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 talk about the following:

the generalized F-signature of A (with Y. Nakajima and with P. Symonds). Examples of G and V such that A is F-rational, but not F-regular. 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).

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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