The Automorphism groups of zero-dimensional monomial algebras

Roberto Díaz Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Ñuñoa, Santiago, Chile. robertodiaz@uchile.cl , Alvaro Liendo Instituto de Matemática y Física, Universidad de Talca, Casilla 721, Talca, Chile. aliendo@utalca.cl , Gonzalo Manzano-Flores Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Ñuñoa, Santiago, Chile. gonzalo.manzano@usach.cl and Andriy Regeta Institut für Mathematik, Friedrich-Schiller-Universität Jena, Jena 07737, Germany andriyregeta@gmail.com

Abstract.

A monomial algebra $B$ is defined as a quotient of a polynomial ring by a monomial ideal, which is an ideal generated by a finite set of monomials. In this paper, we determine the automorphism group of a monomial algebra $B$ , under the assumption that $B$ is a finite-dimensional vector space over a field $\mathbf{k}$ of characteristic zero. We achieve this by providing an explicit classification of the homogeneous locally nilpotent derivations of $B$ . The main body of the paper addresses the more general case of semigroup algebras, with the polynomial ring being a particular case.

2020 Mathematics Subject Classification: 13F55, 13N15, 14L30, 14M25.
Key words: monomial ideal, locally nilpotent derivation, automorphism group.
The first author is supported by the Fondecyt postdoc project 3230406. The second author was partially supported by Fondecyt project 1240101. The third author is supported by the Fondecyt postdoc project 3240191. The fourth author is supported by DFG, project number 509752046.

Introduction

An affine semigroup $S$ is a finitely generated semigroup with an identity element that can be embedded into $\mathbb{Z}^{n}$ . Let $\mathbf{k}$ be a field of characteristic zero and $S$ an affine semigroup. The semigroup algebra $\mathbf{k}[S]$ is defined as $\mathbf{k}[S]=\bigoplus_{m\in S}\mathbf{k}\cdot\mathbf{x}^{m}$ , where $\mathbf{x}^{m}$ are new variables satisfying the multiplication rules $\mathbf{x}^{m}\cdot\mathbf{x}^{m^{\prime}}=\mathbf{x}^{m+m^{\prime}}$ and $\mathbf{x}^{0}=1$ . A common example of a semigroup algebra, which motivates our notation, is the polynomial ring $\mathbf{k}[\mathbf{x}]=\mathbf{k}[x_{1},\ldots,x_{n}]$ obtained with $S=\mathbb{Z}_{\geq 0}^{n}$ , where $\mathbf{x}^{m}$ corresponds precisely to monomials under the usual multi-index notation. By analogy, we refer to all symbols $\mathbf{x}^{m}$ as monomials of $\mathbf{k}[S]$ . An ideal $I\subseteq\mathbf{k}[S]$ is called monomial if it has a generating set composed of monomials. A monomial algebra $\mathbf{k}[S]/I$ is the quotient of $\mathbf{k}[S]$ by a monomial ideal $I$ . A monomial algebra $\mathbf{k}[S]/I$ is zero-dimensional if and only if its dimension as a $\mathbf{k}$ -vector space is finite.

The combinatorial nature of monomial algebras allows them to be identified with objects from other categories, making these algebras suitable for developing examples and verifying general theories. For instance, Stanley-Reisner rings, which are monomial algebras characterized by having all their exponents being 0 or 1, are in bijective correspondence with abstract simplicial complexes [MS05, Theorem 1.7]. Stanley proved the upper bound conjecture for spheres using the Cohen-Macaulay structure of these rings [Sta75], which was established by Reisner [Rei76]. In another example, a special class of monomial algebras known as discrete Hodge algebras was studied in [DCEP82], enabling the examination of homogeneous coordinate rings of Grassmann varieties and their Schubert subvarieties.

Our main goal in this paper is to compute the automorphism groups of zero-dimensional monomial algebras.

Our approach to computing the automorphism group of a zero-dimensional monomial algebra is inspired by Demazure’s foundational paper [Dem70], where he computes the automorphism groups of proper smooth toric varieties. A normal variety is called toric if it admits an algebraic torus action with an open orbit, and affine toric varieties are precisely given by $\operatorname{Spec}\mathbf{k}[S]$ , where $S$ is an affine semigroup [Oda83, Ful93, CLS11]. Although the geometric nature of smooth proper toric varieties is significantly different from that of zero-dimensional monomial algebras, both objects have similar combinatorial descriptions. This similarity allows us to apply some of the techniques from [Dem70] to our context. We refer the reader to the paper [LLA22] by the second named author for a concise modern account of Demazure’s results.

For a $\mathbf{k}$ -algebra $B$ , we denote by $\operatorname{Der}(B)$ the $\mathbf{k}$ -vector space of $\mathbf{k}$ -derivations $\partial\colon B\to B$ . Additionally, if $I\subset B$ is an ideal, we denote by $\operatorname{Der}_{I}(B)$ the subspace of $\operatorname{Der}(B)$ consisting of $\mathbf{k}$ -derivations such that $\partial(I)\subset I$ . All our derivations are $\mathbf{k}$ -derivations, so we will omit $\mathbf{k}$ from the notation. A derivation $\partial\in\operatorname{Der}(B)$ is called locally nilpotent if for every $f\in B$ there exists $\ell\in\mathbb{Z}_{\geq 0}$ such that $\partial^{\ell}(f)=0$ , where $\partial^{\ell}$ denotes the $\ell$ -th composition of $\partial$ .

As a first step towards our goal of computing the automorphism groups of monomial algebras, we describe the set $\operatorname{Der}_{I}(\mathbf{k}[S])$ , where $S$ is an affine semigroup and $I$ is a monomial ideal (see Proposition 1.3). This description has already been provided for the special case where the semigroup algebra $\mathbf{k}[S]$ is a polynomial ring in [BS95, Tad09]. Consider an embedding $S\hookrightarrow M$ where $M\simeq\mathbb{Z}^{n}$ such that the group generated by $S$ in $M$ is $M$ itself. The algebra $\mathbf{k}[S]$ admits an $M$ -grading, and since $I$ is $M$ -graded, $\mathbf{k}[S]/I$ inherits this $M$ -grading. We make extensive use of these $M$ -gradings throughout the paper, and in particular, we derive Proposition 1.3 straightforwardly from [KLL15] by the second named author.

Let $S$ be a monomial algebra and $I$ be a monomial ideal. There is a natural map $\pi\colon\operatorname{Der}_{I}(\mathbf{k}[S])\to\operatorname{Der}(\mathbf{k}[S]/I)$ induced by the quotient. In Theorem 2.2, we show that $\pi$ is surjective for every monomial ideal $I$ if and only if $\mathbf{k}[S]$ is a polynomial ring. A derivation $\overline{\partial}\colon\mathbf{k}[S]/I\to\mathbf{k}[S]/I$ is said to be liftable if there exists a derivation $\partial\colon\mathbf{k}[S]\to\mathbf{k}[S]$ such that $\pi(\partial)=\overline{\partial}$ . In Theorem 3.7, we provide a classification of the liftable locally nilpotent derivations on the monomial algebra $\mathbf{k}[S]/I$ that are homogeneous with respect to the $M$ -grading.

The interest in locally nilpotent derivations of an algebra $B$ stems from their one-to-one correspondence with additive group actions on $B$ ; see, for instance, [Dai03, Section 4]. Specifically, given a locally nilpotent derivation $\partial\colon B\to B$ , the associated $\mathbb{G}_{\mathrm{a}}$ -action on $B$ is obtained via the exponential map:

\displaystyle\exp(s\partial)\colon\mathbb{G}_{\mathrm{a}}\times B\to B\quad\text{defined by}\quad(s,f)\mapsto\sum_{i=0}^{\infty}\frac{s^{i}\partial^{i}(f)}{i!}\,.

(1)

To describe our main result, we let $S=\mathbb{Z}_{\geq 0}^{n}$ so that $B=\mathbf{k}[S]$ is a polynomial ring and we let $I$ be a monomial ideal. Without loss of generality, we may assume that $I$ is full, i.e., no standard basis vector in $\mathbb{Z}^{n}$ is contained in $I$ (see Definition 4.2). Our main result, presented in Theorem 4.17, implies the following.

Theorem.

Let $S=\mathbb{Z}^{n}_{\geq 0}$ , making $B=\mathbf{k}[S]$ a polynomial ring, and let $I\subset B$ be a full monomial ideal such that $B/I$ is a zero-dimensional monomial algebra. If $G=\operatorname{Aut}_{\mathbf{k}}(B/I)$ , then the following statements hold:

$(i)$

The group $G$ is linear algebraic.
$(ii)$

The algebraic torus $T=\operatorname{Spec}\mathbf{k}[\mathbb{Z}^{n}]$ is a maximal torus of $G$ .
$(iii)$

The connected component $G^{0}$ of $G$ is generated by the maximal torus and the images in $G^{0}$ of all the $\mathbb{G}_{\mathrm{a}}$ -actions corresponding to homogeneous locally nilpotent derivations $\partial\colon B/I\to B/I$ .
$(iv)$

The finite group $G/G^{0}$ is generated by the images of automorphisms $B/I\to B/I$ induced by semigroup automorphisms $S\to S$ that map $I$ to itself.

Let now $B$ be an affine domain, and assume that every automorphism of the connected component of the identity $\operatorname{Aut}_{\mathbf{k}}^{0}(B)$ of $\operatorname{Aut}_{\mathbf{k}}(B)$ is algebraic (see [PR24, page 3] for the definition of an algebraic element). Then, $\operatorname{Aut}_{\mathbf{k}}^{0}(B)$ is a solvable group with a derived length of at most two (see [PR24, Theorem 1.1]; see also [PR23, Theorem 1.3]). In contrast, the automorphism group of a zero-dimensional monomial algebra $B$ does not necessarily have a solvable connected component (see Example 4.20).

The article is organized as follows: In Section 1, we present preliminaries on semigroup algebras, monomial ideals, and homogeneous derivations. Section 2 explores the relationship between derivations on a semigroup algebra and its quotient monomial algebras. In Section 3, we classify liftable locally nilpotent derivations of monomial algebras. Finally, in Section 4, we describe the automorphism group of zero-dimensional monomial algebras that are quotients of the polynomial ring.

Acknowledgments

This collaboration began at the AGREGA2 conference, which took place at Universidad Técnica Federico Santa María in March 2023. We extend our gratitude to the institution for its support and hospitality.

1. Preliminaries

In this section, we gather some preliminary results that are essential for the subsequent sections.

1.1. Semigroup algebras, monomial ideals and homogeneous derivations

Throughout this article, we fix a free abelian group $M$ of rank $n$ , i.e., $M\simeq\mathbb{Z}^{n}$ . We also consider $N$ as the dual group, defined by $N=\operatorname{Hom}(M,\mathbb{Z})$ . There exists a natural duality pairing $\langle\ ,\ \rangle\colon M\times N\to\mathbb{Z}$ given by $\langle m,p\rangle=p(m)$ . Let $L$ be any field. We define the $L$ -vector spaces $M_{L}=M\otimes_{\mathbb{Z}}L$ and $N_{L}=N\otimes_{\mathbb{Z}}L$ . The aforementioned duality pairing extends naturally to a pairing $\langle\ ,\ \rangle\colon M_{L}\times N_{L}\to L$ .

An affine semigroup $S$ is a finitely generated semigroup with an identity element that can be embedded into $M$ . The cone $\omega_{S}$ of $S$ is the cone spanned by $S$ inside $M_{\mathbb{R}}$ . We say that an affine semigroup $S$ is saturated if $\omega_{S}\cap M=S$ , and pointed if the only vector space contained in $\omega_{S}$ is $\{0\}$ . Furthermore, $S$ is said to be minimally embedded if the smallest subgroup of $M$ containing $S$ is $M$ . Unless otherwise stated, all semigroups in the sequel are assumed to be affine, saturated, pointed, and minimally embedded.

Let $S$ be a semigroup. We say that a subsemigroup $S^{\prime}\subset S$ is a face of $S$ if for all $m,m^{\prime}\in S$ with $m+m^{\prime}\in S^{\prime}$ , we have $m,m^{\prime}\in S^{\prime}$ . Note that $S^{\prime}\subset S$ is a face if and only if $S^{\prime}=S\cap\tau$ , where $\tau$ is a polyhedral face of $\omega_{S}$ . A face of $S$ isomorphic to $\mathbb{Z}_{\geq 0}$ as a semigroup is called a ray, and a maximal proper face of $S$ is called a facet. The rays of $S$ are given by $S\cap\tau$ with $\tau$ a one-dimensional polyhedral face of $\omega_{S}$ , and the facets of $S$ are given by $S\cap\tau$ with $\tau$ a $(n-1)$ -dimensional polyhedral face of $\omega_{S}$ . A ray of $S$ is uniquely determined by its unique generator as a semigroup; thus, we denote by $S(1)\subset S$ the set of all these generators of all the rays. The rank of a face $F\subset S$ , denoted by $\operatorname{rank}F$ , is the rank of the subgroup of $M$ generated by $F$ . A semigroup is called simplicial if $S(1)$ is an $\mathbb{R}$ -basis of $M_{\mathbb{R}}$ .

