 University and we will talk about Frobenius maps on the rings of invariance. Thank you. Do you hear me? So I thank the organizers for giving me an opportunity to give a talk here at a nice, very nice place, nice sea, very nice institute and very good place. So today's my talk is Frobenius maps on the rings of invariance. Today's talk there is no tight closure but Frobenius things on the rings of invariance. And mainly on the finite group schemes. So let P be a prime number and K a perfect field of characteristic P. So A are positively graded. Positively graded means that it is graded non negatively and moreover A0 is the base field K. And it is finitely generated of course commutative K algebra. And let Fe, FeA be the Frobenius map of, its Frobenius map of A. It is a map from A to A and this target A is written in such a way EA. And FeA is of course A to the P to the E. And the element A to the P to the E viewed as an element right here is have some this E on the shoulder, the left shoulder. And the maximum ideal, the irrelevant maximum ideal of A is denoted by M. And T, let T be a module finite. This one is non commutative because the group is non commutative, non commutative graded A algebra. This is the setting. Now as T is a module finite over A, A is a positively graded, finitely generated K algebra. So the Q group model T, this is the category of Q graded finitely generated T modules. The ring T itself is Z graded, but we need the Q grading because when we take the Frobenius, the grading must be divided by P. So of course Z, we can, maybe we may use this ring, but Q is much simpler. So I use the Q graded. So the Q group model T is crucial. Namely, it is an additive category. Of course, it is an Arbelian category. Additive category with each object can be decomposed into indecomposable object whose endomorphism ring is local. Such a category is called Krull Schmidt and the famous Krull Schmidt theorem holds for this category. Namely, any object can be uniquely affine at direct sum of indecomposable objects. Now let M be a Q graded finite T module. Then the following are equivalent. The first, M is indecomposable in mod T. The second is MM. MM is the maximal ideal of A. So A is a commutative ring, so we can localize. MM is indecomposable in mod TM. The third, M hat, hat is M-addict completion. MM is indecomposable in mod T hat. Mod means the category of finitely generated Q graded modules. So M hat is nothing but it is A finite also. So M hat is A hat tensor AM. And the fourth, M is indecomposable in Q graded mod T. So the indecomposability does not depend on the grading or forgetting. It's the same thing. As a corollary, we have the following equivalence. So let M and N be objects of Q graded mod T. Then the following are equivalent. If M is decomposed into indecomposable objects, say some I, MI, and N is also decomposed into some J, NJ. And then we have the number of indecomposables the same. And after moving the ordering, then N i is isomorphic to M sigma i d i. Sigma is some permutation and d i is the shift of degree. So up to permutation and degree shifting, the indecomposable summands are isomorphic. The second condition is M is isomorphic to N. So you see that, of course, one implies two. This is a very obvious thing. And two implies three. Three is if we localize that M, they are isomorphic. And third, when we even take the compilations, they are isomorphic. So this order is trivial. When M heart is isomorphic to N heart, the number one also holds. So they are equivalent. So we use this factor later. Let's see the additive category. We said bracket C is the Arbillian group generated by all the incoindical isomorphism classes of C. So the generator is the isomorphism class of objects. And we divide it by the relation M equals to M1 plus M2 whenever M is isomorphic to M1 direct sum M2. So moreover, we said CR equals to R tensor ZC. So if the category C is an Arbillian category, which is also a Krull-Schmidt, then this C is freely generated by in decomposable objects. And we write theta star T by q-mode TR. So it is the R vector space freely generated by in decomposable objects. Now I define theta circle T to be the theta star T divided by M bracket minus M bracket, M C bracket. So we ignore the difference of the degree. So if they agree after shifting, then we regard that these objects are the same. This is the theta circle T. Then it is easy to see that theta circle T has in the circle q-mode T as its R basis, where in the circle q, I'm sorry, in the circle q-mode T is by definition in the q-mode T, this is the isomorphism classes in the decomposable object, divided by the relation, they agree when they are isomorphic after the degree shifting. So we ignore the difference of degree. Then for an alpha, alpha