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[5:30pm] Mitsuyasu Hashimoto, Metropolitan University, Sumiyoshiku, Osaka, Japan
 Description:
Virtual Commutative Algebra Seminar.
Speaker: Mitsuyasu Hashimoto, Metropolitan University, Sumiyoshiku,
Osaka, Japan
Date/Time: Friday, 30 December 2022, 5:30 pm
Gmeet link: [1]meet.google.com/yduyqgusxq [2]
Title: Asymptotic behaviors of the Frobenius pushforwards of the ring of
invariants
Abstract: Let k be an algebraically closed field of characteristic p >
0, n a positive integer, and V = k^d. Let G be a finite subgroup of
GL(V) without pseudoreflections. Let S = Sym V be the symmetric algebra
of V, and A = S^G be the ring of invariants. The functor (?)^G gives an
equivalence between the category {}^*Ref(G,S), the category of Qgraded
Sfinite Sreflexive (G,S)modules and the category {}^*Ref(A), the
category of Qgraded Afinite Areflexive Amodules. As the ring of
invariants of the Frobenius pushforward ({}^e S)^G is the Frobenius
pushforward {}^eA, the study of the (G,S)module {}^e S for various e
yields good information on {}^eA. Using this principle, we get some
results on the properties of A coming from the asymptotic behaviors of
{}^eA.
In this talk, we will discuss the following:
(1) The generalized Fsignature of A (with Y. Nakajima and with P.
Symonds).
(2) Examples of G and V such that A is Frational, but not Fregular.
(3) Examples of G and V such that (the completion of) A is not of finite
Frepresentation type (work in progress with A. Singh).
Generalizing finite groups to finite group schemes G, we have that
s(A)>0 if and only if G is linearly reductive, and if this is the case,
s(A)=1/G, where G is the dimension of the coordinate ring k[G] of G,
provided the action of G on Spec S is 'small' (with F. Kobayashi).
For more information and links to previous seminars, visit the website
of VCAS:
https://sites.google.com/view/virtualcommalgebraseminar [3]
Links:

[1] http://goog_9085540/
[2] http://meet.google.com/yduyqgusxq
[3] https://sites.google.com/view/virtualcommalgebraseminar


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