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First, here is my primary question:

Is there an example of a connected graded Noetherian $k$-algebra $R$ such that there is a $\mathfrak{p} \in \mathrm{Ass}_R R$ with $\mathrm{dim}\; R/\mathfrak{p}=d$, and $H^d_{\mathfrak{m} }R =0$? (Here $\mathfrak{m}$ denotes the ideal of positively graded elements, and $H$ denotes local cohomology.)

Here is a bit of context for this question. The smallest $i$ with $H^i_{\mathfrak{m}}R \not= 0$ is equal to the depth of $R$, and we know that if $\mathfrak{p}$ is an associated prime with $\dim R/ \mathfrak{p} =d$, then the depth of $R$ is less than or equal to $d$. Therefore, a naive hope is that having a $d$-dimensional associated prime implies that the $d^{th}$ local cohomology is nonzero. I noticed that this is true for a special class of graded rings, and I would like to know whether or not this is true in general.

By the vanishing and nonvanshing theorems for local cohomology, a ring that is an affirmative answer to my primary question would necessarily have a gap of at least two between its depth and dimension, and it must have an associated prime whose dimension is strictly between the depth and dimension of the ring.

Here is how I have tried to construct an example. The idea is to glue a "fat line" with an even fatter origin to $\mathbb{A}^2$ along the $x$-axis. The resulting ring is $S=k[x,y,z,w]/(w^2,z^2,wx,wy,wz,xz)$. However, this ring is finite over $k[x,y]$, and as a module over $k[x,y]$ we see that $S$ splits up as a direct sum $k[x,y] \oplus \Sigma k \oplus \Sigma k[x,y]/x$, where $\Sigma$ denotes a shift in degree. The free part is the submodule generated by $1$, the copy of $k$ is the submodule generated by $w$, and the copy of $k[x,y]/x$ is the submodule generated by $z$. Of course, because it splits up this way we can compute its local cohomology using the computation for the local cohomology of a polynomial ring, and we see that $H^1$ isn't zero. Therefore an example can't split up in this way.

Bonus question: the $S$ above considered as a $k[x,y]=R$ module has the property that $\dim_R (d(H^i_{\mathfrak{m}} S ))\le i$, where $d$ denotes the Matlis dual functor, which in this situation is just the $k$-linear dual. Is there a setting in which this is always true?

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    For a local noetherian ring $(R,m)$, a finitely generated non-zero $R$-module $M$, and a prime $p\in\mathrm{Supp}(M)$, if $H^i_{pR_p}(M_p)\ne0$ then $H^{i+t}_M(M)\ne 0$, where $t=\dim R/p$. (The result belongs to Sharp.) – 2017-02-08
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    Of course, I mean $H^{i+t}_m(M)\ne 0$. – 2017-02-08
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    @user26857 I didn't know that result, thank you. That answers the question, doesn't it? If $p \in \mathrm{Ass}_R M$, then won't $\Gamma_{p R_p} M_p$ be nonzero? – 2017-02-08
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    Yes, I think so. – 2017-02-08
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    @user26857 Interesting. Any idea of a reference? Is it in Brodman and Sharp? I am still interested in the "bonus question", perhaps I should ask that separately. If you feel like posting your comment as an answer, I'll happily accept it. Thanks! – 2017-02-08
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    It's from a paper of Sharp entitled "Some results on the vanishing of local cohomology modules". (I hope I remember this correctly.) – 2017-02-08
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    I think it's better to ask the second question separately. – 2017-02-08

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