is the following uhhh if if u is an open subset uhhh of x containing p then o x p is the same as o up okay. So, uhhh so this is the important this is the local this is the localness namely uhhh the local ring at a point of a variety uhhh depends only on it it depends depends only on the neighbourhood of the point okay it does not depend on the ambient variety. So, if you take uhhh so if you have a point if the point p of your variety is lying in an open set okay then of course u is an open subset of a variety and therefore you and of course you contains a point p so it is uhhh non-empty alright and it is of course uhhh irreducible okay and the point is that uhhh an open subset of a variety is also a variety okay. An open subset of a variety is also a variety right uhhh whether your uhhh so this is a fact that you can check uhhh using simple topology uhhh an open subset of a variety is automatically a variety right. So, of course if the variety is an affine variety an open subset will become a quasi affine variety if the variety is a projective variety the open subset will become a quasi projective variety if the variety is a quasi affine variety then an open subset of that will continue to be a quasi affine variety if it is a quasi projective variety an open subset will continue to be a quasi projective variety okay. So, in any case an open subset of a variety is again a variety right therefore I can think of the local ring of the point P with respect to the variety U okay and there is no difference between the local ring of the variety local ring at P whether you are considering P as a point of view or P as a point of X okay that is the purpose of the lemma. So well what is the importance of this lemma the importance of this lemma is the following give me any variety you know we proved last time that any variety can be covered by finitely many open subsets each of which is isomorphic to an affine variety. Therefore if you give me any variety and give me a point on a variety okay I can choose an affine open subset I can choose an open subset containing the point which is isomorphic to an affine variety and therefore by this lemma I just have to calculate always the local ring of an affine variety at a point okay that will give me the local rings of any variety at any point okay and when you calculate the local ring of an affine variety at a point there is an explicit formula for that in commutative algebra it is simply given by the localization of the affine coordinate ring at the maximal ideal corresponding to that point okay that is what I will show next okay. So this lemma is very useful namely it tells you what the local ring looks like at least more explicitly in terms of commutative algebra this definition of local ring here is a very general process you could do it in very general circumstances you could take X to be a topological space and in and for functions you could just take continuous real valued functions or continuous complex valued functions or you could take X to be manifold say a domain in Euclidean space and you can take functions to be you know so many times differentiable or infinitely differentiable namely C infinity functions or you could even take X to be a domain in the complex in complex n dimensional space and you could take the functions to be holomorphic functions holomorphic in each variable and you can always do this process and this process will always lead you to a local ring okay so the process is very general so the fact that you are getting a local ring is very beautiful it is something that tells you that local rings come out of geometry by focusing attention at a point but the question is what is this local ring if you ask the question what this local ring is can you write it down as some nice ring that you know in terms of rings that you know then the answer to that comes from this lemma for example if you choose U to be an affine open subset of X that is an open subset of X which is isomorphic to an affine variety then what I am going to say later on will tell you how to write down that local ring okay alright so you see so let us prove this is very easy to prove it is pretty easy to prove so proof is rather proof is rather simple so you know you have OXP so you have OX or rather I should say I have OXP tilde okay and there is OUP tilde okay and the quotient of OXP tilde by the equivalence relation as I have defined here is going to give me the local ring OXP and similarly if I take the quotient by this equivalence relation here I am going to get the local ring OUP okay and what I want to say is to say that these two are isomorphic it is very very simple so what I do is I give a map from here to here okay so namely you give me U, F here you give me an element here so this consists of a pair which has a function F which is regular on the open set U which contains the point P maybe I should not use the same U let me use something else let me use V because U is already fixed let me use V so if you have a pair V, F then I have V an open subset of the point P F is a regular function on V okay now after all I can simply send it to I can just restrict it to V intersection U so I can just V I can just take V intersection U, F restricted to V intersection U I can do this see restriction of a regular function to an open set is continues to be a regular function because the notion of a regular function is local okay that is one thing then the second thing is any two open sets any two non-empty open sets will always intersect because of irreducibility okay so all our varieties are irreducible so any two non-empty open sets will