==== TABLE OF CONTENTS ====
An-algebra-of-infinite-little-finitistic-dimension//01-introduction.tex
An-algebra-of-infinite-little-finitistic-dimension//02-selection-process.tex
An-algebra-of-infinite-little-finitistic-dimension//03-localization-and-lifting.tex
An-algebra-of-infinite-little-finitistic-dimension//04-tensor-realization.tex
An-algebra-of-infinite-little-finitistic-dimension//05-ordinary-simulation.tex
An-algebra-of-infinite-little-finitistic-dimension//06-square-zero-and-conclusion.tex
An-algebra-of-infinite-little-finitistic-dimension//paper.tex
An-algebra-of-infinite-little-finitistic-dimension//preamble.tex
An-algebra-of-infinite-little-finitistic-dimension//references.bib
===========================

==== BEGIN An-algebra-of-infinite-little-finitistic-dimension//01-introduction.tex ====
\section{Introduction}
\label{sec:introduction}

For a finite-dimensional algebra \(A\) over a field \(k\), its
\emph{little finitistic dimension} is
\[
\findim A
=\sup\{\pd_A N:
       N\text{ is a finitely generated left }A\text{-module},
       \ \pd_A N<\infty\}.
\]
The little finitistic-dimension conjecture asserts that this number
is finite for every such algebra.  Thus it asks for a bound on the
lengths of the finite projective resolutions over each fixed algebra,
while allowing other modules to have infinite projective dimension.
The finitistic-dimension questions were publicized by Bass in
1960 \cite{Bass1960}.  The \emph{big} finitistic dimension \(\Findim A\)
is defined by allowing all left modules in the same supremum; the little
dimension is at most the big one.

This paper gives a negative resolution of the little
finitistic-dimension conjecture.

\begin{theorem}\label{thm:main}
There is a finite-dimensional unital complex algebra \(A\) such that,
for every integer \(m\ge1\), there is a finite-dimensional left
\(A\)-module \(N_m\) with
\[
2m-2\le\pd_A N_m<\infty.
\]
In particular, \(\findim A=\infty\).
\end{theorem}

The algebra in \Cref{thm:main} is fixed before \(m\) varies.
Consequently its finitely generated modules admit terminating
projective resolutions of unbounded lengths.  This also makes its
big finitistic dimension infinite and refutes the corresponding
universal finiteness assertion for Artin algebras, since a
finite-dimensional complex algebra is an Artin algebra.

Independently, the companion~\cite[Corollary~1.3]{OpenAITachikawa2026}
obtains finite-dimensional opposite endomorphism algebras \(\Gamma_K\)
of infinite little and big left finitistic dimensions for every field
extension \(K/k\), where \(k=\mathbb F_2(q,H_1,H_2)\), using successive
cokernels in a projective-injective resolution. This is a distinct
characteristic-two example; the algebra in Theorem~\ref{thm:main}
remains the complex example constructed here.

\subsection{Background and consequences}

The equality of the little and big finitistic dimensions is a different
question from their finiteness. Huisgen-Zimmermann disproved that equality:
for every integer \(n\ge2\), she constructed a finite-dimensional monomial
algebra with little finitistic dimension \(n\) and big finitistic dimension
\(n+1\) \cite{HuisgenZimmermann1992}. Those examples have finite finitistic
dimensions and vary with \(n\); they do not give unbounded finite projective
dimensions over a single algebra. Her survey \cite{HuisgenZimmermann1995}
describes the development of the two questions.

Important positive cases for little finitistic finiteness include
finite-dimensional monomial algebras,
by Green, Kirkman, and Kuzmanovich \cite{GreenKirkmanKuzmanovich1991},
and Artin algebras with radical cube zero, by Green and
Huisgen-Zimmermann \cite{GreenHuisgenZimmermann1991}.
Igusa and Todorov introduced numerical invariants of syzygies that give
another proof of the radical-cube-zero case and establish little
finiteness for Artin algebras of representation dimension at most three
\cite[Corollaries~7 and~9]{IgusaTodorov2005}.
More recently, Cummings proved that universal little finitistic
finiteness is equivalent to the assertion that, for every
finite-dimensional algebra, finiteness holds on the left if and only
if it holds on the right \cite[Theorem~A]{Cummings2024}.
These results address bounds within classes or reductions between
finiteness assertions. Our construction instead preserves arbitrarily
long terminating processes while keeping all defining algebra data fixed.

There is also a derived-category consequence. Say that injective left
modules \emph{generate} if the smallest triangulated subcategory of the
unbounded derived category of all left modules that contains them and is
closed under arbitrary coproducts is the whole category. Rickard proved
that injective generation implies finite big finitistic dimension
\cite[Theorem~4.3]{Rickard2019}. Thus injective left \(A\)-modules do not
generate for the algebra of \Cref{thm:main}: its big left finitistic
dimension is infinite. This applies Rickard's theorem to
\(A^{\mathrm{op}}\), matching his right-module convention.

Combining this example with Cummings' auxiliary triangular algebra gives
an extreme left--right asymmetry for both dimensions and for injective
generation.

\begin{corollary}[Extreme left--right asymmetry]
\label{cor:left-right-asymmetry}
There is a finite-dimensional unital complex algebra \(\Lambda\) such that
\[
\findim \Lambda=\Findim \Lambda=\infty,
\qquad
\findim(\Lambda^{\mathrm{op}})=\Findim(\Lambda^{\mathrm{op}})=0.
\]
All four dimensions here use left modules. Thus the little and big
finitistic dimensions of \(\Lambda\) are infinite on the left and zero
on the right. Injective left \(\Lambda\)-modules do not generate the
unbounded derived category of all left \(\Lambda\)-modules, whereas
injective right \(\Lambda\)-modules generate the unbounded derived
category of all right \(\Lambda\)-modules.
\end{corollary}

The algebra \(\Lambda\) is auxiliary: the corollary makes no assertion
about the right finitistic dimensions or right injective generation of
the original algebra \(A\). Passing from \(\Lambda\) to
\(\Lambda^{\mathrm{op}}\) reverses the two sides of the corollary.
We give the proof in \Cref{sec:left-right-asymmetry}, after constructing
\(A\).

\subsection{The construction and its principal mechanisms}

The proof connects a finite presentation with a derived tensor
process and then with ordinary projective resolutions.
We describe the three stages, including the points at which
compatibility of the constructions is needed.

First, \Cref{prop:selection} constructs one finitely presented
complex algebra \(R\), a central idempotent \(e\), and an automorphism
\(\alpha\).  The functor
\[
H(Y)={}_{\alpha}(eY)
\]
selects the direct summand \(eY\) and restricts its action along
\(\alpha\).  It admits finite-dimensional modules \(Y_m\) that
survive exactly \(m-1\) iterations and vanish on the \(m\)th.
The algebra \(R\) is a group algebra.  In the underlying group,
an automorphism shifts an indexed family of central involutions;
finite matrix quotients detect arbitrarily long independent
segments of that family.  Prescribing their characters gives the
modules \(Y_m\).  The group is motivated by matrix groups of
Abels type \cite{Rego2022}; its finite presentation, centrality
relations, quotients, and representations are established here.

The second stage realizes this selection process over a
finite-dimensional algebra.  A quadratic presentation of \(R\)
gives a directed algebra \(B\) with three vertices and
\(\gldim B\le2\), together with modules \(M(Y)\).
The module \(M(Y)\) places the vector space \(Y\) at each vertex and
records the generator actions along the arrows.
The encoding follows the homogenization principle used in
universal localization \cite{NRS2004}.
The principal realization statement, \Cref{thm:realization},
constructs a fixed bounded \(B\)-bimodule complex \(P\), projective
term by term on the right, for which
\[
P\otL_B M(Y)\simeq M(H(Y))\oplus M(H(Y))[3]
\]
for every finite-dimensional \(R\)-module \(Y\).

The principal obstacle is to make this realization uniform in \(Y\).
Inverting the two distinguished arrows of \(B\) makes the presentation
of \(R\) act on a localized category of complexes. The summand selected
by \(e\) need not lift by itself; two mapping cones represent its
direct sum with a copy shifted by three inside that localized category
(\Cref{lem:odd-double}).
We then clear denominators for one finite diagram, before choosing any
test module (\Cref{lem:diagram-lift}). Its relations initially hold only
up to chain homotopy. A three-column differential incorporates those
homotopies into an actual bimodule complex (\Cref{lem:rectification}).
There are no paths of length three, so no further relation overlaps
require higher compatibility data. Finally, the gap between the two
surviving cohomological degrees exceeds the global-dimension bound on
\(B\), so the complex splits as displayed. These steps realize selection
by a single tensor functor over \(B\).

The third stage turns this bounded complex into an ordinary
bimodule.  A directed radical-square-zero chain records the terms
and differential of \(P\).  With a suitable product algebra \(D\)
of finite global dimension and an ordinary \(D\)-bimodule \(X\),
the second iterate of \(F=X\otL_D-\) is isomorphic, on the
\(B\)-factor of \(D\), to \(P\otL_B-\) up to a fixed shift
(\Cref{prop:ordinary-simulation}).  We then take
\[
A=D\ltimes X.
\]
For a \(D\)-module \(N\) inflated to \(A\), the absolute bar
construction decomposes by the number of occurrences of \(X\):
\[
D\otL_A N\simeq\bigoplus_{r\ge0}(F^rN)[r].
\]
\Cref{prop:bar-splitting} proves this decomposition with derived
tensor powers, including the nonflat case. This splitting is also a
consequence of Minamoto and Yamaura's graded-component decomposition
for trivial extensions
\cite[Lemma~4.13(4) and the proof of Theorem~4.17]{MinamotoYamaura}.
Their projective-dimension formula gives a stronger form of the final
transfer. We prove the required splitting directly by the bar construction
and give the minimal-resolution argument used here.
Eventual vanishing of \(F^rN\) yields finite projective dimension,
while a nonzero \(r\)th iterate forces projective dimension at
least \(r\). For \(N_m=M(Y_m)\), the two-step simulation of selection
gives \(F^{2m}N_m\simeq0\) and \(F^{2m-2}N_m\not\simeq0\).
These yield the finiteness and lower bound in \Cref{thm:main}, respectively.

\Cref{thm:realization} applies to any finitely presented
\(k\)-algebra with a central idempotent and a unital endomorphism.
Readers interested in the homological constructions may begin with
\Cref{sec:localization} and use \Cref{prop:selection} as input;
\Cref{sec:selection} supplies the unbounded extinction times needed
for the final counterexample.

In the proof order, \Cref{sec:selection} supplies the selection modules;
\Cref{sec:localization} constructs the fixed lifted diagram; and
\Cref{sec:realization} produces and identifies its bimodule complex.
\Cref{sec:ordinary-simulation} makes the bimodule ordinary, and
\Cref{sec:square-zero} turns its iterates into projective-dimension bounds.

\subsection{Conventions}

All algebras are associative and unital.  Modules are left modules
unless expressly designated right; over a finite-dimensional
\(k\)-algebra, finite dimensionality and finite generation are
equivalent.  Tensor products carrying a superscript \(\mathbf L\)
are derived tensor products.  Complexes have cohomological grading,
with
\[
K[s]^n=K^{n+s},
\qquad d_{K[s]}^n=(-1)^s d_K^{n+s}.
\]
In particular, a module shifted by \(s\) has its cohomology in
degree \(-s\).  Products of paths and endomorphisms are written
in order of composition, from right to left.
An isomorphism denoted \(\simeq\) is taken in the derived category
under discussion.  Splittings used to iterate a tensor functor
are needed only for individual objects; no natural choice of
those splittings is asserted.
==== END An-algebra-of-infinite-little-finitistic-dimension//01-introduction.tex ====

==== BEGIN An-algebra-of-infinite-little-finitistic-dimension//02-selection-process.tex ====
\section{A selection process with unbounded extinction}
\label{sec:selection}

Let $R$ be a unital $k$-algebra, let $e\in R$ be a central idempotent,
and let $\alpha:R\to R$ be a unital endomorphism.  For a left
$R$-module $Y$, define
\begin{equation}
  H(Y)={}_{\alpha}(eY).
  \label{eq:selection-functor}
\end{equation}
Here $eY$ is an $R$-submodule, and the subscript denotes restriction of
scalars along $\alpha$: on the vector space $eY$, an element $r\in R$
acts by the original operator $\alpha(r)$.  This construction defines an
additive functor on left $R$-modules and preserves finite dimensionality.
We write $H^{\circ j}$ for its $j$-fold iterate, with
$H^{\circ 0}$ the identity functor; superscripts without a circle on
$H$ will denote cohomology of a complex.

\begin{proposition}
\label{prop:selection}
There exist a finitely presented unital complex algebra $R$, a central
idempotent $e\in R$, and an automorphism $\alpha$ of $R$ such that, for
every integer $m\geq 1$, a finite-dimensional left $R$-module $Y_m$
satisfies
\begin{equation}
  H^{\circ j}(Y_m)\neq 0\quad(0\leq j<m),
  \qquad H^{\circ m}(Y_m)=0.
  \label{eq:extinction}
\end{equation}
The algebra, idempotent, and automorphism are independent of $m$.
\end{proposition}

We shall take $R=\C G$, where $G$ is a finitely presented group with
central involutions $z_N$ indexed by $N\in\Z$. An automorphism shifting
this family will give $\alpha$, and we shall take $e=(1+z_0)/2$.

