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+compile_latex:
+  image: texlive/texlive:latest
+
+  script:
+    - latexmk -pdf -interaction=nonstopmode -halt-on-error ncr-of-monomial-curves.tex
+
+  artifacts:
+    paths:
+      - ncr-of-monomial-curves.pdf

ncr-of-monomial-curves.tex

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+
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+%\usepackage{parskip}
+
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+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\Spec}{Spec}
+\DeclareMathOperator{\Frac}{Frac}
+\DeclareMathOperator{\spn}{span}
+\DeclareMathOperator{\Min}{Min}
+\DeclareMathOperator{\hgt}{ht}
+\DeclareMathOperator{\depth}{depth}
+\DeclareMathOperator{\gldim}{gl.dim}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\id}{id}
+\DeclareMathOperator{\gp}{gp}
+\newcommand{\qpower}[2][q]{{}^{#1}#2}
+\newcommand{\qmult}[1][q]{\mathbin{\star}_#1}%\mathbin{\vcenter{\hbox{\scalebox{0.8}{$\circledast$}}}}_#1
+\newcommand{\gapset}{\NN_0\setminus\SGR}
+\newcommand{\kk}{\mathsf{k}}
+\newcommand{\SGR}{\mathsf{S}}
+\newcommand{\NN}{\mathbb{N}}
+\newcommand{\ZZ}{\mathbb{Z}}
+\newcommand{\QQ}{\mathbb{Q}}
+\newcommand{\A}{\mathbb{A}}
+\newcommand{\shcong}{\cong_{\mathrm{sh}}}
+\newcommand{\grcong}{\cong_{\mathrm{gr}}}
+\newcommand{\intval}[1]{\llbracket #1 \rrbracket}
+
+\title{Noncommutative Resolutions of Monomial Curves}
+\author{Markus M. R. Tripp}
+\date{}
+
+\begin{document}
+\maketitle
+
+\begin{abstract}
+\end{abstract}
+
+\section{Introduction}
+
+% some results/terminology in section 2 are made independent of the characteristic
+
+\section{Preliminaries}
+
+We write $\NN = \{1, 2, 3, \dots \}$ for the set of natural numbers. For $a$, $b \in \ZZ$, we denote by $\intval{a,b}$ the set of all integers from $a$ to $b$, including both endpoints.
+
+Throughout, we fix an arbitrary ground field $\kk$. We begin by recalling some basic facts about toric algebras, focusing in particular on the one-dimensional case. See~\cite{Cox2011} for further details.
+
+Let $\SGR$ be an affine semigroup, that is, a finitely generated submonoid of $\ZZ^d$ for some $d \in \NN$. The \textbf{toric algebra} associated with $\SGR$ is the semigroup algebra
+\[ R \coloneqq \kk[\SGR] = \spn_\kk \{ x^a \mid a \in \SGR \}. \]
+The algebra $R$ is a finitely generated $\kk$-subalgebra of the Laurent polynomial ring~$\kk[x_1^{\pm 1}, \dots, x_d^{\pm 1}]$. In particular, it is a Noetherian domain and inherits the natural $\ZZ^d$-grading. Its Krull dimension is equal to the rank of $\gp(\SGR)$, where $\gp(\SGR)$ denotes the subgroup of $\ZZ^d$ generated by $\SGR$~\cite[Prop. 1.1.14]{Cox2011}. Consequently, $R$ is one-dimensional if and only if $\gp(\SGR) \cong \ZZ$, in which case we may assume without loss of generality that $\gp(\SGR) = \ZZ$. Every submonoid of $\ZZ$ is either a group, a submonoid of $\NN_0$, or a submonoid of $-\NN_0$~\cite[§ 2]{Geroldinger2025}. Hence, when $R$ is one-dimensional, $\SGR$ is, up to isomorphism of monoids, either $\ZZ$ or a submonoid of $\NN_0$. Accordingly, either $R\cong\kk[x^{\pm 1}]$ or $R \cong \kk[x^{a_1},\dots,x^{a_n}]$ for some $a_1$, \dots, $a_n \in \NN_0$. 
+
+When $\kk$ is algebraically closed, the algebra $R=\kk[\SGR]$ is the coordinate ring of the associated affine toric variety $X = \Spec(R)$. In dimension one, the corresponding affine toric variety is either the one-dimensional torus $\Spec(\kk[x^{\pm1}]) = \A^1_\kk \setminus \{ 0 \}$, or an \textbf{affine monomial curve} $\Spec(\kk[x^{a_1},\dots,x^{a_n}])$, parametrized by
+\begin{equation}\label{eq:para-mon-curve}
+ \varphi \colon \A^1_\kk \to \A^n_\kk, \qquad t \longmapsto (t^{a_1},\dots,t^{a_n}).
+\end{equation}
+
+\subsection{Numerical monoids} 
+
+Let $\SGR$ be a submonoid of $\NN_0$. Writing $\SGR = \gcd(\SGR) \cdot \SGR^\prime$, where $\SGR^\prime$ is a submonoid of $\NN_0$ with $\gcd(\SGR^\prime) = 1$, yields an isomorphism of monoids between $\SGR$ and $\SGR^\prime$. Thus, it suffices to consider submonoids of $\NN_0$ with greatest common divisor $1$, called \textbf{numerical monoids}. Crucially, the gap set $\gapset$ of a numerical monoid $\SGR$ is finite. For further background on numerical monoids, see~\cite{Rosales2009}. 
+
+For a numerical monoid $\SGR=\langle a_1,\dots, a_n\rangle \subseteq \NN_0$, the parametrization $\varphi$ from~\eqref{eq:para-mon-curve} of the associated affine monomial curve $X = \Spec(R)$ restricts to an isomorphism of varieties $\A_\kk^1 \setminus \{ 0 \} \xrightarrow{\sim} X \setminus \{ 0 \}$. Hence $X$ is nonsingular away from the origin. At the origin, it is singular unless $\SGR = \NN_0$, in which case $R=\kk[x]$ and $X=\A_\kk^1$.
+
+\subsection{Normality, seminormality, and weak normality}
+
+A domain $\Lambda$ is called \textbf{normal} if it is integrally closed in its field o.f fractions $\Frac(\Lambda)$, and \textbf{seminormal} if, for every $\lambda \in \Frac(\Lambda)$, $\lambda^2 \in \Lambda$ and $\lambda^3 \in \Lambda$ imply that $\lambda \in \Lambda$. The normalization~$\overline{\Lambda}$ and the seminormalization ${}^+\Lambda$ of $\Lambda$ are, respectively, the intersections of all intermediate normal and seminormal rings $\Gamma$ satisfying $\Lambda \subseteq \Gamma \subseteq \Frac(\Lambda)$ If $\Lambda$ has prime characteristic $p$, then $\Lambda$ is called \textbf{weakly normal} if, for every $\lambda \in \Frac(\Lambda)$, the condition $\lambda^p \in \Lambda$ implies $\lambda\in\Lambda$. Clearly, normality implies seminormality. In prime characteristic $p$, normality implies weak normality, and weak normality in turn implies seminormality. Indeed, suppose that $\Lambda$ is weakly normal and let $\lambda \in \Frac(\Lambda)$ satisfy $\lambda^2$, $\lambda^3 \in \Lambda$. Write $p = 2k + 3\ell$ for some $k$, $\ell \in \NN_0$. Then
+\[
+\lambda^p = (\lambda^2)^k(\lambda^3)^\ell \in \Lambda,
+\]
+and hence $\lambda \in \Lambda$.
