Diagonal Subalgebras of Residual Intersections

TL;DR

This paper investigates the conditions under which diagonal subalgebras of residual intersections are Koszul or Cohen-Macaulay, providing bounds and applications to ideals with linear resolutions.

Contribution

It establishes bounds for parameters ensuring Koszulness of diagonal subalgebras in geometric residual intersections and explores their Cohen-Macaulay properties, with applications to ideals in polynomial rings.

Findings

Bounds for c,e ensuring Koszul property of S_Δ

Conditions for Cohen-Macaulayness of these algebras

All powers of certain linearly presented ideals have linear resolutions

Abstract

Let ${\sf k}$ be a field, $S$ be a bigraded ${\sf k}$-algebra, and $S_\Delta$ denote the diagonal subalgebra of $S$ corresponding to $\Delta = \{ (cs,es) \; | \; s \in \mathbb{Z} \}$. It is know that the $S_\Delta$ is Koszul for $c,e \gg 0$. In this article, we find bounds for $c,e$ for $S_\Delta$ to be Koszul, when $S$ is a geometric residual intersection. Furthermore, we also study the Cohen-Macaulay property of these algebras. Finally, as an application, we look at classes of linearly presented perfect ideals of height two in a polynomial ring, show that all their powers have a linear resolution, and study the Koszul, and Cohen-Macaulay property of the diagonal subalgebras of their Rees algebras.

Equations

$$\left[\begin{array}{cccc}{z}_1& z_2& \dots & z_m\end{array}\right]=\left[\begin{array}{cccc}{x}_1& x_2& \dots & x_n\end{array}\right]\cdot \varphi .$$

$$0⟶S(-m{)}^{\oplus \left(\genfrac{}{}{0ex}{}{m}{m}\right)}\stackrel{{\varphi }^m}{\to }S(-m+1{)}^{\oplus \left(\genfrac{}{}{0ex}{}{m}{m-1}\right)}\stackrel{{\varphi }^{m-1}}{\to }\cdots ⟶S(-2{)}^{\oplus \left(\genfrac{}{}{0ex}{}{m}{2}\right)}\stackrel{{\varphi }^2}{\to }S(-1{)}^{\oplus \left(\genfrac{}{}{0ex}{}{m}{1}\right)}\stackrel{{\varphi }^1}{\to }S\to 0.$$

$${\mathbb{K}}_{\bullet }(\underline{z};S{)}_d:\text{hfill}0⟶S_{d-m}\stackrel{{\varphi }_d^m}{\to }(S_{d-m+1}{)}^{\oplus \left(\genfrac{}{}{0ex}{}{m}{m-1}\right)}⟶\cdots ⟶(S_{d-2}{)}^{\oplus \left(\genfrac{}{}{0ex}{}{m}{2}\right)}\stackrel{{\varphi }_d^2}{\to }(S_{d-1}{)}^{\oplus \left(\genfrac{}{}{0ex}{}{m}{1}\right)}\stackrel{{\varphi }_d^1}{\to }{S}_d.$$

$${\mathbb{K}}_{\bullet }(\underline{z};S{)}_d^{\ast }:0\to S_d^{\ast }\stackrel{({\varphi }_d^1{)}^{\ast }}{\to }(S_{d-1}^{\ast }{)}^{\oplus \left(\genfrac{}{}{0ex}{}{m}{1}\right)}⟶\cdots \stackrel{({\varphi }_d^{m-1}{)}^{\ast }}{\to }(S_{d-m+1}^{\ast }{)}^{\oplus \left(\genfrac{}{}{0ex}{}{m}{m-1}\right)}\stackrel{({\varphi }_d^m{)}^{\ast }}{\to }{S}_{d-m}^{\ast }\to 0.$$

$$\mathbb{EN}:0\to S_{m-n}^{\ast }\stackrel{({\varphi }_{m-n}^1{)}^{\ast }}{\to }(S_{m-n-1}^{\ast }{)}^{\oplus \left(\genfrac{}{}{0ex}{}{m}{1}\right)}⟶\cdots ⟶(S_1^{\ast }{)}^{\oplus \left(\genfrac{}{}{0ex}{}{m}{m-n-1}\right)}\stackrel{({\varphi }_{m-n}^{m-n}{)}^{\ast }}{\to }(S_0^{\ast }{)}^{\left(\genfrac{}{}{0ex}{}{m}{m-n}\right)}\stackrel{ϵ}{\to }{R}_y\to 0$$

$$\begin{array}{rl}{\mathrm{reg}}_x^S(S/J)& =\sup \{a-i|{\beta }_{i,(a,b)}^S(S/J)\ne 0\text{ for some i,b∈Z}\},\\ {\mathrm{reg}}_y^S(S/J)& =\sup \{b-i|{\beta }_{i,(a,b)}^S(S/J)\ne 0\text{ for some i,a∈Z}\}.\end{array}$$

$$\left[\begin{array}{cccc}{z}_1& z_2& \dots & z_m\end{array}\right]=\left[\begin{array}{cccc}{x}_1& x_2& \dots & x_n\end{array}\right]\cdot \varphi .$$

$${\mathbb{T}}_{\bullet }:0\to P_m\to P_{m-1}\to \cdots \to P_n\to Q_{n-1}\oplus P_{n-1}\to \cdots \to Q_1\oplus P_1\to Q_0\to S/J\to 0$$

$$a+b=i-1,$$

$$Q_k$$

$$P_i$$

$$r(i,j)=\left(\genfrac{}{}{0ex}{}{n+i-2j-2}{i-1-j}\right)\left(\genfrac{}{}{0ex}{}{n+i-1-j}{j}\right)\left(\genfrac{}{}{0ex}{}{m}{n+i-1-j}\right).$$

$${\beta }_{i,(a,b)}^S(S/J)=\{\begin{array}{ll}\left(\genfrac{}{}{0ex}{}{m}{i}\right)& 0\le i=a=b\le n-1\\ r(i,a)& 1\le i\le m,b=n+i-1-a,\\ & \max \{0,i-(m-n+1)\}\le a\le \min \{i-1,n-1\}\\ 0& \text{otherwise}\end{array}$$

$$0\to (F_m{)}_{\Delta }\to \cdots \to (F_i{)}_{\Delta }\to \cdots \to (F_1{)}_{\Delta }\to S_{\Delta }\to (S/J{)}_{\Delta }\to 0$$

$${\mathrm{depth}}_{S_{\Delta }}(S/J{)}_{\Delta }\ge \min \{{\mathrm{depth}}_{S_{\Delta }}(F_i{)}_{\Delta }-i|0\le i\le m\}.$$

$${\mathrm{reg}}_{S_{△}}(F_i{)}_{△}=\max \left\{\lceil \frac{a_i^{\max }}{c}\rceil ,\lceil \frac{b_i^{\max }}{e}\rceil \right\},$$

$${\mathrm{reg}}_{S_{△}}(S/J{)}_{△}\le \max \left\{\lceil \frac{a_i^{\max }}{c}\rceil -i,\lceil \frac{b_i^{\max }}{e}\rceil -i|0\le i\le m\right\}.$$

