adapt dash

ncr-of-monomial-curves.tex

@@ -129,13 +129,13 @@ In the case of a numerical monoid $\SGR$, the corresponding toric algebra $R$ sa
 %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}  
+\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 $M \shcong M_1^\prime \oplus \cdots \oplus M_\ell^\prime \oplus M_2 \oplus \cdots \oplus M_k$.
+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