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\title{Multiplicative bases and deformations of representation-finite algebras}
\author[H.\ Enomoto]{Haruhisa Enomoto}
\address{Parakeet Inc., Japan}
\email{haruhisa.enomoto.math@gmail.com}
\subjclass[2020]{16G10, 16G20, 16G60}
\keywords{multiplicative basis, representation-finite algebra,
Auslander--Reiten theory, mesh category, deformation}
\date{October 16, 2026}

\begin{document}

\begin{abstract}
Bautista, Gabriel, Roiter, and Salmer\'on proved that every
finite-dimensional representation-finite algebra over an algebraically
closed field admits a multiplicative basis, in which products of basis
elements are zero or basis elements. Outside characteristic two, they also
proved that such an algebra is standard: its category of indecomposable
modules is presented by its Auslander--Reiten quiver and mesh relations. We
give another proof of both theorems. For a basic algebra, grading morphisms
between indecomposable modules by powers of the categorical radical and
restricting to projectives yields its standard form, a standard algebra with
the same Auslander--Reiten quiver. After constructing a multiplicative basis
of the standard form, we recover the original algebra by filtered
deformation. Positive-degree Hochschild cohomology embeds into cokernels
measuring errors in mesh relations. In characteristic two, lifting
derivations modulo four produces cocycles supported on zero basis products;
the resulting polynomial family contains the original algebra with
coefficients zero or one.
\end{abstract}

\maketitle
\tableofcontents
\clearpage

\section{Introduction}
\label{sec:introduction}

A finite-dimensional algebra is \emph{representation-finite} if it has only
finitely many isomorphism classes of indecomposable finite-dimensional
modules. Although this condition is module-theoretic, it imposes strong
combinatorial restrictions on the multiplication of the algebra itself: over
an algebraically closed field, every representation-finite algebra admits a
\emph{multiplicative basis}, a basis in which the product of any two elements
is either zero or another basis element (Theorem~A below). Consequently, the
entire algebra multiplication is encoded by a finite combinatorial table
with structure constants zero or one.

A companion structural theorem governs the category of indecomposable
modules. A basic representation-finite algebra is called \emph{standard} if
one can choose representatives of the irreducible morphisms so that every
morphism between indecomposable modules is a linear combination of paths in
these representatives, and all relations among such paths follow from the
quadratic mesh relations induced by almost split sequences. Outside
characteristic two, every basic representation-finite algebra over an
algebraically closed field is standard (Theorem~B below). This structure
theory was established in the landmark work of Bautista, Gabriel, Roiter,
and Salmer\'on \cite{BGRS}, abbreviated BGRS below, building on the
construction by Bretscher and Gabriel \cite{BretscherGabriel} of the
\emph{standard form} of a given algebra: a standard algebra having the same
Auslander--Reiten quiver.

Because of the complexity of the original combinatorial case analysis, finding
a more transparent proof has remained a longstanding problem. Bongartz
described the BGRS paper as ``beautiful but hard to read''
\cite[Section~1]{BongartzSurvey} and surveyed subsequent simplifications and
extensions; more recently, Ringel remarked that ``any publication which
provides help, provides improvement would be strongly appreciated''
\cite[Section~3]{RingelVossieck}.

In this paper we give an alternative proof of both classical theorems by
constructing the standard form directly from the radical filtration of the
category of indecomposable modules and studying how its multiplication
deforms to recover the original algebra.

\subsection{The classical results revisited}
\label{subsec:intro-results}

We begin with the existence theorem for multiplicative bases compatible with
primitive idempotents and powers of the Jacobson radical.

\begin{definition}
\label{def:normed-multiplicative-basis}
A basis $\B$ of a finite-dimensional algebra $A$ is \emph{multiplicative} if
$bc\in\B\cup\{0\}$ for all $b,c\in\B$. It is \emph{normed} if it contains
a complete set of primitive orthogonal idempotents and
$\B\cap(\rad A)^n$ is a basis of $(\rad A)^n$ for every $n\geq1$.
\end{definition}

The existence assertion, including the normed condition, is due to BGRS
\cite[Sections~1.2 and~1.12]{BGRS}.

\begin{theoremA}[= Theorem~\ref{thm:final-basis}]
Every finite-dimensional representation-finite algebra over an algebraically
closed field has a normed multiplicative basis.
\end{theoremA}

Using matrix units to account for the multiplicities of indecomposable
projective summands, Lemma~\ref{lem:nonbasic-basis} reduces Theorem~A
immediately to basic algebras. We therefore focus on basic algebras in what
follows.

