I've faced a problem while reading a paper. It is mentioned to be trivial but I couldn't prove it. I'd appreciate if you can lead me to some resources or if you can prove it for me. Thank you.
Let $ S=K[X_1,X_2,X_3] $ be the polynomial ring with standard grading. The $c$th Veronese algebra of $S$ is $S^{(c)}=\bigoplus_{i}S_{ic}$. I know $S^{(c)}$ is a Cohen-Macaulay ring. I need to use Hilbert function to prove that $\operatorname{reg}S^{(c)}\leq2$. Any other approach is also acceptable.
This should be easy once you know few basis facts, such as:
$H^i_{\mathfrak m}(S)^{(c)}\simeq H^i_{\mathfrak m^{(d)}}(S^{(c)})$; see here, Theorem 3.1.1.
$\operatorname{reg}S=0$.
We then have $H^i_{\mathfrak m^{(d)}}(S^{(c)})_j\simeq H^i_{\mathfrak m}(S)_{cj}$. The only $i$ that counts is $i=3$ (why?), and from 2. we have $H^3_{\mathfrak m}(S)_{cj}=0$ unless $cj\le -3$. Since $cj\le -3$ we get $j\le -1$, so $\operatorname{reg}S^{(c)}\le 2$.