+
+For a toric algebra $R = \kk[\SGR]$, the field of fractions in the definitions above may be replaced by $\kk[\gp(\SGR)]$. Indeed, $\gp(\SGR) \cong \ZZ^{d^\prime}$ for some $d^\prime \in \NN_0$, so $\kk[\gp(\SGR)]$ is isomorphic to a Laurent polynomial ring and is therefore normal. Consequently, every element of $\Frac(R)$ that is integral over $R$ already belongs to $\kk[\gp(\SGR)]$.\todo{mention normality, ... for affine monoids?}
+
+In the case of a numerical monoid $\SGR$, the corresponding toric algebra $R$ satisfies ${}^+R=\kk[x]$. Indeed, let $n_1 > \cdots > n_k$ be the elements of $\gapset$. Since $n_1$ is the largest gap of $\SGR$, we have $(x^{n_1})^2$, $(x^{n_1})^3 \in R \subseteq{}^+R$, and hence $x^{n_1}\in{}^+R$ by seminormality.
+%Since $n_1$ is the largest gap of $\SGR$, both $2n_1$ and $3n_1$ belong to $\SGR$. Hence $(x^{n_1})^2$, $(x^{n_1})^3 \in R \subseteq{}^+R$, and seminormality gives $x^{n_1}\in{}^+R$. 
+Repeating this argument successively for the remaining gaps shows that $x^{n_1}$, \dots, $x^{n_k} \in{}^+R$, and therefore $\kk[x] \subseteq{}^+R$. Since $\kk[x]$ is normal, hence seminormal, the reverse inclusion follows. As $\kk[x]$ is normal, it also follows that $\overline{R} = \kk[x]$, and consequently $\Frac(R) = \kk(x)$. Thus, unless $\SGR=\NN_0$, the ring $R$ is neither normal nor seminormal.
+
+\subsection{Graded Krull–Schmidt theorem}  
+
+The category of finitely generated $\QQ^d$-graded $R$-modules satisfies the \textbf{Krull--Schmidt property}. That is, in the graded category, every such module decomposes as a finite direct sum of indecomposable modules, uniquely up to graded isomorphism and permutation of the summands~\cite[§ 2]{Shibuta2017}.
+
+We are interested in the summands of such graded decompositions only up to degree shifts, and therefore introduce the following notation. Let $M$, $N$, $M_1$, \dots,~$M_k$ be $\QQ^d$-graded $R$-modules. We write $M \grcong N$ if $M$ and $N$ are isomorphic as $\QQ^d$-graded $R$-modules. Moreover, we write
+\[ M \shcong M_1 \oplus \cdots \oplus M_k \]
+if there exist $d_i \in \QQ^d$ such that $M\grcong M_1(d_1)\oplus\cdots\oplus M_k(d_k)$, where $M_i(d_i)$ denotes the degree shift of $M_i$ by $d_i$. The notation is compatible with refinements of decompositions. If $M \shcong M_1 \oplus \cdots \oplus M_k$ and $M_1 \shcong M_1^\prime \oplus \cdots \oplus M_\ell^\prime$, then $M \shcong M_1^\prime \oplus \cdots \oplus M_\ell^\prime \oplus M_2 \oplus \cdots \oplus M_k$.
+
+\section{Frobenius maps on toric algebras}
+Let $R = \kk[\SGR]$ be the toric algebra associated with an affine monoid $\SGR \subseteq \ZZ^d$. Fix a positive integer $q$, and write $q^{-1}\SGR = \{ a/q \mid a \in \SGR \} \subseteq  \QQ^d$ for the rescaling of $\SGR$ by the factor $1/q$. Consider the semigroup algebra
+\[ R^{1/q} \coloneqq \kk[q^{-1}\SGR] = \spn_\kk\{ x^{a/q} \mid a \in \SGR \} \]
+regarded as a $\kk$-subalgebra of $\kk[\QQ^d] = \spn_\kk \{ x^\alpha \mid \alpha \in \QQ^d \}$, which carries a natural $\QQ^d$-grading. While the algebra $R^{1/q}$ is defined for arbitrary $q \in \NN$ in any characteristic, the notation is motivated by the case $q = p^e$ with $\kk$ perfect of prime characteristic $p$, where $R^{1/q}$ is the ring of $q$-th roots of elements in $R$. Further, for $r = \sum_{a \in \QQ^d} k_a x^{a} \in \kk[\QQ^d]$, we set $\qpower{r} \coloneqq \sum_{a \in \QQ^d} k_a x^{qa}$, and denote by $\qpower{F}$ the resulting $\kk$-algebra endomorphism 
+\[ \qpower{F} \colon R \to R, \qquad r \longmapsto \qpower{r}, \]
+%of $R$ mapping $r$ to $\qpower{r}$, 
+referred to as the \emph{$q$-th pseudo-Frobenius endomorphism} of $R$.
+
+\begin{remark}
+% PREVIOUS VERSION % Note that even in positive characteristic $p$ with $q = p^e$, the $q$-th pseudo-Frobenius twist need not agree with the usual $q$-th Frobenius twist, where $R$-scalar multiplication is pulled back via the $e$-times iterated Frobenius endomorphism $F^e \colon r \mapsto r^{p^e}$ on $R$. The two agree precisely when the underlying field $\kk$ is finite, in which case $\kk \subseteq R$ is fixed pointwise by $F^e$.
+Note that even in prime characteristic $p$ with $q = p^e$, the $q$-th pseudo-Frobenius endomorphism $\qpower{F}$ need not agree with the usual $e$-times iterated Frobenius endomorphism $F^e \colon R \to R$, mapping $r$ to $r^{q}$. %The two agree precisely when the underlying field $\kk$ is finite of order $p^f$ with $f \mid e$, in which case $\kk \subseteq R$ is fixed pointwise by $F^e$.
+\end{remark}
+
+\begin{proposition}
+Assume that $\kk$ is a field of prime characteristic $p$ and that \mbox{$q = p^e$}. Then the $e$-times iterated Frobenius endomorphism $F^e$ and the $q$-th pseudo-Frobenius endomorphism $\qpower{F}$ agree precisely when $\kk$ is finite of order $p^f$ with $f \mid e$.
+\end{proposition}
+
+\begin{proof}
+First, note that for $r = \sum_{a \in \SGR} k_a x^a \in R$, we have $F^e(r) = \sum_{a \in \SGR} k_a^{q} x^{qa}$. Hence, $\qpower{F}$ and $F^e$ agree precisely when $F^e$ fixes $\kk\subseteq R$ pointwise.
+
+Suppose first that $\kk$ is finite of order $p^f$. By Lagrange's theorem, $k^{p^f} = k$ for every $k \in \kk$. If $f \mid e$, then repeated application of this identity gives $k^{p^e} = k$ for all $k \in \kk$. Now assume that $f \nmid e$. Since the multiplicative group of a finite field is cyclic~\cite[§ 9, Prop. 18]{Dummit2004}, there exists an element $k \in \kk \setminus \{ 0 \}$ of order $p^f -1$. We claim that $k^{p^e} \neq k$. Suppose, for contradiction, that $k^{p^e} = k$. Then $k^{p^e -1 } = 1$, and hence $p^f -1 \mid p^e -1$. It follows that 
+\[ p^f -1 = \gcd(p^e-1,p^f-1) = p^{\gcd(e,f)}-1, \]
+where the last equality is a standard identity; see, for example,~\cite[Example 1.14]{Mollin2008}. Therefore, $f = \gcd(e,f)$, and hence $f \mid e$, contradicting our assumption. Finally, if $\kk$ is infinite, the polynomial $x^q - x \in \kk[x]$ cannot vanish on all of $\kk$, so there exists some $k \in \kk$ such that $k^q \neq k$.