$$\lceil \frac{a_i^{\max }}{c}\rceil -i\le \max \left\{\lceil \frac{i}{c}\rceil -i|1\le i\le (n-1)\right\}\le 1,$$

$$\lceil \frac{b_i^{\max }}{e}\rceil -i\le \max \left\{\lceil \frac{n-1+i}{e}\rceil -i|1\le i\le (m-n+1)\right\}=\lceil \frac{n}{e}\rceil -1\le 1,$$

$$\Phi =\left[\begin{array}{cccc}{x}_1& 0& 0& 0\\ x_2& x_1& 0& 0\\ x_3& x_2& x_1& 0\\ 0& x_3& x_2& x_1\\ 0& 0& x_3& x_2\end{array}\right].$$

$$[z_1\cdots z_4]$$

$$=\left[\begin{array}{cccc}{x}_1{y}_1+x_2{y}_2+x_3{y}_3& x_1{y}_2+x_2{y}_3+x_3{y}_4& x_1{y}_3+x_2{y}_4+x_3{y}_5& x_1{y}_4+x_2{y}_5\end{array}\right]$$

$$=[x_1{x}_2{x}_3]\left[\begin{array}{cccc}{y}_1& y_2& y_3& y_4\\ y_2& y_3& y_4& y_5\\ y_3& y_4& y_5& 0\end{array}\right]=[x_1{x}_2{x}_3]\cdot \varphi .$$

$$\begin{gathered} 0\to S(-2,-4{)}^6\to S(-2,-3{)}^{12}\oplus S(-1,-4{)}^8\to S(-1,-3{)}^{12}\oplus S(0,-4{)}^3\oplus \\ S(-2,-2{)}^6\to S(0,-3{)}^4\oplus S(-1,-1{)}^4\to S. \end{gathered}$$

Full text

Introduction

Let $\displaystyle{\sf k}$ be a field, $\displaystyle c,e\in\mathbb{N}$, and $\displaystyle\Delta=\{(cs,es)\;|\;s\in\mathbb{Z}\}$ be the $\displaystyle(c,e)$-diagonal of $\displaystyle\mathbb{Z}^{2}$. Given a bigraded $\displaystyle{\sf k}$-algebra $\displaystyle S=\bigoplus_{(u,v)\in{\mathbb{Z}^{2}_{\geq 0}}}S_{(u,v)}$, one can associate a graded $\displaystyle{\sf k}$-algebra $\displaystyle S_{\Delta}=\oplus_{s\in\mathbb{Z}}S_{(cs,es)}$, the $\displaystyle(c,e)$-diagonal subalgebra of $\displaystyle S$. In a special case, when $\displaystyle S={\sf k}[x_{1},\dots,x_{n},y_{1},\dots,y_{p}]$ with $\displaystyle\deg x_{i}=(1,0)$ and $\displaystyle\deg y_{j}=(0,1)$, then $\displaystyle S_{\Delta}$ is the homogeneous coordinate ring of the image of the correspondence under the Segre embedding $\displaystyle\mathbb{P}^{n-1}\times\mathbb{P}^{p-1}\hookrightarrow\mathbb{P}^{np-1}$ (cf. [16]).

The motivation for the study of diagonal subalgebras comes from algebraic geometry. Many authors have studied the algebraic properties (for instance, normality, Cohen-Macaulayness, defining equations etc.) of rational surfaces obtained by blowing up projective space $\displaystyle\mathbb{P}^{2}$ at a finite set of points, for instance see [9, 10, 11]. Simis, Trung and Valla introduced diagonal subalgebras to study the normality and the Cohen-Macaulayness of rational surfaces obtained by blowing up projective space $\displaystyle\mathbb{P}^{2}$ along a subvariety (see [16]). Conca, Herzog, Trung and Valla used diagonal subalgebras as an effective tool for the study of normality, Cohen-Macaulayness, Gorensteinness and Koszulness of embedded $\displaystyle(n-1)$-fold obtained by blowing up $\displaystyle\mathbb{P}^{n-1}$ along a subvariety (cf. [6]).

The notion of residual intersections was introduced and initially studied by Artin-Nagata ([2]), and Huneke-Ulrich ([12]). Our motivation for studying geometric residual intersections is the following: Morey and Ulrich ([14]) show that if $\displaystyle I$ is a linearly presented perfect ideal of height two in a polynomial ring, its Rees algebra $\displaystyle\mathcal{R}(I)$ is a geometric residual intersection when $\displaystyle I$ satisfies certain conditions (see 3.1). For such an ideal $\displaystyle I$, the authors in [6] also show that $\displaystyle\mathcal{R}(I)_{\Delta}$ is Cohen Macaulay for $\displaystyle c,e\gg 0$.

The Koszul property of the diagonal subalgebras of the Rees algebra of ideals generated by a regular sequence has been studied in multiple articles. In particular, for an ideal $\displaystyle I$, generated by a homogeneous regular sequence $\displaystyle f_{1},f_{2},f_{3}\in{\sf k}[x_{1},\ldots,x_{n}]$ of the same degree, explicit lower bounds are obtained in [5] and [13], such that for larger values of $\displaystyle c$ and $\displaystyle e$, $\displaystyle\mathcal{R}(I)_{\Delta}$ is Koszul. More generally, the authors in [6] also show that for given any standard bigraded $\displaystyle{\sf k}$-algebra $\displaystyle S$, one has $\displaystyle S_{\Delta}$ is Koszul for $\displaystyle c,e\gg 0$.

Motivated by the above results, we focus on giving explicit bounds for $\displaystyle c,e$ for diagonal subalgebras of certain classes of geometric residual intersections to be Koszul, or Cohen-Macaulay. More specifically, in this article, given positive integers $\displaystyle m\geq n$, we study an ideal $\displaystyle J=\langle z_{1},\ldots z_{m}\rangle+I_{n}(\phi)$ in the polynomial ring $\displaystyle S={\sf k}[x_{1},\ldots,x_{n},y_{1},\ldots,y_{p}]$, where $\displaystyle\phi$ is an $\displaystyle n\times m$ matrix, with entries linear in $\displaystyle\underline{y}$, such that $\displaystyle\underline{z}=\underline{x}\phi$. When $\displaystyle\operatorname{ht}(J)\geq m$, and $\displaystyle\operatorname{ht}(I_{n}(\phi))\geq m-n+1$, $\displaystyle J$ is a geometric residual intersection of $\displaystyle\langle x_{1},\dots,x_{n}\rangle$. For such ideals, Bruns, Kustin and Miller construct an $\displaystyle S$-free resolution of $\displaystyle S/J$ ([4, Theorem 3.6]).