Let $S$ be a semigroup. We say that a subsemigroup $S^{\prime}\subset S$ is a face of $S$ if for all $m,m^{\prime}\in S$ with $m+m^{\prime}\in S^{\prime}$ , we have $m,m^{\prime}\in S^{\prime}$ . Note that $S^{\prime}\subset S$ is a face if and only if $S^{\prime}=S\cap\tau$ , where $\tau$ is a polyhedral face of $\omega_{S}$ . A face of $S$ isomorphic to $\mathbb{Z}_{\geq 0}$ as a semigroup is called a ray, and a maximal proper face of $S$ is called a facet. The rays of $S$ are given by $S\cap\tau$ where $\tau$ is a one-dimensional polyhedral face of $\omega_{S}$ , and the facets of $S$ are given by $S\cap\tau$ where $\tau$ is a $(n-1)$ -dimensional polyhedral face of $\omega_{S}$ . A ray of $S$ is uniquely determined by its unique generator as a semigroup. We denote by $S(1)\subset S$ the set of all these generators of all the rays. The rank of a face $F\subset S$ , denoted by $\operatorname{rank}F$ , is the rank of the subgroup of $M$ generated by $F$ . A semigroup is called simplicial if $S(1)$ is an $\mathbb{R}$ -basis of $M_{\mathbb{R}}$ .

The semigroup algebra $\mathbf{k}[S]$ of $S$ is defined as:

\mathbf{k}[S]=\bigoplus_{m\in S}\mathbf{k}\cdot\mathbf{x}^{m}\quad\mbox{where}\quad\mathbf{x}^{m}\cdot\mathbf{x}^{m^{\prime}}=\mathbf{x}^{m+m^{\prime}}\quad\mbox{and}\quad\mathbf{x}^{0}=1\,.

The symbols $\mathbf{x}^{m}$ , with $m\in M$ are called monomials. Note that since $S$ is affine and saturated, $\mathbf{k}[S]$ is a normal affine domain.

An ideal $I\subset\mathbf{k}[S]$ is called monomial if there exists $l\in\mathbb{Z}_{>0}$ and $\mathbf{a}_{1},\ldots,\mathbf{a}_{l}\in S$ such that $I=(\mathbf{x}^{\mathbf{a}_{1}},\dots,\mathbf{x}^{\mathbf{a}_{l}})$ . The support of a monomial ideal $I\subset\mathbf{k}[S]$ is denoted by $\operatorname{supp}(I),$ and is given by

\operatorname{supp}(I)=\{m\in S\mid\mathbf{x}^{m}\in I\}=\bigcup_{k=1}^{l}(\mathbf{a}_{k}+S)\,.

Let $I\subset\mathbf{k}[S]$ be a monomial ideal. We say that $I$ has cofinite support if $S\setminus\operatorname{supp}(I)$ is finite.

Example 1.1 (The polynomial ring).

Let $M=\mathbb{Z}^{n}$ and let $E=\{e_{1},\ldots,e_{n}\}$ be the canonical basis. Let $S=\mathbb{Z}_{\geq 0}^{n}$ , there exists a natural isomorphism from $\mathbf{k}[S]$ to the polynomial ring $\mathbf{k}[x_{1},\ldots,x_{n}]$ , given by $\mathbf{x}^{e_{i}}\mapsto x_{i}$ , for $1\leq i\leq n$ . Under this isomorphism, for every $m=(m_{1},\ldots,m_{n})\in S$ , the monomial $\mathbf{x}^{m}$ corresponds to $x_{1}^{m_{1}}\cdots x_{n}^{m_{n}}$ .

Let $M=\mathbb{Z}^{n}$ and let $E=\{e_{1},\ldots,e_{n}\}$ be the canonical basis. For $S=\mathbb{Z}_{\geq 0}^{n}$ , there exists a natural isomorphism from $\mathbb{K}[S]$ to the polynomial ring $\mathbb{K}[x_{1},\ldots,x_{n}]$ , given by $\mathbf{x}^{e_{i}}\mapsto x_{i}$ for $1\leq i\leq n$ . Under this isomorphism, for every $m=(m_{1},\ldots,m_{n})\in S$ , the monomial $\mathbf{x}^{m}$ corresponds to $x_{1}^{m_{1}}\cdots x_{n}^{m_{n}}$ .

Now, fixing $n=2$ we have that $\mathbf{k}[S]=\mathbf{k}[x,y]$ . Let $I_{1}=(x^{2}y^{5},x^{3}y^{2},x^{5})\subset\mathbf{k}[S]$ and $I_{2}=(y^{5},x^{3}y^{2},x^{6})\subset\mathbf{k}[S]$ . Note that $I_{2}$ is a monomial ideal with cofinite support while $I_{1}$ is not. The ideal $I_{1}$ is represented in Figure 2 and $I_{2}$ is represented in Figure 2. The solid dots $\bullet$ represent elements in $\operatorname{supp}(I_{i}),$ while the hollow dots $\circ$ represent elements in $m\in S\setminus\operatorname{supp}(I_{i})$ , for $1\leq i\leq 2$ .

Figure 1.

I_{1}=(x^{2}y^{5},x^{3}y^{2},x^{5})

Figure 2.

I_{2}=(y^{5},x^{3}y^{2},x^{6})

The algebra $\mathbf{k}[S]$ is naturally $M$ -graded, and under this grading every monomial ideal $I$ is a graded ideal. Thus, the quotient $\mathbf{k}[S]/I$ inherits an $M$ -grading given by $\deg{\overline{\mathbf{x}}}^{m}=m$ if $m\notin\operatorname{supp}(I)$ , where ${\overline{\mathbf{x}}}^{m}$ denotes the image of $\mathbf{x}^{m}$ within $\mathbf{k}[S]/I$ . Recall that if $m\in\operatorname{supp}(I)$ , then ${\overline{\mathbf{x}}}^{m}=0$ . In the sequel, we will consider $\mathbf{k}[S]/I$ as an $M$ -graded algebra.

Let $B$ be a $\mathbf{k}$ -algebra. A derivation $\partial$ of $B$ , is a $\mathbf{k}$ -linear map satisfying the Leibniz rule, that is,

\partial(fg)=\partial(f)g+f\partial(g),\quad\text{for all }f,g\in B.

We denote by $\operatorname{Der}(B)$ the $\mathbf{k}$ -vector space of derivations of $B$ . Let $I\subset B$ be an ideal. We denote by $\operatorname{Der}_{I}(B)$ the subspace of derivations $\partial\in\operatorname{Der}(B)$ such that $\partial(I)\subset I$ . A derivation $\partial$ is locally nilpotent if for each $f\in B$ there exists $\ell\in\mathbb{Z}_{>0}$ such that $\partial^{\ell}(f)=0$ , where $\partial^{\ell}$ is the composition of $\partial$ $\ell$ -times.

Letting $S$ be a semigroup, we let now $\partial$ be a derivation of $\mathbf{k}[S]$ . We say that $\partial$ is homogeneous if it sends homogeneous elements into homogeneous elements with respect to the $S$ -grading of $\mathbf{k}[S]$ . By [DL24, Lemma 1.1], there exists a unique element $\alpha\in M$ , called the degree of $\partial$ and denoted by $\deg\partial$ , such that for every $\mathbf{x}^{m}\notin\ker\partial$ we have $\partial(\mathbf{x}^{m})=\lambda\mathbf{x}^{m+\alpha}$ for some $\lambda\in\mathbf{k}$ . Furthermore, by [DL24, Proposition 2.1], every derivation in $\operatorname{Der}(\mathbf{k}[S])$ admits a finite decomposition $\partial=\sum\partial_{\alpha}$ , where each $\partial_{\alpha}$ is a homogeneous derivation on $\mathbf{k}[S]$ of degree $\alpha\in M$ . We say that a homogeneous derivation $\partial$ is inner if $\deg\partial\in S$ and outer if $\deg\partial\in M\setminus S$ . By a straightforward application of the Leibniz rule, it follows that every homogeneous derivation on $\mathbf{k}[S]$ is of the form

\displaystyle\partial_{\alpha,p}\colon\mathbf{k}[S]\to\mathbf{k}[S],\quad\mathbf{x}^{m}\mapsto p(m)\cdot\mathbf{x}^{m+\alpha}\,,

(2)

where $\alpha=\deg\partial_{\alpha,p}\in M$ and $p\in N_{\mathbf{k}}=N\otimes_{\mathbb{Z}}\mathbf{k}$ .

Let now $S$ be an affine semigroup and $I\subset\mathbf{k}[S]$ be a monomial ideal. The Lie algebras $\operatorname{Der}(\mathbf{k}[S])$ and $\operatorname{Der}_{I}(\mathbf{k}[S])$ have been studied by several authors, see for instance [BS95, Tad09, Lie10, KLL15, DL24]. We present now the description of derivations of $\mathbf{k}[S]$ given in [KLL15, Proposition 3.1] with the notation adapted to our context. To describe inner derivations, we now let

\mathfrak{g}=\bigoplus_{\alpha\in S}\mathfrak{g}_{\alpha}\subset\operatorname{Der}(\mathbf{k}[S])\quad\mbox{where}\quad\mathfrak{g}_{\alpha}=\left\{\partial_{\alpha,p}\mid p\in N_{\mathbf{k}}\right\}.

For every $\alpha\in S$ , we have that $\mathfrak{g}_{\alpha}$ is a vector space isomorphic to $N_{\mathbf{k}},$ since $\partial_{\alpha,p}+\partial_{\alpha,q}=\partial_{\alpha,p+q},$ for all $p,q\in N_{\mathbf{k}}$ .

To describe outer derivations, which correspond exactly to the homogeneous locally nilpotent derivations in the notation of [KLL15] we need to introduce some notation. For every semigroup $S$ , we define its dual semigroup as

S^{\vee}=\{p\in N\mid p(m)\geq 0,\mbox{ for all }m\in S\}\,.

We observe that $S^{\vee}$ is an affine saturated semigroup, see [CLS11, Proposition 1.2.4]. For every $\rho\in S^{\vee}(1)$ we define the set

\displaystyle\mathcal{R}=\bigcup_{\rho\in S^{\vee}(1)}\mathcal{R}_{\rho}\quad\mbox{where}\quad\mathcal{R}_{\rho}=\left\{\alpha\in M\mid\rho(\alpha)=-1\mbox{ and }\rho^{\prime}(\alpha)\geq 0\mbox{ for all }\rho^{\prime}\in S^{\vee}(1)\setminus\{\rho\}\right\}\,.

(3)

Elements in $\mathcal{R}$ are called the Demazure roots of the semigroup $S$ . Note that by definition, the sets $\mathcal{R}_{\rho}$ are disjoint since $\rho(\alpha)=-1$ for one and only one $\rho\in S^{\vee}(1)$ . We now let

\mathfrak{s}=\bigoplus_{\alpha\in\mathcal{R}}\mathfrak{s}_{\alpha}\quad\mbox{where}\quad\mathfrak{s}_{\alpha}=\left\{\partial_{\alpha,p}\mid p=\lambda\cdot\rho,\mbox{ with }\lambda\in\mathbf{k}\right\}

is the $1$ -dimensional vector space of outer derivations of degree $\alpha$ .

Proposition 1.2 ([KLL15, Proposition 3.1]).

Let $S$ be a saturated affine semigroup. Then

\operatorname{Der}(\mathbf{k}[S])=\mathfrak{g}\oplus\mathfrak{s}\,,

where homogeneous derivations in $\mathfrak{g}$ are inner and homogeneous derivations in $\mathfrak{s}$ are outer.