The auxiliary group used below is motivated by finitely presented
matrix groups of Abels type. In particular, the commutator transfer
and centrality argument below parallels the proof of Proposition~4.9,
equations~(4.9)--(4.10), in \cite{Rego2022}. Our presentation and the
finite quotients used for selection are specified here, and all required
properties are proved directly.

\subsection{The group and its finite presentation}

Write $[x,y]=xyx^{-1}y^{-1}$ and let $I=\{1,2,3\}$.  Define a group
$G$ with commuting generators $T_\ell$, $\ell\in I$, and generators
\[
  U_i(r),\qquad V_i(r),\qquad W_{ij}(r)
  \quad(i,j\in I,\ i\neq j,\ r\in\Z).
\]
Each generator type $S$ has a weight $\lambda_S\in\Z^3$, given by
\begin{equation}
  \lambda_{U_i}=-\varepsilon_i,\qquad
  \lambda_{V_i}=\varepsilon_i,\qquad
  \lambda_{W_{ij}}=\varepsilon_i-\varepsilon_j,
  \label{eq:group-weights}
\end{equation}
where $\varepsilon_1,\varepsilon_2,\varepsilon_3$ is the standard basis.
Impose the conjugation rules
\begin{equation}
  T_\ell S(r)T_\ell^{-1}=S(r+\lambda_{S,\ell}).
  \label{eq:group-conjugation}
\end{equation}
The remaining relations, with all arguments ranging independently
over $\Z$, are
\begin{enumerate}
  \item $U_i(r)^2=1$;
  \item generators of types $U_i,U_j$, of types $V_i,V_j$, and of
    types $U_i,V_j$ commute whenever $i\neq j$;
  \item $W_{ij}$ commutes with $U_h$ for $h\neq i$ and with $V_h$
    for $h\neq j$;
  \item for $i\neq j$,
  \begin{equation}
    [U_i(r),W_{ij}(s)]=U_j(r+s),\qquad
    [W_{ij}(s),V_j(r)]=V_i(s+r).
    \label{eq:group-commutator-relations}
  \end{equation}
\end{enumerate}

\begin{lemma}
\label{lem:group-finite-presentation}
The group $G$ is finitely presented.
\end{lemma}

\begin{proof}
For $n=(n_1,n_2,n_3)\in\Z^3$, put
$T^n=T_1^{n_1}T_2^{n_2}T_3^{n_3}$ and regard a weight as the map
$\lambda_S(n)=\sum_\ell\lambda_{S,\ell}n_\ell$.  Retain only the
fifteen generators
\[
  T_\ell,\quad u_i=U_i(0),\quad v_i=V_i(0),\quad
  w_{ij}=W_{ij}(0)\quad(i\neq j).
\]
Besides commutation of the $T_\ell$, require the following finite
set of centralizer relations.  Each of $u_i,v_i$ commutes with
$T_h$ for $h\neq i$, and $w_{ij}$ commutes with $T_iT_j$ and with
$T_h$ for the unique $h\notin\{i,j\}$.  These are precisely
centralization by a basis of each kernel lattice:
\[
\begin{aligned}
 \ker\lambda_{U_i}=\ker\lambda_{V_i}
   &=\bigoplus_{h\neq i}\Z\varepsilon_h,\\
 \ker\lambda_{W_{ij}}
   &=\Z(\varepsilon_i+\varepsilon_j)
     \oplus\Z\varepsilon_h
       \quad(h\notin\{i,j\}).
\end{aligned}
\]
Since the $T_\ell$ commute, the kernel lattice also centralizes
every $T$-conjugate of its argument-zero generator.  Thus one may define
\begin{equation}
\begin{aligned}
  U_i(r)&=T_i^{-r}u_iT_i^r,&
  V_i(r)&=T_i^rv_iT_i^{-r},&
  W_{ij}(r)&=T_i^rw_{ij}T_i^{-r}.
\end{aligned}
\label{eq:group-finite-generators}
\end{equation}
Equivalently, $S(r)=T^nS(0)T^{-n}$ for any $n$ with
$\lambda_S(n)=r$.  The kernel relations make this independent of
the choice of $n$ and imply all of \eqref{eq:group-conjugation}.

It remains to impose the argument-zero instance of each of the
remaining relation patterns.  For every pair of input types actually
occurring in those patterns, the two weights give a surjection
$\Z^3\to\Z^2$.  The following table supplies integral preimages of
an arbitrary pair $(r,s)$; unmentioned coordinates are zero.
\begin{center}
\small
\begin{tabular}{@{}lll@{}}
\toprule
Input types & Index condition & Coordinates of $n$ \\
\midrule
$U_i,U_j$ & $i\neq j$ & $n_i=-r,\ n_j=-s$ \\
$V_i,V_j$ & $i\neq j$ & $n_i=r,\ n_j=s$ \\
$U_i,V_j$ & $i\neq j$ & $n_i=-r,\ n_j=s$ \\
$W_{ij},U_j$ & $i\neq j$ & $n_j=-s,\ n_i=r-s$ \\
$W_{ij},U_h$ & $h\notin\{i,j\}$ &
  $n_i=r,\ n_j=0,\ n_h=-s$ \\
$W_{ij},V_i$ & $i\neq j$ & $n_i=s,\ n_j=s-r$ \\
$W_{ij},V_h$ & $h\notin\{i,j\}$ &
  $n_i=r,\ n_j=0,\ n_h=s$ \\
$U_i,W_{ij}$ & $i\neq j$ & $n_i=-r,\ n_j=-r-s$ \\
$W_{ij},V_j$ & $i\neq j$ & $n_j=s,\ n_i=r+s$ \\
\bottomrule
\end{tabular}
\end{center}
Conjugating a zero-argument commutation relation by the corresponding
$T^n$ therefore gives every required instance.  For the two relations
in \eqref{eq:group-commutator-relations}, the output weight is the sum
of the input weights:
\[
  -\varepsilon_i+(\varepsilon_i-\varepsilon_j)=-\varepsilon_j,
  \qquad
  (\varepsilon_i-\varepsilon_j)+\varepsilon_j=\varepsilon_i.
\]
Hence the same conjugation also gives the correct output argument
$r+s$.  The square relation propagates from argument zero because
each individual weight is surjective onto $\Z$.

We have obtained all the defining relations from finitely many
relators on fifteen generators.  Conversely, the chosen kernel
centralizers and zero-argument relators hold in the original
presentation by \eqref{eq:group-conjugation}.  The two presentations
are therefore equivalent.
\end{proof}

\subsection{Central involutions and an automorphism}

The finite presentation is now fixed. We next identify central
involutions that the automorphism will shift; their independent images
in finite quotients will let us prescribe the extinction time.

\begin{lemma}
\label{lem:central-shift}
For each $N\in\Z$, the expression
\begin{equation}
  z_N=[U_i(a),V_i(b)],\qquad a+b=N,
  \label{eq:central-involutions}
\end{equation}
is independent of $i,a,b$.  Each $z_N$ is central and satisfies
$z_N^2=1$.  There is an automorphism $\alpha_G$ of $G$ fixing all
$T_\ell,V_i(r),W_{ij}(r)$ and sending $U_i(r)$ to $U_i(r+1)$;
it satisfies $\alpha_G(z_N)=z_{N+1}$.
\end{lemma}

\begin{proof}
Fix distinct $i,j$ and integers $r,q,s$, and set
\[
 x=U_i(r),\quad y=W_{ij}(q),\quad v=V_j(s),\qquad
 u=[x,y]=U_j(r+q),\quad w=[y,v]=V_i(q+s).
\]
The defining commutations give $[x,v]=[w,u]=[w,v]=1$.
Using $xy=uyx$ and $yv=wvy$, reorder $xyv$ in the two ways
\[
  xyv=uwvyx=(xwx^{-1})vuyx.
\]
With $c=[x,w]$, cancellation gives $uwv=cwvu$, and consequently
\[
  c=uwvu^{-1}v^{-1}w^{-1}=uvu^{-1}v^{-1}=[u,v].
\]
The last equality uses only the commutation of $w$ with $u$ and $v$.
We have proved
\begin{equation}
 [U_i(r),V_i(q+s)]=[U_j(r+q),V_j(s)].
 \label{eq:commutator-transfer}
\end{equation}
Given $a+b=c+d=N$ at distinct internal indices, choose
$r=a$, $s=d$, and $q=c-a=b-d$ in
\eqref{eq:commutator-transfer}.  This compares the two expressions
in \eqref{eq:central-involutions}.  Two splittings at the same
internal index can be compared through another index, proving
the claimed independence.

To show that $z_N$ commutes with $U_h(r)$ or $V_h(r)$, express
$z_N$ using an index $i\neq h$.  Both factors of that commutator
commute with the tested generator.  To commute with $W_{hj}(r)$,
use the index outside $\{h,j\}$, so that again both factors commute
with it.  Finally, conjugation by $T_\ell$ changes the arguments
$(a,b)$ in \eqref{eq:central-involutions} to
$(a-\delta_{\ell i},b+\delta_{\ell i})$.  Their sum is unchanged,
so the splitting independence proves invariance under $T_\ell$.
Thus $z_N$ is central.  Since $U_i(a)^2=1$, centrality also gives
\[
  1=[U_i(a)^2,V_i(b)]=[U_i(a),V_i(b)]^2=z_N^2.
\]

Translation of every $U$-argument by $1$ preserves the conjugation,
square, and commutation relations.  The first relation in
\eqref{eq:group-commutator-relations} becomes
\[
  [U_i(r+1),W_{ij}(s)]=U_j(r+s+1),
\]
which is another defining instance; the second relation is unchanged.
This defines an endomorphism $\alpha_G$.  Translation by $-1$
preserves the relations for the same reason and is its two-sided
inverse.  Formula \eqref{eq:central-involutions} then gives
$\alpha_G(z_N)=z_{N+1}$.
\end{proof}

\subsection{Finite quotients detecting independent involutions}

\begin{lemma}
\label{lem:finite-central-quotients}
For every $m\geq1$, there is a finite quotient $F_m$ of $G$ in which
the images of $z_0,\ldots,z_{m-1}$ generate a central subgroup
isomorphic to $(\Z/2\Z)^m$.
\end{lemma}

\begin{proof}
Let $S_m=\F_2[t]/(t^m-1)$.  This ring is finite, and $t$ is a unit
with inverse $t^{m-1}$, so $t^r$ is defined for every $r\in\Z$.
Use matrices indexed by $0,1,2,3,4$.  Send $T_\ell$ to the diagonal
matrix with $t$ in position $\ell$ and $1$ elsewhere, and set
\begin{equation}
\begin{aligned}
 U_i(r)&\longmapsto I_5+t^rE_{0i},&
 V_i(r)&\longmapsto I_5+t^rE_{i4},&
 W_{ij}(r)&\longmapsto I_5+t^rE_{ij}.
\end{aligned}
\label{eq:finite-matrix-images}
\end{equation}
Here $E_{ab}$ denotes a matrix unit.  Every displayed elementary
matrix is invertible, with itself as inverse: its off-diagonal
matrix unit has square zero, and the coefficient ring has
characteristic two.