+\end{proof}
+
+\begin{definition}
+Let $M$ be a $\QQ^d$-graded $R$-module. The \textbf{$q$-th pseudo-Frobenius twist} of $M$, denoted $\qpower{M}$, is the $R$-module obtained by restricting scalars along the \mbox{$q$-th} pseudo-Frobenius endomorphism $\qpower{F}$. That is, $\qpower{M}$ is the $R$-module with the same underlying abelian group as $M$, but with $R$-action given by~$r \qmult m \coloneqq \qpower{r} \cdot m$  
+% EXPLICIT DECLARATION % \[ \Bigl( \sum_{a \in S} k_a x^a \Bigr) \cdot_q \; m \coloneqq \bigl( \sum_{a \in S} k_a x^{aq} \bigr) m \]
+for all $r \in R$ and $m \in M$.
+\end{definition}
+
+\begin{remark}
+The $q$-th pseudo-Frobenius twist of $M$ carries a $\QQ^d$-graded $R$-module structure when endowed with the homogeneous components $(\qpower{M})_\alpha \coloneqq M_{q \alpha}$ for $\alpha \in \QQ^d$. Indeed,
+\[ R_\alpha \qmult (\qpower{M})_\beta \subseteq R_{q\alpha}M_{q \beta} \subseteq M_{q(\alpha + \beta)} = (\qpower{M})_{\alpha + \beta} \]
+for all $\alpha$, $\beta \in \QQ^d$.
+\end{remark}
+
+\begin{lemma}\label{lemma:cannonc-isom}
+There is a cannonical $\QQ^d$-graded $R$-module isomorphism 
+\[ R^{1/q} \xrightarrow{\;\sim\;} \qpower{R}, \]
+defined by $r = \sum_{a \in \SGR} k_a x^{a/q} \mapsto \qpower{r} = \sum_{a \in \SGR} k_a x^a$.
+\end{lemma}
+
+\begin{proof}
+By abuse of notation, we also denote by $\qpower{F}$ the map $R^{1/q} \to \qpower{R}$ defined above. This map is clearly additive, bijective and preserves the $\QQ^d$-grading. To verify that it is compatible with the respective $R$-module structures, let $r = \sum_{a \in \SGR} k_a x^a \in R$ and $m = \sum_{a \in \SGR} \ell_a x^{a/q} \in R^{1/q}$. Then
+\[ \qpower{F}(rm) = \qpower{F}\Bigl(\sum_{a, b \in \SGR} k_a \ell_b x^{a + \frac{b}{q}}\Bigr) = \sum_{a, b \in \SGR} k_a \ell_b x^{qa + b}, \]
+which is precisely $r \qmult \qpower{F}(m)$ in $\qpower{R}$. 
+\end{proof}
+
+\begin{lemma}\label{lemma:qpower-under-shift}
+Let $M$, $M_1$, \dots, $M_k$ be $\QQ^d$-graded $R$-modules. If it holds that \mbox{$M \shcong M_1 \oplus \cdots \oplus M_k$}, then $\qpower{M} \shcong \qpower{M_1} \oplus \cdots \oplus \qpower{M_k}$. 
+\end{lemma}
+
+\begin{proof}
+For simplicity of notation, we prove the result only for $k =2$. By assumption, there are $d_1$, $d_2 \in \QQ^d$ such that $M \grcong M_1(d_1) \oplus M_2(d_2)$. As the construction of the pseudo-Frobenius twist respects direct sums, we have 
+\[ \qpower{M} \grcong \qpower{(M_1(d_1))} \oplus \qpower{(M_2(d_2))} = (\qpower{M_1})(d_1/q) \oplus (\qpower{M_2})(d_2/q), \]
+where the last equality follows from the computation
+\[ \setlength{\arraycolsep}{2pt} \begin{array}{rcccc}
+\left(\qpower{(M_1(d_1))} \oplus \qpower{(M_2(d_2))}\right)_\alpha & = & \bigl(M_1(d_1)\bigr)_{q\alpha} & \oplus & \bigl(M_2(d_2)\bigr)_{q\alpha} \\
+& = & (M_1)_{q\alpha+d_1} & \oplus & (M_2)_{q\alpha+d_2} \\
+& = & (\qpower{M_1})_{\alpha+\frac{d_1}{q}} & \oplus & (\qpower{M_2})_{\alpha+\frac{d_2}{q}} \\
+& = & \left((\qpower{M_1})(d_1/q)\right)_\alpha & \oplus & \left((\qpower{M_2})(d_2/q)\right)_\alpha.
+\end{array} \]
+\end{proof}
+
+\begin{definition}
+We say that $R$ is \textbf{$q$-split} if the pseudo-Frobenius map $\qpower{F} \colon R \to \qpower{R}$ splits as a map of $R$-modules, that is, if there exists an $R$-module homomorphism $\sigma \colon \qpower{R} \to R$ such that $\sigma \circ \qpower{F} = \id_R$. Moreover, we call $R$ \textbf{$q$-weakly normal} if, for every $r \in \kk[\gp(\SGR)]$, $\qpower{r} \in R$ implies $r \in R$.
+\end{definition}
+
+\begin{remark}
+%Recall that $R$ is $q$-split if and only if $R$ is a direct summand of $\qpower{R}$; see, for example,~\cite[§ 10, Prop. 26]{Dummit2004}.
+By~\cite[Prop. 2.3.1]{Nastasescu2004}, the map $\qpower{F}$ splits as an $R$-module homomorphism if and only if it splits in the category of $\QQ^d$-graded $R$-modules. In other words, whenever $\qpower{F}$ splits, the splitting map $\sigma$ may be chosen to be $\QQ^d$-graded. Hence, $R$ is $q$-split if and only if it is a graded direct summand of $\qpower{R}$, that is, if there exists a $\QQ^d$-graded $R$-module $N$ such that $\qpower{R}\grcong R \oplus N$.
+\end{remark}
+
+%\begin{remark}
+%In semigroup-thereotic terms, $R = \kk[\SGR]$ is $q$-weakly normal if and only if, for every $a \in G(\SGR)$, $qa \in S$ implies $a \in S$. In particular, $\SGR$ (or equivalently, $R$) is normal if and only if $R$ is $q$-weakly normal for all $q \in \NN$.
+%\end{remark}
+
+In prime characteristic $p$, the Frobenius endomorphism yields $\qpower[p]{r} \in R$ if and only if $r^p \in R$, from which we obtain the following.
+
+\begin{corollary}
+Assume that $\kk$  is a field of prime characteristic $p$. Then $R$ is $p$-weakly normal 
+% MORE INFORMATION % (or, equivalently, $p^e$-weakly normal for all $e \in \NN$)
+if and only if $R$ is weakly normal. In particular, $R$ is seminormal in this case.
+\end{corollary}
+
+It is known that, in prime characteristic, the Frobenius map splits as a map of $R$-modules if and only if $R$ is weakly normal. The argument of \cite[Prop. 5.3]{Yasuda2009} extends verbatim to the characteristic-independent setting upon replacing the Frobenius map by the pseudo-Frobenius map. For the reader's convenience, we include the argument here.
