commit 56abf2fe8cc5704178c4be85823214346915ab40 Author: Markus Maximilian Rudolf Tripp Date: Wed Aug 12 16:02:02 2026 +0200 first commit diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..8f5414b --- /dev/null +++ b/.gitignore @@ -0,0 +1,259 @@ +## Core LaTeX auxiliary files +*.aux +*.lof +*.log +*.lot +*.fls +*.out +*.toc +*.fmt +*.fot +*.cb +*.cb2 +.*.lb + +## Bibliography +*.bbl +*.bcf +*.blg +*-blx.aux +*-blx.bib +*.run.xml + +## Build tool directories/files +*.fdb_latexmk +*.synctex +*.synctex(busy) +*.synctex.gz +*.synctex.gz(busy) +*.pdfsync + +## Build tool auxiliary files +*.alg +*.loa + +## Auxiliary and intermediate files from other packages +# algorithms +*.alg +*.loa + +# achemso +acs-*.bib + +# amsthm +*.thm + +# beamer +*.nav +*.pre +*.snm +*.vrb + +# changes +*.soc + +# cprotect +*.cpt + +# elsarticle (documentclass of Elsevier journals) +*.spl + +# endnotes +*.ent + +# fixme +*.lox + +# feynmf/feynmp +*.mf +*.mp +*.t[1-9] +*.t[1-9][0-9] +*.tfm + +#(r)(e)ledmac/(r)(e)ledpar +*.end +*.?end +*.[1-9] +*.[1-9][0-9] +*.[1-9][0-9][0-9] +*.[1-9]R +*.[1-9][0-9]R +*.[1-9][0-9][0-9]R +*.eledsec[1-9] +*.eledsec[1-9]R +*.eledsec[1-9][0-9] +*.eledsec[1-9][0-9]R +*.eledsec[1-9][0-9][0-9] +*.eledsec[1-9][0-9][0-9]R + +# glossaries +*.acn +*.acr +*.glg +*.glo +*.gls +*.glsdefs +*.lzo +*.lzs +*.slg +*.slo +*.sls + +# uncomment this for glossaries-extra (will ignore makeindex's style files!) +# *.ist + +# gnuplot +*.gnuplot +*.table + +# gnuplottex +*-gnuplottex-* + +# gregoriotex +*.gaux +*.gtex + +# htlatex +*.4ct +*.4tc +*.idv +*.lg +*.trc +*.xref + +# hyperref +*.brf + +# knitr +*-concordance.tex +*.tikz +*-tikzDictionary + +# listings +*.lol + +# luatexja-ruby +*.ltjruby + +# makeidx +*.idx +*.ilg +*.ind + +# minitoc +*.maf +*.mlf +*.mlt +*.mtc[0-9]* +*.slf[0-9]* +*.slt[0-9]* +*.stc[0-9]* + +# minted +_minted* +*.pyg + +# morewrites +*.mw + +# newpax +*.newpax + +# nomencl +*.nlg +*.nlo +*.nls + +# pax +*.pax + +# pdfpcnotes +*.pdfpc + +# sagetex +*.sagetex.sage +*.sagetex.py +*.sagetex.scmd + +# scrwfile +*.wrt + +# svg +svg-inkscape/ + +# sympy +*.sout +*.sympy +sympy-plots-for-*.tex/ + +# pdfcomment +*.upa +*.upb + +# pythontex +*.pytxcode +pythontex-files-*/ + +# tcolorbox +*.listing + +# thmtools +*.loe + +# TikZ & PGF +*.dpth +*.md5 +*.auxlock + +# titletoc +*.ptc + +# todonotes +*.tdo + +# vhistory +*.hst +*.ver + +# easy-todo +*.lod + +# xcolor +*.xcp + +# xmpincl +*.xmpi + +# xindy +*.xdy + +# xypic precompiled matrices and outlines +*.xyc +*.xyd + +# endfloat +*.ttt +*.fff + +# Latexian +TSWLatexianTemp* + +## Editors / OS junk +*~ +*.swp +*.bak +.DS_Store +Thumbs.db + +## Generated PDF/output — comment these two lines out if you WANT to track the compiled PDF +*.pdf +!final/*.pdf + +## latexmk +*.fdb_latexmk +*.fls + +## Other common temp files +*.tmp +*.log~ diff --git a/ncr-of-monomial-curves.tex b/ncr-of-monomial-curves.tex new file mode 100644 index 0000000..f60c7c2 --- /dev/null +++ b/ncr-of-monomial-curves.tex @@ -0,0 +1,249 @@ +\documentclass[10pt]{amsart} + +\usepackage[T1]{fontenc} +\usepackage{mathtools} +\usepackage{todonotes} +\usepackage{blindtext} +%\usepackage{parskip} + +\usepackage[ + backend=biber, + style=alphabetic, % common in math; change if journal requires + sorting=nyt, + giveninits=true, + eprint=false, url=false, doi=false, isbn=false, + maxbibnames=9 +]{biblatex} +\addbibresource{references.bib} +\AtEveryBibitem{% + \clearfield{month}% + \clearfield{day}% + \clearlist{language}% +} + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheoremstyle{remstyle} + {\medskipamount} % space above (half of standard topsep) + {\medskipamount} % space below (half of standard topsep) + {} % body font (normal/upright) + {} % indent + {\itshape} % head font (italic) + {.} % punctuation after label + {.5em} % space after label + {} % head spec +\theoremstyle{remstyle} +\newtheorem{remark}[theorem]{Remark} +\DeclareMathOperator{\supp}{supp} + +\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} +\newcommand{\qpower}[2][q]{{}^{#1}#2} +\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}} + +\title{Noncommutative Resolutions of Monomial Curves} +\author{Markus M. R. Tripp} +\date{} + +\begin{document} +\maketitle + +\begin{abstract} +\end{abstract} + +\section{Monomial curves} + +\section{Frobenius maps on affine semigroups} +Let $\SGR$ be an affine semigroup, i.e., a finitely generated submonoid of $\ZZ^d$ for some $d \in \NN$, and let $R \coloneqq \kk[\SGR]$ denote the associated semigroup algebra. Fix a positive integer $q$, and write $q^{-1}\SGR \coloneqq \{ a/q \mid a \in \SGR \} \subseteq \QQ^d$ for the affine semigroup $\SGR$ refined by $q$. Consider the semigroup algebra +\[ R^{1/q} \coloneqq \kk[q^{-1}\SGR] = \spn_\kk\{ x^{a/q} \mid a \in \SGR \} \] +regarded as a subring of $\kk[\QQ^d] = \spn_\kk \{ x^\alpha \mid \alpha \in \QQ^d \}$, which carries a natural $\QQ^d$-grading. While the semigroup ring $R^{1/q}$ is defined for arbitrary $q \in \NN$ in any characteristic, the notation is motivated by the case $q = p^e$ with $\kk$ perfect of characteristic $p$, where $R^{1/q}$ is the ring of $q$-th roots of elements in $R = \kk[\SGR]$. Further, for $r = \sum_{a \in \QQ^d} k_a x^{a}$, we set $\qpower{r} \coloneqq \sum_{a \in \QQ^d} k_a x^{aq}$, and write $\qpower{F}$ for the resulting \emph{$q$-th pseudo-Frobenius endomorphism} on $R$, which maps $r$ to $\qpower{r}$. + +\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 positive 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, in which case $\kk \subseteq R$ is fixed pointwise by $F^e$. +\end{remark} + +\begin{definition} +Let $M$ be an $R$-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~$r\cdot m \coloneqq \qpower{r}m$ +% 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{lemma} +There is a cannonical $R$-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 write $\qpower{F}$ for the map $R^{1/q} \to \qpower{R}$ above; it is clearly additive and bijective. To verify compatibility with the respective $R$-module structures, write $r = \sum_{a \in \SGR} k_a x^a \in R$ and $f = \sum_{a \in \SGR} \ell_a x^{a/q} \in R^{1/q}$. Then +\[ \qpower{F}(rf) = \qpower{F}\Bigl(\sum_{a, b \in \SGR} k_a\ell_b x^{a + b/q}\Bigr) = \sum_{a, b \in \SGR} k_a\ell_b x^{aq + b}, \] +which is precisely $r\cdot\qpower{F}(f)$ in $\qpower{R}$. +\end{proof} + +\begin{definition} +We say that $R$ is \textbf{$q$-split} if the Frobenius map $\qpower{F} \colon R \to \qpower{R}$ splits as a map of $R$-modules. Moreover, we call $R$ \textbf{$q$-weakly normal} if, for every $r \in \kk[G(\SGR)]$\todo{$\Frac(R)$?