We check the relations.  Diagonal conjugation multiplies $E_{ab}$
by the ratio of the diagonal entries at $a$ and $b$, which gives
exactly \eqref{eq:group-weights}.  The identity
$E_{ab}E_{cd}=\delta_{bc}E_{ad}$ shows that all products in both
orders vanish for each required commuting pair: two units $E_{0i}$,
two units $E_{i4}$, the pair $E_{0i},E_{j4}$ for $i\neq j$, the pair
$E_{ij},E_{0h}$ for $h\neq i$, and the pair $E_{ij},E_{h4}$ for
$h\neq j$.  The required square relations hold by the same
square-zero calculation.  For $i\neq j$, the nonzero product in the
first noncommutation relation is $E_{0i}E_{ij}=E_{0j}$, and in the
second it is $E_{ij}E_{j4}=E_{i4}$.  The higher products vanish,
giving
\[
\begin{aligned}
 [I_5+t^rE_{0i},I_5+t^sE_{ij}]&=I_5+t^{r+s}E_{0j},\\
 [I_5+t^sE_{ij},I_5+t^rE_{j4}]&=I_5+t^{s+r}E_{i4}.
\end{aligned}
\]
Thus \eqref{eq:finite-matrix-images} defines a homomorphism from
$G$ into the finite group $\operatorname{GL}_5(S_m)$.  Let $F_m$
be its image.

The image of $z_N$ is $I_5+t^NE_{04}$.  It is central in $F_m$:
the unit $E_{04}$ has zero product in both orders with every
off-diagonal unit in \eqref{eq:finite-matrix-images}, and the
diagonal matrices have entry $1$ at both $0$ and $4$.  For
$c_0,\ldots,c_{m-1}\in\{0,1\}$,
\[
 \prod_{q=0}^{m-1}(I_5+t^qE_{04})^{c_q}
   =I_5+\left(\sum_{q=0}^{m-1}c_qt^q\right)E_{04}.
\]
The classes $1,t,\ldots,t^{m-1}$ are an $\F_2$-basis of $S_m$,
because the defining polynomial is monic of degree $m$.
Thus this product is the identity only when all $c_q$ are zero.
This argument also applies when $m$ is even and $S_m$ is
nonreduced.  The claimed independence and orders follow.
\end{proof}

\subsection{The algebra and its test modules}

The quotient in \Cref{lem:finite-central-quotients} supplies the independent
signs. We pass to the group algebra and choose the signs so that each
selection before the last is the identity, while the last is zero.

\begin{proof}[Proof of \Cref{prop:selection}]
Take $R=\C G$.  A finite group presentation yields a finite
presentation of its unital group algebra: for each group generator
adjoin a formal inverse and the two inverse equations, and impose
the finitely many group relators as algebra relations.  Hence $R$
is finitely presented by \Cref{lem:group-finite-presentation}.
Extend $\alpha_G$ linearly to an algebra automorphism $\alpha$ and put
\[
  e=\frac{1+z_0}{2}.
\]
By \Cref{lem:central-shift}, this is a central idempotent.  For
$q\geq0$, write
\[
  e_q=\alpha^q(e)=\frac{1+z_q}{2}.
\]
These are commuting central idempotents of the one fixed algebra $R$.

For the finite quotient in \Cref{lem:finite-central-quotients}, let
$\overline z_q$ be the image of $z_q$ and let
$Z_m=\langle\overline z_0,\ldots,\overline z_{m-1}\rangle$.
Independence defines a character $\chi_m:Z_m\to\{1,-1\}$ by
\[
 \chi_m(\overline z_q)=
 \begin{cases}
   1,&0\leq q<m-1,\\
   -1,&q=m-1.
 \end{cases}
\]
In the complex regular representation of $F_m$, form
\[
  p_{\chi_m}=\frac{1}{|Z_m|}
       \sum_{z\in Z_m}\chi_m(z)^{-1}z,
  \qquad
  Y_m=\C[F_m]p_{\chi_m}.
\]
The element $p_{\chi_m}$ is nonzero, since its identity coefficient
is $1/|Z_m|$, and $z p_{\chi_m}=\chi_m(z)p_{\chi_m}$ for
$z\in Z_m$.  Centrality of $Z_m$ then shows that every vector of
$Y_m$ has this same character.  In particular, $Y_m$ is a nonzero
finite-dimensional left $\C[F_m]$-module.  Regard it as an
$R$-module through the quotient map.

For every left $R$-module $Y$, induction in the definition of $H$
gives an identification
\begin{equation}
  H^{\circ j}(Y)\cong {}_{\alpha^j}
       \bigl(e_0e_1\cdots e_{j-1}Y\bigr),
  \qquad j\geq0,
  \label{eq:selection-iterates}
\end{equation}
where the empty product is $1$.  Indeed, on the vector space at
stage $j$, selection by $e$ acts as the original operator
$\alpha^j(e)=e_j$.  Pulling the resulting action back once more
along $\alpha$ makes $r$ act by $\alpha^{j+1}(r)$, proving the
inductive step.  Centrality of all the $e_q$ makes the subspace in
\eqref{eq:selection-iterates} an $R$-submodule before this pullback.

On $Y_m$, the operators $e_0,\ldots,e_{m-2}$ are the identity and
$e_{m-1}$ is zero.  Consequently \eqref{eq:selection-iterates}
gives \eqref{eq:extinction}, also for $m=1$.  This argument does
not require $\alpha$ to preserve the kernel of the finite quotient:
each pullback uses the same finite-dimensional vector space with
its action precomposed by an automorphism of $R$.  The construction
of $R,e,\alpha$ preceded the choice of $m$, as required.
\end{proof}
==== END An-algebra-of-infinite-little-finitistic-dimension//02-selection-process.tex ====

==== BEGIN An-algebra-of-infinite-little-finitistic-dimension//03-localization-and-lifting.tex ====
\section{Localization and a fixed lifted diagram}
\label{sec:localization}

Let $k$ be a field, let $R$ be a finitely presented unital $k$-algebra,
let $e\in R$ be a central idempotent, and let
$\alpha:R\to R$ be a unital endomorphism.  As in the preceding section,
\[
  H(Y)={}_{\alpha}(eY)
\]
for a left $R$-module $Y$: the underlying vector space is $eY$, and
$r\in R$ acts there by $\alpha(r)$.  Centrality of $e$ ensures that this
action preserves $eY$.  The construction in this section uses these
fixed data and applies to every finite-dimensional $Y$.

We first encode the presentation of $R$ in a finite-dimensional directed
algebra.  This is a three-vertex version of the homogenized quiver
construction in \cite[Section~1, Theorem~1.1 and Proposition~1.2]{NRS2004}.
We then work directly with a Verdier
quotient and its evaluation functors.

\subsection{A quadratic presentation and its directed encoding}

Choose a presentation
\[
  R=k\langle x_1,\ldots,x_d\rangle/(p_1,\ldots,p_t)
\]
in which each $p_j$ has degree at most two, allowing constant and linear
terms.  Such a presentation exists: in any finite presentation, introduce
finitely many auxiliary generators for the successive partial products
in its finitely many monomials, together with their binary-product
definitions.  The resulting relations have degree at most two, and
eliminating the auxiliary generators recovers the original presentation.

Consider the quiver with vertices $0,1,2$ and arrow sets
\[
  \mathcal A_{01}=\{s,a_1,\ldots,a_d\},\qquad
  \mathcal A_{12}=\{s',a'_1,\ldots,a'_d\},\qquad
  \mathcal A=\mathcal A_{01}\sqcup\mathcal A_{12},
\]
where the subscripts specify the source and target.  Products of arrows
are written in the order of composition, from right to left.  Impose
the relations
\begin{equation}\label{eq:quadratic-encoding}
  a'_h s=s'a_h\quad(1\le h\le d),
  \qquad \widetilde p_j=0\quad(1\le j\le t),
\end{equation}
where homogenization replaces
\[
  x_hx_l\longmapsto a'_h a_l,\qquad
  x_h\longmapsto s'a_h,\qquad
  1\longmapsto s's.
\]
Let $B$ be the resulting path algebra with relations, and let
$\mathbf e_0,\mathbf e_1,\mathbf e_2$ be its vertex idempotents.
Write $\mathcal R$ for a fixed vector-space basis of the span of all
the relations in \eqref{eq:quadratic-encoding}, viewed in the free
space of paths of length two.  Thus each $\rho\in\mathcal R$ has a
unique expression
\[
  \rho=\sum_{\substack{a\in\mathcal A_{01}\\b\in\mathcal A_{12}}}
             \rho_{ba}\,ba.
\]
The basis $\mathcal R$ generates the same relation ideal.  There are
no paths of length three, so $B$ is finite-dimensional.

The following elementary bound will also be used for other directed
algebras later.

\begin{lemma}[Directed dimension bound]\label{lem:directed-bound}
Let $\Lambda$ be a finite-dimensional $k$-algebra with complete
orthogonal idempotents $\varepsilon_0,\ldots,\varepsilon_{n-1}$ such that
\[
  1=\sum_{i=0}^{n-1}\varepsilon_i,\qquad
  \varepsilon_i\Lambda\varepsilon_i=k\varepsilon_i,\qquad
  \varepsilon_j\Lambda\varepsilon_i=0\quad(j<i).
\]
Then $\Lambda$ has global dimension at most $n-1$ on both sides.
In particular, the algebra $B$ above has global dimension at most two
on both sides.
\end{lemma}

\begin{proof}
Say that a left module $M$ is supported at indices at least $p$ if
$\varepsilon_jM=0$ for $j<p$.  There is a projective surjection
\[
  \bigoplus_{i=p}^{n-1}
      \Lambda\varepsilon_i\otimes_k\varepsilon_iM
      \longrightarrow M,
  \qquad c\otimes v\longmapsto cv.
\]
The source is projective even when $M$ is not finitely generated,
since each summand is a direct sum of copies of
$\Lambda\varepsilon_i$.  Its components below $p$ vanish, and its
component at $p$ maps isomorphically onto $\varepsilon_pM$:
only the summand $i=p$ contributes there.  Its kernel is therefore
supported at indices at least $p+1$.

Starting with $p=0$ and repeating this construction, the
$(n-1)$st kernel is supported only at the last vertex.  Every such
module is projective, since
$\Lambda\varepsilon_{n-1}=k\varepsilon_{n-1}$.
This gives the left dimension bound, including the case $n=1$.
The same proof for $\Lambda^{\mathrm{op}}$, with the reverse ordering
of the idempotents, gives the right bound.  The path directions and
the homogeneous quadratic relations of $B$ give precisely the
displayed hypotheses with $n=3$.
\end{proof}

For a finite-dimensional left $R$-module $Y$, define $M(Y)$ to be the
left $B$-module with $Y$ at every vertex, with $s,s'$ acting as the
identity, and with $a_h,a'_h$ acting as $x_h$.  The relations of $B$
hold because they evaluate to the defining relations of $R$.

\subsection{The quotient action and evaluation}

Let
\[
  K=\K^b(\proj\text{-}B),\qquad E_i=\mathbf e_iB,
\]
so $K$ is the homotopy category of bounded complexes of finitely
generated projective \emph{right} $B$-modules.  With our path convention,
an arrow $a:i\to j$ belongs to $\mathbf e_jB\mathbf e_i$.
Left multiplication by $a$ is consequently a right-module map
$E_i\to E_j$.  Composing these maps agrees with multiplying paths
in the stated order.

