\documentclass[CRMATH,Unicode,biblatex,XML]{cedram} \TopicFR{Algèbre} \TopicEN{Algebra} \TopicFR{Théorie des représentations} \TopicEN{Representation theory} \addbibresource{CRMATH_Jasso_20240125.bib} \usepackage{amssymb}% \usepackage{mathtools}% \usepackage{tikz}% \usepackage{tikz-cd}% \usetikzlibrary{decorations,snakes}% \newcommand{\kk}{\mathbf{k}}% \NewDocumentCommand{\A}{}{A_\infty}% \RenewDocumentCommand{\AA}{}{\mathcal{A}}% \NewDocumentCommand{\Hom}{O{\AA}D<>{}mm}{\operatorname{Hom}_{#1}^{#2}\!\left(#3,#4\right)}% \NewDocumentCommand{\Ext}{O{\AA}D<>{*}mm}{\operatorname{Ext}_{#1}^{#2}\!\left(#3,#4\right)}% \NewDocumentCommand{\RHom}{O{A}mm}{\operatorname{RHom}_{#1}\!\left(#2,#3\right)}% \RenewDocumentCommand{\mod}{m}{\operatorname{mod}{#1}}% \NewDocumentCommand{\Db}{m}{\operatorname{D}^{\mathrm{b}}\!\left(#1\right)}% \NewDocumentCommand{\len}{m}{\operatorname{len}\!\left(#1\right)}% \NewDocumentCommand{\class}{m}{#1}% \NewDocumentCommand{\dg}{m}{{#1}_{\mathrm{dg}}}% \NewDocumentCommand{\F}{}{\mathcal{F}}% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \makeatletter \let\save@mathaccent\mathaccent \newcommand*\if@single[3]{% \setbox0\hbox{${\mathaccent"0362{#1}}^H$}% \setbox2\hbox{${\mathaccent"0362{\kern0pt#1}}^H$}% \ifdim\ht0=\ht2 #3\else #2\fi } %The bar will be moved to the right by a half of \macc@kerna, which is computed by amsmath: \newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna} %If there's a superscript following the bar, then no negative kern may follow the bar; %an additional {} makes sure that the superscript is high enough in this case: \newcommand*\widebar[1]{\@ifnextchar^{{\wide@bar{#1}{0}}}{\wide@bar{#1}{1}}} %Use a separate algorithm for single symbols: \newcommand*\wide@bar[2]{\if@single{#1}{\wide@bar@{#1}{#2}{1}}{\wide@bar@{#1}{#2}{2}}} \newcommand*\wide@bar@[3]{% \begingroup \def\mathaccent##1##2{% %Enable nesting of accents: \let\mathaccent\save@mathaccent %If there's more than a single symbol, use the first character instead (see below): \if#32 \let\macc@nucleus\first@char \fi %Determine the italic correction: \setbox\z@\hbox{$\macc@style{\macc@nucleus}_{}$}% \setbox\tw@\hbox{$\macc@style{\macc@nucleus}{}_{}$}% \dimen@\wd\tw@ \advance\dimen@-\wd\z@ %Now \dimen@ is the italic correction of the symbol. \divide\dimen@ 3 \@tempdima\wd\tw@ \advance\@tempdima-\scriptspace %Now \@tempdima is the width of the symbol. \divide\@tempdima 10 \advance\dimen@-\@tempdima %Now \dimen@ = (italic correction / 3) - (Breite / 10) \ifdim\dimen@>\z@ \dimen@0pt\fi %The bar will be shortened in the case \dimen@<0 ! \rel@kern{0.6}\kern-\dimen@ \if#31 \overline{\rel@kern{-0.6}\kern\dimen@\macc@nucleus\rel@kern{0.4}\kern\dimen@}% \advance\dimen@0.4\dimexpr\macc@kerna %Place the combined final kern (-\dimen@) if it is >0 or if a superscript follows: \let\final@kern#2% \ifdim\dimen@<\z@ \let\final@kern1\fi \if\final@kern1 \kern-\dimen@\fi \else \overline{\rel@kern{-0.6}\kern\dimen@#1}% \fi }% \macc@depth\@ne \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax}% \macc@set@skewchar\relax \let\mathaccentV\macc@nested@a %The following initialises \macc@kerna and calls \mathaccent: \if#31 \macc@nested@a\relax111{#1}% \else %If the argument consists of more than one symbol, and if the first token is %a letter, use that letter for the computations: \def\gobble@till@marker##1\endmarker{}% \futurelet\first@char\gobble@till@marker#1\endmarker \ifcat\noexpand\first@char A\else \def\first@char{}% \fi \macc@nested@a\relax111{\first@char}% \fi \endgroup } \makeatother \let\oldbar\bar \renewcommand*{\bar}[1]{\mathchoice{\widebar{#1}}{\widebar{#1}}{\widebar{#1}}{\oldbar{#1}}} \let\oldtilde\tilde \renewcommand*{\tilde}[1]{\mathchoice{\widetilde{#1}}{\widetilde{#1}}{\oldtilde{#1}}{\oldtilde{#1}}} %\let\tilde\widetilde %\let\hat\widehat \let\oldhat\hat \renewcommand*{\hat}[1]{\mathchoice{\widehat{#1}}{\widehat{#1}}{\oldhat{#1}}{\oldhat{#1}}} %\let\tilde\widetilde \renewcommand*{\to}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}} \let\oldmapsto\mapsto \renewcommand*{\mapsto}{\mathchoice{\longmapsto}{\oldmapsto}{\oldmapsto}{\oldmapsto}} %jouer avec le \hsize pour avoir la bonne largeur des adresses \makeatletter \def\@setafterauthor{% \vglue3mm% \begingroup\hsize=12cm\advance\hsize\abstractmarginL\raggedright \noindent \begin{minipage}[t]{13cm} \leftskip\abstractmarginL \normalfont\Small \@afterauthor\par \end{minipage} \endgroup \vskip2pt plus 3pt minus 1pt } \makeatother %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand*{\mk}{\mkern -1mu} \newcommand*{\Mk}{\mkern -2mu} \newcommand*{\mK}{\mkern 1mu} \newcommand*{\MK}{\mkern 2mu} \hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red} \newcommand*{\romanenumi}{\renewcommand*{\theenumi}{\roman{enumi}}} \newcommand*{\Romanenumi}{\renewcommand*{\theenumi}{\Roman{enumi}}} \newcommand*{\alphenumi}{\renewcommand*{\theenumi}{\alph{enumi}}} \newcommand*{\Alphenumi}{\renewcommand*{\theenumi}{\Alph{enumi}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title[On a theorem of B.~Keller]{On a theorem of B.~Keller on Yoneda algebras of simple modules}% \alttitle{Sur un théorème de B. Keller sur les algèbres de Yoneda de modules simples} \author{\firstname{Gustavo} \lastname{Jasso}} \address{Lund University, Centre for Mathematical Sciences, Box 118, 22100 Lund, Sweden} \curraddr{Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Cologne, Germany} \thanks{The author's research was supported by the Swedish Research Council (Vetenskapsrådet) Research Project Grant 2022-03748 `Higher structures in higher-dimensional homological algebra.'} \CDRGrant[VR]{2022-03748} \email{gjasso@math.uni-koeln.de} \subjclass{18G70, 16G20, 16G70} \keywords{\kwd{Yoneda algebras} \kwd{simple modules} \kwd{Nakayama algebras} \kwd{$A_\infty$-algebras}} \altkeywords{\kwd{Algèbres de Yoneda} \kwd{modules simples} \kwd{algèbres de Nakayama} \kwd{$A_\infty$-algèbres}} \begin{abstract} A theorem of Keller states that the Yoneda algebra of the simple modules over a finite-dimensional algebra is generated in cohomological degrees $0$ and $1$ as a minimal $A_\infty$-algebra. We provide a proof of an extension of Keller's