If $A$ is a Cohen Macaulay local ring, and $B$ is a quotient ring of $A$ and $B$ is also Cohen Macaulay, Then is $B$ always a quotient by a regular sequence of $A$?



A problem on Cohen Macaulay ring
Here is an even better example:


$A=(k[x,y,z]/(xy-z^2))\_{\mathfrak m}$ with ${\mathfrak m}=(x,y,z)$, $B=k[x]\_{(x)}$.
Here $A$ is Gorenstein and $B$ is regular. The map is given by $x\mapsto x$ and $y,z\mapsto 0$.


Geometrically, $B$ corresponds to a line going through the vertex of a quadric cone corresponding to $A$. It is not a quotient by a regular sequence, because that would have to be just a regular element for dimension reasons, but the line is not a Cartier divisor, so it cannot be defined by a single equation. To see that it is not a Cartier divisor, for example one can compute its self-intersection number which is $\frac 12$.


