Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups

We consider purely inseparable extensions H ↪ H P ∗ \textrm {H}\hookrightarrow \sqrt [\mathscr {P}^*]{\textrm {H}} of unstable Noetherian integral domains over the Steenrod algebra. It turns out that there exists a finite group G ≤ GL ( V ) G\le \textrm {GL}(V) and a vector space decomposition V = W 0 ⊕ W 1 ⊕ ⋯ ⊕ W e V=W_0\oplus W_1\oplus \dotsb \oplus W_e such that H ¯ = ( F [ W 0 ] ⊗ F [ W 1 ] p ⊗ ⋯ ⊗ F [ W e ] p e ) G \overline {\textrm {H}}=(\mathbb {F}[W_0] \otimes \mathbb {F}[W_1]^p\otimes \dotsb \otimes \mathbb {F}[W_e]^{p^e})^G and H P ∗ ¯ = F [ V ] G \overline {\sqrt [\mathscr {P}^*]{\textrm {H}}}=\mathbb {F}[V]^G , where ( − ) ¯ \overline {(-)} denotes the integral closure. Moreover, H \textrm {H} is Cohen-Macaulay if and only if H P ∗ \sqrt [\mathscr {P}^*]{\textrm {H}} is Cohen-Macaulay. Furthermore, H ¯ \overline {\textrm {H}} is polynomial if and only if H P ∗ \sqrt [\mathscr {P}^*]{\textrm {H}} is polynomial, and H P ∗ = F [ h 1 , … , h n ] \sqrt [\mathscr {P}^*]{\textrm {H}}=\mathbb {F}[h_1,\dotsc ,h_n] if and only if \[ H = F [ h 1 , … , h n 0 , h n 0 + 1 p , … , h n 1 p , h n 1 + 1 p 2 , … , h n e p e ] , \textrm {H}=\mathbb {F}[h_1,\dotsc ,h_{n_0},h_{n_0+1}^p,\dotsc ,h_{n_1}^p, h_{n_1+1}^{p^2},\dotsc ,h_{n_e}^{p^e}], \] where n e = n n_e=n and n i = dim F ⁡ ( W 0 ⊕ ⋯ ⊕ W i ) n_i=\dim _{\mathbb {F}}(W_0\oplus \dotsb \oplus W_i) .