theorem to abelian length categories by reducing the problem to a particular class of Nakayama algebras, where the claim can be shown by direct computation. \end{abstract} \begin{altabstract} Un théorème de Keller stipule que l'algèbre de Yoneda des modules simples sur une algèbre de dimension finie est générée en degrés cohomologiques $0$ et $1$ comme une $A_\infty$-algèbre minimale. Nous prouvons une extension du théorème de Keller aux catégories de longueur abélienne en réduisant le problème à une classe particulière d'algèbres de Nakayama, où l'affirmation peut être démontrée par un calcul direct. \end{altabstract} \begin{document} \maketitle We work over an arbitrary field $\kk$. Let $A$ be a finite-dimensional algebra and $S$ the direct sum of a complete set of representatives of the simple (right) $A$-modules. The Yoneda algebra $\Ext[A]{S}{S}$, as a graded algebra, does not determine the algebra $A$ up to Morita equivalence, as the following example shows~(see~\cite[Section~2.1]{Kel02a} or~\cite[Example~B.2.2]{Mad02}): Let $A=\kk[x]/(x^\ell)$, $\ell\geq 3$; then \begin{equation} \label{eq:example} \Ext[A]{S}{S}\cong\kk[u,v]/(u^2),\qquad |u|=1,\ |v|=2, \end{equation} does not depend on $\ell$. On the other hand, the Yoneda algebra of an arbitrary finite-dimensional algebra inherits, via Kadeishvili's Homotopy Transfer Theorem~\cite{Kad82,Mar06}, the structure of a minimal\footnote{Recall that an $\A$-algebra is minimal if its underlying complex has vanishing differential.} $\A$-algebra since the Yoneda algebra is the cohomology of the differential graded algebra $\RHom[A]{S}{S}$. Endowed with this additional $\A$-structure the Yoneda algebra does determine the algebra $A$ up to Morita equivalence~\cite[Section~7.8]{Kel01}. The purpose of this short article is to provide a proof of a minor extension of a theorem of Keller~\cite[Section~2.2, Proposition~1(b)]{Kel02a} that is stated below. A proof of Keller's theorem that utilises the calculus of Massey products was announced by Minamoto in~\cite{Min16}; our proof should have some similarities with his. \begin{theo*}[Keller] Let $\AA$ be a $\kk$-linear abelian length category~\cite{KV18} with only finitely many pairwise non-isomorphic simple objects, for example the category of finite-dimensional modules over a finite-dimensional algebra. Let $S_1,\dots,S_n$ be a complete set of representatives of the simple objects in $\AA$ and set ${S\coloneqq S_1\oplus\cdots\oplus S_n}$. Then, the Yoneda algebra $\Ext{S}{S}$ is generated by its homogeneous components of cohomological degrees $0$ and $1$ as an $\A$-algebra. \end{theo*} \begin{rema*} Since the abelian category $\AA$ is not assumed to have any non-zero projective or injective objects, we interpret the Yoneda algebra $\Ext[\AA]{S}{S} $ in terms of Yoneda equivalence classes of exact sequences~\cite{Yon60}. The results in~\cite[Section~III.3]{Ver96} (see in particular paragraph~III.3.3.2 therein) readily imply that we can identify the Yoneda algebra with the graded algebra $\bigoplus_{k\geq0}\Hom[]{S}{S[k]}$ of endomorphisms of $S$ in the derived category of $\AA$. Thus, we may and we will identify minimal $\A$-algebra structures on the Yoneda algebra with those of the latter graded algebra. \end{rema*} \begin{rema*} The isomorphism~\eqref{eq:example} shows that the conclusion of the theorem may fail if the Yoneda algebra $\Ext{S}{S}$ is considered as a graded algebra only. \end{rema*} \begin{proof}[Proof of the theorem] We use freely the theory of $\A$-categories~\cite{Kel01,Lef03}, as well as the theory of differential graded (=DG) categories and their derived categories~\cite{Kel94,Kel06}. We also assume familiarity with the Auslander--Reiten (=AR) theory of Nakayama algebras, see for example~\cite[Chapter~V]{ASS06}. Let $d\geq1$ and $\class{\delta}\in\Ext{S}{S}$ a Yoneda class represented by an exact sequence \begin{equation} \label{eq:delta} 0\to S_a\to M_1\to M_2\to\cdots\to M_{d+1}\to S_b\to0 \end{equation} between some simple objects; notice that we identify $\Ext{S}{S}=\bigoplus_{i,j=1}^n\Ext{S_i}{S_j}$. Below we prove that $\delta$ is generated by Yoneda classes of degrees $1$ and $d$ under the operations $m_k^{\AA}$ with $k\leq\len{M_1}$. The theorem then follows by induction on $d\geq1$. A straightforward inductive argument shows that the exact sequence \eqref{eq:delta} can be extended, by first choosing composition series of $N_i\coloneqq\operatorname{img}(M_i\to M_{i+1})$, to a commutative diagram of the following form in which all squares are bicartesian, the apparent sequences at the bottom are short exact, and all of the objects along the bottom of the diagram are simple: \begin{center} \begin{resizebox}{\textwidth}{!}{% \begin{tikzpicture}[rotate=-45] \node (000) at (0,0) {$S_a$};% \node (001) at (0,1) {$\bullet$};% \node (002) at (0,2) {$\bullet$};% \node (003) at (0,3) {$\bullet$};% \node (004) at (0,4) {$\bullet$};% \node (005) at (0,5) {$M_1$}; \draw[>->] (000)--(001);% \draw[>->] (001)--(002);% \draw[>->] (002)--(003);% \draw[>->] (003)--(004);% \draw[>->] (004)--(005);% \begin{scope}[shift={(1,1)}] \node (100) at (0,0) {$\bullet$};% \node (101) at (0,1) {$\bullet$};% \node (102) at (0,2) {$\bullet$};% \node (103) at (0,3) {$\bullet$};% \node (104) at (0,4) 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\draw[>->] (1500)--(1501);% \draw[>->] (1501)--(1502);% \end{scope} \begin{scope}[shift={(16,16)}] \node (1600) at (0,0) {$\bullet$};% \node (1601) at (0,1) {$\bullet$};% \draw[>->] (1600)--(1601);% \end{scope} \begin{scope}[shift={(17,17)}] \node (1700) at (0,0) {$S_b$};% \end{scope} \draw[->>] (001)--(100);% \draw[->>] (002)--(101);% \draw[->>] (003)--(102);% \draw[->>] (004)--(103);% \draw[->>] (005)--(104);% \draw[->>] (101)--(200);% \draw[->>] (102)--(201);% \draw[->>] (103)--(202);% \draw[->>] (104)--(203);% \draw[->>] (105)--(204);% \draw[->>] (106)--(205);% \draw[->>] (107)--(206);% \draw[->>] (108)--(207);% \draw[->>] (109)--(208);% \draw[->>] (1010)--(209);% \draw[->>] (201)--(300);% \draw[->>] (202)--(301);% \draw[->>] (203)--(302);% \draw[->>] (204)--(303);% \draw[->>] (205)--(304);% \draw[->>] (206)--(305);% \draw[->>] (207)--(306);% \draw[->>] (208)--(307);% \draw[->>] (209)--(308);% \draw[->>] (301)--(400);% \draw[->>] (302)--(401);% \draw[->>] (303)--(402);% \draw[->>] 