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Let $A$ be an integral Noetherian $\mathbb{C}$-algebra. Denote by $\mathbb{P}^3_A:=\mathbb{P}^3_{\mathbb{C}} \times_{\mathbb{C}} \mathrm{Spec}(A)$. Let $X,Y$ be closed local complete intersection subschemes (not necessarily reduced or irreducible) in $\mathbb{P}^3_A$ of pure dimension $\dim A+1$ flat over $\mathrm{Spec}(A)$. Assume that $X \subset Y$. Denote by $\mathcal{I}$ the ideal sheaf obtained as the ideal quotient $[\mathcal{I}_Y:\mathcal{I}_X]$.

Assume further that $X$ is reduced and the reduced scheme associated to $Y$ is $X$. Note that, in this case, $[\mathcal{I}_Y:\mathcal{I}_X] \subset \mathcal{I}_X$. Then,

1. Is the scheme defined by $\mathcal{I}$ flat over $\mathrm{Spec}(A)$?

2. Does the ideal quotient agree with specialization i.e., for any maximal ideal $m$ of $A$, is $[\mathcal{I}_Y:\mathcal{I}_X] \otimes_A A/m \cong [\mathcal{I}_Y \otimes_A A/m:\mathcal{I}_X \otimes_A A/m]$?

Any idea/reference in this direction will be most appreciated.

Let $A$ be an integral Noetherian $\mathbb{C}$-algebra. Denote by $\mathbb{P}^3_A:=\mathbb{P}^3_{\mathbb{C}} \times_{\mathbb{C}} \mathrm{Spec}(A)$. Let $X,Y$ be closed local complete intersection subschemes (not necessarily reduced or irreducible) in $\mathbb{P}^3_A$ of pure dimension $\dim A+1$ flat over $\mathrm{Spec}(A)$. Assume that $X \subset Y$. Denote by $\mathcal{I}$ the ideal sheaf obtained as the ideal quotient $[\mathcal{I}_Y:\mathcal{I}_X]$.

Assume further that $X$ is reduced and the reduced scheme associated to $Y$ is $X$. Then,

1. Is the scheme defined by $\mathcal{I}$ flat over $\mathrm{Spec}(A)$?

2. Does the ideal quotient agree with specialization i.e., for any maximal ideal $m$ of $A$, is $[\mathcal{I}_Y:\mathcal{I}_X] \otimes_A A/m \cong [\mathcal{I}_Y \otimes_A A/m:\mathcal{I}_X \otimes_A A/m]$?

Any idea/reference in this direction will be most appreciated.

Let $A$ be a reduced Noetherian $\mathbb{C}$-algebra. Denote by $\mathbb{P}^3_A:=\mathbb{P}^3_{\mathbb{C}} \times_{\mathbb{C}} \mathrm{Spec}(A)$. Let $X,Y$ be closed local complete intersection subschemes (not necessarily reduced or irreducible) in $\mathbb{P}^3_A$ of pure dimension $1$ flat over $\mathrm{Spec}(A)$. Assume that $X \subset Y$. Denote by $\mathcal{I}$ the ideal sheaf obtained as the ideal quotient $[\mathcal{I}_Y:\mathcal{I}_X]$. Then,

1. Is the scheme defined by $\mathcal{I}$ flat over $\mathrm{Spec}(A)$?

2. Does the ideal quotient agree with specialization i.e., for any maximal ideal $m$ of $A$, is $[\mathcal{I}_Y:\mathcal{I}_X] \otimes_A A/m \cong [\mathcal{I}_Y \otimes_A A/m:\mathcal{I}_X \otimes_A A/m]$?

I am particularly interested in the case when $X$ is reduced and the reduced scheme associated to $Y$ is $X$. In this case, $[\mathcal{I}_Y:\mathcal{I}_X] \subset \mathcal{I}_X$

Is there any text where I can read on residual schemes of local complete intersections. I know Fulton's "Intersection theory" contains some theory on this but it restricts to cartier divisors. I am also aware of Gorenstein liaisons and complete intersection liaisons.

