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I am working in the context of Noetherian commutative rings with unit. It is well-known that for a field $K$, the ring $R=K[x^4,x^3y,xy^3,y^4]$ is a 2-dimensional standard graded $K$-algebra which not Cohen-Macaulay at its homogeneous maximal ideal $\mathfrak{m}=(x^4,x^3y,xy^3,y^4)$. In fact, one can show that $H^1_\mathfrak{m}(R)$ is supported only in degree 1 as a graded module over $R$.

I am searching for additional examples of non Cohen-Macaulay rings of dimension 2. In particular, I would like examples of a reduced, (standard) graded $K$-algebra $R$ of dimension 2 with homogeneous maximal ideal $\mathfrak{m}$ such that:

  • $H^1_\mathfrak{m}(R)$ is supported in degree $0$ and at least one other degree, and
  • the degree $0$ part is $[H^1_\mathfrak{m}(R)]_0=K$, the residue field.

If $R$ is a domain, then $H^1_\mathfrak{m}(R)$ is finite length and so will be supported in finitely many graded degrees -- this would be the case of most interest to me. Furthermore, I would prefer the example to be independent of the characteristic of $K$, but I primarily care about the case where the characteristic of $K$ is prime. I have already searched the example repository DaRT and could not find an example of behavior like this.

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    $\begingroup$ $\mathbb Z+x\mathbb Q[x]$ is at least 2 dimensional and not C-M, but I guess you have checked that it fails one of your bullet points, huh? I am not familiar with those conditions. If you find an example, I would like to see it! $\endgroup$
    – rschwieb
    Oct 12, 2022 at 0:49
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    $\begingroup$ @rschwieb - I think this example is not a $K$-algebra for any field $K$... My suspicion is that for any maximal ideal $\mathfrak{m}=p\mathbb{Z} + x\mathbb{Q}[x]$ of this ring, that $H^1_\mathfrak{m}$ will be infinite length (the domain implies $H^1$ has finite length part relies on $K$-algebra I think), but I have not worked it out. $\endgroup$
    – walkar
    Oct 12, 2022 at 14:58
  • $\begingroup$ @rschwieb To follow up, it looks like the reference I was thinking for domain implies finite length $H^1$ requires the ring to be the image of a Gorenstein domain to utilize local duality. I have no idea how to check whether $\mathbb{Z}+x\mathbb{Q}[x]$ has such a ring mapping onto it. The reference is Anurag K. Singh and Uli Walther - A connectedness result in positive characteristic - Proposition 2.4. $\endgroup$
    – walkar
    Oct 12, 2022 at 15:59

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I found many examples in the context of monomial ideals if I allow myself to drop the requirement that the example be reduced.

For example, if $S=K[x,y,z,w]/(xyz,xw,yw,zw)$, then $S$ is dimension $2$ and depth $1$, so is not Cohen-Macaulay, and further if $\mathfrak{m}$ is the homogeneous maximal ideal of $S$ we have $H^1_\mathfrak{m}(S)=K$ is concentrated in degree $0$.

By results of Takayama in the paper "Combinatorial characterizations of generalized Cohen-Macaulay monomial ideals," if we adjust to the ring $R=K[x,y,z,w]/(xyz,x^2w^2,y^2w^2,z^2w^2)$, then $H^1_\mathfrak{m}(R)$ will be supported in degree 0 with $[H^1_\mathfrak{m}(R)]_0=K$ and additional degrees, and still be dimension $2$ and depth $1$.

Many examples can be produced this way by adjusting Stanley-Reisner ideals related to simplicial complexes of dimension $1$ which are disconnected.

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