 Thank you sir. And good evening, everybody. Today I'm going to talk on a data modules, So let me start with this question. What is an a data? Yeah, so what is an a data module? Then we'll talk about the category of a data modules. I'll also talk about the last no decent for a data modules. And then I'll talk about hoxade homology and we'll be decent on the hoxade homology when you define the leader in its homology groups. So let us start with the data module. So what is the difference between a modules and a data modules? Yeah, so basically you have a new module and in addition to that you're going to have a map from it's going to be a pair. So it's going to be a module along with a map delta M from M to M which satisfies some compatibility condition with the derivation on A. So your setup is you have a ring, you also have a derivation on that and then you take this double M delta M. So what is that? It's a collection of two things. One you have a name module and then you have a map from M to M delta M which has a compatibility with the derivation on A. So delta M, A delta M acting on A dot M is going to be delta A dot M plus A dot delta M acting on M. So this pair is called an A data module if you have a module with such map. And what are going to be the morphisms on this, this collection of a data modules? Of course, these are going to be the module morphisms, module maps with another compatibility condition which is again going to be an obvious one that you demand that there should be a commutative diagram here. So when you have moving from F to N and you have a delta N here on N, you have a delta M here on M. And of course, then you want it to a commutative diagram and compatibility with the structure which you have on modules and the maps among the, and A module maps between the two modules M and N. So these morphisms, you call them data morphisms. So these are going to be the morphisms. So objects are going to be A data modules and morphisms are going to be A modules map is this compatibility condition. The morphisms between A data modules are called data morphisms. We usually denotes the collection of all data morphisms by home delta M and N. And then the category of A data modules we denoted by a dole term or just like we denote for a mode, the category of all a modules. The first observation about this category is that this is actually a symmetric monoidal category. Of course, one has to define a tensor product here and then you have to see it's symmetric. So you the obvious candidate for the tensor product is going to be you take the tensor product in the module category, and then you try to define a data map on it. And one can observe that if you define delta M tensor and this map on the module M tensor and this tensoring is inside the mode A. This map is actually makes it an A delta module. So what you have given M and N in the A delta mode, you just define a new element in the category A delta mode where M tensor N is coming from the A mode. And then you are defining a delta map here on tensor, which is actually very naturally this is delta M tensor 1 plus 1 tensor delta and which is the natural extension of derivation if you take M and N that how do we extend derivation on the tensor module. And this is how we are going to do it. So it's a very natural way to define a new data on M tensor N and this makes it an A delta module. Of course, the tensor taking place in the A mode. So it's going to be a symmetric monoidal category. And now we define an internal home object here in the category. How do we define the internal home? So given two objects M delta M and N delta N in A delta mode, you set the internal home as what are what is an internal home? It's a home coming from the A mode with the map delta bar. So all the morphisms in the A mode category you take that collection along with this delta bar F, delta bar acting on F is delta N composed with F minus F composed with delta M. So now this, this is actually going to be an object in the A delta mode. So this internal home object we usually denoted by home delta MN and one can observe that it's an A delta module. So it's actually an object in the category A delta mode. Okay, so we have an internal home. So the next thing we will prove is that this is actually a right adjoint to the tensor project. The moment we have that we can claim that it's going to be the close symmetric monoidal category. And we do have that result for a delta modules. So if you take the family, these three, any candidates in the A delta modules category, M delta M and delta N and P delta P, then home of M tensoring over A N comma P delta that means delta morphisms between these two objects is nothing but delta morphism of M comma home, delta N comma P. So basically you have seen it in the mode category or so mode A this, this is going to be left tensor is always left adjoint to home or home is right adjoint to the tensor. So this is an internal home going to be the right adjoint to the tensor. And of course, then we have this result for A delta modules that the category of A delta modules is a close symmetric monoidal category. So it's just outline the statement actually follows for what we have discussed so far about the data models. Now if you have any morphisms, any morphism from an algebra k algebra a to a prime and another derivation delta prime on a prime with the compatibility condition all the time we are writing this compatibility condition of f with delta prime, then you can make a prime delta prime a new module. This object you can make it as a delta module. And it's a very natural way to do it, because in the module category also you call it the extension of a scalars and the restriction of a scalars when you make it to an element of the other category. So if you have an element in, if you have, you can consider a prime data prime mode you can talk about the category of those modules. And on the other hand you have a delta mode category, and you can treat a prime delta prime as a data mode module element so this is going to be an element in that category. And we have these functors between these two categories that naturally so f star is going to be from a data mode to a prime delta prime mode. That's the extension of a scalars. And similarly, extension of the scalars is basically you will do the tensor with a prime delta prime and the restriction of the scalars again because of that compatibility condition you will say, see the natural restriction which you will have to make any a prime a prime module by putting the multiplication a dot m is just f a dot m the natural way you do for the module category you will get the result here as well. And you can see