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I forget to say that U = X-D
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Ariyan Javanpeykar
Ariyan Javanpeykar
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Let $X$ be a regular integral noetherian scheme of dimension 2 and let $D$ be a simple normal crossings divisor in $X$.

EDIT: Let $U = X-D$.

Consider a finite etale morphism $V\longrightarrow U$ with $V$ connected. Let $\pi:Y\longrightarrow X$ be the normalization of $X$ in the function field of $V$. So $Y$ is a $2$-dimensional normal noetherian scheme and $\pi$ is finite.

One can show that $\pi$ is surjective, flat and that $Y$ is CM.

Q: Is $\pi$ a local complete intersection?

Let $X$ be a regular integral noetherian scheme of dimension 2 and let $D$ be a simple normal crossings divisor in $X$.

Consider a finite etale morphism $V\longrightarrow U$ with $V$ connected. Let $\pi:Y\longrightarrow X$ be the normalization of $X$ in the function field of $V$. So $Y$ is a $2$-dimensional normal noetherian scheme and $\pi$ is finite.

One can show that $\pi$ is surjective, flat and that $Y$ is CM.

Q: Is $\pi$ a local complete intersection?

Let $X$ be a regular integral noetherian scheme of dimension 2 and let $D$ be a simple normal crossings divisor in $X$.

EDIT: Let $U = X-D$.

Consider a finite etale morphism $V\longrightarrow U$ with $V$ connected. Let $\pi:Y\longrightarrow X$ be the normalization of $X$ in the function field of $V$. So $Y$ is a $2$-dimensional normal noetherian scheme and $\pi$ is finite.

One can show that $\pi$ is surjective, flat and that $Y$ is CM.

Q: Is $\pi$ a local complete intersection?

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Ariyan Javanpeykar
Ariyan Javanpeykar
  • 9.5k
  • 2
  • 36
  • 79

Is this finite surjective flat morphism of 2 dimensional schemes a local complete intersection

Let $X$ be a regular integral noetherian scheme of dimension 2 and let $D$ be a simple normal crossings divisor in $X$.

Consider a finite etale morphism $V\longrightarrow U$ with $V$ connected. Let $\pi:Y\longrightarrow X$ be the normalization of $X$ in the function field of $V$. So $Y$ is a $2$-dimensional normal noetherian scheme and $\pi$ is finite.

One can show that $\pi$ is surjective, flat and that $Y$ is CM.

Q: Is $\pi$ a local complete intersection?