We now provide a description of $\operatorname{Der}_{I}(\mathbf{k}[S])$ . The image $\partial(I)$ is contained in $I$ for all inner derivations $\partial\in\mathfrak{g}$ . So $\mathfrak{g}\subset\operatorname{Der}_{I}(\mathbf{k}[S])$ . For outer derivations, let $\mathfrak{s}(I)=\{\partial\in\mathfrak{s}\mid\partial(I)\subset I\}$ . Letting $I\subset\mathbf{k}[S]$ be the ideal $I=(\mathbf{x}^{\mathbf{a}_{1}},\dots,\mathbf{x}^{\mathbf{a}_{l}})$ with $\mathbf{a}_{i}\in S$ , we let

\mathcal{R}(I)=\bigcup_{\rho\in S^{\vee}(1)}\mathcal{R}_{\rho}(I)\quad\mbox{where}\quad\mathcal{R}_{\rho}(I)=\left\{\alpha\in\mathcal{R}_{\rho}\mid\mathbf{a}_{i}+\alpha\in\operatorname{supp}(I)\mbox{ for all }\mathbf{a}_{i}\notin\rho^{\bot}\right\}\,.

The following proposition provides a description of $\operatorname{Der}_{I}(\mathbf{k}[S])$ .

Proposition 1.3.

Let $S$ be a saturated affine semigroup and let $I=(\mathbf{x}^{\mathbf{a}_{1}},\dots,\mathbf{x}^{\mathbf{a}_{l}})$ with $\mathbf{a}_{i}\in S$ . Then

\mathfrak{s}(I)=\bigoplus_{\alpha\in\mathcal{R}(I)}\mathfrak{s}_{\alpha}.

In particular $\operatorname{Der}_{I}(\mathbf{k}[S])=\mathfrak{g}\oplus\mathfrak{s}(I).$

Proof.

Let $\partial\in\mathfrak{s}(I).$ Then there exists $r\in\mathbb{Z}_{>0}$ such that $\partial=\sum_{i=1}^{r}\partial_{\alpha_{i}},$ where $\alpha_{i}=\deg\partial_{\alpha_{i}},\>\partial_{\alpha_{i}}\in\mathfrak{s}_{\alpha_{i}}$ and $\partial_{\alpha_{i}}$ is homogeneous for all $i\in\{1,\ldots,r\}.$ Let $\alpha=\alpha_{i},$ for some $i\in\{1,\ldots,r\},$ and let $\rho\in N$ be such that $\alpha\in\mathcal{R}_{\rho}.$ We claim that $\alpha\in\mathcal{R}_{\rho}(I).$ Let $\mathbf{a}\in\{\mathbf{a}_{1},\ldots,\mathbf{a}_{l}\}\setminus\rho^{\perp}.$ Then $\partial(\mathbf{x}^{\mathbf{a}})=\sum\partial_{\alpha_{i}}(\mathbf{x}^{\mathbf{a}})\in I.$ It follows by [HH11, Corollary 1.1.3] that $\partial_{\alpha}(\mathbf{x}^{\mathbf{a}})\in I,$ that is $\mathbf{a}+\alpha\in\operatorname{supp}(I).$ Hence, $\alpha\in\mathcal{R}_{\rho}(I).$ The second statement follows directly from Proposition 1.2. ∎

The above description of $\mathfrak{s}(I)$ was given in [Tad09, Theorem 2.2] for the case where $\mathbf{k}[S]$ is the polynomial ring, as in Example 1.1, see also [BS95, Theorem 2.2.1]. We apply Proposition 1.3 to the polynomial ring in the following example.

Example 1.4 (Derivations of the polynomial ring).

With the notation of Example 1.1, we have that the inner derivations of the polynomial ring are of the form

\partial_{\alpha,p}=x_{1}^{\alpha_{1}}\ldots x_{n}^{\alpha_{n}}\cdot\left(p_{1}x_{1}\frac{d}{dx_{1}}+\cdots+p_{n}x_{n}\frac{d}{dx_{n}}\right),

for $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{Z}_{\geq 0}^{n}$ and $p=(p_{1},\dots,p_{n})\in\mathbf{k}^{n}.$ We now compute the outer derivations. Let $E^{*}=\{e_{1}^{*},\ldots,e_{n}^{*}\}$ be the canonical basis of $N.$ Then $E^{*}$ is the set of generating vectors of the rays of $S^{*}.$ Thus, for $1\leq j\leq n,$ we have

\mathcal{R}_{e_{j}^{*}}=\big{\{}\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{Z}^{n}\mid\alpha_{j}=-1\mbox{ and }\alpha_{i}\geq 0\mbox{ for all }i\neq j\big{\}}.

Hence, $\mathcal{R}=\bigcup_{j=1}^{n}\mathcal{R}_{e_{j}^{*}}$ yields the degree of all the outer derivations on $\mathbf{k}[S].$ Thus, for $1\leq j\leq n,$ the derivation $\partial_{\alpha,p}$ where $p=\lambda\cdot e_{j}^{*},$ for some $\lambda\in\mathbf{k}^{*}$ and $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathcal{R}_{e_{j}^{*}},$ corresponds to

\partial_{\alpha,p}=\lambda\cdot x_{1}^{\alpha_{1}}\cdots\widehat{x_{j}^{\alpha_{j}}}\cdots x_{n}^{\alpha_{n}}\cdot\frac{d}{dx_{j}}\,,

where $\widehat{x_{j}^{\alpha_{j}}}$ means that the factor is excluded from the product.

Example 1.5.

With the notation of Example 1.1, we let $n=2$ . Let $I\subset\mathbf{k}[x,y]$ be the monomial ideal $(x^{2}y^{5},x^{3}y^{2},x^{5})$ . We compute $\mathcal{R}(I).$ The support $\operatorname{supp}(I)$ is presented in Figure 4. Let $\mathbf{a}_{1}=(2,5),\mathbf{a}_{2}=(3,2),\mathbf{a}_{3}=(5,0)\in\mathbb{Z}_{\geq 0}^{2}$ be the exponents of the generators of $I.$ We first note that $\mathcal{R}_{e_{1}^{*}}$ is empty, because $\mathbf{a}_{1}+\alpha=(1,\ell+5)\notin\operatorname{supp}(I),$ for any $\alpha=(-1,\ell),$ where $\ell\in\mathbb{Z}_{\geq 0}.$

We now describe $\mathcal{R}_{e_{2}^{*}}(I).$ Note that $\mathcal{R}_{e_{2}^{*}}$ is the set of elements $\alpha=(\ell,-1),$ where $\ell\in\mathbb{Z}_{\geq 0},$ such that $\alpha+\mathbf{a}_{1}=(2+\ell,4),\alpha+\mathbf{a}_{2}=(3+\ell,1)\in\operatorname{supp}(I),$ because $e_{2}^{*}(\mathbf{a}_{3})=0.$ Thus $\mathcal{R}_{e_{2}^{*}}(I)=\{(\ell,-1)\in\mathbb{Z}^{2}\mid\ell\geq 2\}.$ Hence, we have

\mathfrak{s}(I)=\bigoplus_{\alpha\in\mathcal{R}_{e_{2}^{*}}(I)}\mathfrak{s}_{\alpha}=\bigoplus_{\ell=2}^{\infty}\mathbf{k}\cdot x^{\ell}\frac{d}{dy}=\left(\bigoplus_{\ell=2}^{\infty}\mathbf{k}\cdot x^{\ell}\right)\frac{d}{dy}=x^{2}\mathbf{k}[x]\cdot\frac{d}{dy}.

The set $\mathcal{R}(I)=\mathcal{R}_{e_{2}^{*}}(I)$ is given by the red dots in in Figure 4 represents the degrees of outer derivations $\partial\in\operatorname{Der}_{I}(\mathbf{k}[x,y])$ . Finally, we observe that all red dots represent trivial derivations on $\mathbf{k}[x,y]/I,$ except those of degree $\alpha\in\{(2,-1),(3,-1),(4,-1)\}$ .

Figure 3.

I=(x^{2}y^{5},x^{3}y^{2},x^{5})

Figure 4.

\mathcal{R}(I)=\{(\ell,-1)\mid\ell\geq 2\}

2. Derivations and Monomial Ideals

Let $S$ be an affine semigroup and $I\subset\mathbf{k}[S]$ a monomial ideal. Consider the natural projection map

\displaystyle\pi\colon\operatorname{Der}_{I}(\mathbf{k}[S])\to\operatorname{Der}(\mathbf{k}[S]/I),\quad\partial\mapsto\overline{\partial}\quad\text{defined by}\quad\overline{\partial}(\overline{a})=\overline{\partial(a)}.

(4)

This map $\pi$ is not always surjective. Indeed, the following example, provided by Kraft in his unpublished notes [Kra17, Example 5], demonstrates non-surjectivity in a specific case.

Example 2.1.

Let $M=\mathbb{Z}^{2}$ and let $S$ be the semigroup generated by $\mu_{1}=(1,0)$ , $\mu_{2}=(1,1)$ , and $\mu_{3}=(0,2)$ . Note that $S$ is not saturated because $(0,1)\in\omega_{S}\cap M$ , but $(0,1)\notin S$ . Furthermore, $X=\operatorname{Spec}\mathbf{k}[S]$ is isomorphic to the variety in the affine space $\mathbb{A}^{3}$ defined by the equation $x^{2}z-y^{2}=0$ , by setting $x=\mathbf{x}^{\mu_{1}}$ , $y=\mathbf{x}^{\mu_{2}}$ , and $z=\mathbf{x}^{\mu_{3}}$ (commonly known as Whitney’s Umbrella). It follows from [DL24, Theorem 3.11] that $\mathcal{R}=\{(\ell,-1)\in\mathbb{Z}^{2}\mid\ell\geq 1\}$ . This is represented by the green points in Figure 6. Let $I\subset\mathbf{k}[S]$ be the monomial ideal generated by $\mathbf{a}_{1}=(1,0)$ and $\mathbf{a}_{2}=(1,1)$ , shown in Figure 6. The ideal $I$ corresponds to $I=(x,y)$ under the isomorphism mentioned above, and $\mathbf{k}[S]/I\simeq\mathbf{k}[\overline{z}]$ . Since $\dfrac{d}{d\overline{z}}$ is a homogeneous derivation on $\mathbf{k}[\overline{z}]$ of degree $(0,-2)$ , we conclude that $\pi$ cannot be surjective because $(0,-2)\notin\mathcal{R}$ .

Figure 5.

S=\{(0,2),(1,1),(1,0)\}\mathbb{Z}_{\geq 0}

Figure 6.

I=(x,y)

Inspired by the above example, which concerns a non-saturated affine semigroup, we present the following theorem, which is the aim of this section. In this theorem, we show that non-surjectivity can occur for any saturated affine semigroup that is not isomorphic to the first octant $\mathbb{Z}^{n}_{\geq 0}$ . Recall that, unless stated otherwise, all our semigroups are affine, saturated, pointed and minimally embedded in $M$ .

Theorem 2.2.

Let $S$ be a semigroup. The natural map

\pi\colon\operatorname{Der}_{I}(\mathbf{k}[S])\to\operatorname{Der}(\mathbf{k}[S]/I)

is surjective for every monomial ideal $I$ if and only if $S\simeq\mathbb{Z}^{n}_{\geq 0}$ .

We divide the proof into several lemmas. Let $S$ be a saturated affine semigroup. We say that $m\in S\setminus\{0\}$ is irreducible if $m=m_{1}+m_{2}$ implies $m_{1}=0$ or $m_{2}=0$ . The set $\mathcal{H}$ of all irreducible elements is called the Hilbert basis of $S$ . Since $S$ is pointed, the Hilbert basis $\mathcal{H}$ is the unique minimal generating set of $S$ as a semigroup. Let $\mathcal{H}=\{\mu_{1},\dots,\mu_{l}\}\subset S$ be the Hilbert basis of $S$ . In all cases where $S$ is different from the first octant $\mathbb{Z}_{\geq 0}^{n}$ , we will prove that there exists a non-liftable derivation in $\mathbf{k}[S]/I_{\mathcal{H}}$ , where $I_{\mathcal{H}}$ is the monomial ideal

I_{\mathcal{H}}=\bigoplus_{m\in S\setminus\mathcal{H}}\mathbf{k}\cdot\mathbf{x}^{m}\,.

Let $\rho\in S^{\vee}(1)$ . Note that there exist $m_{1},m_{2}\in\mathcal{H}$ such that $\rho(m_{1})=0$ and $\rho(m_{2})=1$ , since $\mathcal{H}$ generates $S$ .

Lemma 2.3.

Let $S$ be a semigroup. If there exists $\rho\in S^{\vee}(1)$ such that $\{0,1\}$ is a proper subset of $\rho(\mathcal{H})$ , then the natural map $\pi\colon\operatorname{Der}_{I_{\mathcal{H}}}(\mathbf{k}[S])\to\operatorname{Der}(\mathbf{k}[S]/I_{\mathcal{H}})$ is not surjective.

Proof.

