# Non Cohen-Macaulay rings of dimension 2

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.

• $\mathbb Z+x\mathbb Q[x]$ 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! Oct 12, 2022 at 0:49
• @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. Oct 12, 2022 at 14:58
• @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. Oct 12, 2022 at 15:59

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.