\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{amsmath,amssymb,mathtools}
\usepackage[left=26mm,right=26mm,top=27mm,bottom=25mm]{geometry}
\usepackage{microtype}
\usepackage{needspace}
\usepackage[hidelinks]{hyperref}
\hypersetup{pdftitle={Another Counterexample to Little Finitistic Dimension Conjecture}}
\setlength{\parindent}{0pt}
\setlength{\parskip}{4pt plus 1pt minus 1pt}
\setlength{\emergencystretch}{1.5em}
\raggedbottom
\DeclareMathOperator{\Mod}{Mod}
\DeclareMathOperator{\Perf}{Perf}
\DeclareMathOperator{\Ind}{Ind}
\DeclareMathOperator{\Loc}{Loc}
\DeclareMathOperator{\thick}{thick}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\Rees}{Rees}
\DeclareMathOperator{\cofib}{cofib}
\DeclareMathOperator{\Free}{Free}
\DeclareMathOperator{\pd}{pd}
\newcommand{\Cat}{\mathrm{Cat}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\fd}{\mathrm{fd}}
\newcommand{\op}{\mathrm{op}}
\newcommand{\gl}{\mathfrak{gl}}
\newcommand{\slie}{\mathfrak{sl}}
\newcommand{\C}{\mathcal C}
\newcommand{\Pcat}{\mathcal P}
\newcommand{\D}{\mathcal D}

\title{Another Counterexample to\\Little Finitistic Dimension Conjecture}
\author{}
\date{}

\begin{document}
\begin{center}
  {\LARGE Another Counterexample to\\[5pt]
  Little Finitistic Dimension Conjecture}
\end{center}
\vspace{12pt}

\section*{Introduction}

This note gives a characteristic-zero construction of an algebra of infinite
little finitistic dimension, following OpenAI \cite{OAI}. Tang \cite{Tang}
gives refinements of OpenAI's construction and an extension to arbitrary fields.
The argument of Section~3 is adapted from the square-zero-extension step in
OpenAI \cite{OAI}, expressed here using a square-zero monad; see also
\cite[\S6]{Tang}. Related homological-dimension formulas for trivial extensions
were established earlier by Minamoto and Yamaura \cite{MY}.

Sections~1--2 provide a different source of the required nilpotence. Section~1
uses the theory of homogenized enveloping algebras of Le Bruyn and Smith
\cite{LS} and Le Bruyn, Smith, and Van den Bergh \cite{LSV}, together with the
graded Beilinson theorem of Minamoto and Mori \cite{MM}. Section~2 constructs
the endofunctor explicitly and computes its iterates using the classical
Clebsch--Gordan decomposition and the eigenvalues of the Casimir operator.

\section*{Construction}

Let $k$ be a field of characteristic zero, and let
\[
  \Cat_k=\Mod_{\Perf(k)}(\Cat^{\mathrm{perf}})
\]
be the category of small idempotent-complete stable $k$-linear categories and
exact $k$-linear functors. A quotient means a quotient in $\Cat_k$.
All categories are $\infty$-categories, and $\Hom$ denotes the $k$-module of maps.

\section{A smooth proper category from filtered \texorpdfstring{$\gl_2$}{gl2}-modules}

Give $U(\gl_2)$ its Poincar\'e--Birkhoff--Witt filtration $F_\bullet$, where
$F_dU(\gl_2)$ is spanned by products of at most $d$ elements of $\gl_2$, and put
\[
  B=\Rees_F U(\gl_2)
    =\bigoplus_{d\geq0}F_dU(\gl_2)\,s^d
    \subseteq U(\gl_2)[s].
\]
Write $\Mod^{\mathbf Z}(B)$ for the stable category of graded left
$B$-modules, and $\Perf^{\mathbf Z}(B)$ for its compact objects. Let
\[
  \Perf^{\mathbf Z}_{\fd}(B)
  =\left\{M\in\Perf^{\mathbf Z}(B):
       \bigoplus_{r\in\mathbf Z}M_r\in\Perf(k)\right\}.
\]
Thus finite-dimensionality is total over the grading and homotopy
degrees. Define
\[
  \boxed{\C=\Perf^{\mathbf Z}(B)\big/\Perf^{\mathbf Z}_{\fd}(B)},
  \qquad \Pcat=\Ind(\C).
\]
We use $M(r)_d=M_{d+r}$ for grading shifts and $[r]$ for suspension.