+
+\begin{proposition}\textnormal{\cite[Prop. 5.31]{Hochster1976}, \cite[Prop. 5.3]{Yasuda2009}}
+$R$ is $q$-split if and only if it is $q$-weakly normal.
+\end{proposition}
+
+\begin{proof}
+% FAKE PROOF % Suppose $R$ is $q$-split, with $\sigma \colon \qpower{R} \to R$ an $R$-module homomorphism satisfying $\sigma \circ \qpower{F} = \id_R$; in particular, $\sigma(1) = 1$. Now let $r \in \Frac(R)$ be such that $\qpower{r} \in R$. Then $r  = r\sigma(1) = \sigma(\qpower{r}) \in R$. 
+
+Suppose $R$ is $q$-split, with $\sigma \colon \qpower{R} \to R$ an $R$-module homomorphism satisfying $\sigma \circ \qpower{F} = \id_R$. Let $r \in \kk[\gp(\SGR)]$ be such that $\qpower{r} \in R$, and choose $s \in R\setminus \{0 \}$ with $sr \in R$. Then
+\[ sr = \sigma\bigl( \qpower{F}(sr) \bigr) = \sigma(\qpower{s}\qpower{r}) = \sigma(s \qmult (\qpower{r})) = s\sigma(\qpower{r}), \]
+and cancelling $s$ yields $r = \sigma(\qpower{r}) \in R$.
+
+Conversely, suppose that $R$ is $q$-weakly normal. Let $\sigma \colon \qpower{R} \to R$ be the map that sends $f \in \qpower{R}$ to the unique $g \in R$ with $f = \qpower{g}$, if such a $g$ exists, and to $0$ otherwise. Clearly $\sigma \circ \qpower{F} = \id_R$, so it remains to show that $\sigma$ is a homomorphism of $R$-modules. As $\sigma$ is additive, it suffices to check that $\sigma(x^a \qmult x^b) = x^a \sigma(x^b)$ for $a$, $b \in \SGR$. Suppose first that $b = qc$ for some $c \in \SGR$. Then 
+\[ \sigma(x^a \qmult x^b) = \sigma(x^{q(a+c)}) = x^{a+c} = x^a \sigma(x^b).  \]
+Now suppose that $b$ is not of the form $b = qc$ for some $c \in \SGR$, meaning that $\sigma(x^b) = 0$. We claim that $qa + b$ is also not of the form $qa+ b = qc$ for some $c \in \SGR$. Indeed, suppose that there exists a $c \in \SGR$ such that $qa + b = qc$. Then $\qpower{(x^{c-a})} = x^b \in R$. By assumption, this implies that $x^{c-a} \in R$, and hence $c-a \in \SGR$. Consequently, $b = q(c-a)$, contradicting our assumption on $b$. It follows that $\sigma(x^a \qmult x^b) = \sigma(x^{qa+b}) = 0$.
+\end{proof}
+
+\begin{corollary}\textnormal{\cite[Cor. 6.3]{Bruns2006}}
+$R$ is normal if and only if $R$ is $q$-weakly normal for all $q \in \NN$ if and only if $R$ is $q$-split for all $q \in \NN$.
+\end{corollary}
+
+\begin{proof}
+By~\cite[Prop. 1]{Hochster1972}, the toric algebra $R = \kk[\SGR]$ is normal if and only if $\SGR$ is normal, that is, if for every $a \in \gp(\SGR)$ and $q \in \NN$, the condition $qa \in \SGR$ implies $a \in \SGR$. This is precisely the monoid-theoretic formulation of $R$ being $q$-weakly normal for all $q \in \NN$. By the previous proposition, this is equivalent to $R$ being $q$-weakly normal for every $q \in \NN$.
+\end{proof}
+
+\begin{lemma}\textnormal{\cite[Lemma 5.0.3]{Blickle2013}}\label{lemma:q-split}
+Let $q = p_1^{e_1} \cdots p_k^{e_k}$, where the $p_i$ are distinct primes and $e_i \in \NN$. Then $R$ is $q$-split if and only if $R$ is $(p_1^{\ell_1}\cdots p_k^{\ell_k})$-split for all $(\ell_1, \dots, \ell_k) \in \NN_0^k$.
+\end{lemma}
+
+\begin{proof}
+Assume that $R$ is $q$-split, i.e., there exists a $R$-module homomorphism $\sigma \colon \qpower{R} \to R$ such that $\sigma \circ \qpower{F} = \id_R$. We first claim that, for each $i \in \intval{1,k}$, $R$~is $p_i^{\ell_i}$-split for every $\ell_i \in \NN_0$. To this end, fix an $i \in \intval{1,k}$ and consider the $R$-module homomorphism $f_i \colon \qpower[p_i]{R} \to \qpower{R}$, defined by $r \mapsto \qpower[q/p_i]{r}$. Setting $\sigma_i \coloneqq \sigma \circ f_i$, we obtain that 
+\[ \sigma_i(\qpower[p_i]{F}(r)) = \sigma(f_i(\qpower[p_i]{r})) = \sigma(\qpower{r}) = r \] 
+for all $r \in R$, so $R$ is $p_i$-split. Now, an induction on $\ell_i$ shows that, as maps of sets, $\sigma_i^{\ell_i} \circ \qpower[p_i^{\ell_i}]{F} = \id_R$ for all $\ell_i \in \NN_0$, where $\sigma_i^{\ell_i}$ denotes the $\ell_i$-fold composition of $\sigma_i$. Moreover, $\sigma_i^{\ell_i}$, regarded as a map $\qpower[p_i^{\ell_i}]{R} \to R$, is readily verified to be $R$-linear, providing the claim.
+
+Finally, fix $(\ell_1, \dots, \ell_k) \in \NN_0^k$, and set $q^\prime \coloneqq p_1^{\ell_1} \cdots p_k^{\ell_k}$. By the first part, $R$ is $p_i^{\ell_i}$-split for each $i \in \intval{1,k}$. Hence, there exist $R$-module homomorphisms $\sigma_i^\prime \colon \qpower[p_i^{\ell_i}]{R} \to R$ such that $\sigma_i^\prime \circ \qpower[p_i^{\ell_i}]{F} = \id_R$. Setting $\sigma^\prime \coloneqq \sigma_1^\prime \circ \cdots \circ \sigma_k^\prime$, a direct verification shows that $\sigma^\prime$ gives the required splitting of $\qpower[q^\prime]{F}$.
