 Thank you. I would like to begin by thanking the organizers for giving me the opportunity to talk today and of course, collecting us all of here for this wonderful school and workshop. So today I'll be talking about local homology of invariant rings. This is actually a joint work with Jack Jeffries and Anne Ruxing. This is when I was at the University of Utah on an awful bright narrow post-doctoral fellowship. So thanks to Mandira. She has set up a few things for me so that I can brush with them quickly and we can get in track. So for me, things will be a little restrictive in the sense that my ring will be a polynomial ring to begin with. In particular, it's a polynomial ring over a field k and it has dimension d. This means I'm talking about the number of indeterminates. The action that we'll be talking about of the group today over here will be a finite subgroup of gl sub dfk. The action that we are looking for is also a little specific in the sense that we are looking for a linear action of this group on the polynomial ring r. In particular, if you want to understand this in terms of ring homophisms, it's a degree preserving key algebra, automorphism. So once we have this set up, we can talk about the elements of the ring which are invariant under this group action and this is precisely what we call as, collection of all these elements is what we call as the ring of invariance of this group and we denote this by r top g. So once we have this set up, just to give you a quick example of how and what things happens. So let's start with a tiny example so that we can see things clearly. It's a small two cross two matrix sigma and it's acting on a polynomial ring in two variables. In particular, it is doing what? It is just permuting the variables. So once the permutation happens, it's easy to check that x plus y and xy are two obvious elements which remain invariant under this group action and one can prove that the invariant ring is actually the algebra generated by these two elements and in particular, it is a polynomial ring and we can do this in a little more generality in the sense that if we have a polynomial ring in d variables and we have the permutation group sd acting on the polynomial ring, of course again, sd acts by permuting the variables. So in this case, rg is the polynomial ring generated by the elementary symmetric polynomials in the determinants. This is actually one of the classical examples and one of the motivating examples to find more examples where the invariant ring turns out to be a polynomial ring. If not a polynomial ring, what are the other special properties that we can look for will be a question. So in particular, if it's not a polynomial ring, the next obvious characterization that at least I would want to look at in a polynomial, in an invariant ring is if that ring is cone Macaulay or not. So and of course, this is one of the basic questions that one asks in invariant theory of rings is can we have some kind of a bridge between the properties of the group and the properties of the invariant ring to understand each other. Just a quick redefinition of things because Mandir did introduce it. An element of the group is called a soda reflection if it fixes the co-dimension one space. In particular, if you are representing this element of group by a matrix and i is the identity matrix, then what we are really asking for is that the rank of the difference of these two matrices should be less than equal to one. So once we have this, let's look at some classical results where we do have some answers. And when I mean classical results, invariant theory was actually developed over complex numbers. So there are many results which fall into the case of the non-modular category. In particular, where the order of the group was invertible in the field K. And two main and classical results in this field in the non-modular setup are by Sheffield, Todd, Chivalier and Seher where they give a characterization of RG being a polynomial ring. This is if and only if G is generated by soda reflections. The other classic result to characterize the Coen-Makalli properties by Eger and Hochster, they prove that in non-modular case, whatever is happening in the given setup invariant ring will always turn out to be Coen-Makalli. And in the modular case, when the characteristic divides the order of the group as expected, many things fall out and so does these two theorems. I'll quickly state them. So Seher proved that RG being polynomial ring in this setup does imply that the group is generated by soda reflections, but the converse is not true. This example quickly is given by Nakajima in 1979. The characteristic of the field is P, order of the group is P cubed. So we are in the modular setup, G is generated by soda reflections, but Nakajima was able to show that RG is not a polynomial ring. The other result about the Coen-Makalli property, Eger and Hochster showed that RG was Coen-Makalli in the non-modular setup, but in the modular case, this is a very simple example. This is three copies of S2 acting on polynomial ring in six variables. What it does is the following. It just permutes the XIs with the YIs. With just the simple action of three copies of S2 acting on a polynomial ring in six variables, one can check that RG is not a Coen-Makalli ring. So just to give you an insight, there are many things which fall out in the modular case, and that's why this case is more interesting to look at. So our results over here, our project over here was to look at a few more other properties and things, and we kind of concentrated on local homology modules because these modules themselves also preserve a lot more properties of rings. So just to give you a quick insight, if you are not familiar with local homology modules, is the following. Of course, the main target will be the topmost, the topmost local homology module, which is HD of R plus R. Of course, my support is the irrelevant ideal, or the maximal homogenous ideal. This can be seen as a case span of elements of other fractions, if you can say fractions, or cosets of the form one over X1 to the A1, so on, XT to the AD, where the powers are all positive, and one important invariant that we'll be looking for is the A invariant. So what really happens over here is the following. The ring to begin with was a graded ring, in particular, the top local homology module will be a graded module, and this means we can talk about the graded components, in particular, the highest non-zero graded component of this module. And this degree is what we call as the A invariant, or the A invariant. So once we have this set up that, we have a group action on the ring, and we also have R plus to be a G stable ideal, which happens in the case of our Lamel ring. The action of the group extends to the action on the top local homology module, or the local homology modules themselves. So