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I am looking for examples of Artinian Gorenstein subalgebras with the same socle degrees. More precisely, let $A$ be an Artinian Gorenstein $k$-algebra (with $k$ algebraically closed of characteristic $0$) with standard grading (generated in degree $1$) and socle degree, say $d$. Let $B \subsetneq A$ be an Artinian Gorenstein subalgebra of $A$. Is it possible that $B$ has the same socle degree $d$?

Similarly, if $B$ is a quotient of $A$ and $B$ is an Artinian, Gorenstein $k$-algera. Is it possible that $B$ has the same socle degree $d$?

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Let $A=k[x,y]/(x^2,y^2)$. If $\mathrm{char}(k)\neq 2$, then the subalgebra generated by $x+y$ is Gorenstein with the same socle degree. On the other hand, for any graded Artinian algebra $A$, any nonzero ideal $I$ intersects the socle notrivially. Hence if $A$ is Gorenstein then $I$ contains the socle (say of degree $d$), so $A/I$ has no elements of degree $d$.

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  • $\begingroup$ Thank you! Why should any nonzero ideal intersect the socle non-trivially? Could you please give a hint/reference. $\endgroup$
    – Chen
    Commented Jun 30 at 19:21
  • $\begingroup$ Hint: the socle of $I$ is contained in the socle of $A$. $\endgroup$
    – Richard Stanley
    Commented Jun 30 at 19:26

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