Let $\mathcal S$ be the thick subcategory of $K$ generated by the
cones of $s:E_0\to E_1$ and $s':E_1\to E_2$, and form
\[
  q:K\longrightarrow Q=K/\mathcal S.
\]
We use the same notation for objects and arrows after applying $q$.
The maps $s$ and $s'$ are isomorphisms in $Q$.  Put $L=E_0$ and
\[
  \overline x_h=s^{-1}a_h\in\End_Q(L).
\]
The relation $a'_h s=s'a_h$ gives $a'_h=s'a_hs^{-1}$, whence
\[
  (s's)^{-1}a'_ha_l
     =s^{-1}a_hs^{-1}a_l
     =\overline x_h\overline x_l.
\]
Likewise $(s's)^{-1}s'a_h=\overline x_h$ and
$(s's)^{-1}s's=\id_L$.  The homogenized relations therefore give
a unital homomorphism
\begin{equation}\label{eq:quotient-action}
  \theta:R\longrightarrow\End_Q(L),
  \qquad x_h\longmapsto\overline x_h.
\end{equation}

For each finite-dimensional $Y$, tensoring defines an exact functor
\[
  \operatorname{ev}_Y:K\longrightarrow\D^b(k),
  \qquad T\longmapsto T\otimes_B M(Y).
\]
Here $\D^b(k)$ denotes the bounded derived category of finite-dimensional
$k$-vector spaces; termwise projectivity of $T$ computes the derived
tensor product.  Under the natural identifications
\[
  E_i\otimes_B M(Y)\cong\mathbf e_iM(Y)=Y,
\]
an arrow map evaluates to its action on $M(Y)$.  Thus
$\operatorname{ev}_Y$ takes $s,s'$ to identities and their cones to zero.
The kernel of an exact functor is thick, so it contains $\mathcal S$.
Equivalently, every map whose cone belongs to $\mathcal S$ evaluates
to an isomorphism.  The functor consequently factors through $Q$.
We retain the notation $\operatorname{ev}_Y$ for this factorization.
On $L$ the homomorphism \eqref{eq:quotient-action} evaluates to the
given $R$-action on $Y$.

\subsection{An odd double of a formal summand}

The idempotent $\theta(e)$ evaluates to the projection of $Y$ onto
$eY$, but it need not split inside $Q$. We therefore seek an object
of $Q$ whose evaluation is $eY$ plus a shifted copy. The following
lemma gives such an object for a copy shifted by three.

For an additive category $\mathcal T$, its idempotent completion
$\Kar(\mathcal T)$ has objects $(T,p)$, where $p^2=p\in\End(T)$,
and morphisms $u:(T,p)\to(T',p')$ satisfying $u=p'up$.
The embedding $T\mapsto(T,\id_T)$ is fully faithful.
If $\mathcal T$ has a shift functor, it extends
by $(T,p)[1]=(T[1],p[1])$.

We need the following consequence of the triangle axioms.  It is
compatible with the general idempotent-completion result of
Balmer and Schlichting \cite[Theorem~1.5]{BalmerSchlichting2001}; the argument
below uses only triangles already present in $\mathcal T$.

\begin{lemma}[Odd double]\label{lem:odd-double}
Let $\mathcal T$ be a triangulated category, let $L\in\mathcal T$,
and let $\epsilon\in\End_{\mathcal T}(L)$ be idempotent.
Put $U=(L,\epsilon)\in\Kar(\mathcal T)$.
Both $U\oplus U[1]$ and $U\oplus U[3]$ are isomorphic in
$\Kar(\mathcal T)$ to objects of $\mathcal T$.
\end{lemma}

\begin{proof}
First consider a triangle from $\mathcal T$,
\[
  A\xrightarrow{f}B\xrightarrow{g}T\xrightarrow{h}A[1],
\]
and suppose that formal decompositions
$A=I\oplus A_0$, $B=I\oplus B_0$ identify $f$ with the identity
on $I$ and zero elsewhere.  For any $Z\in\Kar(\mathcal T)$,
the long exact sequence $\Hom(Z,-)$ still holds for this triangle:
$Z$ is a retract of an object of $\mathcal T$, and its Hom sequence
is a direct summand of the exact sequence for that object.

The equalities $gf=0$ and $f[1]h=0$ give factorizations
\[
  g=g_0\operatorname{pr}_{B_0},\qquad
  h=\operatorname{in}_{A_0[1]}h_0.
\]
Exactness gives, for every such $Z$, a short exact sequence
\begin{equation}\label{eq:formal-cone-hom}
  0\longrightarrow\Hom(Z,B_0)
   \xrightarrow{(g_0)_*}\Hom(Z,T)
   \xrightarrow{(h_0)_*}\Hom(Z,A_0[1])
   \longrightarrow0.
\end{equation}
Indeed, a map into $B_0$ killed by $g_0$ factors through $f$ after
inclusion into $B$, and projection back to $B_0$ makes it zero.
At the other end, every map into $A_0[1]$ is killed by $f[1]$ after
inclusion into $A[1]$, so it lifts through $h$.
Taking $Z=A_0[1]$ supplies a section $\sigma$ of $h_0$.
It follows from \eqref{eq:formal-cone-hom} that
\[
  (g_0,\sigma):B_0\oplus A_0[1]\longrightarrow T
\]
induces isomorphisms on $\Hom(Z,-)$ for every $Z$, hence is an
isomorphism.

Apply this calculation to a triangle on $1-\epsilon:L\to L$.
Its identity summand is $(L,1-\epsilon)$, and its zero summand on
both sides is $U$.  Its cone is an object $S\in\mathcal T$ with
$S\cong U\oplus U[1]$ in the completion.  Under this identification,
consider the map
\[
  S[1]\cong U[1]\oplus U[2]
     \xrightarrow{\left(\begin{smallmatrix}0&0\\\id&0\end{smallmatrix}\right)}
  U\oplus U[1]\cong S.
\]
Fullness of $\mathcal T\hookrightarrow\Kar(\mathcal T)$ makes
this a morphism between the old objects $S[1]$ and $S$.
Its cone is again in $\mathcal T$.
The preceding calculation, with common identity summand $U[1]$,
source complement $U[2]$, and target complement $U$, identifies
that cone with $U\oplus U[3]$.
\end{proof}

Apply \Cref{lem:odd-double} in $Q$ to $\epsilon=\theta(e)$ and write
\[
  U=(L,\theta(e)),\qquad V=U\oplus U[3].
\]
Fix once and for all an object $V_0\in Q$ and an isomorphism
$V_0\cong V$ in $\Kar(Q)$ supplied by \Cref{lem:odd-double}.
Centrality of $e$ gives a unital action of $R$ on $U$ by
\[
  r\longmapsto\theta(e\alpha(r)e)\in\End_{\Kar(Q)}(U).
\]
Its unit is $\theta(e)=\id_U$, and its products agree with those
in $R$ because $e$ commutes with every $\alpha(r)$.
Use the same action on $U[3]$ and transport the resulting action
on $V$ to $V_0$.

There is now a $B$-diagram in $Q$ with $V_0$ at all three vertices:
$s,s'$ act as identities, and $a_h,a'_h$ act by the endomorphism
just defined for $x_h$.  Its relations hold by the unital
$R$-action.  Here a \emph{$B$-diagram} in an additive $k$-linear
category means objects at the three vertices and arrow maps satisfying
the relations of $B$ in that category.

The category $\D^b(k)$ splits idempotents: every bounded complex of
vector spaces is isomorphic to its cohomology with zero differential,
where splitting can be done degree by degree.  Thus evaluation extends
additively to $\Kar(Q)$.  It takes $U$ to $eY$, and the preceding
action of $r$ to the restriction of $\alpha(r)$ to $eY$.
Consequently the evaluated $B$-diagram has
\begin{equation}\label{eq:target-evaluation}
  eY\oplus(eY)[3]
\end{equation}
at every vertex, identities on $s,s'$, and the action of
$\alpha(x_h)$ on both summands for $a_h,a'_h$.

\subsection{Clearing denominators for the finite diagram}

The quotient diagram is now fixed and has the required evaluation on
every $Y$. We next represent it by one diagram in the homotopy category,
so that the same chain data will work under all those evaluations.

We give the lifting argument together with the fraction rules it uses.
Let
\[
  \Sigma=\{u\in\operatorname{Mor}(K):\Cone(u)\in\mathcal S\}.
\]
The Verdier quotient is the localization of $K$ at $\Sigma$.
The following triangle arguments give its right-fraction description;
see also \cite[Section 13.6]{StacksProject}.
Identities belong to $\Sigma$, and composition preserves $\Sigma$
by the octahedral axiom.  If $u:Y'\to Y$ belongs to $\Sigma$ and
$f:X\to Y$ is any map, complete $u$ to a triangle
\[
  Y'\xrightarrow{u}Y\xrightarrow{c}T\longrightarrow Y'[1],
  \qquad T\in\mathcal S.
\]
Complete $cf$ to a triangle
$X'\xrightarrow{v}X\xrightarrow{cf}T\to X'[1]$.
Then $v\in\Sigma$, and exactness of $\Hom(X',-)$ gives
$g:X'\to Y'$ with $fv=ug$.  This is the required Ore square.

For the cancellation rule, suppose that $u:Y\to Y'$ belongs to
$\Sigma$ and $ug=0$ for $g:X\to Y$.
A triangle $T[-1]\to Y\xrightarrow{u}Y'\to T$ shows that $g$
factors through a map $b:X\to T[-1]$ with $T\in\mathcal S$.
Completing $b$ to a triangle gives a denominator $v:X'\to X$
with $bv=0$, hence $gv=0$.  Applying this to differences gives
the corresponding rule for equality of morphisms.

These rules construct the localization by roofs
\[
  X\xleftarrow{u}X'\xrightarrow{f}Y,
  \qquad u\in\Sigma,
\]
representing $q(f)q(u)^{-1}$; equality is tested after a common source
refinement.  In particular, every morphism $q(X)\to q(Y)$ has this
form, and
\begin{equation}\label{eq:zero-fraction}
  q(g)=0\quad\Longrightarrow\quad
  gv=0\text{ for some }v:X'\to X\text{ in }\Sigma
\end{equation}
for $g:X\to Y$.  For the last assertion compare the roofs
$(\id_X,g)$ and $(\id_X,0)$.
Finite families can be treated with one denominator by collecting
their maps into a single map to a finite direct sum of targets.

\begin{lemma}[Finite diagram lifting]\label{lem:diagram-lift}
Every $B$-diagram in $Q$ is isomorphic to the image under $q$ of a
$B$-diagram in $K$.
\end{lemma}

\begin{proof}
Write the given objects as $W_i$ and its arrow maps as $t_a$.
Choose $C_i\in K$ and isomorphisms $\psi_i:q(C_i)\to W_i$.
First collect the maps
\[
  (\psi_2^{-1}t_b\psi_1)_{b\in\mathcal A_{12}}:
     q(C_1)\longrightarrow
     \bigoplus_{b\in\mathcal A_{12}}q(C_2).
\]
Represent this tuple by one roof, with denominator
$u_1:D_1\to C_1$ and numerator components $f_b:D_1\to C_2$.
Set $D_2=C_2$ and
\[
  \phi_1=\psi_1q(u_1),\qquad \phi_2=\psi_2.
\]
Then $\phi_2q(f_b)=t_b\phi_1$ for every $b\in\mathcal A_{12}$.