is an element of theta circle T. So it can be written as a linear combination of M where M is in the circle q-mode T. It is uniquely a linear combination because it is R basis. Then we define norm of alpha is the sum of absolute value C-M times U-M, where U-M is the length of the M over J-M, where J is radical, the graded radical of T. And length T is the length of a module. Then Rad-Dino's graded radical. Then theta circle T is a vector space and this is a norm. It is easy to check that this is a norm. So it is a normed space. So we can define the limit and the convergence and such a thing. And I would like to define the Froin-Venus limit of a module but the limit is not a module anymore so it is an element of this abelian group. Now I'd like to talk about the finite group scheme. Let G be a finite K group scheme. That is G is a finite K scheme with group operation which are morphisms. Namely, G is a K scheme equipped with the product and the inverse which induces the... For each K scheme X, the X-valued points gets a group with these morphisms. And now the coordinate ring KG of G is a finite dimensional commutative K-hope algebra. Conversely, a finite dimensional commutative K-hope algebra yields a finite K-group scheme so they are the same thing. G-module and KG-commodule it is a hop-algebra so it is a cop-algebra so there is a notion of commodule and KG star. KG star as KG is finite dimensional the dual is finite dimensional hop-algebra again so we can talk about the KG star module so they are one of the same thing. Let G be a finite group then it is a K-hope algebra with the coproduct defined by the group structure of G. I skip the detail. So a finite group is a finite group scheme. This is the first example of a finite group scheme. The second example is so let M be a finite r-billion group then the group algebra KM is a finite dimensional commutative K-hope algebra so it is a commutative algebra because M is commutative so it is a commutative K-algebra then when we define the coproduct of M is given by delta M is M tensor M such an element is called group-like and group-like so letting each element of M group-like KM is K-hope algebra it is commutative and cop-commutative then spec KM is a group scheme and this is not necessarily reduced so there is a non-reduced K-group scheme but only in characteristic P. In characteristic 0 every finite group scheme is et al in particular they are reduced however for example when M is Z over NZ then spec KZ to the NZ is spec KT over TN minus 1 so when N is P when N is P then T to the P minus 1 is T minus 1 to the P so it has a new potent element T minus 1 so it cannot be reduced we write this spec KZ over NZ by mu N so the third example is Frobenius kernels if H is a commutative K-hope algebra then the Frobenius map it is also a hope algebra is an algebra so there is a Frobenius map then we can take the spec spec is a contravariant factor we have a K-morphism Frobenius morphism from GE to G then luckily it is a group homomorphism so we can talk about the kernel of the Frobenius this is called the Frobenius kernel of G and it is always infinitesimal infinitesimal means the spec of an artinial local ring it is only one prime element so it is only one element when we take the read of such a group scheme it is one point, it is a trivial group so it is set theoretically one point but it is not trivial this is an infinitesimal group so Frobenius kernel is always infinitesimal now I'd like to talk about GB modules so let B B is a positively graded finitely generated commutative K-algebra on which G acts such that each BI is a G-sub-module of B we call such an algebra graded G-algebra and we say that M is a GB module if M is a G-module, B-module structures as K vector space is coming from the G-module structure and the B-module structure are the same and the action B times M to M is G linear if G is constant group then this is GBM, GBGM for G in G, B in B and M in M so this describes the rough idea of GB modules and M is a graded GB-module if M is graded as a B-module and each MI is a G-sub-module let H be a K-hop algebra and B a K-module algebra it is a B is a K-algebra H-module such that the product B times B to B is H linear as H is a hop-algebra if B is a H-module then B times B is a H-module again and we require that the product of B is H linear such an object is called K-module algebra we define the smashed product B-sharp H to be B tensor H as a K-vector space the product of B-sharp H is defined by this product and where H is the coproduct or image of the coproduct of H is written as such a swedler's notation what is important here is B-sharp H is a K-algebra and if B is graded then we can make B-sharp H a graded algebra again letting the element H to be of a degree 0 then it