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3 edited body

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 on 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 $\frac 12$.

2 deleted 2 characters in body

Here is an even better example:

$A=(k[x,y,z]/(xy-z^2))_{\mathfrak m}$ with ${\mathfrak m}=(x,y,z)$, $B=k[x]_{(x)}$. 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$corresponds on a line going through the vertex of a quadric cone corresponding to$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$. 1 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 on 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\$.