Let $\mathcal{H}=\{\mu_{1},\dots,\mu_{l}\}$ be the Hilbert basis of $S$ . By hypothesis, there exists $\rho\in S^{*}(1)$ such that the set $\{0,1\}$ is a proper subset of $\rho(\mathcal{H}).$ Without loss of generality, we assume that $\rho(\mu_{1})=\ell>1$ and $\rho(\mu_{2})=0$ . We define the linear map

\overline{\partial}\colon\mathbf{k}[S]/I_{\mathcal{H}}\to\mathbf{k}[S]/I_{\mathcal{H}},\quad\mbox{given by}\quad\overline{\partial}(\overline{\mathbf{x}}^{\mu_{1}})=\overline{\mathbf{x}}^{\mu_{2}}\mbox{ and }\overline{\partial}(\overline{\mathbf{x}}^{\mu_{i}})=0\mbox{ for all }i>1\,.

The map $\overline{\partial}$ is well defined since $\overline{\partial}(I)=0$ . Moreover, it is a derivation. Indeed, the only non-trivial instance of the Leibniz rule is

0=\overline{\partial}(\overline{\mathbf{x}}^{\mu_{1}+\mu_{i}})=\overline{\partial}(\overline{\mathbf{x}}^{\mu_{1}})\overline{\mathbf{x}}^{\mu_{i}}+\overline{\mathbf{x}}^{\mu_{1}}\overline{\partial}(\overline{\mathbf{x}}^{\mu_{i}})=\overline{\mathbf{x}}^{\mu_{1}}\overline{\partial}(\overline{\mathbf{x}}^{\mu_{i}})\,.

Note that $\overline{\partial}(\overline{\mathbf{x}}^{\mu_{i}})$ is equal to $0$ or $\overline{\mathbf{x}}^{\mu_{2}}$ . In both cases, the Leibniz rule is verified since

\overline{\mathbf{x}}^{\mu_{1}}\overline{\partial}(\overline{\mathbf{x}}^{\mu_{i}})=0\,.

The derivation $\overline{\partial}$ is homogeneous and its degree is $\deg\overline{\partial}=\alpha=\mu_{2}-\mu_{1}$ and $\rho(\alpha)=-\ell<-1$ . Assume that there exists $\partial_{\alpha}\in\operatorname{Der}(\mathbf{k}[S])$ homogeneous such that $\pi(\partial_{\alpha})=\overline{\partial}_{\alpha}$ . Since $\rho(\alpha)$ is negative, $\partial_{\alpha}$ must be an outer derivation. Finally, for all outer derivations we have $\rho(\alpha)\geq-1$ by (3). This provides a contradiction. Hence, the lifting $\partial_{\alpha}$ does not exist. ∎

In the proof of Lemma 2.6, we need the following technical lemma.

Lemma 2.4.

Let $M$ be a free abelian group of rank $n$ and let $N$ be the dual group $N=\operatorname{Hom}(M,\mathbb{Z})$ . Let $\beta=\{\mu_{1},\ldots,\mu_{n}\}\subset M$ and $\beta^{\prime}=\{\rho_{1},\ldots,\rho_{n}\}\subset N$ be linearly independent sets. We let $M^{\prime}$ be the subgroup of $M$ generated by $\beta$ and $N^{\prime}$ be the subgroup of $N$ generated by $\beta^{\prime}$ . If $A=(a_{ij})$ is the $n\times n$ matrix with $a_{ij}=\rho_{i}(\mu_{j})$ , then

|M/M^{\prime}|\cdot|N/N^{\prime}|=|\det(A)|\,.

Proof.

The lemma follows directly by taking dual bases $\beta$ and $\beta^{*}$ of $M$ and $N$ , respectively and applying [Bar02, Chapter VII, Section 2, Theorem 2.5] to the subgroups $M^{\prime}\subset M$ and $N^{\prime}\subset N$ on both sides of the duality. ∎

To prove Theorem 2.2, we introduce the following notation that will be convenient in the proof.

Definition 2.5.

Let $S$ be a semigroup minimally embedded in $M$ with $\operatorname{rank}M=n$ . We define a complete flag $\mathcal{F}$ of $S$ as a chain of faces $F_{i}\subseteq S$ such that

\{0\}=F_{0}\subset F_{1}\subset\ldots\subset F_{n}=S\quad\text{with}\quad\operatorname{rank}F_{i}=i\,.

The complete flag $\mathcal{F}$ also defines a chain of reverse inclusions

S^{\vee}=F^{*}_{0}\supset F^{*}_{1}\supset\ldots\supset F^{*}_{n}=\{0\}\quad\text{with}\quad\operatorname{rank}F^{*}_{i}=n-i\,,

where $F^{*}_{i}$ is the face of $S^{\vee}\subset N$ dual to $F_{i}$ , denoted as $F^{*}_{i}=S^{\vee}\cap F_{i}^{\perp}$ .

Given a complete flag, an upper triangular pair of rays is a pair of sets $\beta=\{\mu_{1},\ldots,\mu_{n}\}\subset S(1)$ and $\beta^{\prime}=\{\rho_{1},\ldots,\rho_{n}\}\subset S^{\vee}(1)$ such that

\mu_{i}\in F_{i}\setminus F_{i-1}\quad\text{and}\quad\rho_{i}\in F^{*}_{i-1}\setminus F^{*}_{i}\,,

satisfying

\displaystyle\rho_{i}(\mu_{j})=\begin{cases}0&\text{if}\quad i>j\\ *&\text{if}\quad i\leq j\end{cases}\quad\text{and}\quad\rho_{i}(\mu_{i})>0,\text{ for all }i\,.

(5)

In the following lemma, we show that an upper triangular pair always exists.

Lemma 2.6.

Let $S$ be a semigroup and let $\mathcal{H}$ be its Hilbert basis. Let $\mathcal{F}$ be a complete flag of $S$ . Then, there exists an upper triangular pair $(\beta,\beta^{\prime})$ . Moreover, if $\rho(S(1))=\{0,1\}$ for all $\rho\in S^{\vee}(1)$ , then $\beta$ and $\beta^{\prime}$ are $\mathbb{Z}$ -bases of $M$ and $N$ , respectively.

Proof.

Let $S$ be minimally embedded in $M$ with $\operatorname{rank}M=n$ . The complete flag $\mathcal{F}$ provides us with the inclusions of faces

\{0\}=F_{0}\subset F_{1}\subset\ldots\subset F_{n}=S\quad\text{and}\quad S^{\vee}=F^{*}_{0}\supset F^{*}_{1}\supset\ldots\supset F^{*}_{n}=\{0\}\,.

We will appropriately choose the sets $\beta=\{\mu_{1},\ldots,\mu_{n}\}\subset S(1)$ and $\beta^{\prime}=\{\rho_{1},\ldots,\rho_{n}\}\subset S^{\vee}(1)$ to obtain an upper triangular pair $(\beta,\beta^{\prime})$ . First, we pick $\mu_{i}$ as any ray in $F_{i}\setminus F_{i-1}$ . Then, we choose $\rho_{i}$ to be any ray in $F^{*}_{i-1}$ with $\rho_{i}(\mu_{i})>0$ . Such a ray must exist; otherwise, $\mu_{i}\in F_{i-1}$ , which is a contradiction. This choice satisfies the condition in (5). Indeed, if $i>j$ , then $\rho_{i}$ is a ray in $F^{*}_{i-1}\subseteq F^{*}_{j}$ and $\mu_{j}$ is a ray in $F_{j}$ , so $\rho_{i}(\mu_{j})=0$ . This proves the first statement of the lemma.

To prove the second statement, we assume, moreover, that $\rho(S(1))=\{0,1\}$ . Under these conditions, $\rho_{i}(\mu_{i})>0$ in (5) implies $\rho_{i}(\mu_{i})=1$ . Let $A=(a_{ij})$ be the $n\times n$ matrix with $a_{ij}=\rho_{i}(\mu_{j})$ . By (5) and the fact that $\rho_{i}(\mu_{i})=1$ , we have that $\det(A)=1$ , so $\beta$ and $\beta^{\prime}$ are linearly independent sets. Moreover, by Lemma 2.4, we have that $|M/M^{\prime}|=|N/N^{\prime}|=1$ , where $M^{\prime}$ and $N^{\prime}$ are the submodules spanned by $\beta$ and $\beta^{\prime}$ in $M$ and $N$ , respectively. We conclude that $\beta$ and $\beta^{\prime}$ are $\mathbb{Z}$ -bases of $M$ and $N$ , respectively. ∎

Corollary 2.7.

Let $S$ be a simplicial affine semigroup. If $S$ is not isomorphic to $\mathbb{Z}_{\geq 0}^{n}$ , then the natural map $\pi\colon\operatorname{Der}_{I_{\mathcal{H}}}(\mathbf{k}[S])\to\operatorname{Der}(\mathbf{k}[S]/I_{\mathcal{H}})$ is not surjective.

Proof.

If $S$ is not isomorphic to $\mathbb{Z}_{\geq 0}^{n}$ , then the set $S(1)$ is not a $\mathbb{Z}$ -basis of $M$ . Now, take any complete flag of $S$ . According to Lemma 2.6, we can order the sets $S(1)$ and $S^{\vee}(1)$ , both with $n$ elements, to form an upper triangular pair. Furthermore, by the second statement in Lemma 2.6, there exists $\rho\in S^{\vee}(1)$ such that $\{0,1\}$ is a proper subset of $\rho(S(1))\subset\rho(\mathcal{H})$ , since $S(1)$ is not a $\mathbb{Z}$ -basis of $M$ . Thus, the non-surjectivity of $\pi$ follows from Lemma 2.3. ∎

In the following lemma, we handle the case where $S$ is not simplicial.

Lemma 2.8.

Let $S$ be a semigroup. If $S$ is not simplicial, then the natural map $\pi\colon\operatorname{Der}_{I_{\mathcal{H}}}(\mathbf{k}[S])\to\operatorname{Der}(\mathbf{k}[S]/I_{\mathcal{H}})$ is not surjective.

Proof.

Let $\mathcal{H}=\{\mu_{1},\ldots,\mu_{l}\}$ be the Hilbert basis of $S$ and let $\mathcal{F}$ be a complete flag that gives us the inclusions of faces

\{0\}=F_{0}\subset F_{1}\subset\ldots\subset F_{n}=S\quad\text{and}\quad S^{\vee}=F^{*}_{0}\supset F^{*}_{1}\supset\ldots\supset F^{*}_{n}=\{0\}\,.

By Lemma 2.6, there exists an upper triangular pair $(\beta,\beta^{\prime})$ . Recall that

\mu_{i}\in F_{i}\setminus F_{i-1}\quad\text{and}\quad\rho_{i}\in F^{*}_{i-1}\setminus F^{*}_{i}\,.

Now we have that $F_{n-1}=\rho_{n}^{\perp}\cap S$ and $G=\rho_{n-1}^{\perp}\cap S$ are facets whose intersection

F_{n-1}\cap G=\rho_{n-1}^{\perp}\cap\rho_{n}^{\perp}\cap S=F_{n-2}

is an $(n-2)$ -dimensional face. Since $S$ is not simplicial, up to changing the complete flag $\mathcal{F}$ , we can assume that there exists another ray $\mu\in S(1)$ that is not contained in $F_{n-1}\cup G$ . Hence, $\rho_{n}(\mu)>0$ and $\rho_{n-1}(\mu)>0$ . We now define the linear map

\overline{\partial}\colon\mathbf{k}[S]/I_{\mathcal{H}}\to\mathbf{k}[S]/I_{\mathcal{H}},\quad\text{given by}\quad\overline{\partial}(\overline{\mathbf{x}}^{\mu})=\overline{\mathbf{x}}^{\mu_{1}}\text{ and }\overline{\partial}(\overline{\mathbf{x}}^{\mu^{\prime}})=0\text{ for all }\mu^{\prime}\in\mathcal{H}\setminus\{\mu\}\,,

where $\mu_{1}$ is the first element in the Hilbert basis $\mathcal{H}$ . Using the same argument as in Lemma 2.3, we conclude that $\overline{\partial}$ is a derivation of $\mathbf{k}[S]/I$ .

The degree of $\overline{\partial}$ is $\alpha=\mu_{1}-\mu$ , and so $\rho_{n-1}(\alpha)<0$ and $\rho_{n}(\alpha)<0$ . Assume that there exists $\partial\in\operatorname{Der}(\mathbf{k}[S])$ homogeneous such that $\pi(\partial)=\overline{\partial}$ . The degree of $\partial$ must also be $\alpha$ and since $\rho_{n-1}(\alpha)<0$ and $\rho_{n}(\alpha)<0$ , the derivation $\partial_{\alpha}$ must be an outer derivation. Finally, by (3), for all outer derivations, we have that $\rho(\alpha)$ is negative for one and only one $\rho\in S^{\vee}(1)$ . This provides a contradiction. ∎

We now deal with the case where $\mathbf{k}[S]$ is the polynomial ring.