is not so difficult to see that GB-module and the smashed product B-sharp K-g-star-module are one and the same thing so the category of GB-modules is a kind of a module category so it is an Arbelian category and any filtered direct limit is exactly AB-5 and such a thing holds AB-4-star also holds and such a thing and I'd like to talk about the torsor principle bundle and let GB be a K-group scheme of finite type and F from X to Y be a G invariant chemophism between K-schemes of finite type that is FGX equals FX holds we say that F is a principle G bundle if F is faithfully flat and the map C G times X to X times Y X given by CGX equals GXX is an isomorphism very roughly speaking it is locally but not Zariski locally locally using the FPPF topology it is a trivial G bundle so the base is Y then it is a projection from G times Y to Y not Zariski locally FPPF locally in the flat topology now Grotendick proves that let F from X to Y be a principle G bundle then the pullback F-star from the category of quasi-coherent sheath of Y to the G-equivariant quasi-coherent sheath of X this is an equivalence so very roughly speaking the module category of the quotient is equivalent to the category of G-equivariant modules so the quotient means the speck of the invariant rings in very rough sense and the quasi-inverse is given by this taking the invariance so it's affine version of something like this so when G is speck H and X is speck B and everything is affine then taking the tensor product so basic extension is an equivalence whose quasi-inverse is taking the G-invariance this gives the equivalence between mode A and mode GB and also there is some graded version of it I skip the details but the big problem of this framework is there is no example although a principle bundle is an ideal quotient the quotient map speck B to speck A which appears in invariant theory is scarcely a principle G bundle namely if G is a finite subgroup of G of V acting on the polynomial ring S equals to sin V in a natural way then the canonical map from speck S to speck SG is a principle G bundle if and only if G is a trivial group so there is no example at all but this is because the origin is always fixed by a linear action linear transformation always fix the origin so it is a fixed point the principle bundle requires that you know G acts freely so this is the reason why there is no example now there is a notion of almost principle bundle G equivalent morphism from X to Y is called an almost principle G bundle when we remove I'm sorry this is Y set minus V so there is a big open subset V and U that is the complement it is closed subgroup subsets have co-dimension 2 or more and this big open subset is principle G bundle then we call that it is almost principle G bundle and it is also known as quasi-tosser and we call that closed subgroup scheme G of G of V is called small when the quotient map from V to V over G is an almost principle G bundle as an example if the group is constant then the finite subgroup subgroup of G of V is small if and only if it does not have a should reflection a should reflection is a matrix when we take the difference between the identity matrix the rank is one exactly one then such a matrix is called a should reflection and the action is small if and only if there is no should reflection at all invariant theory of finite groups should reflection is very important things second example is the determinant of variety and let G is GLE act on X equals to home V E times home EW where the dimension of E is smaller than the minimum of dimension V and dimension W and it is greater than equals to one then I'm sorry this is T is dimension one so then doconcini and protezy proves that the determinant of variety is the quotient of this action the G action on X then the quotient map it is not so difficult to prove that it is an almost principle bundle using the fact that almost principle bundle we can prove some important properties of determinant of varieties now we can prove the weaker version of growth and equivalence which was valid in principle G bundles so the restricted version of growth and equivalence also holds for almost principle bundles so let G be an affine K group scheme of finite type and F from X to Y be an almost principle G bundle assume that X and Y are normal then F given by so growth in the growth and equivalence we only have to take the inverse image but we need to take the double in order to make the shift to be reflexive the other side is we only need to take the G invariance I'm sorry this is F star G I'm sorry this is the quasi inverse so with this correspondence two categories are equivalent even if it is not a principle one bundle we only need to assume that it is almost principle bundle using this equivalence