}, $\qpower{r} \in R$ implies $r \in R$. +\end{definition} + +\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 positive 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} +If $\kk$ has characteristic $p >0$, 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 (i.e., for any $r \in \Frac(R)$, $r^p$ implies $r \in R$). In particular, $R$ is seminormal in this case. +\end{corollary} + +\begin{proposition}\textnormal{\cite{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 $r \in \kk[G(\SGR)]$ be such that $\qpower{r} \in R$, and choose $s \in R^\bullet$ with $sr \in R$. Then +\[ sr = \sigma\bigl( \qpower{F}(sr) \bigr) = \sigma(\qpower{s}\qpower{r}) = s\sigma(\qpower{r}), \] +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 $\sigma(x^{aq+b}) = x^a \sigma(x^b)$ for $a$, $b \in \SGR$, which is straightforward using the assumption that $R$ is $q$-weakly normal. % MORE EXPLANATION % Suppose first that $b = qc$ for some $c \in \SGR$. Then $\sigma(x^{aq+b}) = \sigma(x^{q(a+c)}) = x^{a+c} = x^a \sigma(x^b)$. Now suppose there exists no $c \in \SGR$ with $b = qc$, so $\sigma(x^b) = 0$. If $aq+b = qc$ for some $c \in \SGR$, then $\qpower{x^{c-a}} = x^b \in R$, so $x^{c-a} \in R$ by assumption, giving $c-a \in \SGR$ and $b = q(c-a)$, a contradiction. Hence $\sigma(x^{aq+b}) = 0 = x^a\sigma(x^b)$. +\end{proof} + +\begin{corollary}\textnormal{\cite[Cor. 6.3]{Bruns2006}} +$R$ is normal if and only if $R$ is $q$-split for all $q \in \NN$. +\end{corollary} + +\section{$R$-module decomposition of $R^{1/q}$} + +We now specialize to the case where $\SGR = \langle a_1, \dots, a_n \rangle \subseteq \NN_0$ is a numerical semigroup. Following Shibuta~\cite{Shibuta2017}, we sketch how $R^{1/q} = \kk[x^{a_1/q}, \dots, x^{a_n/q}]$ decomposes as an $R$-module. Observe that +\[ +\Gamma_q \coloneqq \left\{ \bigl( x^{a_1/q} \bigr)^{e_1} \cdots \bigl( x^{a_n/q} \bigr)^{e_n} \mid e_i \in [0,q-1] \right\} +\] +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 $\kk[x^q \mid q \in \QQ_{\geq 0}]$ +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 $R$-module decomposition +\[ +R^{1/q} = \bigoplus_{r \in [0,q-1]} I_{q,r}, +\] +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} + +Moreover, writing $m_{q,r}/q$ for the minimal degree occurring among +elements of $\Gamma_{q,r}$, we have +\[ +I_{q,r} = x^{m_{q,r}/q} \cdot \langle x^\alpha \mid \alpha \in \NN_0, \, m_{q,r}/q + \alpha \in \Gamma_{q,r} \rangle_R, +\] +from which it follows that every indecomposable summand of $R^{1/q}$ is isomorphic to $I(A) \coloneqq \langle x^\alpha \mid \alpha \in A \cup \{ 0 \} \rangle_R$ for some $A \subseteq \gapset$\todo{name?