BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Mitsuyasu Hashimoto (Osaka Metropolitan University\, Sumiyoshi-ku\ , Osaka\, Japan) DTSTART:20221230T120000Z DTEND:20221230T130000Z DTSTAMP:20260423T021058Z UID:VCAS/171 DESCRIPTION:Title: A symptotic behaviors of the Frobenius pushforwards of the ring of invariant s\nby Mitsuyasu Hashimoto (Osaka Metropolitan University\, Sumiyoshi-k u\, Osaka\, Japan) as part of IIT Bombay Virtual Commutative Algebra Semi nar\n\n\nAbstract\nLet k be an algebraically closed field of characteris tic 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 symm etric 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 categor y {}^*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 talk about the following:\n\nthe g eneralized F-signature of A (with Y. Nakajima and with P. Symonds).\nExam ples of G and V such that A is F-rational\, but not F-regular.\nExamp les of G and V such that (the completion of) A is not of finite F-re presentation type (work in progress with A. Singh).\nGeneralizing 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 acti on of G on Spec S is ‘small’ (with F. Kobayashi).\n LOCATION:https://researchseminars.org/talk/VCAS/171/ END:VEVENT END:VCALENDAR