Lemma 2.9.

Let $S=\mathbb{Z}_{\geq 0}^{n}$ and let $I$ be a monomial ideal in $\mathbf{k}[S]$ . Then, the natural map $\pi\colon\operatorname{Der}_{I}(\mathbf{k}[S])\to\operatorname{Der}(\mathbf{k}[S]/I)$ is surjective.

Proof.

Recall that $\mathbf{k}[S]=\mathbf{k}[x_{1},\dots,x_{n}]$ as in Example 1.1, and let $\varphi\colon\mathbf{k}[S]\to\mathbf{k}[S]/I$ be the quotient map. For any derivation $\overline{\partial}\in\operatorname{Der}(\mathbf{k}[S]/I)$ , we choose $f_{i}\in\varphi^{-1}(\overline{\partial}(\overline{x}_{i}))$ . Since $\operatorname{Der}(\mathbf{k}[S])$ is freely generated by the partial derivatives $\left\{\dfrac{d}{dx_{1}},\dots,\dfrac{d}{dx_{n}}\right\}$ as a $\mathbf{k}[S]$ -module [Fre17, Section 3.2.2], the map $\partial\colon\mathbf{k}[S]\to\mathbf{k}[S]$ defined by

\partial=\sum_{i=1}^{n}f_{i}\dfrac{d}{dx_{i}}\quad\text{is a derivation of }\mathbf{k}[S]\text{ such that }\partial(x_{i})=f_{i}\,.

Since $\overline{\partial}\circ\varphi=\varphi\circ\partial$ , we have that $\partial(I)\subset I$ . This proves that $\pi(\partial)=\overline{\partial}$ and so $\pi$ is surjective. ∎

We can finally proceed with the proof of Theorem 2.2.

Proof of Theorem 2.2.

The direct implication follows from Corollary 2.7 if $S$ is simplicial and from Lemma 2.8 if $S$ is not simplicial. On the other hand, the converse implication follows from Lemma 2.9. ∎

3. Liftable locally nilpotent derivation of monomial algebras

Let $S$ be a semigroup and let $I\subset\mathbf{k}[S]$ be a monomial ideal. In this section, we compute the homogeneous locally nilpotent derivations of the algebra $\mathbf{k}[S]/I$ , which are obtained as the images of derivations of $\mathbf{k}[S]$ via $\pi$ in (4). We formalize this concept in the following definition.

Definition 3.1.

Let $S$ be a semigroup and let $I$ be a monomial ideal of $\mathbf{k}[S]$ . We say that a derivation $\overline{\partial}$ on $\mathbf{k}[S]/I$ is liftable if there exists a derivation $\partial\in\operatorname{Der}_{I}(\mathbf{k}[S])$ such that $\overline{\partial}=\pi(\partial)$ .

Remark 3.2.

In view of Theorem 2.2, if $S$ is isomorphic to $\mathbb{Z}_{\geq 0}^{n}$ , then all derivations of $\mathbf{k}[S]/I$ are liftable.

If $\overline{\partial}\colon\mathbf{k}[S]/I\to\mathbf{k}[S]/I$ is a liftable homogeneous derivation, then there exists a homogeneous derivation $\partial$ of $\mathbf{k}[S]$ such that $\overline{\partial}=\pi(\partial)$ . According to Proposition 1.3, we have $\partial=\partial_{\alpha,p}$ . Therefore, we conclude that $\overline{\partial}=\pi(\partial_{\alpha,p})$ , and we denote this derivation by $\overline{\partial}_{\alpha,p}$ . Homogeneous derivations in $\mathbf{k}[S]$ come in two types: inner and outer. The case of outer derivations of $\mathbf{k}[S]$ is straightforward since they are all locally nilpotent, and any locally nilpotent derivation that descends to the quotient $\mathbf{k}[S]/I$ remains locally nilpotent. We formalize this in the following lemma for future reference.

Lemma 3.3.

Let $S$ be a saturated affine semigroup, and let $I\subset\mathbf{k}[S]$ be a monomial ideal. Then, any homogeneous derivation $\partial\in\mathfrak{s}(I)$ induces a locally nilpotent derivation of $\mathbf{k}[S]/I$ .

We now turn our attention to inner derivations. Inner derivations are never locally nilpotent on $\mathbf{k}[S]$ , but they may become so in the quotient $\mathbf{k}[S]/I$ . Recall that all outer derivations of $\mathbf{k}[S]$ leave $I$ invariant and, therefore, pass to the quotient $\mathbf{k}[S]/I$ .

Lemma 3.4.

Let $I$ be a monomial ideal of $\mathbf{k}[S]$ and let $\alpha\in S$ . If $(\mathbb{Z}_{\geq 0}\cdot\alpha)\cap\operatorname{supp}(I)\neq\emptyset$ , then the derivation ${\overline{\partial}}_{\alpha,p}\colon\mathbf{k}[S]/I\to\mathbf{k}[S]/I$ is locally nilpotent for all $p\in N_{\mathbf{k}}$ .

Proof.

By hypothesis, there exists $\ell\in\mathbb{Z}_{\geq 0}$ such that $\ell\cdot\alpha\in\operatorname{supp}(I)$ . Let $m\in S$ . A straightforward computation shows that

\displaystyle\partial_{\alpha,p}^{\ell}(\mathbf{x}^{m})=r\cdot\mathbf{x}^{m+\ell\cdot\alpha}\quad\mbox{where}\quad r=\prod_{j=1}^{\ell}\big{(}p(m)+(j-1)p(\alpha)\big{)}\,.

(6)

Now, $\partial_{\alpha,p}^{\ell}(\mathbf{x}^{m})=r\cdot\mathbf{x}^{m+\ell\cdot\alpha}=r\cdot\mathbf{x}^{m}\cdot\mathbf{x}^{\ell\cdot\alpha}$ and $\mathbf{x}^{\ell\cdot\alpha}\in I$ since $\ell\cdot\alpha\in\operatorname{supp}(I)$ . This yields $\partial_{\alpha,p}^{\ell}(\mathbf{x}^{m})\in I$ and so ${\overline{\partial}}_{\alpha,p}^{\ell}(\mathbf{x}^{m})=0$ , proving that ${\overline{\partial}}_{\alpha,p}^{\ell}(\mathbf{x}^{m})$ is locally nilpotent. ∎

Lemma 3.5.

Let $I$ be a monomial ideal on $\mathbf{k}[S]$ and let $\alpha\in S$ . Assume that $(\mathbb{Z}_{\geq 0}\cdot\alpha)\cap\operatorname{supp}(I)=\emptyset$ . Then the derivation ${\overline{\partial}}_{\alpha,p}\colon\mathbf{k}[S]/I\to\mathbf{k}[S]/I$ is locally nilpotent if and only if

\displaystyle(m+\mathbb{Z}_{\geq 0}\cdot\alpha)\cap\operatorname{supp}(I)\neq\emptyset\quad\mbox{for all}\quad m\in S\setminus p^{\bot}\,.

(7)

Proof.

We claim that both propositions of the “if and only if” imply that $p(\alpha)=0$ . Indeed, condition (7) applied to $\alpha$ implies $\alpha\in p^{\bot}$ . On the other hand, if ${\overline{\partial}}_{\alpha,p}$ is locally nilpotent, then $\partial_{\alpha,p}^{\ell}(\mathbf{x}^{\alpha})=\ell!p(\alpha)^{\ell}\mathbf{x}^{(\ell+1)\cdot\alpha}$ by (6) and this expression belongs to the ideal only if $p(\alpha)=0$ .

Now, since $p(\alpha)=0$ , we have that

\displaystyle\partial_{\alpha,p}^{\ell}(\mathbf{x}^{m})=p(m)^{\ell}\cdot\mathbf{x}^{m+\ell\cdot\alpha}\,.

(8)

The derivation ${\overline{\partial}}_{\alpha,p}$ is locally nilpotent if and only if $\partial_{\alpha,p}^{\ell}(\mathbf{x}^{m})\in I$ for all $m\in S$ . By (8) and the same argument in Lemma 3.4, this is the case if and only if

\displaystyle m+\ell\cdot\alpha\in\operatorname{supp}(I)\mbox{ for some }\ell\in\mathbb{Z}_{\geq 0}\quad\mbox{or}\quad p(m)=0\,,

and this last condition is equivalent to (7). ∎

Remark 3.6.

It is sufficient to verify the condition in Lemma 3.5 on a set of generators of $S$ .

We now state our main classification result for liftable homogeneous locally nilpotent derivations on $\mathbf{k}[S]/I$ . It amounts to the recollection of the three previous lemmas.

Theorem 3.7.

Let $S$ be a semigroup, and let $I$ be a monomial ideal on $\mathbf{k}[S]$ . A liftable homogeneous derivation $\overline{\partial}_{\alpha,p}\colon\mathbf{k}[S]/I\to\mathbf{k}[S]/I$ with $p\in N_{\mathbf{k}}$ and $\alpha\in M$ is locally nilpotent if and only if

$(i)$

$\alpha\in\mathcal{R}_{i}(I)$ and $p=\lambda\cdot\rho_{i}$ with $\lambda\in\mathbf{k}$ ; or
$(ii)$

$\alpha\in S$ and $(\mathbb{Z}_{\geq 0}\cdot\alpha)\cap\operatorname{supp}(I)\neq\emptyset$ ; or
$(iii)$

$\alpha\in S$ , $(\mathbb{Z}_{\geq 0}\cdot\alpha)\cap\operatorname{supp}(I)=\emptyset$ , and $(m+\mathbb{Z}_{\geq 0}\cdot\alpha)\cap\operatorname{supp}(I)\neq\emptyset$ for all $m\in S\setminus p^{\bot}$ .

Proof.

The result follows directly from Lemmas Lemma 3.3, Lemma 3.4, and Lemma 3.5. ∎

Remark 3.8.

Let $S$ be a semigroup and let $I\subset\mathbf{k}[S]$ be a monomial ideal. In the case where $S$ is not isomorphic to $\mathbb{Z}_{\geq 0}^{n}$ , by Theorem 2.2 there exist monomial ideals $I$ such that not all derivations are liftable. Some of these non-liftable derivations may be locally nilpotent. Indeed, the non-liftable derivations constructed in the proofs of Lemma 2.3 and Lemma 2.8 are locally nilpotent.

Example 3.9.

With the notation in Example 1.1 for $n=2$ , we present here examples of all the degrees of homogeneous locally nilpotent derivations for different monomial ideals $I$ . Red points indicate the degrees described in Theorem 3.7 $(i)$ . Green points indicate the degrees described in Theorem 3.7 $(ii)$ . Finally, blue points indicate the degrees described in Theorem 3.7 $(iii)$ .

Figure 7.

$I_{1}=(x^{2}y^{5},x^{3}y^{2},x^{5}y)$

Figure 8.

$I_{2}=(xy^{5},x^{3}y^{2},x^{5})$

Figure 9.

$I_{3}=(y^{5},x^{3}y^{2},x^{5})$

Some derivations appearing in Theorem 3.7 are zero derivations. In the following lemma, we provide a criterion to determine when a homogeneous derivation of $\mathbf{k}[S]$ is the trivial derivation of $\mathbf{k}[S]/I$ .

Lemma 3.10.

A homogeneous derivation $\overline{\partial}_{\alpha,p}\colon\mathbf{k}[S]/I\to\mathbf{k}[S]/I$ with $p\in N_{\mathbf{k}}$ and $\alpha\in M$ is trivial if and only if

\big{(}(S\setminus p^{\bot})+\alpha\big{)}\subset\operatorname{supp}(I)\,.

Proof.

The derivation $\partial=\partial_{\alpha,p}$ maps to the trivial derivation via $\pi$ if and only if for every $m\in S$ we have $\partial(\mathbf{x}^{m})\in I$ . This holds if and only if for every $m\in S$ with $p(m)\neq 0$ , we have $m+\alpha$ contained in $\operatorname{supp}(I)$ , so $\mathbf{x}^{m+\alpha}\in I$ . This last statement is equivalent to the one given in the lemma. ∎

Remark 3.11.

It is sufficient to verify the condition in Lemma 3.10 on a set of generators of $S$ .