we can prove that higher dimensional Watanabe's theorem Watanabe's theorem is theorem on invariant subrings the Gorenstein property of invariant subrings under the action of finite groups but we can make the group of the higher cruel dimension so let G and F be as in theorem 6 theorem 6 is this thing and let G add denoted G with a joint G action and let lambda G be the one dimensional representation of G which is obtained by pulling back using the identity element identity element then lambda G is one dimensional representation of G and then omega X corresponds to almost corresponds to omega Y but we can twist it by lambda G so omega Y is almost f star omega X invariant G but we can twist by the dual of lambda G so omega Y is trivial if and only if omega X is isomorphic to OX tensor lambda G not OX so if G is smooth then lambda G is the dead inverse of the joint representation of G so in particular if G is smooth, connected and reductive then the positive root and negative root cancel each other so lambda G is trivial in this case omega Y is OY if and only if OX is omega OX is OX so this is the variant of Watanabe's theorem over for finite group schemes Watanabe's theorem holds let G be a finite K group scheme via finite G module and assumes that the action is small and let S is sin V and A is the invariant subring S G then home S A is isomorphic to S as graded G S modules this was proved by I am sorry I can't pronounce this person's name is anyone know excuse me Calphohan Rojas thank you this person proved I noticed this existence of this theorem on the way very long way from Japan to here so so the theorem is equivalent to say that omega A is isomorphic to omega S G so as a corollary this is the notation B as in theorem 8 in general the A invariant of A is smaller than or equal to AS so we require that the action is small and G is finite group scheme where A denotes the A invariant and moreover the following are equivalent G is a closed subgroup scheme of SLV and the second A is quasi-gorenstein so in modular invariant theory the rings of invariants need not be equal in Macaulay however still this Watanabe's theorem holds for in this form if A is quasi-gorenstein then the A invariant is the same and of course the A invariant of the polynomial ring is the minus of the number of variables so it is minus dim V the equivalence was proved by also by lead K I'm sorry I can't read person's name lead K thank you so this is quasi-gorenstein of course when the group scheme is linearly reductive the rings of invariant is direct sum and subring of the polynomial ring so it is strongly irregular and in this case quasi-gorenstein of course it means gorenstein and on the class group we have the following equivalence it is very easy to see that rank of the module is also preserved by tensoring the polynomial ring taking the double dual does not change the rank of a module so rank one reflexive model corresponds to rank one equivariant modules so there is an isomorphism of the class group and the invariant class group so using this fact we can prove that assume that S is UFD and A is the ring of invariance then the class group is isomorphic to the character group of G and it is isomorphic to the group like elements of KG so in particular UFD in this case if and only if the character group is trivial now I'd like to only 15 minutes so from now K is perfect field of characteristic P and G a finite K group scheme via D-dimensional G module S is the symmetric algebra of V and T is the smash product of S and KG star so the mod T is the category of G-S modules and A is the ring of invariance and we always assume that G is small now I'd like to talk about the Frobenius twist of the representation if G is eta then EG to G is an isomorphism of group schemes so for G module W EG module EW is G module again through this isomorphism so if G is a constant finite group and W is W1WD and G is in G and Aij is the matrix of G on W then the Frobenius twist EW the representation matrix of G on EW is given by you take the one of P ETH roots of each entry then you get a new representation of finite group this is Frobenius twist and using this Frobenius twist we have the mode GB to mode GB the Frobenius twist is a factor from here to here now it induces the group of homomorphism Frobenius twist from EM such that EM is EM so this is the Frobenius twist of theta star GS and theta circle GS now B is called finite F representation type or FFRT when we decompose EB into the direct sum of in the composable modules as B modules then we only need finitely many functions up to degree shifting then we say that B is FFRT in the graded sense now let K be algebraic closure of F3T where T is variable then G is Z over 3Z squared 1 billion group of order 9 and when we