For computations, write $\gl_2=\slie_2\oplus kt$, with $t$ the identity matrix,
and choose the standard generators $E,F,H$ of $\slie_2$. We use the same letters
for their degree-one lifts in $B$. The algebra $B$ is generated by
$E,F,H,t,s$, with $s,t$ central and relations
\begin{equation}\label{eq:relations}
  HE-EH=2sE,\qquad HF-FH=-2sF,\qquad EF-FE=sH.
\end{equation}
The Poincar\'e--Birkhoff--Witt theorem gives that $B$ is a domain, $s$ is regular,
and $B/(s)=k[E,F,H,t]$. The homogenization of $U(\slie_2)$ is noetherian,
Koszul, and Artin--Schelter regular of dimension and Gorenstein parameter four
\cite[\S2]{LS} and \cite[Corollary 2.7 and Remark 2.8]{LSV}. Adjoining the central degree-one
variable $t$ gives these properties for $B$ with dimension and parameter five.

In particular, the $k(r)$ are compact, where $k=B/B_{\geq1}$, and
\[
  \Perf^{\mathbf Z}_{\fd}(B)=\thick\{k(r):r\in\mathbf Z\},
  \qquad
  \Pcat\simeq\Mod^{\mathbf Z}(B)\big/\Loc\{k(r):r\in\mathbf Z\}.
\]
Here $\thick$ denotes closure under finite stable operations and retracts, and
$\Loc$ also allows all colimits. When working in $\Pcat$, we use the same notation for a
graded $B$-module and its image in the quotient. The quotient functor preserves compact objects.

\Needspace{12\baselineskip}
The graded Beilinson theorem gives a compact generator
\[
  p=\bigoplus_{i=0}^{4}B(i)
\]
of $\Pcat$, with
\begin{equation}\label{eq:beilinson}
  \Hom_{\C}\bigl(B(i),B(j)\bigr)\simeq B_{j-i}
  \qquad (j-i\geq-4),
\end{equation}
for $i,j\in\mathbf Z$, where the right side is discrete and $B_d=0$ for $d<0$
\cite[Propositions 4.3--4.4 and Theorem 4.12]{MM}.
Thus $\End_{\C}(p)$ is a finite-dimensional directed algebra of finite global
dimension, and $\C$ is smooth and proper over $k$.

For $n\geq0$, let $V_n=\Sym^n(k^2)$ with its standard $\gl_2$-action $\rho_n$;
in particular, $t$ acts by $n$. For $m\geq0$, define
\[
  M_m=V_m\otimes_k k[s],
\]
with $V_m$ in degree zero and
\[
  a_{M_m}=s\rho_m(a)\qquad(a\in\gl_2).
\]
Thus $t_{M_m}=ms$. These finitely generated graded $B$-modules are perfect,
so $M_m\in\C$. Multiplication by $s$ gives an equivalence in $\C$
\begin{equation}\label{eq:s-local}
  s:M_m(-1)\xrightarrow{\ \sim\ }M_m,
\end{equation}
since its cofiber before taking the quotient is a finite direct sum of copies
of $k$.

The $s$-local object $M_m[s^{-1}]=V_m\otimes_k k[s,s^{-1}]$ is right
orthogonal to every $k(r)$, and
$M_m[s^{-1}]\simeq M_m$ in $\Pcat$ by \eqref{eq:s-local}.
Consequently,
\[
  \Hom_{\Pcat}\bigl(B(i),M_m\bigr)
  \simeq\Hom_{\Mod^{\mathbf Z}(B)}\bigl(B(i),M_m[s^{-1}]\bigr)
  \simeq V_m,
\]
and hence
\begin{equation}\label{eq:evaluation}
  \Hom_{\C}(p,M_m)\simeq V_m^{\oplus5}.
\end{equation}
In particular, $M_m\ne0$, and these mapping $k$-modules are discrete and
finite-dimensional.

\section{The Casimir operator gives unbounded nilpotence orders}

Tensoring with the dual standard representation $V_1^\vee$ defines an
endofunctor $F$ on graded $B$-modules. Explicitly, its underlying graded
$k$-module is $V_1^\vee\otimes_k M$, with
\begin{equation}\label{eq:tensor}
  \begin{aligned}
    a_{F(M)}&=\rho_1^\vee(a)\otimes s_M+1\otimes a_M
        &&(a\in\gl_2),\\
    s_{F(M)}&=1\otimes s_M.
  \end{aligned}
\end{equation}
These actions satisfy \eqref{eq:relations}. Since $t$ acts by $-1$ on
$V_1^\vee$, the central parameter transforms by
\[
  t_{F(M)}=1\otimes(t_M-s_M).
\]
The functor $F$ is exact, preserves colimits, and sends $k(r)$ to $k(r)^{\oplus2}$,
so it descends to $\Pcat$.

