# Normality of certain subrings of polynomial rings in characteristic p

Let $$k$$ be an algebraically closed field of characteristic p. Let $$Z\subset k[x_1,\cdots,x_n]$$ be a graded $$k$$-subalgebra of a polynomial ring, such that for any $$f\in Z,$$ any divisor of $$f$$ (in $$k[x_1,\cdots,x_n]$$) is again an element of $$Z.$$ Then is $$Z[x_1^p,\cdots,x_n^p]$$ a normal ring? For the easiest case of $$Z=k[f]-$$where $$f$$ is an irreducible homogeneous polynomial, the answer is yes, since $$Z[x_1^p,\cdots,x_n^p]$$ in this case is a complete intersection and regular in codimension 2.

• Welcome new contributor. I do not understand your assertion. Consider the case that $n$ equals $2$ and $Z$ equals the subalgebra of $k[x_1,x_2]$ generated by $f=x_1^p + x_2^{p+1}$. Then $Z[x_1^p,x_2^p]$ equals the $k$-subalgebra $k[x_1^p,x_2^p,x_2^{p+1}]$ of $k[x_1,x_2]$. This $k$-subalgebra is not normal, it is not a complete intersection scheme, and it is not regular in codimension $1$, much less in codimension $2$. – Jason Starr Feb 11 at 2:36
• Thank you, I forgot to add that Z is graded subalgebra, I hope this at least saves the question for Z=k[f]. – John Zek. Feb 11 at 2:52
• Welcome new contributor. I still do not understand your assertion. If you insist that $f$ be homogeneous, then please consider the case that $n$ equals $3$ and that $Z$ is the graded $k$-subalgebra of $k[x_1,x_2,x_3]$ generated by $f=x_1^p - x_2^{p-1}x_3$. Then $Z[x_1^p,x_2^p,x_3^p]$ equals $k[x_1^p,x_2^p,x_3^p,x_2^{p-1}x_3]$. This $k$-algebra is not normal, and it is not regular in codimension $1$. – Jason Starr Feb 11 at 12:38
• Thank you again, clearly I've been making a silly mistake. – John Zek. Feb 11 at 16:19