+\end{proof}
+
+\section{$R$-module decomposition of $R^{1/q}$}
+
+We now specialize to the case where $\SGR = \langle a_1, \dots, a_n \rangle \subseteq \NN_0$ %= \{ c_1a_1 + \cdots + c_na_n \mid c_i \in \NN_0\} \subseteq \NN_0 \}
+is a numerical monoid and $R = \kk[\SGR]$ is the corresponding toric algebra. Following Shibuta~\cite{Shibuta2017}, we sketch, for a fixed $q \in \NN$, how $R^{1/q} = \kk[x^{a_1/q}, \dots, x^{a_n/q}]$ decomposes as a~$\QQ_{\geq 0}$-graded $R$-module. Observe that
+\[
+\Gamma_q \coloneqq \left\{ \bigl( x^{a_1/q} \bigr)^{e_1} \cdots \bigl( x^{a_n/q} \bigr)^{e_n} \mid e_i \in \intval{0,q-1} \right\}
+\]
+is a homogeneous generating set for $R^{1/q}$. We partition $\Gamma_q$ by degree modulo $\ZZ$: writing each degree as
+$a/q$ with $a \in \NN_0$, this amounts to grouping elements by the residue of $a$ modulo $q$. This yields
+\[
+\Gamma_q = \Gamma_{q,0} \sqcup \Gamma_{q,1} \sqcup \cdots \sqcup \Gamma_{q,q-1},
+\]
+where $\Gamma_{q,r} \coloneqq \{ x^{a/q} \in \Gamma_q \mid a \equiv r \pmod{q}\}$. Let $I_{q,r}$ be the $R$-submodule of $\kk[\QQ_{\geq 0}] = \kk[x^q \mid q \in \QQ_{\geq 0}]$
+generated by $\Gamma_{q,r}$. As a $\kk$-vector space,
+\begin{equation}\label{eq:ind-summand-as-vector-space}
+I_{q,r} = \spn_\kk \{ x^{a/q} \mid a \in \SGR, \, a \equiv r \pmod{q} \},
+\end{equation}
+and, by construction, summing over $r$ gives the $\QQ_{\geq 0}$-graded $R$-module decomposition
+\[
+R^{1/q} = \bigoplus_{r \in \intval{0,q-1}} I_{q,r},
+\]
+where each summand is indecomposable (see Lemma~\ref{lemma:rank-1}).
+
+% REDUNDANT % \begin{lemma}
+%Each $R$-module $I_{q,r}$ has rank $1$ and is therefore indecomposable.
+%\end{lemma}
+
+%\begin{proof}
+%For simplicity, let $Q \coloneqq \Frac(R)$. The localization $V_{q,r} \coloneqq (R\setminus \{ 0 \})^{-1} I_{q,r}$ is isomorphic to $Q \otimes_R I_{q,r}$ as a $Q$-vector space. For a fixed element $x^{a/q} \in I_{q,r} \subseteq V_{q,r}$, we claim that $V_{q,r}$ is $1$-dimensional over $Q$, generated by $x^{a/q}$.
+
+%To see this, take any $x^{b/q} \in I_{q,r}$. Since $b \equiv a \pmod{q}$, we have $b-a = cq$ for some $c \in \ZZ$. Thus, $x^{b/q} = x^c x^{a/q} \in \spn_Q\{ x^{a/q} \}$. The claim follows because any element of $V_{q,r}$ is a $Q$-linear combination of homogeneous elements of $I_{q,r}$.
+%\todo{note $\Frac{R} = \kk(x)$} 
+%\end{proof}
+
+\begin{definition}\label{def:gap-module}
+Let $A\subseteq \NN$. The (normalized) \textbf{gap module defined by $A$} is the $\NN_0$-graded $R$-submodule of $\overline{R}=\kk[x]$ given by
+\[ I(A) \coloneqq \langle x^\alpha \mid \alpha \in A\cup \{0\} \rangle_R. \]
+\end{definition}
+
+\begin{remark}
+Clearly, in Definition~\ref{def:gap-module}, the set $A$ may always be chosen as a subset of the gap set $\gapset$. In particular, there are only finitely many gap modules. Moreover, we may assume that $A$ consists of elements that are minimal with respect to the partial order $\preceq_\SGR$ induced by $\SGR$, where $\alpha \preceq_\SGR \beta$
+means $\beta - \alpha \in \SGR$. We write
+\[ \Min(\SGR) \coloneqq \{ A \subseteq \gapset \mid A \text{ is minimal with respect to } \preceq_\SGR \}. \]
+\end{remark}
+
+\begin{corollary}\label{cor:indec-summ}
+Every $\QQ_{\geq 0}$-graded indecomposable summand of $R^{1/q}$ is, up to a shift in degree, isomorphic to a gap module.
+\end{corollary}
+
+\begin{proof}
+Let $\frac{m_{q,r}}{q}$ denote the minimal degree occurring among the elements of $\Gamma_{q,r}$, and set
+\[ A_{q,r} \coloneqq \Bigl\{ \alpha \in \NN \, \Bigm| \, \frac{m_{q,r}}{q} + \alpha \in \Gamma_{q,r} \Bigr\}, \]
+from which it follows that $I_{q,r} = x^{\frac{m_{q,r}}{q}} \cdot I(A_{q,r})$.
+%\[
+%I_{q,r} = x^{m_{q,r}/q} \cdot \Big\langle x^\alpha \mid \alpha \in \NN_0, \, \frac{m_{q,r}}{q} + \alpha \in \Gamma_{q,r} \Big\rangle_R.
+%\]
+\end{proof}
+
+In particular, as $q$ ranges over $\NN$, only finitely many isomorphism classes of $\QQ_{\geq 0}$-graded indecomposable summands of $R^{1/q}$ occur, up to shifts in degree. More precisely, for every $q \in \NN$, there exist $m_A \in \NN_0$ for $A \in \Min(\SGR)$ such that
+\[ R^{1/q} \shcong \bigoplus_{A\in\Min(\SGR)} I(A)^{\oplus m_A}. \]
+
+%In particular, as $q$ ranges over $\NN$, only finitely many isomorphism classes of $\QQ_{\geq 0}$-graded indecomposable summands of $R^{1/q}$ occur, up to shifts in degree. More precisely, for every $q \in \NN$, there exist non-negative integers $m_A$ for $A \in \Min(\SGR)$ and non-negative rational numbers $d_{A,j}$ for $j \in [m_A]$ such that
+%\[ R^{1/q} \cong \bigoplus_{A\in\Min(\SGR)} \bigoplus_{j \in [m_A]} x^{d_{A,j}} \cdot I(A) \]
+%as $\QQ_{\geq 0}$-graded modules.
+
+
+\begin{lemma}\label{lemma:rank-1}
+For each $A \in \Min(\SGR)$, the gap module $I(A)$ has rank $1$ and is therefore indecomposable. Moreover, the assignment 
+\[ A \mapsto [I(A)] \]
+defines a bijection between $\Min(\SGR)$ and the set of isomorphism classes of $\NN_0$-graded $R$-submodules of $\overline{R} = \kk[x]$.
+\end{lemma}
+
+\begin{proof}
+For simplicity, set $Q \coloneqq \Frac(R) = \kk(x)$  and fix $A \in \Min(\SGR)$. Observe that $Q \otimes_R I(A)$ is naturally isomorphic, as a $Q$-vector space, to $I(A)_{(0)} \subseteq Q$. Since $I(A)$ contains $1$, it follows that $I(A)_{(0)} = Q$ and $I(A)$ has rank $1$.