Let $A$ be an integral Noetherian $\mathbb{C}$-algebra. Denote by $\mathbb{P}^3_A:=\mathbb{P}^3_{\mathbb{C}} \times_{\mathbb{C}} \mathrm{Spec}(A)$. Let $X,Y$ be closed local complete intersection subschemes (not necessarily reduced or irreducible) in $\mathbb{P}^3_A$ of pure dimension $\dim A+1$ flat over $\mathrm{Spec}(A)$. Assume that $X \subset Y$. Denote by $\mathcal{I}$ the ideal sheaf obtained as the ideal quotient $[\mathcal{I}_Y:\mathcal{I}_X]$.

Assume further that $X$ is reduced and the reduced scheme associated to $Y$ is $X$. Note that, in this case, $[\mathcal{I}_Y:\mathcal{I}_X] \subset \mathcal{I}_X$. Then,

1. Is the scheme defined by $\mathcal{I}$ flat over $\mathrm{Spec}(A)$?

2. Does the ideal quotient agree with specialization i.e., for any maximal ideal $m$ of $A$, is $[\mathcal{I}_Y:\mathcal{I}_X] \otimes_A A/m \cong [\mathcal{I}_Y \otimes_A A/m:\mathcal{I}_X \otimes_A A/m]$?

Any idea/reference in this direction will be most appreciated.

Let $A$ be a reduced Noetherian $\mathbb{C}$-algebra. Denote by $\mathbb{P}^3_A:=\mathbb{P}^3_{\mathbb{C}} \times_{\mathbb{C}} \mathrm{Spec}(A)$. Let $X,Y$ be closed local complete intersection subschemes (not necessarily reduced or irreducible) in $\mathbb{P}^3_A$ of pure dimension $1$ flat over $\mathrm{Spec}(A)$. Assume that $X \subset Y$. Denote by $\mathcal{I}$ the ideal sheaf obtained as the ideal quotient $[\mathcal{I}_Y:\mathcal{I}_X]$. Assume that $[\mathcal{I}_Y:\mathcal{I}_X] \subset \mathcal{I}_X$. Then,

1. Is the scheme defined by $\mathcal{I}$ flat over $\mathrm{Spec}(A)$?

2. Does the ideal quotient agree with specialization i.e., for any maximal ideal $m$ of $A$, is $[\mathcal{I}_Y:\mathcal{I}_X] \otimes_A A/m \cong [\mathcal{I}_Y \otimes_A A/m:\mathcal{I}_X \otimes_A A/m]$?

Is there any text where I can read on residual schemes of local complete intersections. I know Fulton's "Intersection theory" contains some theory on this but it restricts to cartier divisors.

Let $A$ be a reduced Noetherian $\mathbb{C}$-algebra. Denote by $\mathbb{P}^3_A:=\mathbb{P}^3_{\mathbb{C}} \times_{\mathbb{C}} \mathrm{Spec}(A)$. Let $X,Y$ be closed local complete intersection subschemes (not necessarily reduced or irreducible) in $\mathbb{P}^3_A$ of pure dimension $1$ flat over $\mathrm{Spec}(A)$. Assume that $X \subset Y$. Denote by $\mathcal{I}$ the ideal sheaf obtained as the ideal quotient $[\mathcal{I}_Y:\mathcal{I}_X]$. Then,

1. Is the scheme defined by $\mathcal{I}$ flat over $\mathrm{Spec}(A)$?

2. Does the ideal quotient agree with specialization i.e., for any maximal ideal $m$ of $A$, is $[\mathcal{I}_Y:\mathcal{I}_X] \otimes_A A/m \cong [\mathcal{I}_Y \otimes_A A/m:\mathcal{I}_X \otimes_A A/m]$?

I am particularly interested in the case when $X$ is reduced and the reduced scheme associated to $Y$ is $X$. In this case, $[\mathcal{I}_Y:\mathcal{I}_X] \subset \mathcal{I}_X$

Is there any text where I can read on residual schemes of local complete intersections. I know Fulton's "Intersection theory" contains some theory on this but it restricts to cartier divisors. I am also aware of Gorenstein liaisons and complete intersection liaisons.