}. We may always choose $A$ to be minimal with respect to the order $\preceq_\SGR$ induced by $\SGR$, where $\alpha \preceq_\SGR \beta$ means $\beta - \alpha \in \SGR$. Write +\[ \Min(\SGR) \coloneqq \{ A \in \gapset \mid A \text{ is minimal with respect to } \preceq_\SGR \}. \] +In particular, as $q$ ranges over $\NN$, only finitely many indecomposable summands of $R^{1/q}$ occur, up to isomorphism. + +\begin{lemma}\label{lemma:rank-1} +For each $A \in \Min(\SGR)$, the $R$-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 homogeneous $R$-submodules of $\bar{R} = \kk[x]$\todo{this also holds for $\kk[x,x^{-1}]$, interesting?}. +\end{lemma} + +\begin{proof} +For simplicity, set $Q \coloneqq \Frac(R) = \kk(x)$\todo{note $\Frac{R} = \kk(x)$} 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, it is clear that $I(A)$, $A \in \Min(\SGR)$, are homogeneous fractional ideals of $R$, so the map is well-defined. To show injectivity, suppose we are given an $R$-module isomorphism $\varphi \colon I(A) \to I(B)$ for some $A$, $B \in \Min(\SGR)$. This induces an isomorphism of $Q$-vector spaces $\varphi_{(0)} \colon I(A)_{(0)} \to I(B)_{(0)}$, where both sides equal~$Q$. +%and $\varphi_{(0)}(f/1) = \varphi(f)$ for all $f \in I(A)$. +In particular, $\varphi_{(0)}$ (and therefore $\varphi$) is given by multiplication with some $g_0 \in Q$, so $\varphi(f) = g_0 f$ for all $f \in I(A)$. Taking $f=1$ yields $g_0 = \varphi(1) \in I(B)$. Since $\varphi$ is surjective and $1 \in I(B)$, a degree argument yields $g_0 \in \kk\setminus\{ 0 \}$. Consequently, +\[ I(B) = \varphi(I(A)) = g_0 \cdot I(A) = I(A), \] +and because $A$ and $B$ are both minimal with respect to $\preceq_\SGR$, it follows that~$A = B$. Now, let $I \subseteq \bar{R}$ be an arbitrary homogeneous $R$-submodule. Because $R$ is Noetherian and $\bar{R}$ 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 $I \cong I(A^\prime)$. The claim then follows from the fact that $I(A^\prime)=I(A)$, where $A$ denotes the set of minimal elements of $A^\prime \cap (\gapset)$ with respect to $\preceq_\SGR$. +\end{proof} + +\begin{remark} +We now present another perspective on the $R$-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 $\SGR$. +\todo{reference for this terminology?} +\end{remark} + +\begin{corollary} +For each $A \in \Min(\SGR)$, the $R$-module $I(A)$ is maximal Cohen--Macaulay. +\end{corollary} + +\begin{proof} +By the Cohen--Seidenberg theorems, $\dim(R) = \dim(\bar{R}) = \dim(\kk[x]) = 1$. 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. +\end{proof} + +\todo{add direct sums are Cohen--Macaulay (Bruns/Herzog, 1993, Prop. 1.2.9)} + +\begin{lemma} +It holds that $\qpower{\bar{R}} = \bar{R}^{\otimes q}$ as $R$-module. +\end{lemma} + +\begin{proof} +Note that $\bar{R}^{1/q} = \kk[x^{1/q}] = \bigoplus_{r \in [0,q]} x^{r/q} \cdot \kk[x]$ +\end{proof} + +%\begin{example} +%\end{example} + +\begin{proposition} +For sufficiently large $q$, all indecomposable summands of $R^{1/q}$ are isomorphic to the normalization $\bar{R} = \kk[x]$ of $R$. That is, $R^{1/q} \cong \bar{R}^{\oplus q}$. +\end{proposition} + +\begin{proof} +Let $q$ be such that $\ZZ_{\geq q} \subseteq S$, and fix $r \in [0,q-1]$. For simplicity, set +\[ \supp(I_{q,r}) \coloneqq \{ a/q \mid a \in S, \, a \equiv r \pmod{q} \} \] +and