+
+For the second part, each $I(A)$ with $A \in \Min(\SGR)$ is an $\NN_0$-graded $R$-submodule of $\overline{R}$, so the map is well-defined. To prove injectivity, suppose that $\varphi \colon I(A) \to I(B)$ is an $R$-module isomorphism for some $A$, $B \in \Min(\SGR)$. Localizing at $(0)$ yields an isomorphism of $Q$-vector spaces $\varphi_{(0)}\colon I(A)_{(0)} \to I(B)_{(0)}$, where both source and target are equal to $Q$. Hence $\varphi$, and thus also $\varphi_{(0)}$, is given by multiplication by some $g_0 \in Q$, so $\varphi(f) = g_0f$ for all $f \in I(A)$. A simple degree argument shows that $g_0 \in \kk \setminus \{ 0 \}$. Therefore,
+\[ I(B) = \varphi(I(A)) = g_0 \cdot I(A) =I(A). \]
+Since $A$ and $B$ are both minimal with respect to $\preceq_\SGR$, it follows that $A = B$. We now prove that the assignment is surjective. Let $I \subseteq \overline{R}$ be an arbitrary $\NN_0$-graded $R$-submodule. Because $R$ is Noetherian and $\overline{R}$ is finitely generated as an $R$-module, $I$ is finitely generated, i.e, $I = \langle x^\beta \mid \beta \in B\rangle_R$ for some finite subset $B \subseteq \NN_0$. Define $m \coloneqq \min(B)$ and $A^\prime \coloneqq \{ \beta - m \mid \beta \in B \} \setminus \{ 0 \}$. Consequently, $I = x^m \cdot I(A^\prime)$, which implies that $I \cong I(A^\prime)$. Surjectivity then follows from the equality $I(A^\prime)=I(A)$, where $A$ is the set of elements of $A^\prime \cap \gapset$ that are minimal with respect to $\preceq_\SGR$.
+\end{proof}
+
+\begin{remark}
+We now present another perspective on the gap modules $I(A)$. For $A \in \Min(\SGR)$, the $R$-module $I(A)$ admits the $\kk$-vector space description
+\[ I(A)=\spn_\kk \{x^\alpha \mid \alpha\in \SGR_A \}, \]
+where $\SGR_A\coloneqq (A \cup \{ 0 \}) +\SGR$. The set $\SGR_A$ is a normalized relative ideal of $\SGR$, that is, a subset of $\NN_0$ containing $0$ that is closed under addition by elements of $\SGR$. One readily checks that $A \mapsto \SGR_A$ defines a one-to-one correspondence between the elements of $\Min(\SGR)$ and the normalized relative ideals of $\SGR$.
+\end{remark}
+
+\begin{corollary}
+For each $A \in \Min(\SGR)$, the gap module $I(A)$ is maximal Cohen--Macaulay. In particular, $R^{1/q}$ is maximal Cohen--Macaulay.
+\end{corollary}
+
+\begin{proof}
+By the Cohen--Seidenberg theorems, $\dim(R) = \dim(\overline{R}) = \dim(\kk[x]) = 1$. Fix $A \in \Min(\SGR)$ and a maximal ideal $\mathfrak{m}$ of $R$. Since $R$ is a domain of Krull dimension $1$, every maximal ideal has height $1$; in particular $\dim(R_\mathfrak{m}) = \hgt(\mathfrak{m}) = 1$. The localization
+$I(A)_\mathfrak{m}$ is a non-zero $R_\mathfrak{m}$-submodule of $\kk(x)$, hence torsion-free, so $1 \leq \depth(I(A)_\mathfrak{m}) \leq  \dim(R_\mathfrak{m}) = 1$. Hence, $I(A)$ is maximal Cohen--Macaulay.
+
+By the depth lemma~\cite[Prop.~1.2.9]{Bruns1993}, the depth of a finite direct sum is the minimum of the depths of its summands. Hence, a finite direct sum of maximal Cohen--Macaulay modules is again maximal Cohen--Macaulay. By Corollary~\ref{cor:indec-summ}, the module $R^{1/q}$ decomposes as a finite direct sum of gap modules $I(A)$, and the claim follows.
+\end{proof}
+
+\begin{lemma}
+For $q \in \NN$, it holds that $\qpower{\overline{R}} \shcong \overline{R}^{\oplus q}$.
+\end{lemma}
+
+\begin{proof}
+First, note that 
+\[ \overline{R}^{1/q} = \kk[x^{1/q}] = \bigoplus_{r \in \intval{0,q-1}} x^{r/q} \cdot \kk[x] \shcong \overline{R}^{\oplus q} \]
+as $R$-modules. By Lemma~\ref{lemma:cannonc-isom}, $\overline{R}^{1/q}$ and $\qpower{\overline{R}}$ are isomorphic as $\QQ_{\geq 0}$-graded modules over $\overline{R}$, and hence, by restriction of scalars along $R \subseteq \overline{R}$, also as $\QQ_{\geq 0}$-graded modules over $R$.
+\end{proof}
+
+\begin{example}\label{ex:large-powers}
+Let $\SGR = \langle 3,4 \rangle \subseteq \NN_0$ and $R = \kk[\SGR] = \kk[x^3,x^4]$.
+
+\begin{center}
+\begin{figure}[htbp]
+\begin{tikzpicture}[x=0.8cm]
+    \draw[->] (0,0) -- (9.5,0);
+    
+    % labels
+    \foreach \x in {0,1,2,5,6,7,8,9}
+        \draw (\x,0.08) -- (\x,-0.08)
+              node[below=3pt] {$\x$};
+    
+    % generators
+    \foreach \x in {3,4}
+        \draw[red] (\x,0.08) -- (\x,-0.08)
+              node[below=3pt,red] {$\x$};
+
+    % black dots for semigroups elements
+    \foreach \x in {0,6,7,8,9}
+        \fill (\x,0) circle (2.5pt);
+    
+    % red dots for generators
+    \foreach \x in {3,4}
+        \fill[red] (\x,0) circle (2.5pt);
+
+    % bracket for frobenius number
+    \draw[thick] (6,-0.18) -- (6,0.18);
+    \draw[thick] (6,0.18) -- (6.15,0.18);
+    \draw[thick] (6,-0.18) -- (6.15,-0.18);
+\end{tikzpicture}
+\caption{The numerical semigroup $\SGR = \langle 3, 4\rangle$}
+\end{figure}
+\end{center}
+
+The set
+\[ \bigl\{ x^{(3/q)e_1 + (4/q)e_2} : e_i \in \intval{0,q-1} \bigr\} \]
+generates $R^{1/q} = \kk[x^{3/q},x^{4/q}]$ as an $R$-module. For $q =2$, this homogeneous generating set is $\{ 1, x^{3/2}, x^2, x^{7/2} \}$, and grouping the numerators of the exponents by residue modulo $2$ yields the $\QQ_{\geq 0}$-graded $R$-module decomposition
+\[ R^{1/2} = \langle 1, x^2 \rangle_R \oplus x^{3/2} \cdot \langle 1, x^2 \rangle_R \shcong I(\{2 \})^{\oplus 2}  \] 
+For $q = 4$, a direct computation shows $R^{1/4} \shcong \overline{R}^{\oplus 4}$. Using the $\QQ_{\geq 0}$-graded isomorphism between $R^{1/q}$ and $\qpower{R}$, we compute
+\[ R^{1/8} \grcong \qpower[8]{R} = \qpower[2]{(\qpower[4]{R})} \grcong \qpower[2]{(R^{1/4})}. \]
+Hence, by Lemma~\ref{lemma:qpower-under-shift}, it holds that $R^{1/8} \shcong (\qpower[2]{\overline{R}})^{\oplus 4}$. The preceding lemma then yields $R^{1/8} \shcong \overline{R}^{\oplus 8}$. By induction, we obtain that $R^{1/2^e} \shcong \overline{R}^{\oplus 2^e}$ for all \mbox{$e \geq 2$}. Consequently, there exist summands appearing in the $\QQ_{\geq 0}$-graded direct sum decomposition of $R^{1/2^e}$ for some $e \in \NN$ that do \emph{not} appear in the decomposition of $R^{1/2^\ell}$ for large $\ell$. This stands in contrast to the case where $\SGR$ is a normal affine monoid and $R=\kk[\SGR]$ is a normal toric algebra~\cite{Bruns2005}.
