In his book *Commutative Ring Theory*, Matsumura proves that if a local ring is equidimensional, and a quotient of a regular local ring, then its completion is equidimensional.


What is an example of a local ring which does not admit such a presentation?



A local ring not a quotient of a regular local ring
A source available online is this [paper](https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-70/issue-none/Examples-of-bad-Noetherian-local-rings/nmj/1118785530.full) "Examples of bad Noetherian rings" by Marinari (example 2.1).


The reason many of these types of construction work is because of the following vague and counter-intuitive phenomemon:


*It is usually easier than we think for a complete local ring to be a completion of a Noetherian ring with certain properties.*


For example, there is this amazing [theorem](https://www.jstor.org/stable/2154327) by Heitmann that most complete local ring of depth at least $2$ is a completion of a UFD !


So back to Marinari's paper, the example is as follows: start with some local Artinian ring $(Q,m)$ such that $Q$ is not Gorenstein. Then $Q[[X]]$ is complete, and one can find a local domain $R$ such that $\hat R=Q[[X]]$. Now if $R$ is a quotient of a regular local ring, then the comletion of $R$ is generically a complete intersection. But $Q[[X]]$ is not even generically Gorenstein, since $Q$ is not.