Let $A$ be a basic finite-dimensional representation-finite algebra over an
algebraically closed field $k$, let $\ind(\modu A)$ be a skeletal category
of indecomposable finite-dimensional right $A$-modules, and let $\Gamma_A$
be its Auslander--Reiten quiver. Writing compositions of morphisms in path
order from left to right, the components of an almost split sequence
\[
 0\longrightarrow\tau X\xrightarrow{(a_i)_i}
 \bigoplus_iY_i\xrightarrow{(b_i)_i}X\longrightarrow0
\]
satisfy $\sum_i a_i b_i=0$. Let $\JJ$ denote the categorical Jacobson radical
of $\ind(\modu A)$. Choosing a basis of each space $\JJ(X,Y)/\JJ^2(X,Y)$ of
irreducible morphisms—which is at most one-dimensional when $A$ is
representation-finite (Lemma~\ref{lem:square-free})—and expressing the
components $a_i$ and $b_i$ modulo $\JJ^2$ in these bases yields a quadratic
mesh relation at each nonprojective vertex $X$. The \emph{mesh category}
$k(\Gamma_A)$ has the vertices of $\Gamma_A$ as objects and paths in
$\Gamma_A$ modulo these quadratic mesh relations as morphisms. Up to
isomorphism fixing the objects, $k(\Gamma_A)$ is independent of the chosen
bases, and its mesh coefficients can all be rescaled to one
(Lemma~\ref{lem:unit-mesh-coefficients}).

\begin{definition}
\label{def:standardness}
A basic finite-dimensional representation-finite algebra $A$ over an
algebraically closed field $k$ is \emph{standard} if there is a $k$-linear equivalence
\[
 k(\Gamma_A)\simeq\ind(\modu A)
\]
sending each vertex of $\Gamma_A$ to the corresponding indecomposable module.
\end{definition}

\begin{theoremB}[= Corollary~\ref{cor:away-from-two-basis}]
Every basic finite-dimensional representation-finite algebra over an
algebraically closed field of characteristic different from two is standard.
\end{theoremB}

In characteristic two, standardness can fail
\cite[Section~3.7]{BongartzSurvey}, even though Theorem~A remains valid.
Example~\ref{ex:nonstandard-pair} exhibits this distinction in dimension
ten: starting with a standard algebra, we replace a zero basis product
by a nonzero basis element and obtain a nonisomorphic, nonstandard algebra
whose basis is still multiplicative.

To treat all characteristics uniformly, we construct in
Section~\ref{sec:graded-category} a basic representation-finite algebra
$A_0$ whose category of indecomposable modules is equivalent to
$k(\Gamma_A)$: this is the \emph{standard form} of $A$, introduced
classically by Bretscher and Gabriel \cite{BretscherGabriel}. Specifically,
write $P_x=e_xA$ for the indecomposable projective modules associated to
a complete set of primitive orthogonal idempotents $e_x$. Transporting the
categorical radical filtration to $A$ along the left-multiplication
isomorphisms $A(x,y)=e_xAe_y\cong\Hom_A(P_y,P_x)$, we set
$F^nA(x,y)=\JJ^n(P_y,P_x)$; thus $F^nA(x,y)$ consists of sums of composites
of $n$ radical maps from $P_y$ to $P_x$ through arbitrary indecomposable
intermediate modules. In particular, $(\rad A)^n\subseteq F^nA$, and the
inclusion can be strict. We define $A_0=\gr_F A$ as the associated graded
algebra of this filtration. Choosing a filtered vector-space splitting
identifies the underlying vector spaces of $A$ and $A_0$, so that their
multiplications agree in leading degree and differ only by terms of strictly
higher degree; moreover, $A$ is standard if and only if $A\cong A_0$
(Corollary~\ref{cor:graded-standard-form}).