+\end{example}
+
+% NOTE: next proposition holds when q \in \SGR
+
+\begin{proposition}\label{prop:asy-behav}
+It holds that $q \in \SGR$ if and only if every $\QQ^d$-graded indecomposable summand of $R^{1/q}$ is, up to a degree shift, isomorphic to the normalization $\overline{R} = \kk[x]$ of $R$, that is, if $R^{1/q} \shcong \overline{R}^{\oplus q}$. In particular, this holds for all sufficiently large $q$.
+%For $q \in \SGR$, every indecomposable summands of $R^{1/q}$ is isomorphic to the normalization $\overline{R} = \kk[x]$ of $R$. That is, $R^{1/q} \cong \overline{R}^{\oplus q}$. In particular, this holds for all sufficiently large $q$.
+\end{proposition}
+
+\begin{proof}
+Assume that $q \in \SGR$, and fix $r \in \intval{0,q-1}$. For simplicity, set 
+\[ \supp(I_{q,r}) \coloneqq \{ a/q \mid a \in \SGR, \, a \equiv r \pmod{q} \} \]
+and observe that $\supp(I_{q,r}) \subseteq \{ \frac{m_{q,r}}{q} + \alpha \mid \alpha \in \NN_0\}$, where $\frac{m_{q,r}}{q}$ is the minimum of $\supp(I_{q,r})$. Conversely, for each $\alpha \in \NN_0$, the choice of $q$ ensures $\alpha q \in \SGR$, and hence $m_{q,r} + \alpha q \in \SGR$. Since $m_{q,r}+ \alpha q \equiv r \pmod q$, we
+obtain $\frac{m_{q,r}}{q} + \alpha \in \supp(I_{q,r})$, and altogether
+\[ \supp(I_{q,r}) = \{ \frac{m_{q,r}}{q} + \alpha \mid \alpha \in \NN_0\}. \]
+It now follows from~\eqref{eq:ind-summand-as-vector-space} that
+\[ I_{q,r} = \spn_\kk \{ x^{m_{q,r}/q + \alpha} \mid \alpha \in \NN_0 \} = x^{\frac{m_{q,r}}{q}} \cdot \spn_\kk\{ x^\alpha \mid \alpha \in \NN_0 \} \cong k[x] . \]
+
+
+\end{proof}
+
+\todo[inline]{think about what can be said about decompositions of $R^{1/q}$ for $q \notin \SGR$}
+
+\begin{example}
+Revisiting Example~\ref{ex:large-powers}, it remains to determine the decomposition of $R^{1/5}$ in order to obtain a complete description of the decompositions of $R^{1/q}$ for $q\in\NN$. A direct computation yields
+\[ R^{1/5}\cong I(\{2\}) \oplus \overline{R}^{\oplus 4}. \]
+Since $\overline{R} = \kk[x]$ corresponds to $I(\{1,2\})$, the indecomposable summands that occur in $R^{1/q}$ for some $q \in \NN$ are precisely $I(\{2\})$ and $I(\{1,2\})$. On the other hand,
+\[ \Min(\SGR) = \{ \emptyset,\{1\}, \{2\}, \{5\}, \{1,2\} \}, \]
+so not every potential candidate $I(A)$ with $A \in \Min(\SGR)$ occurs as a direct summand.
+\end{example}
+
+\begin{corollary}
+For each $q \geq 2$, $R$ is not $q$-split, meaning that $R$ does not occur as a direct summand of $R^{1/q}$. 
+\end{corollary}
+
+\todo{\emph{note:} this can also be easily proven directly!}
+
+\begin{proof}
+Fix $q \geq 2$, and let $p$ be a prime divisor of $q$. Suppose, for contradiction, that $R$ is $q$-split. By Lemma~\ref{lemma:q-split}, $R$ is then $p^e$-split for every $e \in \NN$. This contradicts Proposition~\ref{prop:asy-behav}, which shows that, for all sufficiently large $e$, $R^{1/p^e}$ decomposes as a direct sum of copies of $\overline{R}$, and therefore cannot contain $R$ as a direct summand.
+\end{proof}
+
+\begin{proposition} 
+Let $p$ be a prime divisor of $q$. If $R^{1/q}$ and $R^{1/(pq)}$ have the same indecomposable summands up to isomorphism, then the same holds for $R^{1/(p^e q)}$ for every $e \in \NN$.
+\end{proposition}
+
+\begin{proof}
+Decomposing $R^{1/q} \cong \bigoplus_{i=1}^k I(A_i)^{\oplus n_i}$ for pairwise distinct $A_i \in \Min(\SGR)$, it holds that
+\[ R^{1/(pq)} \cong \qpower[p]{(R^{1/q})} \cong \bigoplus_{i=1}^k \bigl(\qpower[p]{I(A_i)}\bigr)^{\oplus n_i}. \]
+Thus, by assumption, the indecomposable summands occuring in the $R$-modules $\qpower[p]{I(A_i)}$, $i \in \intval{1,k}$, are, up to isomorphism, precisely $I(A_1)$, \dots $I(A_k)$. Repeated application of this observation shows that $R^{1/(p^e q)}$ has the same indecomposable summands, up to isomorphism, for every $e \in \NN$.
+\end{proof}
+
+\section{Global dimension of endomorphism algebras}
+
+Throughout this section, let $\SGR$ again be a numerical semigroup and $R = \kk[\SGR]$ the corresponding monomial curve.
+
+For a commutative Noetherian ring $\Lambda$ and a finitely generated faithful $\Lambda$-module $M$, the endomorphism algebra $\End_\Lambda(M)$ is called a \textbf{noncommutative resolution of singularities} of $\Lambda$ if it has finite global dimension.
+
+\begin{theorem}
+For $q \in \SGR$, the global dimension of $\End_R(R^{1/q})$ is equal to the Krull dimension of $R$. In particular, this is true for all sufficiently large $q$.
+\end{theorem}
+
+\begin{proof}
+For $q \in \SGR$, Proposition~\ref{prop:asy-behav} gives $R^{1/q}\cong \overline R^{\oplus q}$. By~\cite[Cor. 2.3]{Leuschke2012}, multiplicities of direct summands do not affect the endomorphism ring up to Morita equivalence; hence $\End_R(R^{1/q})$ is Morita equivalent to $\End_R(\overline{R})$. As in the proof of Lemma~\ref{lemma:rank-1}, since $\overline{R}$ has rank $1$, any $R$-module endomorphism of $\overline{R}$ is induced by multiplication with an element of $\overline{R}$, and thus $\End_R(\overline{R}) \cong \overline{R}$. Consequently, we obtain
+\[ \gldim(\End_R(R^{1/q})) = \gldim(\overline{R}) = \gldim(\kk[x]) = 1 = \dim(R), \]
+where the first equality follows from the Morita invariance of global dimension.