Our main tool is to show that this resolution gives a bigraded resolution of $\displaystyle S/J$ over $\displaystyle S$, when $\displaystyle S$ is bigraded with $\displaystyle\deg(x_{i})=(1,0)$ and $\displaystyle\deg(y_{j})=(0,1)$. Using the construction given in [4], we first compute the bigraded shifts in Remark 2.2, and use this to compute the bigraded Betti numbers, $\displaystyle x$-regularity and $\displaystyle y$-regularity of $\displaystyle S/J$ over $\displaystyle S$ in 2.3. In order to understand the Cohen-Macaulay property, we use this to give a bound on the depth of $\displaystyle(S/J)_{\Delta}$ in Theorem 2.5 for all $\displaystyle\Delta$. Moreover, we also show that it is Koszul when $\displaystyle e\geq n/2$, in Theorem 2.7.

We give applications of the results in Section 2 to perfect ideals $\displaystyle I$ of height two, which are linearly presented in a polynomial ring $\displaystyle R={\sf k}[x_{1},\ldots,x_{n}]$, which further satisfy $\displaystyle\mu(I_{\mathfrak{p}})\leq\dim(R_{\mathfrak{p}})$ for every $\displaystyle\mathfrak{p}\in V(I)$ with $\displaystyle\dim(R_{\mathfrak{p}})\leq p-1$. This property can also be reformulated as the ideal $\displaystyle I$ being of linear type in the punctured spectrum. In Theorem 3.3, we first show that all powers of such ideals have a linear resolution, generalising a result of Römer ([15, Corollary 5.11]). We also prove that the diagonal subalgebras of the Rees algebra of $\displaystyle I$ are Cohen-Macaulay when $\displaystyle c>(p-1)e$, and are Koszul when $\displaystyle e\geq n/2$. The Koszul nature of diagonal subalgebras of the Rees algebras of such ideals has not been studied before.

Section 1 contains the definitions, basic observations, and known properties of the main objects appearing in this paper. We end the paper with an example in Section 4 to understand the construction given in Section 2 explicitly, and to apply it to understand our results.

Acknowledgements

We thank the anaonymous referee for comments and suggestions which improved the accuracy of the article greatly.

1. Preliminaries

Notations

1. a)

$\displaystyle c$ and $\displaystyle e$ denote positive integers, $\displaystyle\Delta=\{(ci,ei)\ |\ i\in\mathbb{Z}\}$ is the $\displaystyle(c,e)$-diagonal of $\displaystyle\mathbb{Z}^{2}$, and for a real number $\displaystyle\alpha$, we let $\displaystyle\lceil\alpha\rceil$ denote the least integer greater than or equal to $\displaystyle\alpha$. 2. b)

$\displaystyle{\sf k}$ denotes a field, $\displaystyle\underline{x}=x_{1},\ldots x_{n}$, $\displaystyle\underline{y}=y_{1},\ldots,y_{p}$ denote sets of indeterminates over $\displaystyle{\sf k}$, $\displaystyle R_{x}={\sf k}[\underline{x}]$ and $\displaystyle R_{y}={\sf k}[\underline{y}]$ are polynomial rings. 3. c)

$\displaystyle R$ denotes a graded $\displaystyle{\sf k}$-algebra, i.e., $\displaystyle R=\oplus_{i\geq 0}R_{i}$ is a graded ring with $\displaystyle R_{0}={\sf k}$. Let $\displaystyle R_{+}=\oplus_{i\geq 1}R_{i}$ be the unique homogenous maximal ideal of $\displaystyle R$. We say that $\displaystyle R$ is standard graded if $\displaystyle R_{+}$ is generated by $\displaystyle R_{1}$. 4. d)

Given a graded $\displaystyle R$-module $\displaystyle M$ and $\displaystyle j\in\mathbb{Z}$, the shifted module $\displaystyle M(j)$ is the graded $\displaystyle R$-module with $\displaystyle i$th graded component $\displaystyle M(j)_{i}=M_{i+j}$. In particular, $\displaystyle R(-j)$ is the graded free $\displaystyle R$-module of rank one, with the generator in degree $\displaystyle j$. 5. e)

$\displaystyle S=\bigoplus_{(i,j)\in\mathbb{Z}^{2}_{\geq 0}}S_{(i,j)}$ denotes a bigraded $\displaystyle{\sf k}$-algebra, i.e., $\displaystyle S$ is bigraded and $\displaystyle S_{(0,0)}={\sf k}$. Just as in the graded case, for integers $\displaystyle a,b$, the bigraded free $\displaystyle S$-module of rank one, generated in bidegree $\displaystyle(a,b)$, is denoted $\displaystyle S(-a,-b)$.

Eagon-Northcott Complex

Let $\displaystyle S=R_{x}\otimes_{\sf k}R_{y}={\sf k}[\underline{x},\underline{y}]$ be a polynomial ring over $\displaystyle{\sf k}$. Let $\displaystyle m\geq n$, and $\displaystyle\phi$ be an $\displaystyle n\times m$ matrix with entries in $\displaystyle R_{y}$. Define $\displaystyle z_{1},\ldots,z_{m}\in S$ by

[TABLE]

Consider the Koszul complex $\displaystyle\mathbb{K}_{\bullet}(\underline{z};S)$ on $\displaystyle\underline{z}=z_{1},z_{2},\ldots,z_{m}$:

[TABLE]

The $\displaystyle d$-th graded component, with respect to the $\displaystyle x$-grading, of the above complex gives us a complex of free $\displaystyle R_{y}$-modules:

[TABLE]

Applying $\displaystyle\operatorname{Hom}_{R_{y}}(\_\!\_,R_{y})=(\_\!\_)^{\ast}$ to Equation 1, we get a new complex of $\displaystyle R_{y}$-modules:

[TABLE]

Definition 1.1**.**

[Eagon-Northcott Complex] With the notation as above, the Eagon-Northcott complex of the matrix $\displaystyle\phi$ is given by

[TABLE]

where the map $\displaystyle\epsilon$ is given by the $\displaystyle 1\times{m\choose m-n}$ row matrix whose entries are the $\displaystyle(n\times n)$ - minors of the matrix $\displaystyle\phi$.

Remark 1.2**.**

Notice that the first $\displaystyle m-n$ components of $\displaystyle\mathbb{EN}$ are nothing but $\displaystyle\mathbb{K}_{\bullet}(\underline{z};S)_{m-n}^{\ast}$. The complex $\displaystyle\mathbb{EN}$ is exact when $\displaystyle\operatorname{grade}\left(I_{n}(\phi)\right)=m-n+1$ (e.g., [8, Theorem A2.10]). Observe that the homology at the zeroth stage of the above complex is $\displaystyle R_{y}/(\operatorname{Im}(\epsilon))=R_{y}/I_{n}(\phi)$. Hence, if $\displaystyle\operatorname{grade}I_{n}(\phi)=m-n+1$, we see that $\displaystyle\mathbb{EN}$ gives a minimal free resolution of $\displaystyle R_{y}/I_{n}(\phi)$ over $\displaystyle R_{y}$.