\subsection{The idea of the proof}
\label{subsec:intro-method}

To construct a multiplicative basis of $A_0$, we group the nonzero elements
of each corner $e_xA_0e_y$ into \emph{rays}: two elements belong to the
same ray if they differ by multiplication by units in the local endomorphism
rings $e_xA_0e_x$ and $e_yA_0e_y$. In the graded algebra $A_0$, each nonzero
ray contains the nonzero vectors of a unique homogeneous line, so choosing
one nonzero homogeneous representative per ray gives a basis of $A_0$ in
which every product of two basis elements is either zero or a nonzero scalar
multiple of a basis element. In Section~\ref{sec:scalar-normalization} we
normalize these scalars to one by lifting the ray category to the universal
cover of its walk complex, following the covering methods of Bongartz and
Gabriel \cite{BongartzGabriel} and Bretscher and Gabriel
\cite{BretscherGabriel}.

We next compare the original algebra $A$ with its standard form $A_0$.
Because $A$ is a filtered deformation of $A_0$, any obstruction to an
isomorphism $A\cong A_0$ is governed degree by degree by the positive-degree
Hochschild cohomology $\HH^2(A_0)_{>0}$. On the categorical side, choosing
filtered lifts of the irreducible morphisms in $\ind(\modu A)$ produces
higher-degree error terms in the quadratic mesh relations, and adjusting the
lifts modifies these errors by a linear coboundary operator on the periodic
$\tau$-orbits. Outside characteristic two, this mesh coboundary operator is
surjective in positive degrees, so the irreducible morphisms can be corrected
inductively until the mesh relations hold on the nose; this identifies $A$
with $A_0$ and proves Theorem~B.

In characteristic two, the mesh cokernels need not vanish, and $A$ need not
be isomorphic to $A_0$. Instead, we show that $\HH^2(A_0)_{>0}$ embeds into
the mesh cokernels (Proposition~\ref{prop:hochschild-mesh-restriction}) and
that the entire space of positive-degree cohomology classes is spanned by
two-cocycles supported exclusively on pairs of basis elements whose product
in $A_0$ is zero. We construct these cocycles in the spirit of the classical
zero-product modifications of \cite[Sections~9.2 and~9.5--9.7]{BGRS} by an
integral Bockstein mechanism: we lift a positive-degree derivation of $A_0$
from characteristic two to a cochain over $\Z/4\Z$, evaluate its failure to
satisfy the Leibniz rule, and divide that failure by two. When the lifted
cochain itself satisfies the Leibniz rule on every nonzero product of basis
elements, this failure vanishes on all nonzero products, so the resulting
two-cocycle is supported exclusively on zero products.
Example~\ref{ex:bockstein-pair} illustrates this construction on our
ten-dimensional example.

