Canonical Module and Socle of an Artinian $k$-Algebra

Question

Let $R$ be an Artinian $k$-algebra generated by elements of degree $1$. Denote the canonical module of $R$ by $\omega_R$. By Theorem 3.6.19 in Bruns and Herzog (CMR), we have that $\omega_R = (H_m^d(R))^\vee$, where $M^\vee := \operatorname{Hom}_k(M,k)$ is the graded Matlis-dual of $M$ and $d$ is the dimension of $R$. Since in our case $d=0$, we have $\omega_R = \operatorname{Hom}_k(R,k) \cong R$. The latter is an isomorphism of vector spaces.

Question 1: On top of page 179 B&H imply that $\omega_R$ is the homogeneous socle of $R$, i.e., the homogeneous elements of $R$ that get annihilated by $m$. Where does this come from? In what way is this consistent with what Theorem 3.6.19 gave above? Where does the underlined dimension equality come from?

Question 2: And why if $R$ is Artinian it is the case that $\omega_R$ is generated by homogeneous elements of the same degree only if the socle is concentrated at degree $s$? (see image for notation)

Answer 1

1. On page 179 the authors use Proposition 3.3.11(c)(i) and there is no reason to conclude that the canonical module equals the socle. (In fact, every artinian local ring is an essential extension of its socle (why?). If $\omega_R=\operatorname{Soc}R$, since $\omega_R=E_R(k)$ we get $\operatorname{Soc}R$ injective, so $\operatorname{Soc}R=R$, and therefore $R$ is a field.)

2. A Cohen-Macaulay homogeneous $k$-algebra is a level ring iff its canonical module is generated by $(\omega_R)_{-a(R)}$ as an $R$-module, where $a(R)$ is the $a$-invariant of $R$.
   If $R$ is a homogeneous artinian $k$-algebra, then $a(R)=\max\{i:R_i\ne 0\}$. (This is stated on page 143 and can be easily seen if you recall that $a(R)=\deg H_R(t)$.) In order to keep the notation in the book set $s=a(R)$. The maximal irrelevant ideal of $R$ is $R_1\oplus\cdots\oplus R_s$, and thus $R_s\subseteq{}^*\operatorname{Soc}(R)$.
   From Theorem 4.4.6(a) we have $H_{\omega_R}(t)=H_R(t^{-1})$, so $$\dim_k(\omega_R)_{-s}=H(\omega_R,-s)=H(R,s)=\dim_kR_s.$$ On the other side, $R$ is a level ring iff its canonical module is generated by $(\omega_R)_{-s}$ as an $R$-module, that is, iff $\mu(\omega_R)=\dim_k(\omega_R)_{-s}$ iff $R_s={}^*\operatorname{Soc}(R)$ (since $\mu(\omega_R)=\dim_k{}^*\operatorname{Soc}(R)$).