Does anybody know an example of a Noetherian local ring $(R,m)$ which admits a maximal Cohen-Macaulay module of type one, but the ring $R$ itself is not CM?

If $C$ is a maximal CM module then the type of $C$, denoted by $r(C)$, is defined to be $\dim_{R/m}\mathrm{Ext}^d (R/m, C)$, where $d= \dim(R)$.


There are many examples. Take $A$ be any regular local ring, and B is not a Cohen-Macaulay $A$-module. Now you can check that $R = A \ltimes B$ and $M = A$ is an example.


The Following is my communications with Olgur Celikbas in researchgate:

Ogler:You may want to check: $R=k[[x,y]]/(x^2, xy)$ and $M=R/(x)$

me:does still such an emaple exist if we suppose that R is unmixed of dimension 2?(Unmixed = every associated prime is of height zero)

Ogler:Try this one: the ring of the example is from a paper of Costa, Huneke and Miller, titled "complete local domains of type 2 are CM".

$R=k[[x,y,z]]/(xz,yz)$. Then $dim(R)=2$, $depth(R)=1$ ($x-z$ is a non zero-divisor) and $R$ is reduced (so that associated primes are height zero.) Let $M=R/(z)$. Then $depth(M)=2$, i.e., $M$ is MCM. It looks like $Ext^0(k,M)=Ext^1(k,M)=0$ and $Ext^2(k,M)=k$ so $Type(M)=1$. (Type means the vector space dim of the first non zero $Ext$ w.r.t $R/m$)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.