Segre Product, Veronese and diagonal subalgebras

Definition 1.3**.**

Let $\displaystyle c,e\in\mathbb{N}$, $\displaystyle S$ a bigraded $\displaystyle{\sf k}$-algebra, $\displaystyle W$ a bigraded $\displaystyle S$-module, $\displaystyle A$ and $\displaystyle B$ be graded $\displaystyle{\sf k}$-algebras, $\displaystyle M$ and $\displaystyle N$ graded $\displaystyle A$ and $\displaystyle B$ modules respectively.

1. a)

The Segre product of $\displaystyle A$ and $\displaystyle B$ is defined as $\displaystyle A\underline{\otimes}_{{\sf k}}B=\bigoplus_{i\geq 0}(A_{i}\otimes B_{i})$. The Segre product of $\displaystyle M$ and $\displaystyle N$ is the $\displaystyle A\underline{\otimes}_{{\sf k}}B$-module $\displaystyle M\underline{\otimes}_{{\sf k}}N=\bigoplus_{i\in\mathbb{Z}}(M_{i}\otimes N_{i})$. 2. b)

The $\displaystyle c$-th Veronese subring of $\displaystyle A$ is is the graded $\displaystyle{\sf k}$-algebra $\displaystyle A^{(c)}=\bigoplus_{i\geq 0}A_{ci}$. 3. c)

The $\displaystyle(c,e)$-diagonal subalgebra of $\displaystyle S$ is the graded $\displaystyle{\sf k}$-algebra $\displaystyle S_{\Delta}=\bigoplus_{i\geq 0}S_{(ci,ei)}$, and the $\displaystyle(c,e)$-diagonal module corresponding to $\displaystyle W$ is the graded $\displaystyle S_{\Delta}$-module $\displaystyle W_{\Delta}=\bigoplus_{i\geq 0}W_{(ci,ei)}$.

Remark 1.4**.**

Let the notation be as in the above definition.

1. a)

$\displaystyle(\_\!\_)_{\Delta}$ is an exact functor from the category of bigraded $\displaystyle S$-modules to the category of graded $\displaystyle S_{\Delta}$-modules. However, for integers $\displaystyle a,b$, note that $\displaystyle S(-a,-b)_{\Delta}=\bigoplus_{i\in\mathbb{Z}}S_{(-a+ci,-b+ei)}$ need not be a free $\displaystyle S_{\Delta}$-module. 2. b)

For a bihomogeneous ideal $\displaystyle J$ in $\displaystyle S$, $\displaystyle(S/J)_{\Delta}$ has a natural graded $\displaystyle{\sf k}$-algebra structure by definition. 3. c)

The $\displaystyle{\sf k}$-algebra $\displaystyle A\otimes_{{\sf k}}B$ is naturally bigraded and the Segre product $\displaystyle A\underline{\otimes}_{{\sf k}}B$ is the $\displaystyle(1,1)$-diagonal subalgebra of $\displaystyle A\otimes_{{\sf k}}B$. More generally, the $\displaystyle(c,e)$-diagonal subalgebra of $\displaystyle A\otimes_{{\sf k}}B$ is the Segre product of the Veronese subalgebras $\displaystyle A^{(c)}$ and $\displaystyle B^{(e)}$. 4. d)

Let $\displaystyle S={\sf k}[x_{1},\ldots,x_{n},y_{1}\ldots,y_{p}]$ with bigrading $\displaystyle\deg(x_{i})=(1,0)$ and $\displaystyle\deg(y_{j})=(0,1)$ for all $\displaystyle i$ and $\displaystyle j$. By [6, Lemmas $\displaystyle 3.1$, $\displaystyle 3.3$] we have $\displaystyle\dim\left(S(-a,-b)_{\Delta}\right)=p+n-1$. Finally, [6, Lemma 3.10] shows that $\displaystyle S(-a,-b)_{\Delta}$ is Cohen-Macaulay if and only if the following two conditions are satisfied:

(i) $\displaystyle\left\lfloor\frac{a-n}{c}\right\rfloor<\frac{b}{e}$ and (ii) $\displaystyle\left\lfloor\frac{b-p}{e}\right\rfloor<\frac{a}{c}$.

Residual Intersections and Rees Algebras

Let $\displaystyle R$ be any Cohen-Macaulay local ring, $\displaystyle I\subset R$ be an ideal and $\displaystyle\underline{z}=z_{1},\dots,z_{m}\in I$ with $\displaystyle\langle\underline{z}\rangle\neq I$.

Definition 1.5**.**

We say $\displaystyle J=\langle\underline{z}\rangle:I$ is an $\displaystyle m$-residual intersection of $\displaystyle I$ if $\displaystyle\operatorname{ht}(J)\geq m\geq\operatorname{ht}(I)$. Furthermore, if $\displaystyle I_{\mathfrak{p}}=\langle\underline{z}\rangle_{\mathfrak{p}}$ for all $\displaystyle\mathfrak{p}\in V(I)$ with $\displaystyle\operatorname{ht}(\mathfrak{p})\leq m$, then $\displaystyle J$ is called an $\displaystyle m$-geometric residual intersection of $\displaystyle I$.

Example 1.6**.**

Consider the ring $\displaystyle R=k[x_{1},x_{2},y_{1},\ldots,y_{4}],I=(x_{1},x_{2})$ and $\displaystyle[z_{1}~{}z_{2}]=[x_{1}~{}x_{2}]\cdot\phi$ where $\displaystyle\phi=\begin{bmatrix}y_{1}&y_{3}\\ y_{2}&y_{4}\end{bmatrix}$. Then $\displaystyle\langle z_{1},z_{2}\rangle:I$ is a geometric residual intersection of $\displaystyle I$ [12, Theorem 3.3(i)].

Definition 1.7**.**

Let $\displaystyle I=\langle f_{1},\ldots,f_{p}\rangle$ be a homogeneous ideal in $\displaystyle R_{x}={\sf k}[x_{1},\dots,x_{n}]$. We say that $\displaystyle\Phi$ is a presentation matrix of $\displaystyle I$ if $\displaystyle({\sf k}[\underline{x}])^{m}\overset{\Phi}{\longrightarrow}({\sf k}[\underline{x}])^{p}\rightarrow I\rightarrow 0$ is exact. The Rees algebra of $\displaystyle I$ is the subalgebra of $\displaystyle{\sf k}[\underline{x},t]$ defined as $\displaystyle\mathcal{R}(I)={\sf k}[\underline{x},f_{1}t,\ldots,f_{p}t]$.

Remark 1.8**.**

Let $\displaystyle S=R_{x}\otimes_{\sf k}R_{y}={\sf k}[x_{1},\dots,x_{n},y_{1},\ldots,y_{p}]$.

