Let $k$ be a field, $S = k[x_1,\dots,x_n]$ the polynomial ring, $m = (x_1,\dots,x_n)$ and $I$ a homogeneous ideal contained in $m^2$. Define $R = S/I$. For $p \in \mathbb{N}$ we say that $R$ is $p$-regular if $H_m^q(R)_{p-q} = 0$ for all $q>0$ and $H_m^0(R)_r=0$ for all $r \ge 0$. I want to prove that if $R$ is $p$-regular and $p' >p$ then $R$ is also $p'$-regular.
I have proved that not all elements of $m$ in $R / H_m^0(R)$ are zero-divisors. In particular some $x_i$ is not a zero-divisor, say $i=1$. Then $x_1^d$ is not a zero divisor for any positive $d$. This gives an exact sequence $0 \rightarrow \frac{R}{H_m^0(R)} (-d) \stackrel{x_1^d}{\rightarrow} \frac{R}{H_m^0(R)} \rightarrow \frac{R}{H_m^0(R)+(a)} \rightarrow 0$.
Question 1: How can i use this sequence to show that $H_m^q(R)_{p+d-q} = 0$?
If i take the long exact cohomology sequence corresponding to the exact sequence above, i encounter a difficulty:
Question 2: How does $H_m^q(R/H_m^0(R))$ relate to $H_m^q(R)$? Intuitively, it seems that these two are equal for $q>0$ but how can we prove that if this is the case?
The answer to question 2 would be positive if $H_m^0(R)$ is a *injective module. This motivates
Question 3: Is $H_m^0(R)$ an injective module in the category of graded modules?