\Needspace{8\baselineskip}
The graded $B$-linear map
\[
  B\otimes_k V_1^\vee\longrightarrow F(B),
  \qquad b\otimes v\longmapsto b\cdot(v\otimes1)
\]
is an isomorphism modulo $s$. Both modules are bounded below in degree and $s$-torsion-free, so induction on degree proves that the
map is an isomorphism. Consequently,
$F(B(j))\simeq B(j)^{\oplus2}$, and $F$ preserves compact objects.

\Needspace{11\baselineskip}
The element
\[
  u=H^2+2sH+4FE-t(t+2s)\in B_2
\]
is central. Its image modulo $s$ is $H^2+4FE-t^2\ne0$, so $u$ is a
non-zero-divisor. Define an exact colimit-preserving endofunctor of $\Pcat$ by
\begin{equation}\label{eq:K}
  K(X)=\cofib\bigl(u:F(X)\longrightarrow F(X)(2)\bigr).
\end{equation}
It preserves compact objects and therefore restricts to $\C$.
For $0\leq i,j\leq4$, formula \eqref{eq:beilinson} gives
\begin{equation}\label{eq:K-evaluation}
  \begin{aligned}
  \Hom_{\C}\bigl(B(i),KB(j)\bigr)
    &\simeq\cofib\bigl(u:B_{j-i}^{\oplus2}
                    \longrightarrow B_{j+2-i}^{\oplus2}\bigr)\\
    &\simeq\bigl(B/(u)\bigr)_{j+2-i}^{\oplus2}.
  \end{aligned}
\end{equation}
In particular, $\Hom_{\C}(p,Kp)$ is discrete and finite-dimensional.

On $V_n$, the Casimir $H^2+2H+4FE$ acts by $n(n+2)$.
For $m\geq1$, the Clebsch--Gordan decomposition, as an $\slie_2$-representation,
is
\[
  V_1^\vee\otimes_k V_m\simeq V_{m+1}\oplus V_{m-1}.
\]
On $F(M_m)$, the parameter $t$ acts by $(m-1)s$, so $u$ acts on these two
summands by
\[
  \begin{array}{c|c}
    \text{summand}&u\\ \hline
    V_{m+1}\otimes k[s]&\bigl((m+1)(m+3)-(m-1)(m+1)\bigr)s^2=4(m+1)s^2\\[2pt]
    V_{m-1}\otimes k[s]&0.
  \end{array}
\]
The first cofiber is finite-dimensional and vanishes in $\Pcat$. The second
summand is $M_{m-1}$, so its cofiber becomes
$M_{m-1}(2)\oplus M_{m-1}[1]$, which is
$M_{m-1}\oplus M_{m-1}[1]$ by
\eqref{eq:s-local}. For $m=0$, only the first summand occurs, with $u=4s^2$.
Thus
\begin{equation}\label{eq:lowering}
  K(M_0)=0,\qquad
  K(M_m)\simeq M_{m-1}\otimes_k\bigl(k\oplus k[1]\bigr)
  \quad(m\geq1).
\end{equation}
For $0\leq j\leq m$, formula \eqref{eq:lowering} gives
\begin{equation}\label{eq:iterates}
  K^jM_m\simeq M_{m-j}\otimes_k\bigl(k\oplus k[1]\bigr)^{\otimes j}.
\end{equation}
These objects are compact, and
\begin{equation}\label{eq:nilpotence}
  K^{m+1}M_m=0,\qquad K^mM_m\ne0.
\end{equation}

\Needspace{14\baselineskip}
\section{Transforming nilpotence to projective dimension using a square-zero monad}

Form the square-zero monad
\[
  T=\id_{\Pcat}\oplus K
\]
and its category of modules $\D=\Mod_T(\Pcat)$. Write $\Free$ and $U$ for the
free and forgetful functors, and let $z:\Pcat\to\D$ equip objects with the zero
action. Put $L_m=zM_m$.
For every $y\in\Pcat$,
\begin{equation}\label{eq:maps}
  \Hom_{\D}(L_m,zy)\simeq
  \bigoplus_{j=0}^{m}\Hom_{\Pcat}(K^jM_m,y)[-j].
\end{equation}
Moreover, $L_m$ is compact.