4. Automorphism group of monomial algebras

In this section, we apply our classification of homogeneous locally nilpotent derivations on $\mathbf{k}[S]/I$ to describe $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ in the case where $I$ is a monomial ideal with cofinite support. Our approach is inspired by Demazure’s description of $\operatorname{Aut}(X)$ for a complete smooth toric variety $X$ [Dem70], which was recently revisited in modern terms by one of the authors [LLA22]. In the following proposition, we prove that the automorphism group $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ is linear algebraic.

Proposition 4.1.

Let $S$ be a semigroup and let $I$ be a monomial ideal with cofinite support. Then the automorphism group of $\mathbf{k}[S]/I$ is linear algebraic.

Proof.

Since the monomial algebra $\mathbf{k}[S]/I$ is finite-dimensional as a vector space over $\mathbf{k}$ and every automorphism $\varphi\colon\mathbf{k}[S]/I\to\mathbf{k}[S]/I$ is also an invertible linear map, we have that $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ embeds in $\operatorname{GL}(\mathbf{k}[S]/I)$ . Moreover, $\varphi\in\operatorname{GL}(\mathbf{k}[S]/I)$ belongs to $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ if and only if $\varphi(\overline{\mathbf{x}}^{m+m^{\prime}})=\varphi(\overline{\mathbf{x}}^{m})\cdot\varphi(\overline{\mathbf{x}}^{m^{\prime}})$ for all $m,m^{\prime}\in S$ , which is a closed condition. We conclude that $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ is a closed subgroup of $\operatorname{GL}(\mathbf{k}[S]/I)$ and thus is linear algebraic. ∎

Let now $T=\operatorname{Spec}\mathbf{k}[M]$ . Then, $T$ acts faithfully on $\mathbf{k}[S]$ via the usual toric action given by the comorphism

\mathbf{k}[S]\to\mathbf{k}[M]\otimes\mathbf{k}[S]\quad\text{defined by}\quad\mathbf{x}^{m}\mapsto\mathbf{x}^{m}\otimes\mathbf{x}^{m}.

A closed point $t\in T$ corresponds to a maximal ideal of $\mathbf{k}[M]$ and, in turn, corresponds to a homomorphism $t\colon M\rightarrow\mathbf{k}^{*}$ . Indeed, $t$ induces a homomorphism $\mathbf{k}[M]\rightarrow\mathbf{k}$ via $\mathbf{x}^{m}\mapsto t(m)$ , and the maximal ideal defining the point $t$ is the kernel of this homomorphism. With this notation, the action of a closed point $t\in T$ on $\mathbf{k}[S]$ is given by

\displaystyle\mathbf{k}[S]\to\mathbf{k}[S]\quad\text{defined by}\quad\mathbf{x}^{m}\mapsto t\cdot\mathbf{x}^{m}=t(m)\mathbf{x}^{m}.

(9)

In particular, the $T$ -action preserves every monomial ideal $I$ and thus defines a $T$ -action on $\mathbf{k}[S]/I$ .

Definition 4.2.

Let $S$ be a semigroup, and let $I$ be a monomial ideal in $\mathbf{k}[S]$ . We say that $I$ is full if no ray generator of $S$ is contained in $\operatorname{supp}(I)$ .

In the case that a monomial ideal $I$ in $\mathbf{k}[S]$ is not full, let $S^{\prime}$ be the smallest subsemigroup of $S$ containing $S\setminus\operatorname{supp}(I)$ , and let $I^{\prime}=I\cap\mathbf{k}[S^{\prime}]$ . Then, we have that $\mathbf{k}[S]/I=\mathbf{k}[S^{\prime}]/I^{\prime}$ and $I^{\prime}$ is full.

Example 4.3.

If $S$ is the first octant as in Example 1.1, then the action of $T$ is given by component-wise multiplication. Furthermore, a monomial ideal $I\subset\mathbf{k}[S]$ is full if and only if no variable $x_{i}$ belongs to $I$ .

Lemma 4.4.

Let $S$ be a semigroup and let $I$ be a monomial ideal. If $I$ is full, then $T$ acts faithfully on $\mathbf{k}[S]/I$ .

Proof.

Let $t\in T$ act as the identity on $\mathbf{k}[S]/I$ . By (9), we have $t.\mathbf{x}^{m}=t(m)\mathbf{x}^{m}=\mathbf{x}^{m}$ for all $m\in S\setminus\operatorname{supp}(I)$ . Thus, $t(m)=1$ for all $m\in S\setminus\operatorname{supp}(I)$ . If $I$ is full, then $S\setminus\operatorname{supp}(I)$ spans $M_{\mathbb{R}}$ . Therefore, $t(m)=1$ for all $m\in M$ , and hence $t$ is the identity in $T$ . ∎

Under the condition of Lemma 4.4, we denote the image of $T$ inside $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ also by $T$ . In the next lemma, we show that, in certain cases, $T$ is a maximal torus in $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ .

Lemma 4.5.

Letting $S$ be the first octant, we let $I$ be a full monomial ideal with cofinite support. We also let $T=\operatorname{Spec}\mathbf{k}[M]\subset\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ . Then, the centralizer of $T$ equals $T$ . In particular, $T$ is a maximal torus of $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ .

Proof.

Let $g\in\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ be defined by

g(\overline{\mathbf{x}}^{m^{\prime}})=\sum_{m\in S\setminus\operatorname{supp}(I)}a_{m^{\prime},m}\cdot\overline{\mathbf{x}}^{m}\quad\mbox{for all}\quad m^{\prime}\in S\setminus\operatorname{supp}(I)\,.

Assume that $g$ belongs to the centralizer of $T$ , that is, $t\circ g(\overline{\mathbf{x}}^{m^{\prime}})=g\circ t(\overline{\mathbf{x}}^{m^{\prime}})$ , for all $t\in T$ . This implies

\sum_{m\in S\setminus\operatorname{supp}(I)}t(m)\cdot a_{m^{\prime},m}\cdot\overline{\mathbf{x}}^{m}=\sum_{m\in S\setminus\operatorname{supp}(I)}t(m^{\prime})\cdot a_{m^{\prime},m}\cdot\overline{\mathbf{x}}^{m},\quad\mbox{for all }t\in T\,.

This equality can only hold if $a_{m^{\prime},m}=0$ for all $m\neq m^{\prime}$ . Letting $a_{m,m}=a_{m}$ , we conclude

\displaystyle g(\overline{\mathbf{x}}^{m})=a_{m}\overline{\mathbf{x}}^{m}\quad\mbox{for all}\quad m\in S\setminus\operatorname{supp}(I)\,.

(10)

Finally, since the basis vector $e_{i}$ belongs to $S\setminus\operatorname{supp}(I)$ as $S$ is the first octant and $I$ is full, for every $m=(m_{1},\ldots,m_{n})\in M$ , we have that $a_{m}=\prod a_{e_{i}}^{m_{i}}$ and so $g\in T$ by (9). This proves, in particular, that the centralizer of $T$ is $T$ itself, thus $T$ is a maximal torus of $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ . ∎

The hypothesis that $S$ is the first octant in the above lemma is essential, as the following example shows.

Example 4.6.

Let $S$ be the semigroup $\mathbb{Z}_{\geq 0}\{(1,0),(1,1),(1,2)\}$ . We consider the monomial ideal $I\subset\mathbf{k}[S]$ given by $I=(\mathbf{x}^{(2,0)},\mathbf{x}^{(2,1)},\mathbf{x}^{(2,2)},\mathbf{x}^{(2,3)},\mathbf{x}^{(2,4)})$ . Thus, $\mathbf{k}[S]/I\simeq\mathbf{k}\cdot\overline{\mathbf{x}}^{(0,0)}\oplus\mathbf{k}\cdot\overline{\mathbf{x}}^{(1,0)}\oplus\mathbf{k}\cdot\overline{\mathbf{x}}^{(1,1)}\oplus\mathbf{k}\cdot\overline{\mathbf{x}}^{(1,2)}$ . If $g$ is an element of the centralizer of $(\mathbf{k}^{*})^{2}$ , then, using the same argument as in the proof of Lemma 4.5 up to (10), we conclude that

g(\overline{\mathbf{x}}^{(1,0)})=a_{(1,0)}\overline{\mathbf{x}}^{(1,0)},\quad g(\overline{\mathbf{x}}^{(1,1)})=a_{(1,1)}\overline{\mathbf{x}}^{(1,1)},\quad\text{and}\quad g(\overline{\mathbf{x}}^{(1,2)})=a_{(1,2)}\overline{\mathbf{x}}^{(1,2)}.

However, these three coefficients $a_{(1,0)}$ , $a_{(1,1)}$ , and $a_{(1,2)}$ are algebraically independent since the relation $a_{(1,0)}\cdot a_{(1,2)}=a^{2}_{(1,1)}$ does not hold in the quotient. This implies that there is a 3-dimensional torus acting faithfully on $\mathbf{k}[S]/I$ that is an extension of $T$ . In particular, the centralizer of $T$ is larger than $T$ .

Furthermore, a straightforward verification shows that $\mathbf{k}[S]/I$ is isomorphic to $\mathbf{k}[S^{\prime}]/I^{\prime}$ with $S^{\prime}$ being the first octant in $M^{\prime}=\mathbb{Z}^{3}$ and $I^{\prime}=(\mathbf{x}^{(2,0,0)},\mathbf{x}^{(0,2,0)},\mathbf{x}^{(0,0,2)},\mathbf{x}^{(1,1,0)},\mathbf{x}^{(1,0,1)},\mathbf{x}^{(0,1,1)})$ . Now, Lemma 4.5 shows that a maximal torus of $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S^{\prime}]/I^{\prime})\simeq\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ is indeed 3-dimensional.

Definition 4.7 (See [RvS21, Definition 7.1]).

Let $G$ be a linear algebraic group, and let $T\subset G$ be a maximal torus. An algebraic subgroup $U\subset G$ of dimension $m$ , isomorphic to $(\mathbb{G}_{a})^{m}$ , is called a generalized root subgroup with respect to $T$ if there exists a character $\mathbf{x}^{\alpha}$ of $T$ , referred to as the weight of $U$ , such that

t\circ\varepsilon(s)\circ t^{-1}=\varepsilon(\mathbf{x}^{\alpha}(t)\cdot s)\quad\mbox{for all }t\in T\mbox{ and all }s\in(\mathbb{G}_{a})^{m}\,,

where $\varepsilon\colon(\mathbb{G}_{a})^{m}\to U$ is a fixed isomorphism. The weight of a generalized root subgroup $U$ does not depend on the choice of the isomorphism $\varepsilon$ . A generalized root subgroup is said to be maximal if it is maximal by inclusion among generalized root subgroups.

With this definition, we arrive at the following well-known result. In lack of a reference, we provide a proof.

Proposition 4.8.

Let $G$ be a connected linear algebraic group, and let $T$ be a maximal torus of $G$ . Then $G$ is generated as a group by $T$ and all maximal generalized root subgroups.

Proof.

Since the characteristic of $\mathbf{k}$ is zero, there exists a Levi subgroup $L\subset G$ , i.e., a reductive subgroup such that $G=L\ltimes R$ , where $R$ is the unipotent radical of $G$ [Bor91, Definition 11.22]. Furthermore, we can assume that $T\subset L$ . Moreover, all generalized root subgroups of $L$ with respect to $T$ are one-dimensional, and together with $T$ , they generate $L$ [Spr09, Proposition 8.1.1.]. Now, let $g\in R$ . Since $G=L\ltimes R$ and $T\subset L$ , we have that $R$ is a direct product of root subgroups $R_{1},\dots,R_{k}$ with respect to $T$ with some weight $\mathbf{x}^{\alpha_{i}}$ , $i=1,\dots,k$ . Thus, each $R_{i}$ belongs to the maximal generalized root subgroup corresponding to $\mathbf{x}^{\alpha_{i}}$ . This completes the proof of the proposition. ∎

In the next lemma, we show that elements in generalized root subgroups correspond to homogeneous locally nilpotent derivations.

Lemma 4.9.

Let $S$ be the first octant, and let $I$ be a full monomial ideal with cofinite support. Suppose $g\in\operatorname{Aut}^{0}_{\mathbf{k}}(\mathbf{k}[S]/I)$ and that $g$ is contained in a generalized root subgroup with weight $\mathbf{x}^{\alpha}$ with respect to $T$ . Then $g=\exp\overline{\partial}$ , where $\overline{\partial}$ is a locally nilpotent derivation of degree $\alpha$ .

Proof.

