Virtual Commutative Algebra Seminar.

Speaker: Mitsuyasu Hashimoto, Metropolitan University, Sumiyoshi-ku, 
Osaka, Japan

Date/Time: Friday, 30 December 2022, 5:30 pm

Gmeet link:  [1]meet.google.com/ydu-yqgu-sxq [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 Q-graded 
S-finite S-reflexive (G,S)-modules and the category {}^*Ref(A), the 
category of Q-graded A-finite A-reflexive A-modules. 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 F-signature of A (with Y. Nakajima and with P. 
Symonds).

(2) Examples of G and V such that A is F-rational, but not F-regular.

(3) Examples of G and V such that (the completion of) A is not of finite 
F-representation 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/virtual-comm-algebra-seminar [3]

Links:
------
[1] http://goog_9085540/
[2] http://meet.google.com/ydu-yqgu-sxq
[3] https://sites.google.com/view/virtual-comm-algebra-seminar