intersect any non-empty open set will be irreducible it will be dense okay that is the that is the coarseness of the Zariski topology right so there will always be an intersection and of course the intersection will contain the point P because P is already in U P is also in V so it is P is also in V intersection U and so this is a well defined map like this alright and what you must understand is that this map also this map will also respect the equivalence namely if two functions on X if two regular functions in neighborhood of the point P in coincide in a smaller neighborhood they will also do so when you restrict them to when you restrict these functions to U the intersection with U okay so this so what this will tell you is that this will induce a map like this so that this diagram comes okay you will get a map like this okay and it is easy to check that this map is actually an isomorphism okay in fact you can check that this map is a natural identification okay so what you do is so the fact that you need a map like this boils down to checking that this map respects the equivalence relation okay so which means that if two things are equivalent here then the images there are equivalent once you have that this map goes down to a map like this okay and then which means that this map is not just a map of sets but it is also a map which preserves the equivalence relation on these two sets okay so it goes down to a map of the equivalence classes okay so you have a map like this and then what I want to say is that this map is you can it is you can explicitly check that this map is both injective and surjective because you know see if I have a germ of a function V, F well the map that is going to be here is going to be I am going to send V, F to VU intersection V, F restricted to U intersection V and germ okay this is going to be the map here alright and because this map is induced by this map alright now you see now it is very easy why is this it is very clear that this is going to preserve both addition and this map is going to preserve your addition multiplication and so on so this will be a K algebra homomorphism it is going to preserve addition multiplication it is going to take 0 to 0 0 element to 0 element okay and it is going to take 1 to 1 okay so this map is you can almost see that you know the germ of F at the point P where F lives on V is going to be literally the same as a germ of F restricted to U intersection V because after all it is the same function you have just restricted it and it should define literally the same germ so in fact this should be thought of as actually an identification but if you want to formally say it you have to say it is a K algebra homomorphism it is injective and surjective so the fact is why is it injective this is 0 okay if this germ is 0 it means that the function F restricted to U intersection V is equal to the 0 function on a smaller open subset of U intersection V which contains a point P but then by but if 2 regular functions equal on an open set then they are equal everywhere so it will tell you that F itself is identically the 0 function on V okay so the moral the story is that if something goes to 0 then this is 0 that tells you that this is injective okay and then how will you get surjectivity if you give me if you give me if you give me anything here actually the beautiful thing is anything here is also an element here so actually there is a map in this direction also namely you give me any if you give me any function defined on an open neighbourhood inside U if you give me a regular function on an open neighbourhood inside U inside U of P that is also an element here okay so there is a reverse map okay and that will introduce a reverse map here alright so that is that is the reason why it is surjective so the fact is that O X P to O U P is an isomorphism this is an isomorphism so it is so you know so it is obviously an isomorphism but you know we tend to treat it as an equality we tend to treat it as an equality with the with the understanding that you are literally taking the germ of the same function but restricted to a smaller neighbourhood okay so so that is the proof of this lemma okay now so what is the corollary to this very important corollary to this is if X is a if X is a if X if X is a variety and P belongs to X choosing U to be an open subset of X isomorphic to an affine variety containing P E C that all possible local rings all possible local rings are can be computed up to K algebra isomorphism as local rings of points of affine varieties so this uses the important fact that I stated in the previous lecture lecture that any variety affords a finite open cover a cover by finitely many open sets each of which is isomorphic to an affine variety it uses that fact so so finally so there is another thing that has that has gone into in between the lines which I should have said but I will say that now so I want to I want to say the following thing I want to say that this local ring this is a this is an invariant it is an invariant of the point on the variety okay so so what I am trying to say is that it does not change under isomorphism so you know if you have a variety and you have a point then you have it the local ring at that point if this if this variety is isomorphic to another affine another variety okay and under this isomorphism this point goes to another point then this isomorphism will automatically induce an isomorphism of the local ring of the source variety at the source point with the local ring of the target variety at the target point okay so you will get you will get it automatically