+\end{proof}
+
+\todo[inline]{is there a notion of crepant for non-normal, non-Gorenstein rings?}
+
+\todo[inline]{for $\A \coloneqq \oplus_{A \in \Min(\SGR)} I(A)$, think about $\gldim(\End_R(\A))$ and/or $\gldim(\End_R(R^{1/q}))$ for small $q$}
+
+Following the argument of Faber, Muller, and Smith~\cite[Theorem~6.13]{Faber2019} for the corresponding result on normal toric varieties, we obtain the following:
+
+\begin{theorem}
+For a perfect field $\kk$ of prime characteristic, the ring $D_\kk(R)$ of differential operators on $R$ has finite global dimension.
+\end{theorem}
+
+\todo[inline]{is there machinery to show equality for the global dimension of $D_\kk(R)$ and $\dim(R)$?}
+
+\printbibliography
+
+\end{document}

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+  author    = {Bruns, Winfried},
+  title     = {Conic divisor classes over a normal monoid algebra},
+  booktitle = {Commutative Algebra and Algebraic Geometry},
+  series    = {Contemporary Mathematics},
+  volume    = {390},
+  pages     = {63--71},
+  publisher = {American Mathematical Society},
+  location  = {Providence, RI},
+  year      = {2005},
+}
+
+
+@incollection{Blickle2013,
+	location = {New York, {NY}},
+	title = {$p^{-1}$-Linear Maps in Algebra and Geometry},
+	isbn = {978-1-4614-5291-1 978-1-4614-5292-8},
+	url = {https://link.springer.com/10.1007/978-1-4614-5292-8_5},
+	doi = {10.1007/978-1-4614-5292-8_5},
+	pages = {123--205},
+	booktitle = {Commutative Algebra},
+	publisher = {Springer New York},
+	author = {Blickle, Manuel and Schwede, Karl},
+	editor = {Peeva, Irena},
+	urldate = {2026-09-01},
+	date = {2013},
+	langid = {english},
+}
+
+@inbook{Leuschke2012,
+url = {https://doi.org/10.1515/9783110250404.293},
+title = {Non-commutative Crepant Resolutions: Scenes From Categorical Geometry},
+booktitle = {Progress in Commutative Algebra 1},
+booktitle = {Combinatorics and Homology},
+author = {Graham J. Leuschke},
+editor = {Christopher Francisco and Lee C. Klingler and Sean Sather-Wagstaff and Janet C. Vassilev},
+publisher = {De Gruyter},
+address = {Berlin, Boston},
+pages = {293--364},
+doi = {doi:10.1515/9783110250404.293},
+isbn = {9783110250404},
+year = {2012},
+lastchecked = {2026-09-02}
+}
+
+@article{Faber2019,
+title = {Non-commutative resolutions of toric varieties},
+journal = {Advances in Mathematics},
+volume = {351},
+pages = {236-274},
+year = {2019},
+issn = {0001-8708},
+doi = {https://doi.org/10.1016/j.aim.2019.04.021},
+url = {https://www.sciencedirect.com/science/article/pii/S0001870819301823},
+author = {Eleonore Faber and Greg Muller and Karen E. Smith},
+keywords = {Toric variety, Non-commutative resolution, Finite global dimension, Non-commutative crepant resolution, Rings of differential operators, Strong F-regularity},
+abstract = {Let R be the coordinate ring of an affine toric variety. We prove, using direct elementary methods, that the endomorphism ring EndR(A), where A is the (finite) direct sum of all (isomorphism classes of) conic R-modules, has finite global dimension equal to the dimension of R. This gives a precise version, and an elementary proof, of a theorem of Špenko and Van den Bergh implying that EndR(A) has finite global dimension. Furthermore, we show that EndR(A) is a non-commutative crepant resolution if and only if the toric variety is simplicial. For toric varieties over a perfect field k of prime characteristic, we show that the ring of differential operators Dk(R) has finite global dimension.}
+}
+
+@book{Dummit2004,
+author    = {Dummit, David S. and Foote, Richard M.},
+title     = {Abstract Algebra},
+edition   = {3},
+publisher = {John Wiley \& Sons},
+location  = {Hoboken, NJ},
+year      = {2004},
+}
+
+@book{Mollin2008,
+location = {Boca Raton},
+edition = {2nd ed},
+title = {Fundamental number theory with applications},
+isbn = {978-1-4200-6659-3},
+series = {Discrete mathematics and its applications},
+pagetotal = {369},
+publisher = {Chapman \& Hall/{CRC}},
+author = {Mollin, Richard A.},
+date = {2008},
+}
+
+@book{Nastasescu2004,
+location = {Berlin ; New York},
+title = {Methods of graded rings},
+isbn = {978-3-540-20746-7},
+series = {Lecture notes in mathematics},
+pagetotal = {304},
+number = {1836},
+publisher = {Springer},
+author = {Năstăsescu, Constantin and Oystaeyen, Freddy Van},
+date = {2004},
+keywords = {Associative rings, Graded rings},
+}
+
+@article{Hochster1972,
+ ISSN = {0003486X, 19398980},
+ URL = {http://www.jstor.org/stable/1970791},
+ author = {Melvin Hochster},
+ journal = {Annals of Mathematics},
+ number = {2},
+ pages = {318--337},
+ publisher = {[Annals of Mathematics, Trustees of Princeton University on Behalf of the Annals of Mathematics, Mathematics Department, Princeton University]},
+ title = {Rings of Invariants of Tori, Cohen-Macaulay Rings Generated by Monomials, and Polytopes},
+ urldate = {2026-09-10},
+ volume = {96},
+ year = {1972}
+}
+
+@book{Cox2011,
+author    = {Cox, David A. and Little, John B. and Schenck, Henry K.},
+title     = {Toric Varieties},
+series    = {Graduate Studies in Mathematics},
+volume    = {124},
+publisher = {American Mathematical Society},
+address   = {Providence, RI},
+year      = {2011},
+isbn      = {978-0-8218-4819-2},
+doi       = {10.1090/gsm/124}
+}
+
+
+@article{Geroldinger2025,
+title = {On algebraic properties of power monoids of numerical monoids},
+volume = {265},
+issn = {0021-2172, 1565-8511},
+url = {https://link.springer.com/10.1007/s11856-024-2683-0},
+doi = {10.1007/s11856-024-2683-0},
+pages = {867--900},
+number = {2},
+journaltitle = {Israel Journal of Mathematics},
+shortjournal = {Isr. J. Math.},
+author = {Bienvenu, Pierre-Yves and Geroldinger, Alfred},
+urldate = {2026-09-16},
+date = {2025-02},
+}
+
+@book{Rosales2009,
+location = {New York, {NY}},
+title = {Numerical Semigroups},
+volume = {20},
+rights = {https://www.springernature.com/gp/researchers/text-and-data-mining},
+isbn = {978-1-4419-0159-0 978-1-4419-0160-6},
+url = {https://link.springer.com/10.1007/978-1-4419-0160-6},
+doi = {10.1007/978-1-4419-0160-6},
+series = {Developments in Mathematics},
+publisher = {Springer New York},
+author = {Rosales, J.C. and García-Sánchez, P. A.},
+urldate = {2026-09-16},
+date = {2009},
+langid = {english},
+}
+
+@book{Bruns1993,
+location = {Cambridge [England] ; New York},
+title = {Cohen-Macaulay rings},
+isbn = {978-0-521-41068-7},
+series = {Cambridge studies in advanced mathematics},
+pagetotal = {403},
+number = {39},
+publisher = {Cambridge University Press},
+author = {Bruns, Winfried and Herzog, Jürgen},
+date = {1993},
+keywords = {Cohen-Macaulay rings},
+}