1. a)

The ring $\displaystyle S$ maps naturally onto $\displaystyle\mathcal{R}(I)$ by $\displaystyle x_{i}\mapsto x_{i}$ and $\displaystyle y_{j}\mapsto f_{j}t$. The kernel $\displaystyle\mathcal{K}$ of this map is called the defining ideal of the Rees algebra $\displaystyle\mathcal{R}(I)$. 2. b)

Let $\displaystyle[z_{1}~{}\cdots~{}z_{m}]=[y_{1}~{}\cdots y_{p}]\cdot\Phi$, where $\displaystyle\Phi$ is a $\displaystyle p\times m$ presentation matrix of $\displaystyle I$. Then we have $\displaystyle\langle z_{1},\dots,z_{m}\rangle\subset\mathcal{K}$. If $\displaystyle\mathcal{K}=\langle z_{1},\dots,z_{m}\rangle$, then we say that $\displaystyle I$ is an ideal of linear type. 3. c)

Assume that $\displaystyle I^{e}$ is generated by forms of degree atmost $\displaystyle c-1$. By [6, Lemma $\displaystyle 1.2$ and $\displaystyle 1.3$], we get that $\displaystyle\dim\left(\mathcal{R}(I)_{\Delta}\right)=n$. 4. d)

If $\displaystyle\Delta=(c,e)=(1,1)$, then by [16, Section 3, Proposition 2.3], we have $\displaystyle\dim\left(\mathcal{R}(I)_{\Delta}\right)=n$.

Regularity and the Koszul property

Definition 1.9**.**

Let $\displaystyle R$ be a standard graded $\displaystyle{\sf k}$-algebra, and $\displaystyle M$ be a finitely generated graded $\displaystyle R$-module with $\displaystyle(i,j)$-th graded Betti number $\displaystyle\beta_{ij}^{R}(M)=\dim_{{\sf k}}\left(\operatorname{Tor}_{i}^{R}(M,{\sf k})_{j}\right)$.

1. a)

The Castelnuovo-Mumford regularity of $\displaystyle M$ over $\displaystyle R$ is $\displaystyle\operatorname{reg}_{R}(M)=\sup_{i\geq 0}\left\{j-i\;|\;\beta_{ij}^{R}(M)\neq 0\right\}$. 2. b)

Let $\displaystyle I$ be a homogeneous ideal in $\displaystyle R$ generated in degree $\displaystyle d$. Then $\displaystyle I$ has a linear resolution if $\displaystyle\operatorname{reg}_{R}(I)=d$, i.e., if for all $\displaystyle i$, $\displaystyle\beta_{ij}^{R}(I)=0$ for $\displaystyle j\neq i+d$. 3. c)

We say that $\displaystyle R$ is a Koszul algebra if $\displaystyle\operatorname{reg}_{R}({\sf k})=0$, i.e., if for all $\displaystyle i$, we have $\displaystyle\beta_{ij}^{R}({\sf k})=0$ for $\displaystyle j\neq i$.

Definition 1.10**.**

Let $\displaystyle S={\sf k}[\underline{x},\underline{y}]$ with $\displaystyle\deg(x_{i})=(1,0)$ and $\displaystyle\deg(y_{j})=(0,1)$ be a bigraded polynomial ring, and $\displaystyle J\subset S$ a bigraded ideal. If $\displaystyle\beta_{i,(a,b)}^{S}(S/J)=\dim_{{\sf k}}\left(\operatorname{Tor}_{i}^{S}(S/J,{\sf k})_{(a,b)}\right)$ denote the bigraded Betti numbers of $\displaystyle S/J$, we define the $\displaystyle x$-regularity and $\displaystyle y$-regularity of $\displaystyle S/J$ to be:

[TABLE]

Remark 1.11**.**

Let $\displaystyle R$ be a standard graded $\displaystyle{\sf k}$-algebra, and $\displaystyle S=R_{x}\otimes_{{\sf k}}R_{y}$ be naturally bigraded by setting $\displaystyle\deg(x_{i})=(1,0)$ and $\displaystyle\deg(y_{j})=(0,1)$.

1. a)

Given an exact sequence $\displaystyle 0\longrightarrow N_{m}\longrightarrow\cdots\longrightarrow N_{2}\longrightarrow N_{1}\longrightarrow N_{0}\longrightarrow M\longrightarrow 0$ of graded $\displaystyle R$-modules, repeated application of the depth lemma (e.g., [8, Corollary 18.16] ) and the regularity lemma (e.g., [8, Corollary $\displaystyle 20.19$]) respectively yield the following:

$\displaystyle\operatorname{depth}_{R}(M)\geq\min\{\operatorname{depth}_{R}(N_{i})-i\ |\ 0\leq i\leq m\}$ and $\displaystyle\operatorname{reg}_{R}(M)\leq\max\{\operatorname{reg}_{R}(N_{i})-i\ |\ 0\leq i\leq m\}$. 2. b)

[3, Theorem 4] The Segre product of two Koszul algebras are Koszul. Moreover, Veronese subrings of Koszul algebras are Koszul. In particular, $\displaystyle S_{\bigtriangleup}=(R_{x})^{(c)}\underline{\otimes}(R_{y})^{(e)}$ is Koszul. 3. c)

[6, Lemma $\displaystyle 6.5$]

$\displaystyle\operatorname{reg}_{\left(R_{x}^{(c)}\underline{\otimes}R_{y}^{(e)}\right)}\left(R_{x}(-a)^{(c)}\underline{\otimes}\;R_{y}(-b)^{(e)}\right)=\max\left\{\operatorname{reg}_{R_{x}^{(c)}}\left(R_{x}(-a)^{(c)}\right),\operatorname{reg}_{R_{y}^{(e)}}\left(R_{y}(-b)^{(e)}\right)\right\}$. 4. d)

By [1, Theorem $\displaystyle 2.1$], we have $\displaystyle\operatorname{reg}_{R_{x}^{(c)}}\left(R_{x}(-a)^{(c)}\right)=\left\lceil\frac{a}{c}\right\rceil$.

Hence by (c), we get $\displaystyle\operatorname{reg}_{S_{\Delta}}\left(S(-a,-b)_{\Delta}\right)=\max\left\{\left\lceil\frac{a}{c}\right\rceil,\left\lceil\frac{b}{e}\right\rceil\right\}$. 5. e)

[6, Lemma $\displaystyle 6.6$] Let $\displaystyle I$ be a homogeneous ideal in $\displaystyle R$ with $\displaystyle\operatorname{reg}_{R}(R/I)\leq 1$. If $\displaystyle R$ is Koszul, then so is $\displaystyle R/I$. 6. f)

[15, Theorem $\displaystyle 5.3$] Let $\displaystyle I=\langle f_{1},\ldots,f_{p}\rangle\subset R_{x}$ be a graded ideal generated in degree $\displaystyle d$. Write the Rees algebra of $\displaystyle I$ as $\displaystyle\mathcal{R}(I)\cong S/\mathcal{K}$ (as in Remark 1.8 (a)). Then $\displaystyle\operatorname{reg}_{R_{x}}(I^{s})\leq sd+\operatorname{reg}_{x}^{S}\left(\mathcal{R}(I)\right)$. In particular, if $\displaystyle\operatorname{reg}_{x}(\mathcal{R}(I))=0$, then $\displaystyle I^{s}$ has a linear resolution for all $\displaystyle s$.