that I have a star is a left adjoint to f sub star. With all this you can have certain direct results for the localization and completion. So you will have, if you take any multiplicatively close set the localization you know that you can extend your derivation to the localization. And that's going to be an a delta module. Similarly, for any I add a completion a had delta had we know that we can extend delta to its I add a completion and then he had delta hat. And again the localization all these are going to be a delta modules. Now I'll prove a decent result for a delta module involving localization and completion in the ring so what it what decent result. We have. See, we'll take a particular element which is going to be a non zero divisor. And then we take the ideal I is this the idols generated by this element and the multiplicative close that is going to be one SS square. Then what we have is suppose I have the elements in the as data. So I have the category of a delta modules. Similarly, I can talk about the category of as the test modules and I can also talk about the category of a had data hat modules. So what we have here in the, in this statement, you take a module p delta p in the category as data s, and a module q delta q in the category of a had data hat which is the completion of a with respect to this as addict apology. If we have an isomorphism that says that completion of P in this as addict apology is going to be same as the localization of q with respect to the multiplicative set. So if we have the this isomorphism in the category of as had data set modules. This is one extra condition that the element has to be regular for q, then there exists an a delta module m delta m in the category of a delta such that its localization is going to be p delta p and its completion is going to be q delta q. So basically you can obtain those two modules from the element in mode a delta mode. So that's how we do it. This is going to be just, we have seen it for the a module. So you can obtain the module p delta p as the localization of m delta m and the element which was in the completion you can obtain it as a completion of the same element in the category of a had delta hat. So this is the existence of having such an element in the a delta mode category. Another thing which we have this element is going to be unique, unique up to the so if you have such an element m delta m and you have these maps which we're talking about the localization is going to be p delta p and the completion is q delta q. If you have such maps then they are unique up to the isomorphism that means in the in the category of a delta mode up to isomorphism you are going to have this element unique. So basically we have a unique element m delta m in the category of a delta mode which gives as a localization it will give you the element in the as delta s and on completion it will give you the element in the a had delta hat. Now let us talk about just recalling what is the Huxild homology although we know what it is. So you take a commutative algebra over a commutative ring k and then its Huxild groups are going to be defined the Huxild homology group of the chain complex where the nth term in the chain group on the chain complex is given by this tensor right and plus one components here and the different cells or maps they are given by this expression. So be acting on a 0 tensor a 1 tensor a n this is going to be minus 1 to the power i sum i varying from 0 to n minus 1 and you take the product and the at the last term you bring it to the first place. And this is how the differences are defined and then because you have taken a to be a commutative ring. So H h h n a is going to have any modular structure further if you have a derivation delta on a then it will give you a leaner operator on the Huxild homology groups and this this is not this is already known it is nothing new you define a delta on the chain complex this is how you define it. So, delta acting on a 0 to a n you just take the someone make the data acting on the nth component and take all the sum this is going to be your lead derivative on the chain complex and then you verify that it commute with the differences and you will finally get any map on the Huxild homology. A delta L delta commutes with differential and induces a map on the Huxild homology further you can also observe that this is actually making it an a delta modulus. So, I'll delta acting on an element of the group multiplied with a is actually delta a that element plus a dot L delta a 0 to a. So, basically satisfy the condition of being a H h h n H h n a will be an a delta modulus right. So, that's what we have a phase commutative k algebra if you with the k linear derivation delta then for every n the pair as it's an a L delta n is a delta modulus. This was actually the known example this was a kind of motivation for talking about these eight a delta module category this is already known that given a derivation you can always have a lead derivative on the Huxild homology and it makes it an a delta module. Now, we we prove it already sent on the Huxild homology group as well. So, suppose you have a multiplicatively close set and delta s is the extension of the derivation eight to the localization. Then for each and you localize the Huxild homology group your tensor you understand now this tensor meaning in the a delta module category this is actually going to be the localization of homology Huxild homology group of localization with respect to the delta s. That's all I think these are the references. See this, the one way to see it that whenever you have it, you said that it relates in between the delta modules and the derivations. Yeah, of course from the definition itself, you know, they bold the derivation but you can see another way of looking at it if you have understood that, given a derivation you can always talk about this you call it normally a XD, right, define a multiplication x ax plus T is known. So basically, another way of looking at a data modules could be you see the modules over a XD this to fall in on the ring. In some sense, so the module basic module structure remains same but when we talk about the homological aspect and the categorical things. There's no difference because even with the basic step when you talk about this trivial derivation and you look at the modules over a x they are not. Their tensor internal home is not going to be the way you define the home for a x the normal polynomial ring, but on the ground level when you talk about the a data module they are actually having the same structure as the derivation over a XD, and this is vice versa, so you can always relate the two.