once we have this, of course, we'll go back to the non-modular case and see, do we have a result over there? Yes, we do, because of the Reynolds operator, we have that RG is the direct demand of the ring R, and in particular, one can prove that in the non-modular case, if we look at the fixed points of the action on the top local homology module, which is, sorry, which is the first thing here, this is actually equal or rather isomorphic to the top local homology module of RG. So the obvious question is, does it happen in the modular case? And as expected, as we saw before, things don't happen as we expect them to be. So just a quick example, if we have A3, the alternating group in three variables, acting on polynomial ring in three variables, the characteristic is three. And I'm denoting delta, which is a so-called discriminant, by x square y plus y square z plus x z square. Once we have this set up, one can check that the ring of invariance, RG, is actually this quotient ring, where the sigmas are the elementary symmetric polynomials, delta is the discriminant, and it's a color ring. And one can check that once we look at this element, it's an element of a degree minus three in the top local homology module of RG. One can check that this is a non-zero element in this local homology module. But once we look at the same element, it's a midge in H3M of R, the top local homology module of the ring. This element turns out to be zero. In particular, delta can be written as an alina combination of the sigmas. So this means whatever result we were looking at in the non-modular setup doesn't follow through. So the next question will be, can we have some kind of isomorphism, some kind of R-module maps between these two objects, which behave nicely? And this is what led us to the following result. So our setup, as usual, is as it was in the beginning. G is a finite group. It's acting linearly on the polynomial ring over a field in D number of variables. And we are also assuming the group does not have any pseudo-reflections. Once we have the setup, we prove that the following complex of RG modules is actually exact. So what really happens in this complex? We have a surjection at the right most end. This is via the transfer map. This is just the elements going to the sum of their orbits. And the first map is the sigma map, which is basically summation over one minus g where g varies over the elements of the group. One quick observation here, instead of having g varying over all the elements of the group, one can also restrict to the generators of the group. So this is a little complex in the sense that the proof requires a lot many things. So instead of looking at the proof of this result, I thought let's look at the consequences because even they are equally interesting. So once we have this complex of RG modules, which turns out to be exact once we do not have pseudo-reflections, we do get interesting results about the A and variance and how they can characterize a few more other things. But once you look at the statement, one obvious question will be, why do you need this assumption on lack of pseudo-reflections? Can we really drop this off? So just to give you a quick example, we cannot drop the assumption on absence of pseudo-reflections. So let's quickly look at that. So we have a group generated by one, two, acting on F2XY. This is of course just permitting the variables. We look at this element in the minus two graded component of H2 of R, which is X over sigma one, sigma two. Then one can check that once we look at the right hand side here, the transfer map, this element goes to zero. In particular, it's in the kernel of the transfer map, but one can check that this element is not in the image of the map sigma. In particular, the exactness at the middle stage fails when we have pseudo-reflections. So once we have this, let's look at the consequences. The main result is the following. Once we have the finite group acting linearly on the polynomial ring, the A invariant of the invariant ring is equal to the A invariant of the ring if and only if she has no pseudo-reflections and it is a sub-group of the special linear group. And once we, and of course, yeah, one interesting result which also was required in proving theorem A is this lemma, where we look at sub-groups of the group G because the sub-group will also put an action on the ring, but we wanted to see if we can compare the A invariants of RG and RH. And there are some results in the past. I guess one is in the thesis of Jack Jeffries that he proves this result, but it is in a very specific case, but we were able to prove this in generality that if we have any finite sub-group acting in our setup and we take any H sub-group of G, then the A invariant of RH will only increase. It will always be the bigger than or equal to A invariant of RG. So that was one interesting lemma and an interesting result in itself that we saw. The other interesting result to come back to is this, because we are now comparing the ranks of the local homology modules that is coming back to the question that I started with. If we have a finite cyclic group with no pseudo-reflections, then we compared the Hilbert series of top local homology module of RG and the fixed points of the top local homology module of R, and we were able to prove that if we have no pseudo-reflections, the group is cyclic group, then the Hilbert series of these two modules, they go inside. This is kind of direct consequence of the exactness of the complex we saw in the last page. But again, what happens if we do not have the cyclic nature of the group in the last result? So we wanted here the group to be a finite cyclic group with no pseudo-reflections. We have already seen that we cannot drop pseudo-reflection part, but can we drop the cyclic nature of the group? Just a quick example that it may not be possible at least without any extra assumptions. This is the representation of a client for group over F2 and that's the representation that we are looking at. These two are the elements which generate it. One can clearly see that there are no pseudo-reflections here because minus the identity rank will be two. So there are no pseudo-reflections here. We have two generators, so it's not cyclic. The action is actually on polynomial in six variables. It's a six cross six matrix. One can check that once we look at the minus seventh graded pieces of the respective local homology modules and the fixed points, their ranks differ. So in particular, the cyclic nature of the group could not be dropped in the last result that we saw. So yeah, that's it. This is actually an ongoing project. So this is what we have till now, but we are still looking at a few more questions and a few more ways to generalize more things. So if you have any questions, please do let me know later. So that we can work on them. Thank you.