Since $g$ is contained in a generalized root subgroup of $\operatorname{Aut}^{0}_{\mathbf{k}}(\mathbf{k}[S]/I)$ , $g$ is unipotent, so the smallest algebraic subgroup $U$ containing $g$ is a root subgroup isomorphic to $\mathbb{G}_{\mathrm{a}}$ with weight $\mathbf{x}^{\alpha}$ . The injection $U\hookrightarrow\operatorname{Aut}^{0}_{\mathbf{k}}(\mathbf{k}[S]/I)$ induces a $\mathbb{G}_{\mathrm{a}}$ -action on $\mathbf{k}[S]/I$ . By (1), we have $g=\exp(s\overline{\partial})$ for some locally nilpotent derivation $\overline{\partial}\colon\mathbf{k}[S]/I\to\mathbf{k}[S]/I$ and some $s\in\mathbb{G}_{\mathrm{a}}$ . Since $s\overline{\partial}$ is again locally nilpotent, we can assume $g=\exp\overline{\partial}$ by replacing $\overline{\partial}$ with $s\overline{\partial}$ . According to Definition 4.7,

t\circ\exp(s\overline{\partial})\circ t^{-1}=\exp(\mathbf{x}^{\alpha}(t)s\overline{\partial})\quad\text{for all}\quad t\in T\text{ and all }s\in\mathbb{G}_{a}.

Since conjugation commutes with the exponential, we obtain

\displaystyle t\circ\overline{\partial}\circ t^{-1}=\mathbf{x}^{\alpha}(t)\cdot\overline{\partial}\quad\text{for all}\quad t\in T.

(11)

Now, letting $m^{\prime}\in S\setminus\operatorname{supp}(I)$ , we write

\partial(\overline{\mathbf{x}}^{m^{\prime}})=\sum_{m\in S\setminus\operatorname{supp}(I)}a_{m}\overline{\mathbf{x}}^{m}.

Applying (11) to $\overline{\mathbf{x}}^{m^{\prime}}$ , we get

t^{-1}(m^{\prime})\sum_{m\in S\setminus\operatorname{supp}(I)}t(m)\cdot a_{m}\overline{\mathbf{x}}^{m}=t(\alpha)\sum_{m\in S\setminus\operatorname{supp}(I)}a_{m}\overline{\mathbf{x}}^{m}.

This equality can only hold if $a_{m}=0$ for all but possibly one value of $m$ . This implies that $\partial(\overline{\mathbf{x}}^{m^{\prime}})=a_{m}\overline{\mathbf{x}}^{m}$ , and thus $\overline{\partial}$ is homogeneous. ∎

Remark 4.10.

Non-trivial locally nilpotent derivations on $\mathbf{k}[S]/I$ were classified in Theorem 3.7 and Lemma 3.10. Note that in the case where $I$ has cofinite support, only conditions $(i)$ and $(ii)$ can hold in Theorem 3.7 since $(\mathbb{Z}_{\geq 0}\cdot\alpha)\cap\operatorname{supp}(I)$ is never empty for $\alpha\in S$ .

Let $\mathcal{R}(S,I)$ denote the subset of $M$ consisting of $\alpha$ for which there exists a non-zero homogeneous locally nilpotent derivation $\overline{\partial}_{\alpha,p}$ . According to Lemma 4.9, these are precisely the weights of the non-trivial generalized root subgroups of $\operatorname{Aut}^{0}_{\mathbf{k}}(\mathbf{k}[S]/I)$ . Moreover, for each $\alpha\in\mathcal{R}(S,I)$ , we define

G(\alpha)=\{p\in N_{\mathbf{k}}\mid\overline{\partial}_{\alpha,p}\text{ is locally nilpotent}\}.

Next, we define

\displaystyle\iota_{\alpha}\colon G(\alpha)\to\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)\quad\text{by}\quad p\mapsto\exp\overline{\partial}_{\alpha,p}\,.

(12)

As shown in Lemma 4.9, the image of each $\iota_{\alpha}$ is the maximal generalized root subgroup associated with the weight $\mathbf{x}^{\alpha}$ . Since $\exp\overline{\partial}=\operatorname{id}$ if and only if $\overline{\partial}=0$ , it follows that $\iota_{\alpha}$ is injective, implying that the maximal generalized root subgroup corresponding to the weight $\mathbf{x}^{\alpha}$ is isomorphic to the group structure of the vector space $G(\alpha)$ .

Now, we can present our main result regarding the connected component of $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ , which directly follows from Lemma 4.9 and the aforementioned considerations.

Proposition 4.11.

Let $S$ be the first octant and $I$ be a full monomial ideal with cofinite support. Also, let $T=\operatorname{Spec}\mathbf{k}[M]\subset\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ . Then, the neutral component $\operatorname{Aut}^{0}_{\mathbf{k}}(\mathbf{k}[S]/I)$ is spanned by $T$ and the images of $\iota_{\alpha}$ for all $\alpha\in\mathcal{R}(S,I)$ .

Remark 4.12.

When $X$ is a complete toric variety, all the generalized root subgroups of $\operatorname{Aut}^{0}(X)$ are 1-dimensional [Dem70] (see also [LLA22, Theorem 2 $(i)$ ]). However, this property does not hold for the automorphism group of monomial algebras, as illustrated by the following example.

Example 4.13.

Let $M=\mathbb{Z}^{2}$ and $S=\mathbb{Z}^{2}_{\geq 0}$ . Consider the ideal $I=(y^{2},x^{2}y,x^{3})$ . By Theorem 3.7 and Lemma 3.10, we have that $\alpha=(1,0)\in\mathcal{R}(S,I)$ and $G(\alpha)=N_{\mathbf{k}}$ . Furthermore, every $p=(p_{1},p_{2})\in G(\alpha)$ defines the following derivation and automorphisms, respectively.

\begin{array}[]{ccc}\begin{array}[]{rcl}\overline{\partial}_{p,\alpha_{1}}\colon\mathbf{k}[S]/I&\to&\mathbf{k}[S]/I\\ \overline{x}&\mapsto&p_{1}\overline{x}^{2}\\ \overline{y}&\mapsto&p_{2}\overline{x}\overline{y}\\ \end{array}&\quad\mbox{so that}&\begin{array}[]{rcl}\operatorname{exp}(\overline{\partial}_{p,\alpha_{1}})\colon\mathbf{k}[S]/I&\to&\mathbf{k}[S]/I\\ \overline{x}&\mapsto&\overline{x}+p_{1}\overline{x}^{2}\\ \overline{y}&\mapsto&\overline{y}+p_{2}\overline{x}\overline{y}\\ \end{array}\\ \end{array}\,.

Since $\iota_{\alpha}$ is injective, the maximal generalized root subgroup with weight $\mathbf{x}^{\alpha}$ is 2-dimensional.

Definition 4.14.

Let $S$ and $S^{\prime}$ be semigroups and let $I\subset S$ and $I^{\prime}\subset S^{\prime}$ be monomial ideals. A $\mathbf{k}$ -algebra homomorphism $g\colon\mathbf{k}[S]/I\to\mathbf{k}[S^{\prime}]/I^{\prime}$ is called a toric morphism if there exists a semigroup homomorphism $\widehat{g}\colon S\to S^{\prime}$ with $\widehat{g}(\operatorname{supp}(I))\subset\operatorname{supp}(I^{\prime})$ such that $g(\overline{\mathbf{x}}^{m})=\overline{\mathbf{x}}^{\widehat{g}(m)}$ . A toric automorphism of $\mathbf{k}[S]/I$ is a toric endomorphism that admits an inverse which is also toric. We denote the set of toric automorphisms of $\mathbf{k}[S]/I$ by $\operatorname{Aut}(S,I)$ . An automorphism $g\in\operatorname{Aut}(S,I)$ corresponds to an automorphism $\widehat{g}\colon S\to S$ such that $\widehat{g}(\operatorname{supp}(I))=\operatorname{supp}(I)$ .

Lemma 4.15.

The group $\operatorname{Aut}(S,I)$ is finite.

Proof.

Let $S$ be a semigroup. Consider $g\in\operatorname{Aut}(S,I)$ , which corresponds to the semigroup automorphism $\widehat{g}\colon S\to S$ . The map $\widehat{g}$ induces a permutation of the set of rays $S(1)$ . Consequently, we obtain a homomorphism from $\operatorname{Aut}(S,I)$ to the symmetric group on $S(1)$ . Furthermore, since $S(1)$ spans $M_{\mathbb{R}}$ as a vector space, the action of $\widehat{g}$ on $S(1)$ uniquely determines $\widehat{g}$ , implying that the homomorphism is injective. ∎

Proposition 4.16.

If $I$ is a full monomial ideal with cofinite support, then $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)/\operatorname{Aut}_{\mathbf{k}}^{0}(\mathbf{k}[S]/I)$ is a finite group generated by the images of toric morphisms.

Proof.

Since $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ is a linear algebraic group by Proposition 4.1, it follows that the quotient group $F=\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)/\operatorname{Aut}_{\mathbf{k}}^{0}(\mathbf{k}[S]/I)$ is finite. Let $g\in\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ and let $[g]$ be its image in $F$ . As all maximal tori are conjugate in a linear algebraic group, there exists $h\in\operatorname{Aut}_{\mathbf{k}}^{0}(\mathbf{k}[S]/I)$ such that

(g\circ h)^{-1}\circ T\circ(g\circ h)=h^{-1}\circ(g^{-1}\circ T\circ g)\circ h=T\,.

Since $[g]=[g\circ h]$ , we can replace $g$ with $g\circ h$ , assuming that $g$ normalizes $T$ .

Now, for $m^{\prime}\in S\setminus\operatorname{supp}(I)$ , we have

g(\overline{\mathbf{x}}^{m^{\prime}})=\sum_{m\in S\setminus\operatorname{supp}(I)}a_{m}\overline{\mathbf{x}}^{m}\,.

The set $V=\mathbf{k}\cdot\overline{\mathbf{x}}^{m^{\prime}}\subset\mathbf{k}[S]/I$ is a 1-dimensional vector subspace of $\mathbf{k}[S]/I$ . Applying $g\circ T=T\circ g$ to $V$ , we obtain

\displaystyle\mathbf{k}\cdot\sum_{m\in S\setminus\operatorname{supp}(I)}a_{m}\overline{\mathbf{x}}^{m}=\mathbf{k}\cdot\left\{\sum_{m\in S\setminus\operatorname{supp}(I)}t(m)a_{m}\overline{\mathbf{x}}^{m}\mid t\in T\right\}\,.

The right-hand side is a $\mathbf{k}$ -vector space, while the left-hand side is a $\mathbf{k}$ -vector space if and only if $a_{m}=0$ for all but at most one $m\in S\setminus\operatorname{supp}(I)$ . This implies that $g(\overline{\mathbf{x}}^{m^{\prime}})=a_{m}\overline{\mathbf{x}}^{m}$ for some $m\in S\setminus\operatorname{supp}(I)$ . Replacing $g$ by $g\circ t$ with an appropriate choice of $t\in T$ , we obtain $[g]=[g\circ t]$ and $g(\overline{\mathbf{x}}^{m^{\prime}})=\overline{\mathbf{x}}^{m}$ . Finally, since $I$ is full, the basis vectors $e_{i}\in S=\mathbb{Z}_{\geq 0}^{n}$ are not in the support of $I$ . Hence, we can define a unique homomorphism $\widehat{g}\colon S\to S$ given by $\widehat{g}(e_{i})=m_{i}$ where $g(\overline{\mathbf{x}}^{e_{i}})=\overline{\mathbf{x}}^{m_{i}}$ . Since $g$ is an automorphism, $\widehat{g}$ is also an automorphism, and thus $g$ is a toric automorphism. ∎

We can now present our main theorem, which summarizes all the results in this section and provides a description of the automorphism group of certain monomial algebras. Recall that the map $\iota_{\alpha}$ is defined in (12).

Theorem 4.17.

Let $S$ be the first octant, and let $I$ be a full monomial ideal with cofinite support. Also, let $T=\operatorname{Spec}\mathbf{k}[M]\subset\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ .

$(i)$

The automorphism group $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ is linear algebraic, and $T$ is a maximal torus of $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ .
$(ii)$

The neutral component $\operatorname{Aut}^{0}_{\mathbf{k}}(\mathbf{k}[S]/I)$ is generated by $T$ and the image of $\iota_{\alpha}$ for all $\alpha\in\mathcal{R}(S,I)$ .
$(iii)$

The quotient group $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)/\operatorname{Aut}^{0}_{\mathbf{k}}(\mathbf{k}[S]/I)$ is generated by the images of toric automorphisms.

Proof.

Statement $(i)$ is contained in Proposition 4.1 and Lemma 4.5. Statement $(ii)$ is in Proposition 4.11. Finally, statement $(iii)$ is in Proposition 4.16. ∎

We finish the paper providing several examples illustrating our results.

Example 4.18.

