Looking for an example of a local homomorphism between regular and CM rings

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I am looking for a concrete example of a regular local ring $(R,m)$,
a Cohen-Macaulay local ring $(S,n)$ which is not Gorenstein, and a local homomorphism $(R,m) to (S,n)$ which is not flat.




commutative-algebra homological-algebra




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  • Take $R=k[[x_1,x_2,x_3,x_4]]$, $k$ any field. Take $S=k[[t^3,t^2u, tu^2,u^3]]$ with the map $x_1mapsto t^3, x_2mapsto t^2u$ etc.
    – Mohan
    Jul 30 at 15:05










  • Also, one can take $R = k[[x,y]]$, where $k$ is a field, and $S = k[[x,y]]/(x^2,xy,y^2)$.
    – Youngsu
    Jul 30 at 19:22














up vote
0
down vote

favorite
1












I am looking for a concrete example of a regular local ring $(R,m)$,
a Cohen-Macaulay local ring $(S,n)$ which is not Gorenstein, and a local homomorphism $(R,m) to (S,n)$ which is not flat.




commutative-algebra homological-algebra




share|cite|improve this question



















  • Take $R=k[[x_1,x_2,x_3,x_4]]$, $k$ any field. Take $S=k[[t^3,t^2u, tu^2,u^3]]$ with the map $x_1mapsto t^3, x_2mapsto t^2u$ etc.
    – Mohan
    Jul 30 at 15:05










  • Also, one can take $R = k[[x,y]]$, where $k$ is a field, and $S = k[[x,y]]/(x^2,xy,y^2)$.
    – Youngsu
    Jul 30 at 19:22












up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
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I am looking for a concrete example of a regular local ring $(R,m)$,
a Cohen-Macaulay local ring $(S,n)$ which is not Gorenstein, and a local homomorphism $(R,m) to (S,n)$ which is not flat.




commutative-algebra homological-algebra




share|cite|improve this question











I am looking for a concrete example of a regular local ring $(R,m)$,
a Cohen-Macaulay local ring $(S,n)$ which is not Gorenstein, and a local homomorphism $(R,m) to (S,n)$ which is not flat.





commutative-algebra homological-algebra





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 30 at 13:53









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  • Take $R=k[[x_1,x_2,x_3,x_4]]$, $k$ any field. Take $S=k[[t^3,t^2u, tu^2,u^3]]$ with the map $x_1mapsto t^3, x_2mapsto t^2u$ etc.
    – Mohan
    Jul 30 at 15:05










  • Also, one can take $R = k[[x,y]]$, where $k$ is a field, and $S = k[[x,y]]/(x^2,xy,y^2)$.
    – Youngsu
    Jul 30 at 19:22
















  • Take $R=k[[x_1,x_2,x_3,x_4]]$, $k$ any field. Take $S=k[[t^3,t^2u, tu^2,u^3]]$ with the map $x_1mapsto t^3, x_2mapsto t^2u$ etc.
    – Mohan
    Jul 30 at 15:05










  • Also, one can take $R = k[[x,y]]$, where $k$ is a field, and $S = k[[x,y]]/(x^2,xy,y^2)$.
    – Youngsu
    Jul 30 at 19:22















Take $R=k[[x_1,x_2,x_3,x_4]]$, $k$ any field. Take $S=k[[t^3,t^2u, tu^2,u^3]]$ with the map $x_1mapsto t^3, x_2mapsto t^2u$ etc.
– Mohan
Jul 30 at 15:05




Take $R=k[[x_1,x_2,x_3,x_4]]$, $k$ any field. Take $S=k[[t^3,t^2u, tu^2,u^3]]$ with the map $x_1mapsto t^3, x_2mapsto t^2u$ etc.
– Mohan
Jul 30 at 15:05












Also, one can take $R = k[[x,y]]$, where $k$ is a field, and $S = k[[x,y]]/(x^2,xy,y^2)$.
– Youngsu
Jul 30 at 19:22




Also, one can take $R = k[[x,y]]$, where $k$ is a field, and $S = k[[x,y]]/(x^2,xy,y^2)$.
– Youngsu
Jul 30 at 19:22















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