Next collect the adjusted maps
\[
  (\phi_1^{-1}t_a\psi_0)_{a\in\mathcal A_{01}}:
     q(C_0)\longrightarrow
     \bigoplus_{a\in\mathcal A_{01}}q(D_1).
\]
A single roof gives a denominator $u_0:D_0\to C_0$ and maps
$f_a:D_0\to D_1$.  Put $\phi_0=\psi_0q(u_0)$.
All the arrow squares now commute:
\begin{equation}\label{eq:lift-arrow-squares}
  \phi_jq(f_a)=t_a\phi_i\qquad(a:i\to j).
\end{equation}

For $\rho\in\mathcal R$, form the morphism in $K$
\[
  g_\rho=\sum_{b,a}\rho_{ba}f_bf_a:D_0\longrightarrow D_2.
\]
The relations for $t_a$ and \eqref{eq:lift-arrow-squares} give
$\phi_2q(g_\rho)=0$, so $q(g_\rho)=0$.
Apply \eqref{eq:zero-fraction} to the tuple $(g_\rho)_\rho$.
There is one denominator $v:D'_0\to D_0$ such that
$g_\rho v=0$ in $K$ for every $\rho$.
Replace $D_0$ by $D'_0$, every map $f_a$ with source vertex $0$ by
$f_av$, and $\phi_0$ by $\phi_0q(v)$.
The arrow squares still commute, and each relation becomes
\[
  \sum_{b,a}\rho_{ba}f_b(f_av)=g_\rho v=0
  \quad\text{in }K.
\]
There are no arrows into vertex zero and no longer paths requiring
further compatibility.  Thus the new data form a $B$-diagram in $K$,
and the three $\phi_i$ give an isomorphism from its image to the
entire prescribed diagram in $Q$.
\end{proof}

\subsection{The chain data for tensor realization}

Apply \Cref{lem:diagram-lift} to the diagram on $V_0$ constructed
above.  The resulting relations hold in the homotopy category.
Choosing representatives converts them into a finite list of
specified null-homotopies, which are the input for the next section.

\begin{proposition}[Fixed lifted diagram]\label{prop:lifted-diagram}
For the fixed data $R,e,\alpha$ and the algebra $B$ above, there are
bounded complexes $D_0,D_1,D_2$ of finitely generated projective
right $B$-modules, chain maps
\[
  f_a:D_i\longrightarrow D_j\qquad(a:i\to j),
\]
and right-$B$-linear maps $h_\rho:D_0\to D_2$ of degree $-1$
for $\rho\in\mathcal R$, such that
\begin{equation}\label{eq:relation-homotopies}
  d_{D_2}h_\rho+h_\rho d_{D_0}
      =\sum_{b,a}\rho_{ba}f_bf_a.
\end{equation}
For every finite-dimensional left $R$-module $Y$, the diagram
whose objects and arrows in $\D^b(k)$ are
\begin{equation}\label{eq:evaluated-diagram}
  D_i\otimes_B M(Y),\qquad f_a\otimes_B\id_{M(Y)}
\end{equation}
is isomorphic to the diagram having $eY\oplus(eY)[3]$ at all three
vertices, identities on $s,s'$, and the action of $\alpha(x_h)$
on both summands for $a_h,a'_h$.
All the complexes, maps, and homotopies in this statement are fixed
before $Y$ is chosen.
\end{proposition}

\begin{proof}
The presentation, the quotient, the two cones, their isomorphisms in
the completion, and the action on $V_0$ have all been chosen using
only $R,e,\alpha$.  Apply the finite lifting construction once to
this fixed diagram.  Its objects lie in $K$, so choose bounded
right-projective complex representatives $D_i$ and chain-map
representatives $f_a$.  Each relation is zero in $K$, hence its
representative is null-homotopic.  Choose one right-linear
homotopy $h_\rho$ for each of the finitely many relations.
This gives \eqref{eq:relation-homotopies}.

The isomorphisms $\phi_i$ from the lifting construction are fixed
morphisms in $Q$ and satisfy all the fixed arrow squares
\eqref{eq:lift-arrow-squares}.  Every $\operatorname{ev}_Y$ factors
through this same quotient.  Applying it to those squares and then
to the fixed identification $V_0\cong V$ gives the asserted
isomorphism of diagrams, by \eqref{eq:target-evaluation}.
Thus the same choices work for all $Y$.
\end{proof}

The construction has used the homomorphism
\eqref{eq:quotient-action} and the evaluation functors alone.
It requires neither an identification of $Q$ with a derived category
of $R$ nor a stable-flatness assertion about a universal localization.
The maps $f_a$ satisfy the relations up to the chosen homotopies
\eqref{eq:relation-homotopies}; the next section incorporates those
homotopies into an actual complex of bimodules.
==== END An-algebra-of-infinite-little-finitistic-dimension//03-localization-and-lifting.tex ====

==== BEGIN An-algebra-of-infinite-little-finitistic-dimension//04-tensor-realization.tex ====
\section{Realization by a bimodule complex}
\label{sec:realization}

We retain the algebra \(B\), its relation basis \(\mathcal R\), and the
module \(M(Y)\) from the preceding section. The fixed lifted diagram
will now give a single tensor functor realizing the selection process.

\begin{theorem}\label{thm:realization}
Let \(k\) be a field, let \(R\) be a finitely presented \(k\)-algebra,
let \(e\in R\) be a central idempotent, and let
\(\alpha:R\to R\) be a unital endomorphism.
Use the finite-dimensional algebra \(B\) and the modules \(M(Y)\)
constructed in the preceding section, and put
\(H(Y)={}_{\alpha}(eY)\).
There is a single bounded complex \(P\) of finite-dimensional
\(B\)-bimodules, termwise projective on the right, such that for
every finite-dimensional left \(R\)-module \(Y\) one has
\begin{equation}\label{eq:realization}
 P\otL_B M(Y)
 \simeq M(H(Y))\oplus M(H(Y))[3]
 \qquad\text{in }\D^b(B\text{-}\modu).
\end{equation}
All choices defining \(P\) are made independently of \(Y\).
\end{theorem}

To prove the theorem, we first replace the diagram of
\Cref{prop:lifted-diagram} by a complex of \(B\)-bimodules.
Homotopy actions of algebras do not admit such replacements in general:
higher coherence conditions can obstruct them; see
\cite[Sections~2 and~6]{Keller2000}.  Here the three-vertex shape allows
us to verify the construction directly. Keller's homotopy-corrected
complexes and bar construction \cite[Sections~3 and~7]{Keller2000}
provide the general methodological context; the finite construction
below uses only the displayed relation homotopies.

\begin{lemma}\label{lem:rectification}
Let \(D_0,D_1,D_2\) be bounded complexes of finitely generated projective
right \(B\)-modules.  Suppose that each arrow \(a:i\to j\) of \(B\)
is assigned a chain map \(f_a:D_i\to D_j\), and that each relation
\[
 \rho=\sum_{b,a}\rho_{ba}\,ba\in\mathcal R
\]
is assigned a right \(B\)-linear map \(h_\rho:D_0\to D_2\) of degree
\(-1\) satisfying
\[
 d_{D_2}h_\rho+h_\rho d_{D_0}
 = f_\rho,\qquad
 f_\rho:=\sum_{b,a}\rho_{ba}f_bf_a .
\]
There is a bounded complex \(P\) of finite-dimensional \(B\)-bimodules,
termwise projective as a right \(B\)-module, together with homotopy
equivalences of right-module complexes
\[
 \iota_i:D_i\longrightarrow \mathbf e_iP \qquad (i=0,1,2),
\]
such that, for every arrow \(a:i\to j\), the two maps
\(a\iota_i\) and \(\iota_jf_a\) from \(D_i\) to \(\mathbf e_jP\)
are chain homotopic.
\end{lemma}

\begin{proof}
We construct the differential, compare its vertex complexes with the
given ones, and finally compare the arrow actions. All tensor products
in this construction are over \(k\). Form the
complexes of bimodules
\begin{align*}
 L'_0&=\bigoplus_{i=0}^2 B\mathbf e_i\otimes_k D_i,\\
 L'_1&=\bigoplus_{a:i\to j}B\mathbf e_j\otimes_k D_i,\\
 L'_2&=\bigoplus_{\rho\in\mathcal R}B\mathbf e_2\otimes_k D_0.
\end{align*}
The left \(B\)-action is on the first tensor factor and the right
\(B\)-action is on the second.  Let
\(d_0,d_1,d_2\) denote the respective internal differentials of these
three complexes.

On the summand of \(L'_1\) indexed by \(a:i\to j\), define
\[
 p(c\otimes v)
  =(ca\otimes v)_i-(c\otimes f_a(v))_j .
\]
Here subscripts indicate summands of \(L'_0\).  On the summand of
\(L'_2\) indexed by \(\rho\), define
\[
 q(c\otimes v)
  =\sum_{b,a}\rho_{ba}
       \bigl((cb\otimes v)_a+(c\otimes f_a(v))_b\bigr).
\]
These are bimodule chain maps of degree zero.  Define also the
bimodule map \(h:L'_2\to L'_0\) of degree \(-1\) by
\[
 h(c\otimes v)=(c\otimes h_\rho(v))_2
\]
on the relation summand \(\rho\).  Expanding the composite \(pq\)
gives
\begin{align*}
 pq(c\otimes v)
 &=\sum_{b,a}\rho_{ba}
   \bigl(
     (cba\otimes v)_0-(cb\otimes f_a(v))_1\\
 &\hspace{40mm}
     +(cb\otimes f_a(v))_1-(c\otimes f_bf_a(v))_2
   \bigr)\\
 &=-(c\otimes f_\rho(v))_2.
\end{align*}
The last equality uses \(\rho=0\) in \(B\).

Set
\[
 P=L'_0\oplus L'_1[1]\oplus L'_2[2]
\]
as a graded bimodule, and equip it with the differential
\[
 d_P=
 \begin{pmatrix}
   d_0 & p    & h\\
   0   & -d_1 & q\\
   0   & 0    & d_2
 \end{pmatrix}.
\]
For example, \(P^n=(L'_0)^n\oplus(L'_1)^{n+1}
\oplus(L'_2)^{n+2}\), so \(p,q,h\) each have total degree one
in this matrix.  The diagonal signs agree with our shift convention.
Since \(p\) and \(q\) commute with the internal differentials,
the \((0,1)\) and \((1,2)\) entries of \(d_P^2\) vanish.
Its remaining off-diagonal entry is
\[
 d_0h+pq+hd_2=0,
\]
by the displayed calculation and the identity for \(h_\rho\).
The diagonal entries vanish as well.  Thus \(P\) is a complex of
bimodules.

Every term \(B\mathbf e_j\otimes_k D_i^n\), viewed as a right
\(B\)-module, is a finite direct sum of copies of \(D_i^n\).
Consequently all terms of \(P\) are finite dimensional and projective
on the right.  There are finitely many summands, and the complexes
\(D_i\) are bounded; hence \(P\) is bounded.

It remains to recover the given vertex complexes and arrow maps from
this strict bimodule complex.

Define
\[
 \iota_i(v)=(\mathbf e_i\otimes v)_i
       \in \mathbf e_iL'_0\subseteq \mathbf e_iP .
\]
This is a chain map and is split injective as a map of graded right
modules.  We prove that its quotient is acyclic.

Call \(j\) the \emph{source index} of a summand whose second tensor
factor is \(D_j\), in any of the three columns.  Give
\(\mathbf e_iP\) the decreasing filtration
\[
 F^r(\mathbf e_iP)
  =\text{the sum of its summands with source index at least }r
  \qquad (0\leq r\leq 3).
\]
These are subcomplexes.  Indeed, the internal differentials and the
path-multiplication parts of \(p,q\) preserve the source index;
the \(f_a\) terms increase it; and \(h\) increases it from \(0\)
to \(2\).  In particular \(F^0=\mathbf e_iP\) and \(F^3=0\).
The quotient by \(\iota_i(D_i)\) inherits this finite filtration.

On the associated graded, only the internal differentials and the
path-multiplication parts remain.  The source-\(i\) part consists
solely of \(\iota_i(D_i)\), since the only path from \(i\) to itself
is the trivial path.  It disappears in the quotient.  Source indices
larger than \(i\) give zero.  If \(j<i\) are adjacent vertices, the
source-\(j\) part is the signed total complex obtained by tensoring
\(D_j\) with the exact sequence
\[
 0\longrightarrow
 k^{\{a:j\to i\}}
 \xrightarrow{\ a\mapsto a\ }
 \mathbf e_iB\mathbf e_j
 \longrightarrow 0.
\]
There are no relations of length one, so the displayed map is an
isomorphism.

For the remaining pair \((j,i)=(0,2)\), that part is the signed total
complex of \(D_0\) tensored with
\[
 0\longrightarrow k^{\mathcal R}
 \longrightarrow k^{\{(b,a):a:0\to1,\ b:1\to2\}}
 \longrightarrow \mathbf e_2B\mathbf e_0
 \longrightarrow 0.
\]
The three nonzero horizontal terms have degrees \(-2,-1,0\).
The first map sends the basis vector indexed by \(\rho\) to
\(\sum_{b,a}\rho_{ba}(b,a)\); the second sends \((b,a)\) to \(ba\).
The first map is injective because \(\mathcal R\) is a basis of the
relation space.  Its image is the kernel of the second map: all
defining relations have length two, and no path of positive length
can enter vertex \(0\) or leave vertex \(2\).  Thus the ideal they
generate, in this path space, is exactly their linear span.  The
second map is surjective because length-two paths span
\(\mathbf e_2B\mathbf e_0\).

These exact complexes of vector spaces are split, and tensoring
with a bounded complex \(D_j\) gives an acyclic total complex.
All associated graded pieces of the quotient are therefore acyclic.
Induction through the finite filtration proves that the quotient
itself is acyclic.  Thus \(\iota_i\) is a quasi-isomorphism.
Its cone is a bounded acyclic complex of projective right modules,
and is contractible: starting in the largest nonzero degree, split
the surjection onto its projective last term and continue downward.
It follows that \(\iota_i\) is a homotopy equivalence.

