ncr-of-monomial-curves/ncr-of-monomial-curves.tex

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\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 one-dimensional toric algebra. To exclude the trivial case, we assume that $\SGR$ is proper, that is, $\SGR\subsetneq\NN_0$. 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 \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 $R$-module 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 monoid $\SGR = \langle 3, 4\rangle$}
\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 $\supp(I_{q,r}) \subseteq \bigl\{ \frac{m_{q,r}}{q} + \alpha \mid \alpha \in \NN_0\bigr\}$, 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
\[ \supp(I_{q,r}) = \Bigl\{ \frac{m_{q,r}}{q} + \alpha \mid \alpha \in \NN_0\Bigr\}. \]
It now follows from~\eqref{eq:ind-summand-as-vector-space} that
\[ I_{q,r} = \spn_\kk \bigl\{ x^{\frac{m_{q,r}}{q} + \alpha} \, \big| \, \alpha \in \NN_0 \bigr\} = x^{\frac{m_{q,r}}{q}} \cdot \spn_\kk\{ x^\alpha \mid \alpha \in \NN_0 \} \cong k[x] . \]

Conversely, suppose that every indecomposable $\QQ^d$-graded summand of $R^{1/q}$ is, up to a degree shift, isomorphic to $\overline{R}$. In particular, $I_{q,0} \shcong \overline{R}$. Since both are gap modules, Lemma~\ref{lemma:rank-1} yields $I_{q,0}=\overline{R}=\kk[x]$. The description of $I_{q,0}$ then implies that $q \in \SGR$.
\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 $\QQ_{\geq 0}$-graded 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
\[ R^{1/5} \shcong I(\{2\}) \oplus \overline{R}^{\oplus 4}. \]
Since $\overline{R} = \kk[x]$ corresponds to $I(\{1,2\})$, the indecomposable $\QQ_{\geq 0}$-graded summands occurring in $R^{1/q}$ for some $q\in\NN$ are, up to degree shift, precisely $I(\{2\})$ and $I(\{1,2\})$. On the other hand,
\[ \Min(\SGR)=\{\emptyset,\{1\},\{2\},\{5\},\{1,2\}\}, \]
so not every potential summand $I(A)$ with $A \in \Min(\SGR)$ actually occurs.
\end{example}

\begin{corollary}
Let $\SGR \subsetneq \NN_0$. Then, for every $q \in \NN_{\geq 2}$, the toric algebra $R$ is not $q$-split, or equivalently, $R$ does not occur as a $\QQ_{\geq 0}$-graded direct summand of $R^{1/q}$.
\end{corollary}

\todo{\emph{note:} this can also be easily proven directly!}

\begin{proof}
Fix $q \in \NN_{\geq 2}$, 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 shifted copies of $\overline{R}$, and therefore cannot contain $R$ as a $\QQ_{\geq 0}$-direct summand.
\end{proof}

\begin{proposition} 
Let $p \in \NN$. If the indecomposable $\QQ_{\geq 0}$-graded summands of $R^{1/q}$ and $R^{\frac{1}{pq}}$ coincide up to degree shift, the the same is holds for $R^{1/q}$ and $R^{\frac{1}{p^e q}}$ for every $e \in \NN$.
\end{proposition}

\begin{proof}
Decompose $R^{1/q} \shcong \bigoplus_{i \in \intval{1,k}} I(A_i)^{\oplus m_i}$, where $A_1$, \dots, $A_k \in \Min(\SGR)$ are pairwise distinct and $m_i \in \NN$. By Lemma~\ref{lemma:qpower-under-shift}, it follows that
\[  R^{\frac{1}{pq}} \grcong \qpower[p]{(R^{1/q})} \shcong \bigoplus_{i \in \intval{1,k}} \left(\qpower[p]{I(A_i)}\right)^{\oplus m_i}. \]
Thus, by assumption, the indecomposable $\QQ_{\geq 0}$-graded direct summands occurring in $\qpower[p]{I(A_i)}$, for $i \in \intval{1,k}$, are, up to degree shift, precisely $I(A_1)$, \dots, $I(A_k)$. Iterating this argument shows that $R^{\frac{1}{p^e q}}$ has the same indecomposable $\QQ_{\geq 0}$-graded direct summands, up to degree shift, for every $e \in \NN$. 
\end{proof}

\section{Global dimension of endomorphism algebras}

Throughout this section, let $\SGR$ again be a numerical monoid and $R = \kk[\SGR]$ the corresponding toric algebra.

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}