because the isomorphism will be induced by the by the pullback of functions okay so so let me write that maybe I should say that maybe I should have said that before this so let me say that the local ring of X at P is an invariant so what is an invariant an invariant is something an invariant of a certain object geometric object or an invariant of a mathematical object is something that does not change or change is only up to isomorphism if you change the object up to an isomorphism okay so what I am trying to say is that you know if you if you change X and P up to isomorphism then the local ring will change only up to an isomorphism okay so that is if Phi from X to Y is an isomorphism of varieties then we get an isomorphism Phi of K algebras of O Y Phi P with O X P okay so so you know the situation is that you are having you have X, P this is called a pointed variety that is a variety with a point okay and you have this this map Phi which is an isomorphism that carries X, P to Y, Phi P okay so when I write a pair like this it is a variety and a point on it okay and the map is just not supposed to respect the variety I mean it is not supposed to just go to this variety to that variety but it is also supposed to take this point to that point which is what it does and the point is that the local ring at of X at P will be isomorphic to the local ring of Y at Phi P because of this isomorphism and the answer is the reason is very very simple see because you see what is going to happen is you have X you have Y okay and you have Phi so this Phi will induce by pullback of regular functions so you know Phi is a morphism of varieties basically Phi is of course an isomorphism but if Phi is just even a morphism of varieties it is supposed to pull back regular functions to regular functions okay so Phi is going to induce a pullback so the pullback is going to be Phi upper star and what it is going to do is that you know it is going to go in this direction you give me regular function on an open subset Y of V okay then I am going to get a regular function on the open subset Phi inverse of V of X okay so if you have Y and V an open subset of Y then Phi inverse V is going to be an open subset of X okay and Phi will restrict to Phi inverse V and take it into V this Phi restricted to Phi inverse V will be a morphism of the variety Phi inverse V into V okay and give me any regular function on V by composing it with Phi I will get a regular function on Phi inverse V that is the pullback map so this map is just if you give me a G here I am going to get the pullback which is just G followed by first apply Phi then apply G this is Phi this is the pullback okay this is part of the definition of morphism the definition of a morphism is continuous map which pulls back given any target given any open subset of a target and given a regular function on it namely given a regular function on an open subset of the target then it pulls it back to give you a regular function of the inverse image of that open set on the source okay so this is so this is what happens and you can see that once you have this then you know you know O you have O Y Phi P tilde whose quotient by the equivalence relation as I defined earlier is going to give you the local ring O Y Phi P and you have O X P tilde this is whose quotient by the corresponding equivalence relation is going to give you the local ring of X at P okay and the fact is that there is a map like this this is induced by Phi upper star okay namely what it does is if you give me a V, G it will send it to Phi inverse V, G circle Phi it is going to send a pair here to a pair here okay and what is going to happen is that this is going to of course respect the equivalence relations okay and therefore you are going to get a map like this okay so this is this is this is V R Phi star okay and this also V R Phi star okay. So what this diagram shows is that the moment you have morphism of varieties then you know you take a point P if it goes to a point Phi P then you immediately have a K-algebra homomorphism of local rings in the opposite direction okay so this is a K-algebra homomorphism this diagram first commutes so who does this okay. So you know you must think of this as an infinitesimal version of the following fact see whenever you had morphism from one affine variety to another affine variety it induced a pullback from the regular functions on the target to the regular functions on the source okay when you have a morphism from one affine variety to another affine variety it induced a pullback function pullback of regular functions from the regular functions on the target to the regular functions on the source if you take the infinitesimal version what the infinitesimal version is is going to be this it is going to pull back of regular functions on the target at a point to germs of regular functions of the at the source at the point of the source which corresponds to the target point ok. So it is very simple you have a regular function on V you composite with phi you get a regular function on phi inverse V. So the germ of a regular function of on V at phi P is going to after composition with P is going to give you eventually is going to define the germ of a regular function at P itself ok. So a regular function in a neighbourhood of phi P is after composition with phi is going to give you a regular function at P ok regular function in any board of P and therefore you are going to get a map on germs you are going to get this. But the point is that if phi is an isomorphism ok then you also have a map in the other direction and you can check that these