Finally, for \(a:i\to j\) define a right-linear map of degree \(-1\)
\[
 H_a:D_i\longrightarrow\mathbf e_jP,\qquad
 H_a(v)=(\mathbf e_j\otimes v)_a
          \in\mathbf e_jL'_1[1].
\]
The internal differential on this summand is \(-d_{D_i}\);
its contribution cancels that of \(H_ad_{D_i}\).  The remaining
component is \(p\), giving
\[
 d_PH_a+H_ad_{D_i}
   =a\iota_i-\iota_jf_a .
\]
This is the required arrow homotopy.
\end{proof}

\begin{remark}
The proof also isolates the reason that higher relation homotopies
are unnecessary here.  Every relation runs from vertex \(0\) to
vertex \(2\), and cannot be composed with another nonidentity path.
The three-column differential has only the single square condition
involving \(h\) that was checked above.  There is no further column
or relation overlap that could introduce another coherence equation.
\end{remark}

\begin{proof}[Proof of \Cref{thm:realization}]
Choose the fixed complexes \(D_i\), arrow maps \(f_a\), and relation
homotopies \(h_\rho\) of \Cref{prop:lifted-diagram}, and apply
\Cref{lem:rectification}.  Since \(P\) and the \(D_i\) are bounded
complexes of projective right modules, their ordinary tensor
products with \(M(Y)\) compute the derived tensor products.
Moreover, tensoring preserves the homotopy equivalences \(\iota_i\)
and their arrow homotopies.  Thus the vertex diagram of
\[
 T_Y:=P\otL_B M(Y)
\]
in the derived category of \(k\)-vector spaces is isomorphic to the
evaluated diagram of \Cref{prop:lifted-diagram}.

At each vertex this diagram is \(eY\oplus eY[3]\).  The arrows
\(s,s'\) act by identity maps, and \(a_h,a'_h\) act by
\(\alpha(x_h)\) on each summand.  Applying cohomology, with its
induced arrow maps, therefore gives isomorphisms of left
\(B\)-modules
\[
 H^n(T_Y)\cong
 \begin{cases}
  M(H(Y)),& n=0,-3,\\
  0,& n\ne0,-3.
 \end{cases}
\]
In particular these identifications specify the full \(B\)-action,
not just the vector spaces at the vertices.

Write \(A_{-3}=H^{-3}(T_Y)\) and \(A_0=H^0(T_Y)\).
The canonical truncation triangle is
\[
 A_{-3}[3]\longrightarrow T_Y\longrightarrow A_0
 \xrightarrow{\ \delta\ } A_{-3}[4].
\]
The connecting morphism belongs to
\[
 \Hom_{\D^b(B\text{-}\modu)}(A_0,A_{-3}[4])
   =\Ext_B^4(A_0,A_{-3})=0,
\]
where the vanishing follows from the bound
\(\gldim B\leq2\) in \Cref{lem:directed-bound}.
The triangle consequently splits, giving
\(T_Y\simeq A_0\oplus A_{-3}[3]\), which is
\eqref{eq:realization}.
The complexes, chain maps, and homotopies were fixed before choosing
\(Y\), as was the rectification constructed from them.
\end{proof}

\begin{corollary}
Put \(\Phi=P\otL_B-\).  For every integer \(j\geq0\) and every
finite-dimensional left \(R\)-module \(Y\), there is an isomorphism
\[
 \Phi^j(M(Y))
 \simeq
 \bigoplus_{q=0}^j
   \bigl(M(H^{\circ j}(Y))[3q]\bigr)^{\oplus\binom{j}{q}} .
\]
\end{corollary}

\begin{proof}
The case \(j=0\) is immediate.  For the induction step, apply
\(\Phi\) to an isomorphism for \(j\); derived tensor preserves
isomorphisms, finite direct sums, and shifts.  Apply
\eqref{eq:realization} with \(H^{\circ j}(Y)\) in place of \(Y\) to each
summand, and combine the two resulting multiplicities using
Pascal's identity.  Each \(H^{\circ j}(Y)\) is finite dimensional.
This argument uses the isomorphisms only one object at a time;
no natural choice of the truncation splittings is required.
\end{proof}
==== END An-algebra-of-infinite-little-finitistic-dimension//04-tensor-realization.tex ====

==== BEGIN An-algebra-of-infinite-little-finitistic-dimension//05-ordinary-simulation.tex ====
\section{Simulation by an ordinary bimodule}
\label{sec:ordinary-simulation}

We now construct a larger finite-dimensional algebra and an ordinary
bimodule whose second derived tensor iterate recovers \(P\otL_B-\),
up to a fixed shift. All choices in this section depend only on the
fixed complex \(P\) of \Cref{thm:realization}.

Choose integers \(a_*\le 0\le b_*\) such that \(P^n=0\) outside
\([a_*,b_*]\), enlarging the interval if necessary, and put
\(l=b_*-a_*\).
Let \(C\) be the algebra of the quiver
\[
0\xrightarrow{c_1}1\xrightarrow{c_2}\cdots
\xrightarrow{c_l}l
\]
modulo all paths of length two.
Write \(\varepsilon_i\) for its vertex idempotents, and let \(W\)
be the simple \emph{right} \(C\)-module at vertex \(l\).
There is a projective resolution of right \(C\)-modules
\begin{equation}
\label{eq:right-chain-resolution}
0\longrightarrow \varepsilon_0C
\xrightarrow{c_1\cdot}\varepsilon_1C
\xrightarrow{c_2\cdot}\cdots
\xrightarrow{c_l\cdot}\varepsilon_lC
\longrightarrow W\longrightarrow0.
\end{equation}
Indeed, for \(i>0\), the right projective \(\varepsilon_iC\)
has basis \(\varepsilon_i,c_i\), and its radical is the right
simple at \(i-1\).  Left multiplication by \(c_i\) maps
\(\varepsilon_{i-1}\) to \(c_i\) and, when \(i>1\), maps
\(c_{i-1}\) to zero.  This proves exactness, including injectivity
at the first term.  For \(l=0\), the resolution consists of
the isomorphism \(\varepsilon_0C\to W\).

Define a \((B\otimes_k C,B)\)-bimodule \(O\) by
\[
\varepsilon_iO=P^{a_*+i},
\qquad
c_i\big|_{\varepsilon_{i-1}O}=d_P^{a_*+i-1}.
\]
Here the two \(B\)-actions are those on the terms of \(P\).
The relation \(d_P^2=0\) makes this a representation of \(C\),
and the bimodule linearity of \(d_P\) makes the \(C\)-action
commute with both \(B\)-actions.  Thus \(O\) is an ordinary
finite-dimensional bimodule.  Set
\[
D=B\times(B\otimes_k C),\qquad
T=B\otimes_k W,\qquad X=O\oplus T.
\]
Regard \(O\) as having its left action from the second factor
of \(D\) and its right action from the first.  Regard \(T\)
as having its left action from the first factor and its
right action from the second, with the regular \(B\)-actions
on its \(B\)-factor.  These prescriptions define an ordinary
finite-dimensional \(D\)-bimodule \(X\).

The two summands of \(X\) move complexes between the factors of \(D\),
as shown in \Cref{fig:simulation}. Thus two tensor steps return to the
first factor; the next proposition identifies their composite.

\begin{figure}[htbp]
\centering
\begin{tikzcd}[column sep=huge]
 \D^b(B\text{-}\modu)
 \arrow[r,bend left=22,"O\otL_B-"] &
 \D^b((B\otimes_k C)\text{-}\modu)
 \arrow[l,bend left=22,"T\otL_{B\otimes_k C}-"]
\end{tikzcd}
\caption{The two factors of \(D\) and the tensor functors supplied by
the off-diagonal summands of \(X\). Starting on the left, the composite
is \(P[b_*]\otL_B-\) by \Cref{prop:ordinary-simulation}.}
\label{fig:simulation}
\end{figure}

\begin{proposition}
\label{prop:ordinary-simulation}
The algebra \(D\) has finite global dimension on both sides.
Let \(F=X\otL_D-\).  For every complex \(Z\) of left
\(B\)-modules, regarded as a complex on the first factor of
\(D\), there is an isomorphism
\begin{equation}
\label{eq:compression}
F^2(Z)\simeq P[b_*]\otL_B Z,
\qquad
T\otL_{B\otimes_k C}O\simeq P[b_*].
\end{equation}
The second isomorphism is an isomorphism of complexes of
\(B\)-bimodules, and the first is supported on the first
factor of \(D\).
\end{proposition}

\begin{proof}
The vertices of \(B\otimes_k C\) are pairs \((i,j)\), where
\(0\le i\le2\) and \(0\le j\le l\).  Its diagonal vertex
corners are \(k\).  A nonzero component between distinct
vertices moves in the product of the two directed orders.
Choosing a linear extension of that product order permits
an application of \Cref{lem:directed-bound}, on both sides.
In particular,
\[
\gldim(B\otimes_k C)\le3(l+1)-1,
\qquad
\gldim\bigl((B\otimes_k C)^{\mathrm{op}}\bigr)
\le3(l+1)-1.
\]
\Cref{lem:directed-bound} also gives the bound \(2\) for \(B\) and
\(B^{\mathrm{op}}\).  Modules over a finite product split
according to its factors, so
\[
\gldim D,\ \gldim D^{\mathrm{op}}\le3l+2.
\]

Tensor \eqref{eq:right-chain-resolution} over \(k\) with
the regular \(B\)-bimodule, placing its last projective term
in degree zero.  This gives a bounded resolution \(Q_T\to T\)
by right \(B\otimes_k C\)-projectives, preserving the left
\(B\)-action.  Also, \(O\) is projective as a right
\(B\)-module, being the finite direct sum of the terms of \(P\).
Thus \(Q_X=O\oplus Q_T\), with the prescribed block actions,
is a bounded right-\(D\)-projective resolution of \(X\) retaining
its left \(D\)-action.  Ordinary tensor with \(Q_X\) computes
\(F\) on every complex.  On the first factor, the block supports
therefore give
\[
F^2(Z)\simeq
Q_T\otimes_{B\otimes_k C}(O\otimes_B Z)
\cong
\bigl(Q_T\otimes_{B\otimes_k C}O\bigr)\otimes_B Z.
\]
In degree \(n\), for \(-l\le n\le0\), tensoring \(Q_T\)
with \(O\) gives
\[
\bigl(B\otimes_k\varepsilon_{l+n}C\bigr)
\otimes_{B\otimes_k C}O
\cong\varepsilon_{l+n}O
=P^{a_*+l+n}=P^{b_*+n}.
\]
The induced differential is \(d_P\).
In comparison, the standard differential of \(P[b_*]\) is
\((-1)^{b_*}d_P\).  Multiplication by
\((-1)^{b_*n}\) in degree \(n\) defines a chain isomorphism
to \(P[b_*]\); it respects both \(B\)-actions.
Since this complex is termwise right-\(B\)-projective, its
ordinary tensor with any \(Z\) computes the derived tensor.
This proves the second isomorphism in
\eqref{eq:compression}, and hence the first.
\end{proof}
==== END An-algebra-of-infinite-little-finitistic-dimension//05-ordinary-simulation.tex ====

==== BEGIN An-algebra-of-infinite-little-finitistic-dimension//06-square-zero-and-conclusion.tex ====
\section{A square-zero extension and projective dimension}
\label{sec:square-zero}

We first establish the homological statement for an arbitrary
split square-zero extension over a field.  Its derived tensor
powers cannot in general be replaced by ordinary tensor powers.
The decomposition below will let us prove two separate facts: eventual
extinction gives a finite projective resolution, while survival of the
\(r\)th iterate rules out any projective resolution of length less than \(r\).

