A note on the hit problem for the polynomial algebra in the case of odd primes and its application

2025/10/19 by Phuc, Dang Vo
#55R12 #55S05 #55S10 #55T15 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.17908

Abstract

Let P_h = \mathbbF_p[t₁,…,t_h] be the polynomial algebra over \mathbbF_p (p prime). We consider the hit problem: finding a minimal generating set for P_h as a module over the mod p Steenrod algebra \mathscrA_p, or equivalently, determining a basis for \mathbbF_p ⊗_\mathscrA_p P_h. This problem is related to the \mathscrA_p-module structure of H^*(V; \mathbbF_p) ≅ Λ(V^\sharp) ⊗ P_h, where V is an elementary abelian p-group of rank h. Information about the hit problem aids in studying the Singer algebraic transfer Tr_h^\mathscrA_p, a homomorphism from GL(h, \mathbbF_p)-coinvariants related to H^*(V; \mathbbF_p) to \rm Ext_\mathscrA_p^h,h+*(\mathbbF_p, \mathbbF_p), which helps analyze Ext groups. This work studies \mathscrA_p-generators for P_h when p is an odd prime. As an application, we investigate the third algebraic transfer (h=3) in certain generic degrees. Our main result shows that this transfer is an isomorphism in these degrees.

Citations

• The affirmative answer to Singer's conjecture on the algebraic transfer of rank four

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