Constructing such lifts in general requires the local factorization geometry
of projective rays developed in
Sections~\ref{sec:local-ray-geometry}--\ref{sec:integral-partial-lifts} and
Appendix~\ref{app:local-rays}. For each primitive residual weight $\chi$,
the derivation vanishes on identity and unshiftable rays and sends each
shiftable nonidentity ray $r\colon x\to y$ to a multiple of its
$\chi$-successor $r^+$, which equals one or both of the endpoint products
$\widehat q_xr$ and $r\widehat q_y$, where $\widehat q=\sum_x\widehat q_x$ is
the sum of the local generators of weight $\chi$. Passing to the centralizer
$C_\chi=\operatorname{Cent}_{A_0}(\widehat q)$ and its quotient $B_\chi$ by
the annihilator of $\widehat q$ reduces the two-sided part of the
nonzero-product Leibniz equations to an additive weight equation on the
smaller algebra $B_\chi$, while the exclusion of four-point crowns in the
faithful projective and injective support posets forces the edge labels of
the bipartite graph of one-sided arrow variables to be constant on each
connected component, allowing the solution to be extended to all rays.
Proceeding by simultaneous induction on $\dim_k A$ (set up in
Subsection~\ref{subsec:table-induction}), we assume for strictly smaller
algebras both Theorem~A and the freeness of the abelian group of additive
ray weights. Once these cocycles supported on zero products span
$\HH^2(A_0)_{>0}$, successive filtered changes of basis place $A$ inside the
polynomial deformation family they parameterize
(Theorem~\ref{thm:finite-family-normalization}), and a final diagonal
rescaling reduces the nonzero parameters to one, establishing Theorem~A and
closing the induction on the ray weight group
(Theorem~\ref{thm:final-standardness}).

\subsection{Relation to earlier proofs}
\label{subsec:intro-comparison}

Bretscher and Gabriel \cite[Section~3.1]{BretscherGabriel} construct the
standard form from the projective vertices in the mesh category. We recover
this description by proving that the category obtained from the radical
grading is equivalent to the category of indecomposable modules over the
opposite endomorphism algebra of its projective objects
(Propositions~\ref{prop:graded-realization}
and~\ref{prop:projective-evaluation}). For the multiplicative basis of $A_0$,
we combine the mesh covering of Bongartz and Gabriel
\cite[Section~4]{BongartzGabriel} with the extension of bases from
\cite[Section~2.6]{BretscherGabriel}. These three
ingredients—the standard form, the mesh covering, and the extension of
bases—are classical; Part~\ref{part:graded-form} assembles them through the
radical grading and projective evaluation, while the filtered deformation and
integral lifting arguments in Parts~\ref{part:mesh-deformations}
and~\ref{part:characteristic-two} provide the new route to the basis theorem.

In characteristic two, BGRS \cite[Sections~9.2--9.7]{BGRS} obtain the
nonstandard algebras by replacing specified zero products in the standard
multiplication table with nonzero basis elements. In our framework these
modifications arise as Hochschild two-cocycles supported on zero products,
and the modulo-four Bockstein construction together with the inductive
lifting argument shows that their cohomology classes span $\HH^2(A_0)_{>0}$.
Riedtmann \cite[Sections~3.3--3.9]{Riedtmann1984} likewise uses successive
degree-by-degree corrections of irreducible morphisms and identifies a
coefficient-parity obstruction in characteristic two.
Table~\ref{tab:method-comparison} summarizes how each step compares with the
classical proofs.

\begin{table}[htbp]
\centering
\small
\renewcommand{\arraystretch}{1.15}
\begin{tabularx}{\textwidth}{@{}>{\raggedright\arraybackslash}p{.19\textwidth}
 >{\raggedright\arraybackslash}X >{\raggedright\arraybackslash}X@{}}
\toprule
Task & Classical constructions & Present organization \\
\midrule
Construct the standard form &
Take the endomorphism algebra of the projective vertices of the mesh
category \cite{BretscherGabriel}; alternatively, build an algebra from the
multiplication table of basis elements up to scalars \cite{BGRS}. &
Restrict the radical grading of $\ind(\modu A)$ to projective objects,
then normalize the basis products on a covering category. \\
\addlinespace
Prove standardness ($\charac k\ne2$) &
Compare distinct paths representing the same morphism \cite{BGRS}, or correct the mesh
relations on stable components step by step \cite{Riedtmann1984}. &
Eliminate higher-degree error terms in the mesh relations degree by degree
by modifying the chosen irreducible morphisms. \\
\addlinespace
Treat characteristic two &
Replace specified zero products by nonzero basis elements in explicit normal
forms \cite{BGRS}. &
Construct cocycles supported on zero products by induction on dimension,
then recover $A$ in the resulting polynomial family. \\
\addlinespace
Exclude configurations incompatible with finite representation type &
Exhibit representation-infinite subcategories and compare paths with the
same value locally
\cite{BGRS}; see also the shorter proofs comparing such paths in
\cite{BongartzMild}. &
Bound the mesh errors on periodic components and exclude factorizations
that would produce representation-infinite algebras. \\
\bottomrule
\end{tabularx}
\caption{Comparison between the classical proofs and the present approach.}
\label{tab:method-comparison}
\end{table}