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(802)--(901);% \draw[->>] (803)--(902);% \draw[->>] (804)--(903);% \draw[->>] (805)--(904);% \draw[->>] (806)--(905);% \draw[->>] (807)--(906);% \draw[->>] (808)--(907);% \draw[->>] (901)--(1000);% \draw[->>] (902)--(1001);% \draw[->>] (903)--(1002);% \draw[->>] (904)--(1003);% \draw[->>] (905)--(1004);% \draw[->>] (906)--(1005);% \draw[->>] (907)--(1006);% \draw[->>] (1001)--(1100);% \draw[->>] (1002)--(1101);% \draw[->>] (1003)--(1102);% \draw[->>] (1004)--(1103);% \draw[->>] (1005)--(1104);% \draw[->>] (1006)--(1105);% \draw[->>] (1101)--(1200);% \draw[->>] (1102)--(1201);% \draw[->>] (1103)--(1202);% \draw[->>] (1104)--(1203);% \draw[->>] (1105)--(1204);% \draw[->>] (1201)--(1300);% \draw[->>] (1202)--(1301);% \draw[->>] (1203)--(1302);% \draw[->>] (1204)--(1303);% \draw[->>] (1205)--(1304);% \draw[->>] (1301)--(1400);% \draw[->>] (1302)--(1401);% \draw[->>] (1303)--(1402);% \draw[->>] (1304)--(1403);% \draw[->>] (1401)--(1500);% \draw[->>] (1402)--(1501);% \draw[->>] (1403)--(1502);% \draw[->>] (1501)--(1600);% \draw[->>] (1502)--(1601);% \draw[->>] (1601)--(1700);% \draw [ thick, decoration={ brace, mirror, raise=0.5cm }, decorate ] (0,0) -- (5,5) node [pos=0.5,anchor=north,yshift=-0.55cm] {$\ell=\operatorname{len}(M_1)$}; \end{tikzpicture} }% \end{resizebox} \end{center} Consider now the Nakayama algebra $B=\kk Q_B/I$ with Gabriel quiver \[ \textstyle Q_B\colon\quad 1\to2\to\dots\to p,\qquad p=\sum_{i=1}^{d+1}\len{M_i}-\sum_{j=1}^d\len{N_j}, \] and Kupisch series \begin{align*} (&1,2,\dots,\len{M_1},\len{N_1}+1,\len{N_1}+2,\dots,\len{M_2},\\ &\dots,\len{N_{d-1}}+1,\len{N_{d-1}}+2,\dots,\len{M_d},\len{M_{d+1}}). \end{align*} Observe that the above commutative diagram is indexed by the AR quiver of the category $\mod{B}$ of finite-dimensional (right) $B$-modules. In other words, we can extend the exact sequence \eqref{eq:delta} to an exact functor $F\colon\mod{B}\to\AA$ that sends the simple $B$-modules to simple objects of $\AA$. The exact functor $F$ lifts to an $\A$-functor \[ \tilde{\F}\colon\dg{\Db{\mod{B}}}\to\dg{\Db{\AA}} \] between the corresponding bounded derived DG categories, for example because $\dg{\Db{\mod{B}}}$ is the pre-triangulated hull of $\mod{B}$~\cite[Theorem~6.1 and Example~6.2]{Che23}. The many-objects version of the Homotopy Transfer Theorem yields minimal models of $\dg{\Db{\mod B}}$ and $\dg{\Db{\AA}}$ so that we may consider the induced $\A$-functor \[ \begin{tikzcd} H^*(\dg{\Db{\mod{B}}})\rar[dotted]{\F}\dar[swap]{\wr}&H^*(\dg{\Db{\AA}})\\ \dg{\Db{\mod{B}}}\rar{\tilde{\F}}&\dg{\Db{\AA}}\uar[swap]{\wr} \end{tikzcd}\qquad \begin{tikzcd} H^*(\dg{\Db{\mod{B}}})\rar[dotted]{\F_1}\dar[swap]{\mathrm{id}}&H^*(\dg{\Db{\AA}})\\ H^*(\dg{\Db{\mod{B}}})\rar{H^*(\tilde{\F})}&H^*(\dg{\Db{\AA}})\uar[swap]{\mathrm{id}} \end{tikzcd} \] We obtain, in particular, an $\A$-morphism \[ \F\colon\Ext[B]{S}{S}\to\Ext{S}{S}, \] where the Yoneda algebras are now endowed with minimal $\A$-algebra structures. We claim that the underlying morphism of graded algebras \begin{equation} \label{eq:F1(gamma)} \F_1\colon\Ext[B]{S}{S}\to\Ext{S}{S},\qquad \gamma\longmapsto\delta, \end{equation} maps the Yoneda class $\gamma$ of an augmented minimal projective resolution of the simple $B$-module concentrated at the vertex $p$ of the quiver $Q_B$ to the class $\class{\delta}$. Indeed, $\gamma=\alpha_{d+1}\alpha_2\cdots\alpha_1$ is the product of Yoneda classes of short exact sequences between certain indecomposable $B$-modules and, therefore, \[ \F_1(\gamma)=\F_1(\alpha_{d+1}\alpha_d\cdots\alpha_1)=\F_1(\alpha_{d+1})\F_1(\alpha_d)\cdots\F_1(\alpha_1)=\delta. \] Here, the classes $\F_1(\alpha_i)$, $i=1,\dots,d+1$, satisfy the equality on the right-hand side by construction since the restriction of the graded functor $\F_1\colon H^*(\dg{\Db{\mod{B}}})\to H^*(\dg{\Db{\AA}})$ to the graded subcategories spanned by $\mod{B}$ and $\AA$, respectively, is induced by the exact functor $F\colon\mod{B}\to\AA$ above. Since Nakayama algebras are monomial algebras, an explicit description of a minimal model of the Yoneda algebra $\Ext[B]{S}{S}$ is available due to independent work of Chuang and King (unpublished) and of Tamaroff~\cite[Theorem~4.9]{Tam21}.\footnote{The computation of the explicit product that we need is straightforward using the HTT; see also~\cite{Her21}.} In particular, we can assume that\footnote{Compare with~\cite[Theorem~1]{Min16} where the equivalent of the Massey product $\langle\eta_1,\dots,\eta_{\ell-1},\eta\rangle\ni\pm\gamma$ appears.} \begin{equation} \label{eq:m_ell} m_\ell^B(\eta_1\otimes\eta_2\otimes\dots\otimes\eta_{\ell-1}\otimes\eta)=\gamma,\qquad\ell=\len{M_1}, \end{equation} where $\eta_i\in\Ext[B]<1>{S_{i+1}}{S_i}\cong\kk$ is the Yoneda class of an AR sequence and $\eta\in\Ext[B]{S_p}{S_\ell}\cong\kk$ is the Yoneda class of an exact sequence of the form \[ 0\to S_\ell\to M_2'\to\cdots\to M_{d+1}\to S_p\to0, \] with $M_2'$ the injective hull of the simple $B$-module $S_{\ell}$ and $M_{d+1}$ the projective cover of $S_p$, as indicated in the above diagram that we now interpret as the AR quiver of $\mod{B}$. Moreover, an elementary computation shows that \begin{align} \label{eq:m-vanishing} m_{j-i+1}^B(\eta_i\otimes \eta_{i+1}\otimes \cdots\otimes \eta_j)&\in\Ext[B]<2>{S_{j+1}}{S_i}=0 \intertext{whenever $1\leq i{S}{S}})+\omega, \end{align*} where \[ \omega=\sum_{\substack{r+1+t=k>1\\r+s+t=\ell}}\pm\F_k(\underbrace{\phantom{a}\cdots\phantom{a}}_{=0})+\sum_{\substack{2\leq{k}<\ell\\i_1+\cdots+i_k=\ell}} \pm m_k^{\AA}(\underbrace{\phantom{abc}\cdots\phantom{abc}}_{\in\Ext<1,d>{S}{S}}). \] To argue the vanishing of the first term we use that the inputs involve one the higher products \eqref{eq:m-vanishing}-\eqref{eq:m-vanishing-x} that we know vanish, and for the condition on the degrees of the inputs in the second term we note that \begin{align*} \F_{j-i+1}(\eta_i\otimes\cdots\otimes\eta_j)&\in\Ext<1>{S}{S}&&1\leq i{S}{S}&&1