consider the three-dimensional representation V alpha given by the matrices G1 goes to this Jordan canonical form and G2 go to this matrix then for different alpha V alpha are not isomorphic each other is easy to prove then when we twist V equals to VT by the Frobenius then the first Frobenius twist is V to the T P over 1 and so they are all different using this fact we can prove that for this V equals to VT above there is no sugar reflection so it is small action and the ring of invariance is not FFRT next what about the K is algebraic over the prime field FP so for example the group of order 3 and we we only consider this then we don't know it is of FFRT or not I don't know so the permutation action of the A3 on the three-dimensional vector space I don't know if it is FFRT in characteristic 3 next I'd like to talk about the Frobenius limit so the Frobenius limit is we take for alpha in this group theta circle we take the Frobenius twist of each object so we can take the limit so this makes sense of course we don't know the existence but when it exists we call it Frobenius limit and what we proved is with Peter we proved that for constant finite groups and with Fumiya Kobayashi we proved that for the etal groups and let GB etal and FB and S-finite, S-reflexive Q-graded GS model of some S-finite S-free Q-graded GS model then the Frobenius limit exists and we have that Frobenius limit of F is model the rank it is essentially this model so it converges into this one model so the rank also of course preserved but except for the rank it converges into one model by Frobenius using this let GB any finite K-group scheme not necessarily etal and FB as in the theorem then the Frobenius limit of FG can be described in the linear combination of in decomposable models where MI is defined as follows when we take the tensor product of S with the in decomposable projective model of in decomposable projective G-module then S-tensor PR is GS-module then we take the G-invariance then MI is in decomposable model over the ring of invariance in this way when V1, VR are simple G-modules the complete list of simple G-modules then taking the projective cover P1 to PR is the complete list of in decomposable G-modules and only these models M1 to MR appears in the Frobenius limit so although the ring of invariance may not be FF-RT asymptotically the Frobenius limit is FF-RT I think so now let B be a positively graded commutative K algebra and alpha is some in the this is an element of the theta so called GS then we define the sum M alpha to be CM the coefficient of M if the Frobenius limit of alpha exists then SBM alpha is sum FL alpha sum M alpha alpha this one so this is a generalized I'm sorry this is a generalized F signature of M alpha now the the original F signature is SBBB namely you take the Frobenius twist of B, EB and you count the number of free summands and divide by P to the DE and take the limit this is F signature then Frobenius F signature exists it is proved by Tucker and it lies between 0 to 1 and SB is positive if and only if B is strongly F regular and SB is one if and only if B is regular so the F signature is deeply connected to the ring theoretic property of B now it is proved by Kavahau-Rohas again this is the F signature of the ring of invariance A is 1 over DMKKG if G is linearly reductive this was first proved by Watanabe and Yoshida for constant groups for relative minimal Hilbert-Kunst multiplicity which agrees with the F signature and later if G is not linearly reductive namely the group is not linearly reductive then the subring A is not direct the summand of the polynomial ring S this is equivalent to say that A is not strongly F regular and again it is equivalent to say that A is 0 this was proved by Blohr and independently by Yasuda these I always talk about this and this and this and so and Kavahau-Rohas-Swed Taka is for etal group schemes for this case now we can prove this theorem using this Frobenes limit contains the information complete information of F signature so using this if A appears here then the Frobenes limit is this coefficient and if A does not appear at all then the F signature is 0 using this idea we can prove this formula on the F signature 1 over the dimension of Kg if linearly reductive and 0 if not linearly reductive again using this idea I skip the proof now I'd like to remark that if G is not etal then we cannot EV maybe this is impossible because EG and G are not isomorphic so we cannot consider that EV is G model nevertheless we can take the information on the invariant rings not up stairs but on the downstairs of the ring of invariance we can talk about the Frobenes limits using this idea so that's all thank you are there any questions if there are no any questions I would like very much