From d9f274e028aa3a715b9e5add77a295d037253ab7 Mon Sep 17 00:00:00 2001 From: Markus Maximilian Rudolf Tripp Date: Thu, 24 Sep 2026 17:09:50 +0200 Subject: [PATCH] adapt dash --- ncr-of-monomial-curves.pdf | Bin 371159 -> 371159 bytes ncr-of-monomial-curves.tex | 4 ++-- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/ncr-of-monomial-curves.pdf b/ncr-of-monomial-curves.pdf index 119077f0d234ec24f7658129997fd5e37b8e1bf3..c29a9c8ba859376b8c9f947b3dfefe9c152d6781 100644 GIT binary patch delta 116 zcmcb9S?u~{v4$4L7N!>F7M3lnix;t)8(11xOkc5xRSv@3etHqB7>l!^o13GHfrYD^ lp&1Zc8d_Q!o4Yuhn!6eqS-6-wn%ODX5K^*TWjX6+CIAykAXfkY delta 116 zcmcb9S?u~{v4$4L7N!>F7M3lnix;t)85tT`PG7NzRSv@3etHqB7>l#3rK!1*xv8V6 nrL(28lcRyFk%^h9qp_Qdfq{vElcTGhf(;=h+f|mcZe{`i1eqWm diff --git a/ncr-of-monomial-curves.tex b/ncr-of-monomial-curves.tex index 20d1694..79acae7 100644 --- a/ncr-of-monomial-curves.tex +++ b/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