2. Geometric Residual Intersections and their Diagonal Subalgebras

Let $\displaystyle S=R_{x}[y_{1},\ldots,y_{p}]$ be a polynomial ring over $\displaystyle R_{x}$, and $\displaystyle\mathfrak{m}=\langle\underline{x}\rangle$. Define a bigrading on $\displaystyle S$ by setting $\displaystyle\deg(x_{i})=(1,0),\deg(y_{j})=(0,1)$ for $\displaystyle 1\leq i\leq n$ and $\displaystyle 1\leq j\leq p$.

As in the definition of the Eagon-Northcott complex, let $\displaystyle m\geq n$, and $\displaystyle\phi$ be an $\displaystyle n\times m$ matrix with linear entries in $\displaystyle R_{y}$, and let $\displaystyle\underline{z}=z_{1},\ldots,z_{m}\in S$ be given by

[TABLE]

We study the bigraded $\displaystyle S$-ideals of the form $\displaystyle J=\langle\underline{z}\rangle+I_{n}(\phi)$ where $\displaystyle\operatorname{grade}(I_{n}(\phi))\geq m-n+1$. Suppose $\displaystyle\operatorname{ht}(J)\geq m$. Since $\displaystyle J\subseteq\langle\underline{z}\rangle:\mathfrak{m}$ (e.g., by Cramer’s Rule), we see that $\displaystyle\langle\underline{z}\rangle:\mathfrak{m}$ is an $\displaystyle m$-residual intersection of $\displaystyle\mathfrak{m}$. If we further assume that $\displaystyle\langle\underline{z}\rangle:\mathfrak{m}$ is a geometric $\displaystyle m$-residual intersection of $\displaystyle\mathfrak{m}$, then it is shown in the proof of [4, Theorem 4.8] that equality holds, i.e., $\displaystyle J=\langle\underline{z}\rangle:\mathfrak{m}$.

With $\displaystyle J$ as above, the authors in [4] construct an $\displaystyle S$-free resolution of $\displaystyle S/J$. We show that this is a bigraded resolution, and begin with a brief review of the construction in [4] below.

Bigraded resolution of $\displaystyle S/J$

With notation as above, let $\displaystyle\mathbb{K}_{\bullet}(\underline{x};S)$ be the Koszul complex on the sequence $\displaystyle x_{1},\dots,x_{n}$. As in the construction of the Eagon-Northcott complex, let $\displaystyle\mathbb{K}_{\bullet}(\underline{x};S)_{d}$ be the corresponding $\displaystyle d$-th graded component with differentials $\displaystyle\psi^{i}_{d}$. Set $\displaystyle K_{a}^{b}=\ker((\psi_{a+n-b}^{n-b+1})^{*})$.

Remark 2.1**.**

In [4], the authors have denoted the kernel $\displaystyle K_{a}^{b}$ as $\displaystyle K_{a}^{b}(G)$ and the map $\displaystyle(\psi_{a+n-b}^{n-b+1})^{*}$ by $\displaystyle\eta_{a}^{b}$, hence the need for the notation of $\displaystyle K_{a}^{b}$ presented above. In (cf. [4, Proposition 1.13]), the authors also show that each $\displaystyle K_{a}^{b}$ is a free $\displaystyle R_{y}$-module with $\displaystyle\operatorname{rank}_{R_{y}}(K_{a}^{b})=\binom{n+a-1-b}{a}\binom{n+a}{b}$ .

Consider the following bi-complex $\displaystyle\mathbb{B}_{\bullet}$ of free $\displaystyle S$-modules:

[TABLE]

Under the given conditions, it was proved in [4, Theorem 3.6] that the total complex $\displaystyle\mathbb{T}_{\bullet}$ of the bicomplex $\displaystyle\mathbb{B}_{\bullet}$ is a free resolution of $\displaystyle S/J$. Thus, we see that

[TABLE]

is an $\displaystyle S$-free resolution of $\displaystyle S/J$, with $\displaystyle Q_{i}=S^{m\choose i}$ for $\displaystyle 0\leq i\leq n-1$, and $\displaystyle P_{i}=\bigoplus K_{a}^{b}\otimes S^{m\choose n+a}$, $\displaystyle 1\leq i\leq m$, where the direct sum is over all $\displaystyle(a,b)$ such that

[TABLE]

We show that this is a bigraded resolution of $\displaystyle S/J$ over $\displaystyle S$ by computing the bigraded shifts in $\displaystyle\mathbb{T}_{\bullet}$. In order to do this, we first compute the bigraded shifts in the bicomplex $\displaystyle\mathbb{B}_{\bullet}$.

Remark 2.2**.**

The bigraded shifts appearing in $\displaystyle\mathbb{B}_{\bullet}$ are as follows:

1. a)

The column on the right of the above diagram is the truncation of the Koszul complex $\displaystyle\mathbb{K}_{\bullet}(\underline{z};S)$. Thus the maps on the last column are the differentials $\displaystyle\phi^{i}$, which are bigraded maps of bidegree $\displaystyle(1,1)$. 2. b)

Using [4, Lemma 2.5] the maps appearing in the other columns are induced by $\displaystyle\mathbb{K}_{\bullet}(\underline{x};S)$ and hence are bigraded maps of bidegree $\displaystyle(1,0)$. 3. c)

The horizontal map between the last two columns is the augmentation map $\displaystyle\epsilon^{b}$ as explained in [4, Corollary 2.7]. Thus this augmentation map $\displaystyle K_{0}^{b}\otimes S^{m\choose n}\rightarrow S^{m\choose b}$ is a bigraded map of bidegree $\displaystyle(0,n-b)$. In particular, when $\displaystyle b=0$, since $\displaystyle K_{0}^{0}=R$, the map $\displaystyle K_{0}^{0}\otimes S^{m\choose n}\rightarrow S$ is induced by the $\displaystyle\epsilon$ map appearing in the Eagon-Northcott complex (1.1). 4. d)

Using [4, Lemma 2.6] the other/remaining horizontal maps in the rows are bigraded maps of bidegree $\displaystyle(0,1)$. In particular, since $\displaystyle K^{0}_{a}\cong(S_{a})^{*}$, the maps in the last row of the bicomplex $\displaystyle\mathbb{B}_{\bullet}$ are induced by the maps in the Eagon-Northcott complex (1.1).

The bigraded shifts of each free module appearing in the bicomplex $\displaystyle\mathbb{B}_{\bullet}$ are determined by the fact that each homomorphism has bidegree $\displaystyle(0,0)$. Hence, the domain of a bigraded map of bidegree $\displaystyle(a,b)$ is shifted by $\displaystyle(-a,-b)$. For ease of reference, we rewrite the bicomplex $\displaystyle\mathbb{B}_{\bullet}$ with only the bigraded shifts:

[TABLE]

Thus, we get the following:

Proposition 2.3**.**

The bigraded shifts appearing in the $\displaystyle S$-free resolution (14) of the residual intersection $\displaystyle S/J$ are as follows:

[TABLE]

where $\displaystyle\;\max\{0,i-(m-n+1)\}\;\leq\;j\;\leq\;\min\{i-1,n-1\}$ and

[TABLE]

In particular, we have $\displaystyle\operatorname{reg}_{x}(S/J)=0$, $\displaystyle\operatorname{reg}_{y}(S/J)=(n-1)$, and the bigraded Betti numbers of $\displaystyle S/J$ are given by

[TABLE]

We conclude this section with the following remark, which is used to discuss the Koszul, and Cohen-Macaulay properties, of $\displaystyle(S/J)_{\Delta}$ in the next two subsections.

