A non-graded Gorenstein Artin $k$-algebra such that its associated graded ring is also Gorenstein
I am wondering if there is a non-graded Gorenstein Artin $k$-algebra $R$ such that its associated graded ring, $mathrm{gr}(R)$, is also Gorenstein.
All the non-graded Gorenstein rings I tried so far have not Gorenstein associated graded rings.
Thanks for your help.
commutative-algebra graded-rings gorenstein
edited Dec 1 '18 at 9:04
user26857
39.2k123983
asked Jun 21 '12 at 19:46
eileen
463
add a comment |
I am wondering if there is a non-graded Gorenstein Artin $k$-algebra $R$ such that its associated graded ring, $mathrm{gr}(R)$, is also Gorenstein.
All the non-graded Gorenstein rings I tried so far have not Gorenstein associated graded rings.
Thanks for your help.
commutative-algebra graded-rings gorenstein
edited Dec 1 '18 at 9:04
user26857
39.2k123983
asked Jun 21 '12 at 19:46
eileen
463
add a comment |
I am wondering if there is a non-graded Gorenstein Artin $k$-algebra $R$ such that its associated graded ring, $mathrm{gr}(R)$, is also Gorenstein.
All the non-graded Gorenstein rings I tried so far have not Gorenstein associated graded rings.
Thanks for your help.
commutative-algebra graded-rings gorenstein
edited Dec 1 '18 at 9:04
user26857
39.2k123983
asked Jun 21 '12 at 19:46
eileen
463
I am wondering if there is a non-graded Gorenstein Artin $k$-algebra $R$ such that its associated graded ring, $mathrm{gr}(R)$, is also Gorenstein.
All the non-graded Gorenstein rings I tried so far have not Gorenstein associated graded rings.
Thanks for your help.
commutative-algebra graded-rings gorenstein
commutative-algebra graded-rings gorenstein
edited Dec 1 '18 at 9:04
user26857
39.2k123983
asked Jun 21 '12 at 19:46
eileen
463
edited Dec 1 '18 at 9:04
user26857
39.2k123983
asked Jun 21 '12 at 19:46
eileen
463
edited Dec 1 '18 at 9:04
user26857
39.2k123983
edited Dec 1 '18 at 9:04
user26857
39.2k123983
edited Dec 1 '18 at 9:04
user26857
39.2k123983
39.2k123983
asked Jun 21 '12 at 19:46
eileen
463
asked Jun 21 '12 at 19:46
eileen
463
asked Jun 21 '12 at 19:46
eileen
463
463
add a comment |
add a comment |
2 Answers
2
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oldest
votes
Let $R=k[x,y]/(xy^2, x^3-y^2)$. Then $text{gr}(R)=k[x,y]/(x^4,y^2)$.
$R$ is nongraded Gorenstein and $mathrm{gr}(R)$ is also Gorenstein.
edited Dec 1 '18 at 9:06
user26857
39.2k123983
answered Jun 22 '12 at 21:39
eileen
463
add a comment |
Let R be a field. ${}{}{}{}{}{}{}$
answered Feb 18 '15 at 17:36
Mariano Suárez-Álvarez
110k7155280
A field is graded, isn't it?
– user26857
Dec 1 '18 at 9:06
add a comment |
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2 Answers
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active
oldest
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2 Answers
2
active
oldest
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active
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Let $R=k[x,y]/(xy^2, x^3-y^2)$. Then $text{gr}(R)=k[x,y]/(x^4,y^2)$.
$R$ is nongraded Gorenstein and $mathrm{gr}(R)$ is also Gorenstein.
edited Dec 1 '18 at 9:06
user26857
39.2k123983
answered Jun 22 '12 at 21:39
eileen
463
add a comment |
Let $R=k[x,y]/(xy^2, x^3-y^2)$. Then $text{gr}(R)=k[x,y]/(x^4,y^2)$.
$R$ is nongraded Gorenstein and $mathrm{gr}(R)$ is also Gorenstein.
edited Dec 1 '18 at 9:06
user26857
39.2k123983
answered Jun 22 '12 at 21:39
eileen
463
add a comment |
Let $R=k[x,y]/(xy^2, x^3-y^2)$. Then $text{gr}(R)=k[x,y]/(x^4,y^2)$.
$R$ is nongraded Gorenstein and $mathrm{gr}(R)$ is also Gorenstein.
edited Dec 1 '18 at 9:06
user26857
39.2k123983
answered Jun 22 '12 at 21:39
eileen
463
Let $R=k[x,y]/(xy^2, x^3-y^2)$. Then $text{gr}(R)=k[x,y]/(x^4,y^2)$.
$R$ is nongraded Gorenstein and $mathrm{gr}(R)$ is also Gorenstein.
edited Dec 1 '18 at 9:06
user26857
39.2k123983
answered Jun 22 '12 at 21:39
eileen
463
edited Dec 1 '18 at 9:06
user26857
39.2k123983
edited Dec 1 '18 at 9:06
user26857
39.2k123983
edited Dec 1 '18 at 9:06
user26857
39.2k123983
39.2k123983
answered Jun 22 '12 at 21:39
eileen
463
answered Jun 22 '12 at 21:39
eileen
463
answered Jun 22 '12 at 21:39
eileen
463
463
add a comment |
add a comment |
Let R be a field. ${}{}{}{}{}{}{}$
answered Feb 18 '15 at 17:36
Mariano Suárez-Álvarez
110k7155280
A field is graded, isn't it?
– user26857
Dec 1 '18 at 9:06
add a comment |
Let R be a field. ${}{}{}{}{}{}{}$
answered Feb 18 '15 at 17:36
Mariano Suárez-Álvarez
110k7155280
A field is graded, isn't it?
– user26857
Dec 1 '18 at 9:06
add a comment |
Let R be a field. ${}{}{}{}{}{}{}$
answered Feb 18 '15 at 17:36
Mariano Suárez-Álvarez
110k7155280
Let R be a field. ${}{}{}{}{}{}{}$
answered Feb 18 '15 at 17:36
Mariano Suárez-Álvarez
110k7155280
answered Feb 18 '15 at 17:36
Mariano Suárez-Álvarez
110k7155280
answered Feb 18 '15 at 17:36
Mariano Suárez-Álvarez
110k7155280
answered Feb 18 '15 at 17:36
Mariano Suárez-Álvarez
110k7155280
110k7155280
A field is graded, isn't it?
– user26857
Dec 1 '18 at 9:06
add a comment |
A field is graded, isn't it?
– user26857
Dec 1 '18 at 9:06
A field is graded, isn't it?
– user26857
Dec 1 '18 at 9:06
A field is graded, isn't it?
– user26857
Dec 1 '18 at 9:06
add a comment |
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