\subsection{The derived bar decomposition}

The splitting below also follows from the graded-component decomposition
of Minamoto and Yamaura
\cite[Lemma~4.13(4) and the proof of Theorem~4.17]{MinamotoYamaura},
after forgetting the grading and passing to opposite algebras.
The direct proof identifies each derived tensor factor and its shift.

\begin{proposition}
\label{prop:bar-splitting}
Let \(D\) be a finite-dimensional \(k\)-algebra, let \(X\) be
a finite-dimensional \(D\)-bimodule, and let
\(A=D\ltimes X\) be the algebra with multiplication
\[
(d,x)(d',x')=(dd',dx'+xd').
\]
Let \(N\) be a left \(D\)-module, inflated to \(A\) through
the projection \(A\to D\), and put \(F=X\otL_D-\).
In the derived category of left \(D\)-modules there is an
isomorphism
\begin{equation}
\label{eq:bar-splitting}
D\otL_A N
\simeq
\bigoplus_{r\ge0}(F^rN)[r].
\end{equation}
Here \(F^0N=N\), and every \(F^r\) denotes an iteration
of the derived tensor functor.
\end{proposition}

\begin{proof}
Use the ordinary unnormalized bar resolution of \(N\) over
the field \(k\).  It is a free left \(A\)-resolution with
degree \(-n\) term \(A\otimes_k A^{\otimes_k n}\otimes_k N\).
Its augmentation is a quasi-isomorphism: the augmented
bar complex has the usual \(k\)-linear contraction obtained
by inserting the unit.  After tensoring with the right
\(A\)-module \(D\), it yields a model \(\mathcal B\) for
\(D\otL_AN\), with
\[
\mathcal B^{-n}=D\otimes_k A^{\otimes_k n}\otimes_k N
\qquad(n\ge0).
\]
The differential is the alternating sum of the faces
that multiply adjacent algebra entries or apply the two
endpoint actions.

Expand each internal copy of \(A\) as \(D\oplus X\).
Every nonzero face preserves the number of entries from
\(X\).  Indeed, a product involving exactly one \(X\)-entry
remains in \(X\), a product of two \(X\)-entries is zero,
and an endpoint action by \(X\) is zero on both \(D\) and
\(N\).  Thus there is an actual decomposition of complexes
\[
\mathcal B=\bigoplus_{r\ge0}\mathcal B_r,
\]
where \(\mathcal B_r\) consists of words with precisely
\(r\) entries from \(X\).

For fixed \(r\), write \(i_0,\ldots,i_r\) for the lengths
of the blocks of \(D\)-entries before, between, and after
the \(X\)-entries.  The corresponding tensor is
\begin{equation}
\label{eq:bar-block}
D\otimes_kD^{\otimes_k i_0}\otimes_k X
\otimes_kD^{\otimes_k i_1}\otimes_k\cdots
\otimes_k X\otimes_kD^{\otimes_k i_r}\otimes_k N,
\end{equation}
in degree \(-r-\sum_{j=0}^r i_j\).
We identify this with the multiple two-sided bar complex
for
\begin{equation}
\label{eq:derived-multibar}
D\otL_D
\underbrace{X\otL_D\cdots\otL_D X}_{r\ \text{factors}}
\otL_D N,
\end{equation}
shifted by \(r\).

First consider the signs.  Within block \(j\), a bar face
has alternating-sign offset
\[
j+\sum_{h<j}i_h.
\]
In the ordinary multibar total differential, ordered from
left to right, the corresponding offset is
\(\sum_{h<j}i_h\); shifting the total complex by \(r\)
adds the factor \((-1)^r\).
Rescale the component \eqref{eq:bar-block} by
\[
(-1)^{\sum_{j=0}^r(r-j)i_j}.
\]
When a face reduces \(i_j\) by one, this factor changes by
\((-1)^{r-j}\), precisely the discrepancy between the
two differentials.  This gives the required chain
isomorphism.  When \(r\ge1\) and \(i_j=0\), there is no
nonzero face in that block: the prospective face multiplies
two \(X\)-entries or applies an endpoint \(X\)-action.
For \(r=0\), this is the ordinary two-sided bar complex
for \(D\otL_DN\).  At \(i_0=0\) its differential is absent;
the map \(D\otimes_kN\to N\) is the separate augmentation.

We next justify that the multibar computes the genuinely
derived tensor in \eqref{eq:derived-multibar}, without any
flatness assumption on \(X\) over \(D\).
For a bounded-above complex \(Z\) of left \(D\)-modules,
the total left-module bar resolution has terms obtained
from
\[
D\otimes_k D^{\otimes_k i}\otimes_k Z^q
\]
in degree \(q-i\).  Its terms are free left \(D\)-modules
because \(k\) is a field, and its augmentation to \(Z\)
is a quasi-isomorphism.  For each total degree there are
only finitely many possible \(i\), since \(Z\) is bounded
above.  Thus it is a bounded-above complex of free
modules and can be used to compute a derived tensor.

Starting at the right, resolve \(N\) in this way and
tensor with \(X\).  Resolve the resulting left
\(D\)-complex in the same way and tensor with the next
\(X\), continuing for the fixed finite number \(r\)
of factors.  A final bar resolution with the leading
factor \(D\) gives exactly the multibar
\eqref{eq:bar-block}, before its shift.
All exterior \(D\)-actions are preserved.
This establishes its identification with
\eqref{eq:derived-multibar}, which is isomorphic to
\(F^rN\).

Finally, in degree \(-n\) of \(\mathcal B\) only
\(0\le r\le n\) occur, and for each such \(r\) there
are finitely many tuples satisfying
\(r+\sum_j i_j=n\).
All totalizations above therefore use direct sums that
are finite in each degree.  There is no passage to
products or completed tensor powers.
Direct sums are exact for modules, so the componentwise
quasi-isomorphisms give \eqref{eq:bar-splitting}.
The argument uses no connectedness assumption on \(D\);
in particular it applies to the product algebra of
\Cref{sec:ordinary-simulation}.
\end{proof}

\subsection{Detecting finite projective dimension and lower bounds}

\begin{corollary}
\label{cor:pd-detection}
In the setting of \Cref{prop:bar-splitting}, suppose that
\(D\) has finite global dimension on both sides and that
\(N\) is finite-dimensional.
\begin{enumerate}
\item If \(F^tN\simeq0\) for some \(t\ge1\), then
      \(\pd_A N<\infty\).
\item If \(F^rN\not\simeq0\), then \(\pd_A N\ge r\),
      with infinite projective dimension allowed.
\end{enumerate}
\end{corollary}

\begin{proof}
Every fixed iterate \(F^rN\) has bounded,
finite-dimensional cohomology.
For example, if \(g\) bounds the right global dimension
of \(D\), then for every finite-dimensional left
\(D\)-module \(V\) the groups
\(\Tor_i^D(X,V)\) are finite-dimensional and vanish for
\(i>g\).  Applying this observation to successive
truncation triangles shows inductively that
\[
H^q(F^rN)=0\quad\text{unless}\quad -rg\le q\le0.
\]
In particular, all iterates lie in \(\D^{\le0}(D)\).

For the first assertion, vanishing at \(t\) implies
vanishing at every \(r\ge t\).
By \eqref{eq:bar-splitting},
\(Z=D\otL_AN\) then has bounded cohomology.
Since \(X\) is a nilpotent ideal, it is contained in
\(\rad A\), and
\[
\rad A=(\rad D)\oplus X,
\qquad A/\rad A\cong D/\rad D=:S.
\]
Derived associativity gives
\[
S\otL_A N
\simeq S\otL_D(D\otL_A N)
=S\otL_D Z.
\]
The right \(D\)-module \(S\) has a finite projective
resolution, so this complex also has bounded cohomology.

Choose a minimal projective resolution of \(N\) over
the finite-dimensional algebra \(A\), using successive
projective covers.  Its terms are finitely generated,
and every differential has image in the radical of
its target.  Tensoring with \(S=A/\rad A\) therefore
annihilates every differential.  If \(Q\) is a nonzero
finitely generated projective \(A\)-module, then
\[
S\otimes_A Q\cong Q/(\rad A)Q\ne0
\]
by Nakayama's Lemma.  Consequently bounded cohomology
of \(S\otL_AN\) forces the minimal resolution to have
only finitely many nonzero terms.  This proves
\(\pd_A N<\infty\).

For the second assertion, choose \(q\le0\) with
\(H^q(F^rN)\ne0\).  Its direct summand in
\eqref{eq:bar-splitting} gives
\[
H^{q-r}(D\otL_A N)\ne0,\qquad q-r\le-r.
\]
A projective \(A\)-resolution of \(N\) of length
strictly less than \(r\) would make the derived
tensor zero in this degree.  Hence
\(\pd_A N\ge r\), as asserted.
\end{proof}

\begin{remark}
For inflated modules, the projective-dimension formula
of Minamoto and Yamaura
\cite[Corollary~4.11]{MinamotoYamaura}, after passage
to the opposite algebra to match our left-module
convention, gives the stronger equality
\[
\pd_A N
=\sup_{r\ge0}\bigl\{\pd_D(F^rN)+r\bigr\},
\]
where projective dimension on the right is understood
for derived complexes, with \(\pd_D(0)=-\infty\).
The graded and ungraded dimensions
agree by \cite[Proposition~2.5 and Lemma~3.6]{MinamotoYamaura}.
Their
\cite[Theorem~4.17]{MinamotoYamaura} also gives the
corresponding perfectness criterion.
The direct bar and minimal-resolution arguments above
prove the assertions needed here.
\end{remark}

\subsection{The fixed algebra and its modules}

\begin{proof}[Proof of \Cref{thm:main}]
Work over \(k=\C\).
Fix the algebra \(R\), its central idempotent \(e\),
the automorphism \(\alpha\), and the selection functor
\(H\) supplied by \Cref{prop:selection}.
Apply \Cref{thm:realization} to these fixed data,
obtaining \(B\) and \(P\) satisfying
\eqref{eq:realization}.
Choose the support interval \([a_*,b_*]\) once, and
construct \(C,W,O,D,X\) as in
\Cref{sec:ordinary-simulation}.
Finally fix the ordinary finite-dimensional unital
algebra
\[
A=D\ltimes X.
\]
These choices are made independently of \(m\), as summarized in
\Cref{tab:fixed-varying}.

\begin{table}[htbp]
\centering
\begin{tabular}{@{}p{0.24\textwidth}p{0.68\textwidth}@{}}
\toprule
Choice & Objects \\
\midrule
Fixed once & \(R,e,\alpha\); the presentation and algebra \(B\);
the lifted diagram and complex \(P\); the interval \([a_*,b_*]\);
\(C,W,O,T,D,X\); and \(A=D\ltimes X\). \\
Varying with \(m\) & The quotient \(F_m\), its central character,
the selection module \(Y_m\), and the inflated module
\(N_m=M(Y_m)\). \\
\bottomrule
\end{tabular}
\caption{The quantifier order in the construction of \(A\). Varying the
test module changes neither the tensor functor nor the final algebra.}
\label{tab:fixed-varying}
\end{table}

For a finite-dimensional left \(R\)-module \(Y\),
regard \(M(Y)\) as supported on the first factor
of \(D\).  By \eqref{eq:compression} and
\eqref{eq:realization},
\[
F^2M(Y)
\simeq M(H(Y))[b_*]\oplus M(H(Y))[b_*+3].
\]
Induction, using preservation of finite direct sums
and shifts by derived tensor, gives
\begin{equation}
\label{eq:even-iterates}
F^{2j}M(Y)
\simeq
\bigoplus_{q=0}^{j}
M(H^{\circ j}(Y))[jb_*+3q]^{\oplus\binom{j}{q}}
\qquad(j\ge0).
\end{equation}
The induction applies the realization isomorphism
separately to each module \(H^{\circ j}(Y)\); it requires
no splitting natural in \(Y\).