Remark 2.4**.**

The exactness of $\displaystyle(\_\!\_)_{\Delta}$ shows that

[TABLE]

is exact.

Thus, by Remark 1.11(a), we see that $\displaystyle\operatorname{depth}_{S_{\Delta}}(S/J)_{\Delta}\geq\min\{\operatorname{depth}_{S_{\Delta}}(F_{i})_{\Delta}-i\ |\ 0\leq i\leq m\}$, and $\displaystyle\operatorname{reg}_{S_{\Delta}}(S/J)_{\Delta}\leq\max\{\operatorname{reg}_{S_{\Delta}}(F_{i})_{\Delta}-i\ |\ 0\leq i\leq m\}.$

Depth and the Cohen-Macaulay Property of $\displaystyle(S/J)_{\Delta}$

In this subsection, we retain the setup of the previous one, and identify lower bounds on the depth of $\displaystyle(S/J)_{\Delta}$, which gives sufficient conditions for it to be Cohen-Macaulay.

Theorem 2.5**.**

Suppose $\displaystyle p>m\geq n$, then $\displaystyle\operatorname{depth}(S/J)_{\Delta}\geq p+n-(m+1)$ for all $\displaystyle\Delta$.

Proof.

By Remark 1.11(a) and Remark 2.4, we have

[TABLE]

We see that $\displaystyle\dim\left((F_{i})_{\Delta}\right)=p+n-1$, since $\displaystyle\dim\left(S(-a,-b)_{\Delta}\right)=p+n-1$ for any pair $\displaystyle(-a,-b)$ by Remark 1.4(d). The bigraded shifts $\displaystyle(-a,-b)$, described in 2.3, satisfy $\displaystyle 0\leq a<n$, and $\displaystyle 0\leq b\leq m$. Hence they satisfy the conditions in Remark 1.4(d), and therefore, $\displaystyle S(-a,-b)_{\Delta}$ is Cohen-Macaulay for these shifts. This implies that $\displaystyle\left(F_{i}\right)_{\Delta}$ is Cohen-Macaulay for all $\displaystyle i$.

Thus, $\displaystyle\operatorname{depth}\left((F_{i})_{\Delta}\right)=p+n-1$ for each $\displaystyle i$, and hence we see that $\displaystyle\operatorname{depth}(S/J)_{\Delta}\geq p+n-(m+1)$, proving the result. ∎

Corollary 2.6**.**

Suppose $\displaystyle p>m\geq n$ and $\displaystyle\dim(S/J)_{\Delta}\leq p+n-(m+1)$, then $\displaystyle(S/J)_{\Delta}$ is Cohen-Macaulay.

Regularity and the Koszul property of $\displaystyle(S/J)_{\Delta}$

The following is the main theorem of this subsection.

Theorem 2.7**.**

With notations as before, $\displaystyle(S/J)_{\Delta}$ is Koszul for all $\displaystyle c\geq 1$ and $\displaystyle e\geq\frac{n}{2}$.

Proof.

Note that $\displaystyle S_{\bigtriangleup}$ is Koszul by Remark 1.11(b). Using Remark 1.11(e), it is enough to prove that $\displaystyle\operatorname{reg}_{S_{\bigtriangleup}}(S/J)_{\bigtriangleup}\leq 1$. Since $\displaystyle\operatorname{reg}_{S_{\Delta}}(S/J)_{\Delta}\leq\max\{\operatorname{reg}_{S_{\Delta}}(F_{i})_{\Delta}-i\ |\ 0\leq i\leq m\}$, we compute $\displaystyle\operatorname{reg}_{S_{\Delta}}((F_{i})_{\Delta})$ for all $\displaystyle i$.

By Remark 1.11(d),

[TABLE]

where $\displaystyle a_{i}^{\max}=\max\{a\ |\ S(-a,-b)\text{ is a direct summand of }F_{i}\}$ and $\displaystyle b_{i}^{\max}$ is defined similarly.

Therefore we can write

[TABLE]

By Proposition 2.3, notice that $\displaystyle a^{\max}_{i}=i$ for $\displaystyle i=0,1,\ldots,(n-2)$, and $\displaystyle a^{\max}_{i}=(n-1)$ for $\displaystyle i=(n-1),\ldots,m$. Therefore

[TABLE]

for all $\displaystyle c\geq 1$.

Similarly, when $\displaystyle e\geq\frac{n}{2}$, we can show that $\displaystyle\left\lceil\frac{b^{\max}_{i}}{e}\right\rceil-i\leq 1$ for all $\displaystyle i$. For example, when $\displaystyle m\leq 2n-1$, we have $\displaystyle b^{\max}_{i}=n-1+i$ for $\displaystyle i=1,\ldots,(m-n)$ and $\displaystyle b^{\max}_{i}=m$ for $\displaystyle i=(m-n+1),\ldots,m$, and hence

[TABLE]

since $\displaystyle e\geq\frac{n}{2}$. The calculations when $\displaystyle m>2n-1$ are similar, proving the result. ∎

Corollary 2.8**.**

If $\displaystyle n=2$, then the diagonal subalgebra $\displaystyle(S/J)_{\Delta}$ is always Koszul.

3. Applications to perfect ideals of height two

A natural source of geometric residual intersections are the Rees algebras of certain classes of perfect ideals of height two, which are linearly presented in a polynomial ring. We use the following:

Setup 3.1**.**

Let $\displaystyle I$ be a homogeneous perfect ideal of height two in a polynomial ring $\displaystyle R_{x}={\sf k}[x_{1},\dots,x_{n}]$ with a presentation matrix $\displaystyle\Phi$ (1.7). We assume that $\displaystyle I$ satisfies the following properties:

1. (1)

$\displaystyle\mu(I)=p>n$. 2. (2)

$\displaystyle I$ satisfies $\displaystyle\mu(I_{\mathfrak{p}})\leq\operatorname{ht}\mathfrak{p}$ for every $\displaystyle\mathfrak{p}\in V(I)\backslash\{\mathfrak{m}\}$. Equivalently, $\displaystyle\operatorname{ht}(I_{p-i}(\Phi))>i$ for $\displaystyle 0\leq i<n$ (e.g., [8, Proposition 20.6]). 3. (3)

The presentation matrix $\displaystyle\Phi$ of $\displaystyle I$ is linear in the entries of $\displaystyle R_{x}$.