The calculations in Appendices~\ref{app:periodic-mesh}
and~\ref{app:residual-mesh} analyze the components of the Auslander--Reiten
quiver on which the translation is periodic, determining which higher-degree
errors in the mesh relations can be removed by adjusting the chosen
irreducible morphisms. This linear problem reduces to spaces of paths modulo
cyclic rotation and recovers special cases of Schedler's computations of the
integral zeroth Hochschild homology of preprojective algebras
\cite[Theorem~13.1.1]{Schedler}, together with twisted variants in which
rotation is followed by an automorphism of the underlying tree.

\phantomsection\label{subsec:reading-routes}
\phantomsection\label{subsec:intro-organization}
\noindent\textbf{Organization.}
Part~\ref{part:graded-form} constructs the standard form and proves
standardness outside characteristic two. Part~\ref{part:mesh-deformations}
reduces the remaining case to constructing suitable Hochschild cocycles;
Part~\ref{part:characteristic-two} constructs them and recovers the algebra.
The table below
indicates the sections required for each main result.

\begin{center}
\small
\renewcommand{\arraystretch}{1.15}
\begin{tabularx}{\textwidth}{@{}>{\raggedright\arraybackslash}p{.26\textwidth}
 >{\raggedright\arraybackslash}X@{}}
\toprule
Reading goal & Route and conclusion \\
\midrule
Construct the standard form &
Sections~\ref{sec:graded-category}--\ref{sec:scalar-normalization},
culminating in Corollary~\ref{cor:graded-standard-form}. \\
\addlinespace
Prove Theorems~A and~B outside characteristic two &
Continue through Section~\ref{sec:mesh-deformations} to
Corollary~\ref{cor:away-from-two-basis} (with Lemma~\ref{lem:nonbasic-basis}
covering nonbasic algebras). Appendix~\ref{app:periodic-mesh} proves
Propositions~\ref{prop:periodic-tree-model}
and~\ref{prop:mesh-correction-spaces}, whose statements suffice for the main
text. \\
\addlinespace
Prove Theorem~A in characteristic two &
After Sections~\ref{sec:graded-category}--\ref{sec:scalar-normalization},
read Subsections~\ref{subsec:mesh-derivative}
and~\ref{subsec:brick-hypothesis} together with
Sections~\ref{sec:positive-deformations}--\ref{sec:binary-families}, ending
at Theorem~\ref{thm:final-basis}. Appendices~\ref{app:periodic-mesh}--\ref{app:local-rays}
supply the proofs of the periodic calculations and local restrictions used
on this route.
\\
\bottomrule
\end{tabularx}
\end{center}

\medskip
\noindent\textbf{Conventions and notation.}
Throughout the paper, $k$ denotes an algebraically closed field unless
explicitly stated otherwise. All algebras and modules are finite-dimensional,
and modules are right modules. We assume familiarity with standard
Auslander--Reiten theory, Morita equivalence \cite{ARS}, and basic
homological algebra. We write $D_k=\Hom_k(-,k)$ for standard vector-space
duality and $\tau X$ for the starting term of the almost split sequence
ending at a nonprojective indecomposable module $X$.