observe that $\supp(I_{q,r}) \subseteq \{ m_{q,r}/q + n \mid n \in \NN_0\}$, where $m_{q,r}/q$ is the minimum of $\supp(I_{q,r})$. Conversely, for each $n \in \NN_0$, the choice of $q$ ensures $nq \in S$, and hence $m_{q,r} + nq \in S$. Since $m_{q,r}+nq \equiv r \pmod q$, we +obtain $m_{q,r}/q + n \in \supp(I_{q,r})$, and altogether +\[ \supp(I_{q,r}) = \{ m_{q,r}/q + n \mid n \in \NN_0\}. \] +It now follows from~\eqref{eq:ind-summand-as-vector-space} that +\[ I_{q,r} = \spn_\kk \{ x^{m_{q,r}/q + n} \mid n \in \NN_0 \} = x^{m_{q,r}/q} \cdot \spn_\kk\{ x^n \mid n \in \NN_0 \} \cong k[x] . \] +\end{proof} + +%\begin{proposition} +% R is not q-split +%\end{proposition} + +%\begin{proposition} +% if two consecutive powers have same indecomposable summands, then so do all larger powers +%\end{proposition} + +\section{Global dimension of endomorphism algebras} + +\todo[inline]{For $\A \coloneqq \oplus_{A \in \Min(S)} I(A)$, think about $\gldim(\End_R(\A))$ and/or $\gldim(\End_R(R^{1/q}))$ for small $q$.} + +\printbibliography + +\end{document} \ No newline at end of file diff --git a/references.bib b/references.bib new file mode 100644 index 0000000..85d4ac8 --- /dev/null +++ b/references.bib @@ -0,0 +1,59 @@ +@article{Shibuta2017, + title = {Affine semigroup rings are of finite F-representation type}, + volume = {45}, + issn = {0092-7872, 1532-4125}, + url = {https://www.tandfonline.com/doi/full/10.1080/00927872.2017.1313425}, + doi = {10.1080/00927872.2017.1313425}, + pages = {5465--5470}, + number = {12}, + journaltitle = {Communications in Algebra}, + shortjournal = {Communications in Algebra}, + author = {Shibuta, Takafumi}, + urldate = {2026-08-03}, + date = {2017-12-02}, + langid = {english}, +} + +@article{Hochster1976, +title = {The purity of the Frobenius and local cohomology}, +journal = {Advances in Mathematics}, +volume = {21}, +number = {2}, +pages = {117-172}, +year = {1976}, +issn = {0001-8708}, +doi = {https://doi.org/10.1016/0001-8708(76)90073-6}, +url = {https://www.sciencedirect.com/science/article/pii/0001870876900736}, +author = {Melvin Hochster and Joel L Roberts} +} + +@article{Yasuda2009, +title = {On monotonicity of F-blowup sequences}, +volume = {53}, +issn = {0019-2082}, +url = {https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-53/issue-1/On-monotonicity-of-F-blowup-sequences/10.1215/ijm/1264170841.full}, +doi = {10.1215/ijm/1264170841}, +number = {1}, +journaltitle = {Illinois Journal of Mathematics}, +shortjournal = {Illinois J. Math.}, +author = {Yasuda, Takehiko}, +urldate = {2026-08-11}, +date = {2009-01-01}, +} + +@article{Bruns2006, +title = {On seminormal monoid rings}, +volume = {302}, +rights = {https://www.elsevier.com/tdm/userlicense/1.0/}, +issn = {00218693}, +url = {https://linkinghub.elsevier.com/retrieve/pii/S0021869305006605}, +doi = {10.1016/j.jalgebra.2005.11.012}, +pages = {361--386}, +number = {1}, +journaltitle = {Journal of Algebra}, +shortjournal = {Journal of Algebra}, +author = {Bruns, Winfried and Li, Ping and Römer, Tim}, +urldate = {2026-08-12}, +date = {2006-08}, +langid = {english}, +} \ No newline at end of file