Skip to main content
MathOverflow

Return to Answer

added a comment for $n>3$
Source Link
Vladimir Dotsenko
Vladimir Dotsenko
  • 16.9k
  • 1
  • 55
  • 114

Existence of such an example follows from the same result of Asanuma that is crucial for Gupta's work, see the article

Teruo Asanuma, "Polynomial fibre rings of algebras over noetherian rings", Inventiones mathematicae 87 (1987), 101–127 (DOI link).

It follows from Corollary 5.3 of that article that if
$$ A=\mathbb{Z}[x,y,z]/(-x^{p^e}+y+y^{sp}+pz), $$ where $p$ is a prime number and $e,s$ are positive integers such that $p^e\not\mid sp$, $sp\not\mid p^e$, then $A$ is not isomorphic to a polynomial ring, but $B=A[t]$ is isomorphic to a polynomial ring in three variables. Of course, $A$ is a retract of $B$ via the evaluation at $t=0$.

EDIT: Moreover, consider $$ A_n:=\mathbb{Z}[x,y,z,x_1,\ldots,x_n]/(-x^{p^e}+y+y^{sp}+px_1\cdots x_nz), $$ then the proof of the same Corollary 5.3 (in the case $R=\mathbb{Z}[x_1,\ldots,x_n]$) can be adapted to show that $A_n[t]$ is isomorphic to the polynomial ring in $n+3$ variables, but $A_n$ is not isomorphic to the polynomial ring in $n+2$ variables.

Existence of such an example follows from the same result of Asanuma that is crucial for Gupta's work, see the article

Teruo Asanuma, "Polynomial fibre rings of algebras over noetherian rings", Inventiones mathematicae 87 (1987), 101–127 (DOI link).

It follows from Corollary 5.3 of that article that if
$$ A=\mathbb{Z}[x,y,z]/(-x^{p^e}+y+y^{sp}+pz), $$ where $p$ is a prime number and $e,s$ are positive integers such that $p^e\not\mid sp$, $sp\not\mid p^e$, then $A$ is not isomorphic to a polynomial ring, but $B=A[t]$ is isomorphic to a polynomial ring in three variables. Of course, $A$ is a retract of $B$ via the evaluation at $t=0$.

Existence of such an example follows from the same result of Asanuma that is crucial for Gupta's work, see the article

Teruo Asanuma, "Polynomial fibre rings of algebras over noetherian rings", Inventiones mathematicae 87 (1987), 101–127 (DOI link).

It follows from Corollary 5.3 of that article that if
$$ A=\mathbb{Z}[x,y,z]/(-x^{p^e}+y+y^{sp}+pz), $$ where $p$ is a prime number and $e,s$ are positive integers such that $p^e\not\mid sp$, $sp\not\mid p^e$, then $A$ is not isomorphic to a polynomial ring, but $B=A[t]$ is isomorphic to a polynomial ring in three variables. Of course, $A$ is a retract of $B$ via the evaluation at $t=0$.

EDIT: Moreover, consider $$ A_n:=\mathbb{Z}[x,y,z,x_1,\ldots,x_n]/(-x^{p^e}+y+y^{sp}+px_1\cdots x_nz), $$ then the proof of the same Corollary 5.3 (in the case $R=\mathbb{Z}[x_1,\ldots,x_n]$) can be adapted to show that $A_n[t]$ is isomorphic to the polynomial ring in $n+3$ variables, but $A_n$ is not isomorphic to the polynomial ring in $n+2$ variables.

changed to perennial DOI link (also now pointing to Springerlink)
Source Link
YCor
YCor
  • 63.9k
  • 5
  • 187
  • 286

Existence of such an example follows from the same result of Asanuma that is crucial for Gupta's work, see the article

Teruo Asanuma, "Polynomial fibre rings of algebras over noetherian rings", Inventiones mathematicae 87 (1987), 101–127 (https://link.springer.com/article/10.1007%2FBF01389155DOI link).

It follows from Corollary 5.3 of that article that if
$$ A=\mathbb{Z}[x,y,z]/(-x^{p^e}+y+y^{sp}+pz), $$ where $p$ is a prime number and $e,s$ are positive integers such that $p^e\not\mid sp$, $sp\not\mid p^e$, then $A$ is not isomorphic to a polynomial ring, but $B=A[t]$ is isomorphic to a polynomial ring in three variables. Of course, $A$ is a retract of $B$ via the evaluation at $t=0$.

Existence of such an example follows from the same result of Asanuma that is crucial for Gupta's work, see the article

Teruo Asanuma, "Polynomial fibre rings of algebras over noetherian rings", Inventiones mathematicae 87 (1987), 101–127 (https://link.springer.com/article/10.1007%2FBF01389155).

It follows from Corollary 5.3 of that article that if
$$ A=\mathbb{Z}[x,y,z]/(-x^{p^e}+y+y^{sp}+pz), $$ where $p$ is a prime number and $e,s$ are positive integers such that $p^e\not\mid sp$, $sp\not\mid p^e$, then $A$ is not isomorphic to a polynomial ring, but $B=A[t]$ is isomorphic to a polynomial ring in three variables. Of course, $A$ is a retract of $B$ via the evaluation at $t=0$.

Existence of such an example follows from the same result of Asanuma that is crucial for Gupta's work, see the article

Teruo Asanuma, "Polynomial fibre rings of algebras over noetherian rings", Inventiones mathematicae 87 (1987), 101–127 (DOI link).

It follows from Corollary 5.3 of that article that if
$$ A=\mathbb{Z}[x,y,z]/(-x^{p^e}+y+y^{sp}+pz), $$ where $p$ is a prime number and $e,s$ are positive integers such that $p^e\not\mid sp$, $sp\not\mid p^e$, then $A$ is not isomorphic to a polynomial ring, but $B=A[t]$ is isomorphic to a polynomial ring in three variables. Of course, $A$ is a retract of $B$ via the evaluation at $t=0$.

Source Link
Vladimir Dotsenko
Vladimir Dotsenko
  • 16.9k
  • 1
  • 55
  • 114

Existence of such an example follows from the same result of Asanuma that is crucial for Gupta's work, see the article

Teruo Asanuma, "Polynomial fibre rings of algebras over noetherian rings", Inventiones mathematicae 87 (1987), 101–127 (https://link.springer.com/article/10.1007%2FBF01389155).

It follows from Corollary 5.3 of that article that if
$$ A=\mathbb{Z}[x,y,z]/(-x^{p^e}+y+y^{sp}+pz), $$ where $p$ is a prime number and $e,s$ are positive integers such that $p^e\not\mid sp$, $sp\not\mid p^e$, then $A$ is not isomorphic to a polynomial ring, but $B=A[t]$ is isomorphic to a polynomial ring in three variables. Of course, $A$ is a retract of $B$ via the evaluation at $t=0$.