two maps are inverses of each other ok. So whatever I have written here for phi I can also write for phi inverse because I can do that if I know phi is an isomorphism. So whatever I wrote here I can write everything for phi inverse so I will get a map like this ok. And then you can check that these two maps are inverses of each other ok these two maps are inverses of each other and therefore these two that will give you an isomorphism of this with this ok. So phi star so you know this map via phi by phi star I will call this as phi star this is phi star going down to the point phi P so I will call this via phi star ok let me not complicate the notation. So phi star and phi inverse star which is from O X P to O Y so phi inverse is going to be in this direction so it is going to be from O X P to O Y phi P ok are inverses to each other and hence we get that phi star is an isomorphism of K algebras. So this is the fact that I mean this is what this remark says that you know the local ring is an invariant if you change the variety up to isomorphism the local ring will not change it will change only up to K algebras isomorphism ok. And of course we do not distinguish between we try not to distinguish between our varieties that are isomorphic in the geometric sense and we do not try to distinguish between K algebras which are isomorphic in the commutative algebraic sense ok. So well alright so this is what I am trying to say here if X is a variety and P is a point of X and you choose U to be an open subset of X which is isomorphic to an affine variety then under that isomorphism the local ring of X at P which is the same as the local ring of U at P according to this lemma will be isomorphic to the local ring of that affine variety to which U is isomorphic at the point which the isomorphism carries P2 ok. So it somehow this has to be thought of before one can understand that properly. Now I come to the so the big deal is how do you calculate the what is the formula commutative algebraically for the local ring of an affine variety at a point ok. So this is the so here is the so here is the theorem theorem if X is an affine variety and P and X is a point corresponding to ok. So let me make a small then OXP is canonical isomorphic as K algebras to AX localize at MP as K algebras so this isomorphic as K algebras where MP is the maximal ideal of AX AX is the same as OX corresponding to P ok. So here is the formula for the local ring at a point of an affine variety. So local ring is just gotten by taking the affine coordinate ring of the affine variety which you know is also the same as the ring of regular functions ok these two are the same for an affine variety. And then you localize at the maximal ideal that corresponds to this point ok. So of course localizing at an ideal means that you should for this to make sense you have to localize at either certainly at a prime ideal because localizing at a prime ideal or a maximal ideal means you invert everything outside all the elements outside the prime ideal or the maximal ideal. So this means take the ring AX and invert all the elements which are not in the maximal ideal corresponding to the point P ok. Then the other thing is of course so now I want to you to recall that you know according to the Nullstell and such ok the points of X the points of the affine variety X are in one to one correspondence with the maximal ideals in AX ok. And this is just because of you know the Nullstell and such that is more general that can be more generally applied to the affine space in which X is embedded in ok. So here is a proof of the theorem is also pretty easy proof is well so you have OX which is AX ok and from OX you have the following map into OXP so here is my map you give me any regular function F you just send it to the pair X, F and the germ of that so simply send every regular function to its germ ok. So F is in AX means that F is a regular function on all of X, AX is same as OX ok so the whenever you write a pair U, phi has to be a regular function on U ok. So here F is a regular function on X so I can it make sense to write the pair X, F and then I am taking the germ that is why I put this square bracket ok. This is of course a K-algebra homomorphism this K-algebra homomorphism that is easy to check and well you can also see that is injective because you see you know if the germ of F at the point P is 0 it means that F coincides with this 0 function in neighborhood of P but then it has to coincide with 0 on all of X ok. So if this is 0 then this is 0 so this is an injective homomorphism ok then you also have the localization map so you have the localization map this is the localization of this ring AX at the maximal ideal corresponding to the point P ok. So of course you know MP is all those MP is all those functions in AX which vanish at P that is all ok it is a maximal ideal right and so you know the fact is that you get a map alpha like this with because of the universal property of localization ok. See if you start with G in OX such that G of P is not 0 ok suppose you start with the regular function global regular function on X G and element of OX is same as AX and suppose G is not going to vanish at the point P then this is equivalent to saying that G does not belong to MP so this is this implies that you know G does not belong to MP and if G does not belong to MP its image in the localization will become a unit because localization has this property that it inverts all the elements which are outside MP and since G is not in MP is going to be inverted so G under this localization map goes to a unit in MP ok and so