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A more general series of examples (when this fails) is given by choosing $B$ to be Cohen-Macaulay, but not Gorenstein and $A$ to be Gorenstein.

An explicit example is when $B=(k[x,y,z]/(xy,yz,zx)_{\mathfrak B=(k[x,y,z]/(xy,yz,zx))_{\mathfrak m}$ is the affine coordinate ring of three lines in $\mathbb A^3$ meeting in one point, but not contained in a plane and $A=k[x,y,z]_{\mathfrak m}$ where ${\mathfrak m}=(x,y,z)$.
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A more general series of examples (when this fails) is given by choosing $B$ to be Cohen-Macaulay, but not Gorenstein and $A$ to be Gorenstein.

An explicit example is when $B=k[x,y,z]/(xy,yz,zx)$ B=(k[x,y,z]/(xy,yz,zx)_{\mathfrak m}$ is the affine coordinate ring of three lines in $\mathbb A^3$ meeting in one point, but not contained in a plane and $A=k[x,y,z]$ A=k[x,y,z]_{\mathfrak m}$ where ${\mathfrak m}=(x,y,z)$.
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A more general series of examples (when this fails) is given by choosing $B$ to be Cohen-Macaulay, but not Gorenstein and $A$ to be Gorenstein.

An explicit example is when $B=k[x,y,z]/(xy,yz,zx)$ is the affine coordinate ring of three lines in $\mathbb A^3$ meeting in one point, but not contained in a plane and $A=k[x,y,z]$