Indeed, for $x\in\Pcat$, there is a cofiber sequence
\begin{equation}\label{eq:free-sequence}
  z(Kx)\longrightarrow\Free(x)\longrightarrow zx.
\end{equation}
The forgetful map
\[
  U_*:\Hom_{\D}(zx,zy)\longrightarrow\Hom_{\Pcat}(x,y)
\]
has a section induced by $z$, since $Uz\simeq\id_{\Pcat}$. Consequently,
\eqref{eq:free-sequence} gives
\[
  \Hom_{\D}(zx,zy)\simeq
  \Hom_{\Pcat}(x,y)\oplus\Hom_{\D}(zKx,zy)[-1].
\]
Since $K^{m+1}M_m=0$, iteration proves \eqref{eq:maps}. Since $U$ preserves
filtered colimits, $\Free$ preserves compact objects; backward induction in
\eqref{eq:free-sequence}, using the compactness of the $K^jM_m$, proves that
$L_m$ is compact.
The object $g=\Free(p)$ is a compact generator of $\D$, with an equivalence of
$k$-modules
\begin{equation}\label{eq:algebra}
  \End_{\D}(g)\simeq\End_{\C}(p)\oplus\Hom_{\C}(p,Kp).
\end{equation}
Thus $\End_{\D}(g)$ is discrete and finite-dimensional by
\eqref{eq:beilinson} and \eqref{eq:K-evaluation}. Moreover,
\[
  \Hom_{\D}(g,L_m)\simeq V_m^{\oplus5}
\]
is discrete and finite-dimensional by \eqref{eq:evaluation}.

Equation~\eqref{eq:iterates} exhibits $M_0[m]$ as a direct summand of $K^mM_m$.
Hence \eqref{eq:maps}, with $y=M_0$, exhibits
\[
  \Hom_{\C}(M_0[m],M_0)[-m]\simeq\End_{\C}(M_0)[-2m]
\]
as a direct summand of $\Hom_{\D}(L_m,L_0)$. Since $M_0\ne0$, its identity gives
\[
  \pi_{-2m}\Hom_{\D}(L_m,L_0)\ne0.
\]

Finally, set $R=\End_{\D}(g)^{\op}$ and identify $\D\simeq\Mod(R)$.
Then $R$ is a discrete finite-dimensional $k$-algebra, and the $L_m$ are
discrete finite-dimensional perfect left $R$-modules satisfying
\[
  \boxed{2m\leq\pd_R(L_m)<\infty.}
\]
In particular, the little finitistic dimension of $R$ is infinite.

\renewcommand{\refname}{References}
\begin{thebibliography}{Tang}
\bibitem[OAI]{OAI}
OpenAI, \emph{An algebra of infinite little finitistic dimension},
OpenAI Math preprint, September 23, 2026.
\href{https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf}{Preprint}.

\bibitem[Tang]{Tang}
Qiyue Tang, \emph{A note of refinements to an algebra of infinite little
finitistic dimension}, Hexagon, 2026.
\href{https://hexagonmath.org/2610.00140v4}{hexagon:2610.00140v4}.

\bibitem[MY]{MY}
Hiroyuki Minamoto and Kota Yamaura, \emph{Homological dimension formulas for
trivial extension algebras}, Journal of Pure and Applied Algebra
\textbf{224} (2020), no.~8, 106344.
\href{https://arxiv.org/abs/1710.01469}{arXiv:1710.01469}.

\bibitem[LS]{LS}
L.~Le Bruyn and S.~P.~Smith, \emph{Homogenized $\slie_2$},
Proceedings of the American Mathematical Society \textbf{118} (1993),
725--730, \S2.
\href{https://sites.math.washington.edu/~smith/Research/Homog-sl2.pdf}{Author's PDF}.

\bibitem[LSV]{LSV}
L.~Le Bruyn, S.~P.~Smith, and M.~Van den Bergh,
\emph{Central extensions of three dimensional Artin--Schelter regular algebras},
Mathematische Zeitschrift \textbf{222} (1996), 171--212, Corollary~2.7 and Remark~2.8.
\href{https://sites.math.washington.edu/~smith/Research/Central-extns.pdf}{Author's PDF}.

\bibitem[MM]{MM}
H.~Minamoto and I.~Mori, \emph{Structures of AS-regular algebras},
Propositions~4.3--4.4 and Theorem~4.12.
\href{https://www.math.kyoto-u.ac.jp/preprint/2010/06Minamoto.pdf}{Author's preprint}.
\end{thebibliography}

\end{document}