Letting $M=\mathbb{Z}$ and $S=\mathbb{Z}_{\geq 0}$ , we let $I=(x^{4})$ . The group of toric automorphisms is trivial so

\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)=\operatorname{Aut}^{0}_{\mathbf{k}}(\mathbf{k}[S]/I)=\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[x]/(x^{4}))\,.

By Theorem 3.7 and Lemma 3.10, we have that only $\alpha_{1}=1$ and $\alpha_{2}=2$ give rise to non-trivial homogeneous locally nilpotent derivations different from the zero derivation on $\mathbf{k}[S]/I$ . For $p\in N_{\mathbf{k}}=G(\alpha_{i})=\mathbf{k}$ , for $i=1,2$ , we obtain the derivations

\begin{array}[]{ccc}\begin{array}[]{rcl}\overline{\partial}_{p,\alpha_{1}}\colon\mathbf{k}[S]/I&\to&\mathbf{k}[S]/I\\ \overline{x}&\mapsto&p\overline{x}^{2}\\ \end{array}&\quad\mbox{and}&\begin{array}[]{rcl}\overline{\partial}_{p,\alpha_{2}}\colon\mathbf{k}[S]/I&\to&\mathbf{k}[S]/I\\ \overline{x}&\mapsto&p\overline{x}^{3}\\ \end{array}\\ \end{array}\,.

Applying the exponential map, we obtain

\begin{array}[]{ccc}\begin{array}[]{rcl}\exp(\overline{\partial}_{p,\alpha_{1}})\colon\mathbf{k}[S]/I&\to&\mathbf{k}[S]/I\\ \overline{x}&\mapsto&\overline{x}+p\overline{x}^{2}+p^{2}\overline{x}^{3}\\ \end{array}&\quad\mbox{and}&\begin{array}[]{rcl}\exp(\overline{\partial}_{p,\alpha_{2}})\colon\mathbf{k}[S]/I&\to&\mathbf{k}[S]/I\\ \overline{x}&\mapsto&\overline{x}+p\overline{x}^{3}\\ \end{array}\\ \end{array}\,.

Letting $e_{1}=1,e_{2}=\overline{x},e_{3}=\overline{x}^{2},e_{4}=\overline{x}^{3}$ we have that $\mathbf{k}[S]/I=\mathbf{k}\cdot e_{1}\oplus\mathbf{k}\cdot e_{2}\oplus\mathbf{k}\cdot e_{3}\oplus\mathbf{k}\cdot e_{4}$ . Moreover, $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)\subset\operatorname{GL}_{4}(\mathbf{k})$ by the proof of Proposition 4.1. And by Theorem 4.17, the automorphism group $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ is generated by the matrices

\begin{array}[]{cccc}\begin{pmatrix}1&0&0&0\\ 0&t&0&0\\ 0&0&t^{2}&0\\ 0&0&0&t^{3}\\ \end{pmatrix},&\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&r&1&0\\ 0&r^{2}&2r&1\\ \end{pmatrix},&\mbox{and}&\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&s&0&1\\ \end{pmatrix},\end{array}

with $t\in\mathbb{G}_{\mathrm{m}}$ , $r\in\mathbb{G}_{\mathrm{a}}$ and $s\in\mathbb{G}_{\mathrm{a}}$ .

Example 4.19.

Letting $M=\mathbb{Z}^{2}$ and $S=\mathbb{Z}_{\geq 0}^{2}$ we let $I=(x^{2},y^{2})$ . The group of toric automorphisms is $F\simeq\mathbb{Z}/2\mathbb{Z}$ with generator given by the permutation $\overline{x}\mapsto\overline{y}$ and $\overline{y}\to\overline{x}$ . Hence, $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)$ is generated by $\operatorname{Aut}_{\mathbf{k}}^{0}(\mathbf{k}[S]/I)$ and $F$ .

Figure 10.

I=(x^{2},y^{2})

By Theorem 3.7 and Lemma 3.10, we have that only $\alpha_{1}=(1,0)$ and $\alpha_{2}=(0,1)$ give rise to non-trivial homogeneous locally nilpotent derivations with $G(\alpha_{1})=\mathbf{k}\cdot(0,1)$ and $G(\alpha_{2})=\mathbf{k}\cdot(1,0)$ . Letting $p=(p_{1},p_{2})\in N_{\mathbf{k}}$ , this yields

\begin{array}[]{cc}\begin{array}[]{rcl}\overline{\partial}_{p,\alpha_{1}}\colon\mathbf{k}[x,y]/I&\to&\mathbf{k}[x,y]/I\\ \overline{x}&\mapsto&0\\ \overline{y}&\mapsto&p_{2}\overline{x}\overline{y}\\ \end{array}\par\quad\mbox{and}&\begin{array}[]{rcl}\overline{\partial}_{p,\alpha_{2}}\colon\mathbf{k}[x,y]/I&\to&\mathbf{k}[x,y]/I\\ \overline{x}&\mapsto&p_{1}\overline{x}\overline{y}\\ \overline{y}&\mapsto&0\\ \end{array}\\ \end{array}\,.

Applying the exponential map, we obtain

\begin{array}[]{cc}\begin{array}[]{rcl}\exp(\overline{\partial}_{p,\alpha_{1}})\colon\mathbf{k}[x,y]/I&\to&\mathbf{k}[x,y]/I\\ \overline{x}&\mapsto&\overline{x}\\ \overline{y}&\mapsto&\overline{y}+p_{2}\overline{x}\overline{y}\\ \end{array}\par\quad\mbox{and}&\begin{array}[]{rcl}\exp(\overline{\partial}_{p,\alpha_{2}})\colon\mathbf{k}[x,y]/I&\to&\mathbf{k}[x,y]/I\\ \overline{x}&\mapsto&\overline{x}+p_{1}\overline{x}\overline{y}\\ \overline{y}&\mapsto&\overline{y}\\ \end{array}\\ \end{array}\,.

Letting $e_{1}=1,e_{2}=\overline{y},e_{3}=\overline{x},e_{4}=\overline{x}\overline{y}$ we have that $\mathbf{k}[S]/I=\mathbf{k}\cdot e_{1}\oplus\mathbf{k}\cdot e_{2}\oplus\mathbf{k}\cdot e_{3}\oplus\mathbf{k}\cdot e_{4}$ . Moreover $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[x,y]/I)\subset\operatorname{GL}_{4}(\mathbf{k})$ by the proof of Proposition 4.1. And by Theorem 4.17, $\operatorname{Aut}_{\mathbf{k}}^{0}(\mathbf{k}[S]/I)$ is generated by the matrices

\begin{array}[]{ccc}\begin{pmatrix}1&0&0&0\\ 0&t_{2}&0&0\\ 0&0&t_{1}&0\\ 0&0&0&t_{1}t_{2}\\ \end{pmatrix},&\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&r&0&1\\ \end{pmatrix},\quad\mbox{and}&\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&s&1\\ \end{pmatrix},\end{array}

with $(t_{1},t_{2})\in T=\mathbb{G}_{\mathrm{m}}^{2}$ , $r\in\mathbb{G}_{\mathrm{a}}$ and $s\in\mathbb{G}_{\mathrm{a}}$ .

Finally, all matrices are lower triangular, so the generator of $F$ is not contained in $\operatorname{Aut}_{\mathbf{k}}^{0}(\mathbf{k}[S]/I)$ . We conclude that

\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)\simeq\operatorname{Aut}_{\mathbf{k}}^{0}(\mathbf{k}[S]/I)\ltimes F\,.

Example 4.20.

Letting $M=\mathbb{Z}^{2}$ and $S=\mathbb{Z}_{\geq 0}^{2}$ we let $I=(x^{2},xy,y^{2})$ .

Figure 11.

I=(x^{2},xy,y^{2})

The group of toric automorphisms $F$ is again isomorphic to $\mathbb{Z}/2\mathbb{Z}$ with generator given by the permutation $\overline{x}\mapsto\overline{y}$ and $\overline{y}\mapsto\overline{x}$ . In this case, as we will show by the end of this example that $F\subset\operatorname{Aut}_{\mathbf{k}}^{0}(\mathbf{k}[S]/I)$ . Hence,

\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[S]/I)=\operatorname{Aut}_{\mathbf{k}}^{0}(\mathbf{k}[S]/I).

By Theorem 3.7 and Lemma 3.10, we have that only $\alpha_{1}=(1,-1)$ and $\alpha_{2}=(-1,1)$ give rise to non-trivial homogeneous locally nilpotent derivations with $G(\alpha_{1})=\mathbf{k}\cdot(0,1)$ and $G(\alpha_{2})=\mathbf{k}\cdot(1,0)$ . Letting $p=(p_{1},p_{2})\in N_{\mathbf{k}}$ , this yield

\begin{array}[]{cc}\begin{array}[]{rcl}\overline{\partial}_{p,\alpha_{1}}\colon\mathbf{k}[x,y]/I&\to&\mathbf{k}[x,y]/I\\ \overline{x}&\mapsto&0\\ \overline{y}&\mapsto&p_{2}\overline{x}\\ \end{array}\par\quad\mbox{and}&\begin{array}[]{rcl}\overline{\partial}_{p,\alpha_{2}}\colon\mathbf{k}[x,y]/I&\to&\mathbf{k}[x,y]/I\\ \overline{x}&\mapsto&p_{1}\overline{y}\\ \overline{y}&\mapsto&0\\ \end{array}\\ \end{array}\,.

Applying the exponential map, we obtain:

\begin{array}[]{cc}\begin{array}[]{rcl}\exp(\overline{\partial}_{p,\alpha_{1}})\colon\mathbf{k}[x,y]/I&\to&\mathbf{k}[x,y]/I\\ \overline{x}&\mapsto&\overline{x}\\ \overline{y}&\mapsto&\overline{y}+p_{2}\overline{x}\\ \end{array}\par\quad\mbox{and}&\begin{array}[]{rcl}\exp(\overline{\partial}_{p,\alpha_{2}})\colon\mathbf{k}[x,y]/I&\to&\mathbf{k}[x,y]/I\\ \overline{x}&\mapsto&\overline{x}+p_{1}\overline{y}\\ \overline{y}&\mapsto&\overline{y}\\ \end{array}\\ \end{array}\,.

Letting now $e_{1}=1,e_{2}=\overline{y}$ and $e_{3}=\overline{x}$ we have that $\mathbf{k}[S]/I=\mathbf{k}\cdot e_{1}\oplus\mathbf{k}\cdot e_{2}\oplus\mathbf{k}\cdot e_{3}$ . Moreover $\operatorname{Aut}_{\mathbf{k}}(\mathbf{k}[x,y]/I)\subset\operatorname{GL}_{3}(\mathbf{k})$ by the proof of Proposition 4.1. And by Theorem 4.17, $\operatorname{Aut}_{\mathbf{k}}^{0}(\mathbf{k}[S]/I)$ is generated by the matrices

\begin{array}[]{ccc}\begin{pmatrix}1&0&0\\ 0&t_{2}&0\\ 0&0&t_{1}\\ \end{pmatrix},&\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&r&1\\ \end{pmatrix},\quad\mbox{and}&\begin{pmatrix}1&0&0\\ 0&1&s\\ 0&0&1\\ \end{pmatrix},\end{array}

with $(t_{1},t_{2})\in T=\mathbb{G}_{\mathrm{m}}^{2}$ , $r\in\mathbb{G}_{\mathrm{a}}$ and $s\in\mathbb{G}_{\mathrm{a}}$ . The group $\operatorname{Aut}_{\mathbf{k}}^{0}(\mathbf{k}[S]/I)$ is isomorphic to $\operatorname{GL}_{2}(\mathbf{k})$ taking the lower right $2\times 2$ blocks of the above matrices, c.f. [MS05, Corollary 2.2].

In this case, $F\subset\operatorname{Aut}_{\mathbf{k}}^{0}(\mathbf{k}[S]/I)$ since the permutation $\overline{x}\mapsto\overline{y}$ , $\overline{y}\mapsto\overline{x}$ is obtained as $\varphi^{-1}\circ\psi\circ\varphi$ where $\varphi,\psi\in\operatorname{Aut}_{\mathbf{k}}^{0}(\mathbf{k}[S]/I)$ are defined via $\varphi(\overline{x},\overline{y})=(\overline{x},\overline{x}-\overline{y})$ and $\psi(\overline{x},\overline{y})=(\overline{x}-\overline{y},\overline{y})$ . This phenomenon where some toric morphisms are contained in $\operatorname{Aut}_{\mathbf{k}}^{0}(\mathbf{k}[S]/I)$ can only arise when there exist non-trivial root subgroups with weights $\alpha$ and $-\alpha$ .

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