you have this fact alright and you should see that G will also go to unit there if I take the image of G in OX here also it will go to unit why because you see G will go to the germ of X, G ok but the germ of X, G will not belong to MXP which is an because MXP is supposed to be the germ of all those regular functions which vanish at P so but G does not vanish at P therefore the germ of G in the local ring corresponds to an invertible element a unit. So you see every element which is outside the maximal ideal MP is going to unit here and therefore this there is a universal property of localization which says that whenever you have a ring homomorphism from a given ring to another ring with the property that it inverts all the elements that are inverted by the localization process then that ring then that homomorphism has to factor from the localization the localizing the localized ring is in some sense the initial object ok so this is called universal property of localization. So there exists alpha there is a unique alpha by universal property of localization and you know it is very easy to write down what that alpha is see alpha see an element here will look like F by G ok an element in the localization will look like F by G where F is a regular function G is a regular function but G is not in MP that means G does not vanish at P that is how elements here will look like ok and alpha of F by G is going to be very simple the it is going to be just you just take the image under this so you get the germ of F times you see the germ of G is invertible so it will be this times germ of G inverse ok this is what it will be and what is this this is you know if you think about it this is just you know this is just X F germ of F times germ of G inverse is what I mean G does not vanish at the point P so G will not vanish in neighbourhood of the point P which means a DG the set of all points where G does not vanish inside X so DG will contain P and it will be an open set which contains P ok and you can write 1 by G as a regular function on DG. So this will be just X intersection DG, 1 by G this makes sense this is what it is and so and you know so this is how the map is defined this is how the map is defined so in fact if you now if you take the product you have to take the product on the intersection so it will be just X intersection DG F by G this is the this is the map because you see F by G is certainly a regular function on X intersection DG and it is a germ of this function so this is the map ok. Now you can see very easily that this map is both injective and subjective ok why is it injective well you know if F by G vanishes ok if the regular function F by G vanishes in a neighbourhood of the point then you know see what is going to tell you is that it is going to tell you that yeah so what will happen is that it will tell you that F by G and 0 and the 0 function they are equal on this open set X intersection DG which is in you know dense open subset of X ok so that will tell you that F has to be F has to be 0 ok so the moral of the story is that you will easily see that this is both this is injective and why it is subjective is because well you start with the germ of a regular function at the point P ok then in a neighbourhood of the point P it is going to be it is going to look like F by G and it is going to be the image of under this same F by G under alpha so it is trivially surjective it is trivially surjective just by the definition of a regular function regularity at a point ok what is an element here an element here is germ of a regular function at the point ok but what is a regular function at a point it is supposed to be given by it is a regular function is something that has to locally look like a quotient of polynomial so regular function at this point is in a neighbourhood of that point is going to look like F by G ok where F and G are regular functions where F and G are regular functions F and G are polynomials in fact ok polynomials polynomial functions on the affine space in which X is embedded ok so but then F by so that but then that F by G will also be here it makes sense of an as an element here and alpha will carry it on to your germ so it is surjective it is injective it is surjective so it is an isomorphism so clearly alpha is an isomorphism so it is just a just a movements thought will tell you that alpha is an isomorphism so in other words you have this formula ok so the moral of the story is that whenever you want to calculate whenever you want to go down to commutative algebra and try to focus attention at a point all you will be doing is always trying to look at this ring which is you know you take you will have some finitely generated K algebra which is an integral domain these are how affine coordinate rings of varieties look like and then you will have to localize it at a maximal ideal ok and so these are the rings that you have to study in commutative algebra if you want to get what happens geometrically at a point on a variety so the study of the you know geometrically the study of functions in a neighbourhood of a point on a variety that is reduced to the commutative algebra of trying to study localizations of finitely generated K algebra which are integral domains such rings which are affine coordinate rings at maximal ideals ok so this is what one has to do ok so this is how local rings enter into geometry ok and but interestingly enough the local rings were produced by geometry ok they were defined by geometry and then here is how they come commutative algebraically ok at least in the case of algebraic geometry I will stop here.