Since $\displaystyle I$ is a perfect ideal of height two, the presentation matrix $\displaystyle\Phi$ of $\displaystyle I$ is of size $\displaystyle p\times(p-1)$ by the Hilbert-Burch theorem (e.g., [8, Theorem 20.15]). This also shows that the generators of $\displaystyle I$ are of degree $\displaystyle p-1$. Let $\displaystyle S=R_{x}[y_{1},\dots,y_{p}]$ with bigrading $\displaystyle\deg x_{i}=(1,0)$, and $\displaystyle\deg y_{j}=(0,1)$. With notation as in Remark 1.8(b), let $\displaystyle m=p-1$, and $\displaystyle[z_{1}\cdots z_{p-1}]=[y_{1}\cdots y_{p}]\ \Phi$. We can rewrite this as $\displaystyle[z_{1}\cdots z_{p-1}]=[x_{1}\cdots x_{n}]\cdot\phi$ where $\displaystyle\phi$ is a $\displaystyle n\times(p-1)$ matrix with entries in $\displaystyle R_{y}$. Since $\displaystyle\Phi$ is linear in $\displaystyle{\sf k}[x_{1},\dots,x_{n}]$, we have $\displaystyle\phi$ is linear in $\displaystyle R_{y}$.

Remark 3.2**.**

With the notation as in 3.1, and $\displaystyle\phi$ as above, we get the following from

[14, Theorems 1.2, 1.3] (and their proofs): The Rees algebra of $\displaystyle I$ can be written as $\displaystyle\mathcal{R}(I)\cong S/J$, where $\displaystyle J=\langle z_{1},\dots,z_{p-1}\rangle+I_{n}(\phi)$. Furthermore, $\displaystyle\operatorname{ht}(I_{n}(\phi))\geq m-n+1$, $\displaystyle J=\langle z_{1},\dots,z_{p-1}\rangle:\langle\underline{x}\rangle$, and in particular, $\displaystyle J$ is a geometric $\displaystyle m$-residual intersection of $\displaystyle\langle\underline{x}\rangle$.

Thus 2.3 gives a free resolution of $\displaystyle\mathcal{R}(I)$ over $\displaystyle S$. We can now apply the results from Section 2 to $\displaystyle\mathcal{R}(I)$.

Theorem 3.3**.**

With the notation as in 3.1, we have the following:

1. a)

$\displaystyle I^{s}$* has a linear resolution for all $\displaystyle s\in\mathbb{N}$.* 2. b)

$\displaystyle(\mathcal{R}(I))_{\Delta}$* is Cohen-Macaulay for all $\displaystyle\Delta=(c,e)$ with $\displaystyle c>(p-1)e$ or $\displaystyle c=e=1$.* 3. c)

$\displaystyle(\mathcal{R}(I))_{\Delta}$* is Koszul for all $\displaystyle\Delta=(c,e)$ with $\displaystyle c\geq 1$ and $\displaystyle e\geq\frac{n}{2}$.*

Proof.

a) This follows from Remark 1.11(f), since $\displaystyle\operatorname{reg}_{x}\left(\mathcal{R}(I)\right)=0$ by 2.3.

b) Recall that $\displaystyle m=p-1$ and, by Remark 1.8 (c) and (d) , we have $\displaystyle\dim\mathcal{R}(I)_{\Delta}=n$ in both cases. Thus, by Theorem 2.5 and 2.6, we have $\displaystyle\operatorname{depth}(\mathcal{R}(I)_{\Delta})=\dim(\mathcal{R}(I)_{\Delta})=n$ for all $\displaystyle\Delta$.

c) This is follows immediately from Theorem 2.7. ∎

Remark 3.4**.**

1. a)

The special case of Theorem 3.3(a) when $\displaystyle p=n+1$ is proved in [15, Corollary 5.11]. 2. b)

In [6, Corollary 3.15], the authors show that the $\displaystyle{\mathcal{R}(I)}_{\Delta}$ is Cohen-Macaulay for $\displaystyle{{\Delta}}$ such that $\displaystyle c,e\gg 0$. In Theorem 3.3(b), we give an explicit range for $\displaystyle c,e$. 3. c)

If $\displaystyle c>(p-1)e$, then $\displaystyle{\mathcal{R}(I)}_{\Delta}\cong{\sf k}[(I^{e})_{c}]$ by [6, Lemma 1.2]. Thus by Theorem 3.3, $\displaystyle{\sf k}[(I^{e})_{c}]$ is Cohen-Macaulay. In [7, Corollary 4.4], the authors show that there exists a positive integer $\displaystyle f$ such that $\displaystyle{\sf k}[(I^{e})_{c}]$ is Cohen-Macaulay when $\displaystyle c\geq fe$. In Theorem 3.3(b), we give an explicit values of $\displaystyle f$.

4. Example

Let $\displaystyle I$ be an in $\displaystyle{\sf k}[x_{1},x_{2},x_{3}]$ whose presentation matrix is

[TABLE]

Notice that $\displaystyle\operatorname{ht}(I_{4}(\Phi))\geq 2$ (as $\displaystyle x_{1}^{4},x_{2}^{4}-3x_{1}x_{2}^{2}x_{3}+x_{1}^{2}x_{3}^{2}\in I_{4}(\Phi)$). By the Hilbert-Burch theorem (e.g., [8, Theorem 20.15]), $\displaystyle I$ is a perfect ideal of height two in $\displaystyle{\sf k}[x_{1},x_{2},x_{3}]$. One can easily check (2) of 3.1 by noticing that $\displaystyle\operatorname{ht}(I_{4}(\Phi))\geq 2,\operatorname{ht}(I_{3}(\Phi))=3$. Since $\displaystyle\Phi$ is linear in $\displaystyle{\sf k}[x_{1},x_{2},x_{3}]$, all the hypothesis of 3.1 is satisfied.

Now let $\displaystyle S={\sf k}[y_{1},\dots,y_{5},x_{1},x_{2},x_{3}]$ with bigrading $\displaystyle\deg y_{i}=(0,1)$ and $\displaystyle\deg x_{i}=(1,0)$. As mentioned in Remark 3.2, the Rees algebra $\displaystyle\mathcal{R}(I)\cong S/J$ where $\displaystyle J=\langle z_{1},\dots,z_{4}\rangle+I_{3}(\phi)$ where $\displaystyle[z_{1}\cdots z_{4}]=[y_{1}\cdots y_{5}]\cdot\Phi=[x_{1}~{}x_{2}~{}x_{3}]\cdot\phi$ and $\displaystyle\phi$ can be obtained as follows

[TABLE]

The discussion in Remark 3.2 shows that $\displaystyle J$ is a geometric residual intersection with $\displaystyle\operatorname{ht}\phi\geq 2$. Using 2.3 we can compute the complete resolution of $\displaystyle S/J$ as follows:

[TABLE]

By Theorem 3.3, we have that $\displaystyle\mathcal{R}(I)_{\bigtriangleup}\cong(S/J)_{\Delta}$ is Cohen-Macaulay for all $\displaystyle\Delta$ with $\displaystyle c>4e$ or $\displaystyle c=e=1$, and is Koszul for $\displaystyle e\geq\frac{3}{2}$.