[Paper Review] A $D$-module approach on the equations of the Rees algebra

Yairon Cid‐Ruiz|arXiv (Cornell University)|Jun 19, 2017

Commutative Algebra and Its Applications1 references6 citations

TL;DR

This paper introduces a novel $D$-module approach to characterize the defining equations of the Rees algebra of a height two ideal in $\mathbb{F}[x_1,x_2]$, minimally generated by three homogeneous polynomials of equal degree. By applying the Fourier transform to the symmetric algebra relations, it shows that the kernel $\mathcal{K}$ of the canonical map from the symmetric to the Rees algebra is isomorphic to the solution space of a system of differential equations, with its bigraded structure fully determined by the integral roots of associated $b$-functions.

ABSTRACT

Let $I \subset R = \mathbb{F}[x_1,x_2]$ be a height two ideal minimally generated by three homogeneous polynomials of the same degree $d$, where $\mathbb{F}$ is a field of characteristic zero. We use the theory of $D$-modules to deduce information about the defining equations of the Rees algebra of $I$. Let $\mathcal{K}$ be the kernel of the canonical map $α: ext{Sym}(I) ightarrow ext{Rees}(I)$ from the symmetric algebra of $I$ onto the Rees algebra of $I$. We prove that $\mathcal{K}$ can be described as the solution set of a system of differential equations, that the whole bigraded structure of $\mathcal{K}$ is characterized by the integral roots of certain $b$-functions, and that certain de Rham cohomology groups can give partial information about $\mathcal{K}$.

Motivation & Objective

• To understand the defining equations of the Rees algebra of a height two ideal in $\mathbb{F}[x_1,x_2]$ generated by three homogeneous polynomials of equal degree.
• To characterize the kernel $\mathcal{K}$ of the canonical map from the symmetric algebra to the Rees algebra.
• To use $D$-module theory to describe $\mathcal{K}$ as the solution set of a system of differential equations.
• To relate the bigraded structure of $\mathcal{K}$ to the integral roots of $b$-functions associated with the differential operators.
• To explore connections between de Rham cohomology and the structure of $\mathcal{K}$.

Proposed method

• Apply the Fourier transform to the generators $g_1, g_2$ of the symmetric algebra relations to obtain differential operators $L_1 = \mathcal{F}(g_1)$, $L_2 = \mathcal{F}(g_2)$ in the Weyl algebra.
• Define the solution space $\mathrm{Sol}(L_1, L_2; S)$ as the set of elements in $S = R[T_1,T_2,T_3]$ annihilated by $L_1$ and $L_2$, and show $\mathcal{K} \cong \mathrm{Sol}(L_1, L_2; S)_{\mathcal{F}}(-2, -d+2)$.
• Use the twisted $\mathcal{T}$-module structure on $S$ and the dualizing complex to relate $\mathcal{K}$ to $\mathrm{Tor}$ and $\mathrm{Ext}$ functors over $\mathcal{T}$.
• Construct a bigraded $S$-module structure on $\mathcal{K}$ via the bidegrees $\mathrm{bideg}(x_i) = (0,1)$, $\mathrm{bideg}(T_i) = (1,0)$, and $\mathrm{bideg}(t) = (1,-d)$.
• Compute the $b$-function of the $D$-module associated to the differential system using Macaulay2, leveraging Gröbner deformations and the Weyl algebra.
• Use the $b$-function's integral roots to determine the non-vanishing bigraded components of $\mathcal{K}$.

Experimental results

Research questions

• RQ1 How can $D$-module theory be applied to describe the defining equations of the Rees algebra of a height two ideal in $\mathbb{F}[x_1,x_2]$?
• RQ2 What is the precise relationship between the kernel $\mathcal{K}$ of the map $\mathrm{Sym}(I) \to \mathrm{Rees}(I)$ and the solution space of a system of differential equations?
• RQ3 How do the integral roots of the $b$-functions of the associated $D$-modules determine the bigraded structure of $\mathcal{K}$?
• RQ4 Can de Rham cohomology groups provide partial information about the structure of $\mathcal{K}$?
• RQ5 To what extent can the $b$-function computation predict the minimal generators of the defining ideal of the Rees algebra?

Key findings

• The kernel $\mathcal{K}$ of the canonical map $\mathrm{Sym}(I) \to \mathrm{Rees}(I)$ is isomorphic to the solution space of a system of differential equations: $\mathcal{K} \cong \mathrm{Sol}(L_1, L_2; S)_{\mathcal{F}}(-2, -d+2)$, where $L_1, L_2$ are the Fourier transforms of the symmetric algebra relations.
• The bigraded structure of $\mathcal{K}$ is completely determined by the integral roots of the $b$-functions associated with the differential operators $L_1$ and $L_2$, with $\mathcal{K}_{p,q} \neq 0$ if and only if $q$ lies within the range determined by these roots.
• For the monomial ideal $I = (x^5, x^2y^3, y^5)$, the $b$-functions for $p=2,3,4$ factor as $s(s+1)(s+2)$, indicating that $\mathcal{K}_{p,q} \neq 0$ precisely for $1 \leq q \leq 3$, and for $p=5$, the $b$-function is $s(s+1)(s+2)(s+3)$, so $\mathcal{K}_{5,q} \neq 0$ for $0 \leq q \leq 3$.
• In the case $\mu=1$, $d=7$, the $b$-functions for $p=2$ to $7$ are $s$, $s(s+1)$, $s(s+1)(s+2)$, $s(s+1)(s+2)(s+3)$, $s(s+1)(s+2)(s+3)(s+4)$, and $s(s+1)(s+2)(s+3)(s+4)(s+5)$, respectively, confirming that $\mathcal{K}_{p,q} \neq 0$ for $q$ in the expected range.
• The $\mathcal{T}$-module structure and the use of $\mathrm{Tor}$ and $\mathrm{Ext}$ functors yield an isomorphism $\mathcal{K} \cong \{w \in Q \mid \partial_1 \bullet w = 0, \partial_2 \bullet w = 0\}$, where $Q$ is a certain $\mathcal{T}$-module, providing a dual characterization of $\mathcal{K}$.