For a basic algebra $A$ with a chosen complete set of primitive orthogonal
idempotents $\{e_x\}$, we set $A(x,y)=e_xAe_y$ and $P_x=e_xA$. Products of
paths and morphisms are written in path order from left to right, whereas
$\circ$ always denotes ordinary function composition: thus for morphisms
$f:X\to Y$ and $g:Y\to Z$, their path-order product is $fg=g\circ f$. An
arrow $a:x\to y$ of the quiver of $A$ lies in $A(x,y)=e_xAe_y$. Under the
canonical identification $A(x,y)\cong\Hom_A(P_y,P_x)$, write
$\operatorname{lm}_a:P_y\to P_x$ for left multiplication by $a\in A(x,y)$.
For $b\in A(y,z)$ we then have
$\operatorname{lm}_{ab}=\operatorname{lm}_a\circ\operatorname{lm}_b:P_z\to P_x$.
Because endomorphism algebras use the path-order product, we have
$A\cong\End_A(A)^{\op}$.

For a $k$-linear category $\mathcal E$, we write $\add\mathcal E$ for
its formal completion under finite direct sums and splitting of idempotents. Any $k$-linear category $C$ with
finitely many objects and finite-dimensional morphism spaces is identified
with its category algebra $\bigoplus_{i,j} C(i,j)$, where $C(i,j)$ denotes
morphisms from $i$ to $j$. The product of
$f\in C(i,j)$ and $g\in C(j,l)$ is $fg=g\circ f$; the product is zero when the morphisms are not
composable. Thus $C(i,j)$ is the $(i,j)$-corner of the
category algebra. In particular, the category algebra of the
indecomposable projective $A$-modules is $A^{\op}$. We write $\JJ$ for the categorical Jacobson radical of a
category of indecomposable modules and $\rad A$ for the Jacobson radical of
an algebra $A$.

\begin{formalization}
A Lean~4 formalization of Theorems~A and~B, built on Mathlib, is available at
\url{https://github.com/haruhisa-enomoto/multiplicative-bases-deformations}.
It proves every result it uses, including the Auslander--Reiten theory and
the results cited in our proofs. It follows the strategy of this paper, with
shorter arguments at some steps.
\end{formalization}

\begin{aiuse}
The proof was developed by GPT-6-Astra under the author's direction.
The manuscript was written with GPT-6-Astra, followed by Claude Opus~5.5
and Gemini~4 Argon. GPT-6-Astra and Claude Opus~5.5 were used for
formalization. The author directed the research and revisions and is
responsible for the mathematical content and the final manuscript.
\end{aiuse}

\clearpage
\part{The standard form and the proof outside characteristic two}
\label{part:graded-form}

In this part we construct the standard form $A_0$ of a basic
representation-finite algebra $A$ in arbitrary characteristic, equip $A_0$
with a normed multiplicative basis, and prove Theorems~A and~B when
$\charac k\ne2$.

\input{graded-category}
\input{projective-rays}
\input{scalar-normalization}
\input{mesh-deformations}

\clearpage
\part{Deformations in characteristic two}
\label{part:mesh-deformations}

In this part we assume $\charac k=2$. Viewing $A$ as a filtered
deformation of its standard form $A_0$ whose corrections strictly increase degree, we show that first-order
deformation classes of $A_0$ embed into the mesh cokernels on the
$\tau$-periodic components of $\Gamma_A$, and we reduce Theorem~A to the
construction of homogeneous Hochschild two-cocycles supported on zero
products (Proposition~\ref{prop:family-criterion}).

\input{positive-deformations}
\input{weights-and-parameters}
\input{characteristic-two-construction}

\clearpage
\part{Construction of multiplicative bases in characteristic two}
\label{part:characteristic-two}

In this final part we construct the cocycles required by
Proposition~\ref{prop:family-criterion}, form the polynomial family, and
complete the induction on $\dim_k A$ described in
Subsection~\ref{subsec:table-induction}.

\input{local-ray-geometry}
\input{primitive-ray-equations}
\input{integral-partial-lifts}
\input{integral-deformations}
\input{binary-families}
\input{main-theorems}

\addtocontents{toc}{\protect\medskip}
\clearpage
\appendix
\input{periodic-mesh-calculations}
\clearpage
\input{characteristic-two-mesh-calculations}
\clearpage
\input{local-ray-restrictions}

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\end{thebibliography}
\end{document}