For each \(m\ge1\), let \(Y_m\) be the finite-dimensional
module supplied by \eqref{eq:extinction}, and let
\(N_m=M(Y_m)\), first as a left \(D\)-module on its
first factor and then as an inflated left \(A\)-module.
It is finite-dimensional.
Since \(H^{\circ m}(Y_m)=0\) and \(H^{\circ(m-1)}(Y_m)\ne0\),
\eqref{eq:even-iterates} yields
\[
F^{2m}N_m\simeq0,\qquad
F^{2m-2}N_m\not\simeq0.
\]
The latter nonvanishing follows because a finite
direct sum of shifts of the nonzero module
\(M(H^{\circ(m-1)}(Y_m))\) is nonzero. Shifts preserve both extinction
and nonvanishing; the lower bound now uses the iterate index
\(r=2m-2\) in \Cref{cor:pd-detection}, not a new estimate on the shifts.
The algebra \(D\) has finite global dimension on both
sides by \Cref{prop:ordinary-simulation}.
Applying \Cref{cor:pd-detection} proves
\[
2m-2\le\pd_A N_m<\infty
\qquad(m\ge1).
\]
All \(N_m\) are finitely generated over the single
fixed algebra \(A\), while these lower bounds are
unbounded.  Therefore \(\findim A=\infty\), which
refutes the asserted finiteness for all
finite-dimensional algebras.
\end{proof}

\subsection{The auxiliary left--right asymmetry}
\label{sec:left-right-asymmetry}

\begin{proof}[Proof of \Cref{cor:left-right-asymmetry}]
Let \(A\) be the algebra of \Cref{thm:main}, and choose a basic
finite-dimensional complex algebra \(C\) Morita equivalent to
\(A^{\mathrm{op}}\). The right-module
little and big finitistic dimensions of \(C\) are infinite, and its
injective right modules do not generate: these are the conclusions
for left \(A\)-modules in \Cref{sec:introduction}, transported through
Morita equivalence. Indeed, this equivalence preserves projective dimensions
and finite generation, and the induced equivalence of unbounded derived
categories preserves injectives and coproducts.

Choose primitive idempotents \(e_1,\ldots,e_n\) representing the simple
right \(C\)-modules \(S_C(e_i)=e_iC/e_i\rad C\). In Cummings'
Construction~2.1 \cite{Cummings2024}, set
\[
E=\prod_{i=1}^n\C[\varepsilon_i]/(\varepsilon_i^2),
\qquad U=\bigoplus_{i=1}^n S_C(e_i),
\qquad
T=\begin{pmatrix}C&0\\ {}_E U_C&E\end{pmatrix}.
\]
The left \(E\)-action on \(U\) factors through
\(E/\rad E\cong\C^n\), with the \(i\)th factor acting on \(S_C(e_i)\).
This is a finite-dimensional complex algebra.

Write \(\findim_r\) and \(\Findim_r\) for the right-module dimensions,
to display Cummings' convention. By
\cite[Propositions~3.2--3.3]{Cummings2024},
\[
\begin{gathered}
\findim_r C\le\findim_r T,\\
\Findim_r C\le\Findim_r T,\\
\findim_r(T^{\mathrm{op}})=\Findim_r(T^{\mathrm{op}})=0.
\end{gathered}
\]
His Proposition~4.3 says that injective generation for right
\(T\)-modules implies it for right \(C\)-modules, while
Proposition~4.4 establishes it for right \(T^{\mathrm{op}}\)-modules
\cite[Propositions~4.3--4.4]{Cummings2024}. Hence both right finitistic
dimensions of \(T\) are infinite and its injective right modules do not
generate; for
\(T^{\mathrm{op}}\), both right dimensions are zero and its injective
right modules generate. Taking \(\Lambda=T^{\mathrm{op}}\) translates
these statements into our left-module convention.
\end{proof}
==== END An-algebra-of-infinite-little-finitistic-dimension//06-square-zero-and-conclusion.tex ====

==== BEGIN An-algebra-of-infinite-little-finitistic-dimension//paper.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{An algebra of infinite little finitistic dimension}
\author{OpenAI}
\date{September 23, 2026}
\begin{document}
\maketitle
\begin{abstract}
We construct a finite-dimensional complex algebra whose finite-dimensional
modules have unbounded finite projective dimensions. This disproves the little
finitistic-dimension conjecture.
\end{abstract}
\input{sections/01-introduction}
\input{sections/02-selection-process}
\input{sections/03-localization-and-lifting}
\input{sections/04-tensor-realization}
\input{sections/05-ordinary-simulation}
\input{sections/06-square-zero-and-conclusion}
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END An-algebra-of-infinite-little-finitistic-dimension//paper.tex ====

==== BEGIN An-algebra-of-infinite-little-finitistic-dimension//preamble.tex ====
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==== END An-algebra-of-infinite-little-finitistic-dimension//preamble.tex ====

==== BEGIN An-algebra-of-infinite-little-finitistic-dimension//references.bib ====
@article{Bass1960,
  author  = {Bass, Hyman},
  title   = {Finitistic dimension and a homological generalization of semi-primary rings},
  journal = {Transactions of the American Mathematical Society},
  volume  = {95},
  number  = {3},
  year    = {1960},
  pages   = {466--488},
  doi     = {10.1090/S0002-9947-1960-0157984-8},
  url     = {https://doi.org/10.1090/S0002-9947-1960-0157984-8},
  note    = {\href{https://doi.org/10.1090/S0002-9947-1960-0157984-8}{Publisher DOI}}
}

@article{Rego2022,
  author  = {Santos Rego, Yuri},
  title   = {On the finiteness length of some soluble linear groups},
  journal = {Canadian Journal of Mathematics},
  volume  = {74},
  number  = {5},
  year    = {2022},
  pages   = {1209--1243},
  doi     = {10.4153/S0008414X21000213},
  eprint  = {1901.06704},
  archivePrefix = {arXiv},
  primaryClass = {math.GR},
  url     = {https://arxiv.org/abs/1901.06704v3},
  note    = {\href{https://doi.org/10.4153/S0008414X21000213}{Publisher DOI};
             \href{https://arxiv.org/abs/1901.06704v3}{arXiv:1901.06704v3}}
}

@article{NRS2004,
  author  = {Neeman, Amnon and Ranicki, Andrew and Schofield, Aidan},
  title   = {Representations of algebras as universal localizations},
  journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
  volume  = {136},
  number  = {1},
  year    = {2004},
  pages   = {105--117},
  doi     = {10.1017/S030500410300700X},
  eprint  = {math/0205034},
  archivePrefix = {arXiv},
  url     = {https://arxiv.org/abs/math/0205034v2},
  note    = {\href{https://doi.org/10.1017/S030500410300700X}{Publisher DOI};
             \href{https://arxiv.org/abs/math/0205034v2}{arXiv:math/0205034v2}}
}

@article{BalmerSchlichting2001,
  author  = {Balmer, Paul and Schlichting, Marco},
  title   = {Idempotent completion of triangulated categories},
  journal = {Journal of Algebra},
  volume  = {236},
  number  = {2},
  year    = {2001},
  pages   = {819--834},
  doi     = {10.1006/jabr.2000.8529},
  url     = {https://doi.org/10.1006/jabr.2000.8529},
  note    = {\href{https://doi.org/10.1006/jabr.2000.8529}{Publisher DOI};
             \href{https://webhomes.maths.ed.ac.uk/~v1ranick/papers/balmschl.pdf}{full text}}
}

@article{Keller2000,
  author  = {Keller, Bernhard},
  title   = {Bimodule complexes via strong homotopy actions},
  journal = {Algebras and Representation Theory},
  volume  = {3},
  number  = {4},
  year    = {2000},
  pages   = {357--376},
  doi     = {10.1023/A:1009954126727},
  eprint  = {math/9910178},
  archivePrefix = {arXiv},
  url     = {https://arxiv.org/abs/math/9910178v1},
  note    = {\href{https://doi.org/10.1023/A:1009954126727}{Publisher DOI};
             \href{https://arxiv.org/abs/math/9910178v1}{arXiv:math/9910178v1}}
}

@article{MinamotoYamaura,
  author  = {Minamoto, Hiroyuki and Yamaura, Kota},
  title   = {Homological dimension formulas for trivial extension algebras},
  journal = {Journal of Pure and Applied Algebra},
  volume  = {224},
  number  = {8},
  year    = {2020},
  pages   = {106344},
  doi     = {10.1016/j.jpaa.2020.106344},
  eprint  = {1710.01469},
  archivePrefix = {arXiv},
  primaryClass = {math.RA},
  url     = {https://arxiv.org/abs/1710.01469v1},
  note    = {30 pp.
             \href{https://doi.org/10.1016/j.jpaa.2020.106344}{Publisher DOI};
             \href{https://arxiv.org/abs/1710.01469v1}{arXiv:1710.01469v1}.
             All result numbers in citations to this paper refer to this arXiv version}
}

@misc{StacksProject,
  author = {{The Stacks Project Authors}},
  title  = {The {Stacks} project},
  year   = {2026},
  howpublished = {\url{https://stacks.math.columbia.edu}},
  url    = {https://stacks.math.columbia.edu},
  note   = {Tags \href{https://stacks.math.columbia.edu/tag/04VI}{04VI},
            \href{https://stacks.math.columbia.edu/tag/04VJ}{04VJ},
            \href{https://stacks.math.columbia.edu/tag/05RG}{05RG}, and
            \href{https://stacks.math.columbia.edu/tag/05RJ}{05RJ};
            accessed 23 September 2026}
}

@article{HuisgenZimmermann1992,
  author = {Zimmermann Huisgen, Birge},
  title = {Homological domino effects and the first {Finitistic Dimension Conjecture}},
  journal = {Inventiones Mathematicae},
  volume = {108},
  year = {1992},
  pages = {369--383},
  doi = {10.1007/BF02100610},
  note = {\href{https://doi.org/10.1007/BF02100610}{Publisher DOI}}
}

@article{GreenKirkmanKuzmanovich1991,
  author = {Green, Edward L. and Kirkman, Ellen E. and Kuzmanovich, James},
  title = {Finitistic dimensions of finite dimensional monomial algebras},
  journal = {Journal of Algebra},
  volume = {136},
  number = {1},
  year = {1991},
  pages = {37--50},
  doi = {10.1016/0021-8693(91)90062-D},
  note = {\href{https://doi.org/10.1016/0021-8693(91)90062-D}{Publisher DOI}}
}

@article{GreenHuisgenZimmermann1991,
  author = {Green, Edward L. and Zimmermann Huisgen, Birge},
  title = {Finitistic dimension of {Artinian} rings with vanishing radical cube},
  journal = {Mathematische Zeitschrift},
  volume = {206},
  number = {4},
  year = {1991},
  pages = {505--526}
}

@incollection{IgusaTodorov2005,
  author = {Igusa, Kiyoshi and Todorov, Gordana},
  title = {On the finitistic global dimension conjecture for {Artin} algebras},
  booktitle = {Representations of Algebras and Related Topics},
  series = {Fields Institute Communications},
  volume = {45},
  publisher = {American Mathematical Society},
  address = {Providence, RI},
  year = {2005},
  pages = {201--204}
}

@article{Rickard2019,
  author = {Rickard, Jeremy},
  title = {Unbounded derived categories and the finitistic dimension conjecture},
  journal = {Advances in Mathematics},
  volume = {354},
  year = {2019},
  pages = {106735},
  doi = {10.1016/j.aim.2019.106735},
  eprint = {1804.09801},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1804.09801v1},
  note = {\href{https://doi.org/10.1016/j.aim.2019.106735}{Publisher DOI};
    \href{https://arxiv.org/abs/1804.09801v1}{arXiv:1804.09801v1}}
}

@incollection{HuisgenZimmermann1995,
  author = {Zimmermann Huisgen, Birge},
  title = {The finitistic dimension conjectures---a tale of 3.5 decades},
  booktitle = {Abelian Groups and Modules},
  editor = {Facchini, Alberto and Menini, Claudia},
  series = {Mathematics and Its Applications},
  volume = {343},
  publisher = {Kluwer Academic Publishers},
  address = {Dordrecht},
  year = {1995},
  pages = {501--517},
  doi = {10.1007/978-94-011-0443-2_41},
  eprint = {1407.2383},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1407.2383v1},
  note = {\href{https://doi.org/10.1007/978-94-011-0443-2_41}{Publisher DOI};
    \href{https://arxiv.org/abs/1407.2383v1}{arXiv:1407.2383v1}}
}

@article{Cummings2024,
  author = {Cummings, Charley},
  title = {Left--right symmetry of finite finitistic dimension},
  journal = {Bulletin of the London Mathematical Society},
  volume = {56},
  number = {2},
  year = {2024},
  pages = {624--633},
  doi = {10.1112/blms.12954},
  eprint = {2211.04394},
  archivePrefix = {arXiv},
  note = {\href{https://doi.org/10.1112/blms.12954}{Publisher DOI}}
}

@misc{OpenAITachikawa2026,
  author = {{OpenAI}},
  title = {{A counterexample to Tachikawa's second conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-Tachikawas-second-conjecture-September-23-2026}},
  year = {2026}
}
==== END An-algebra-of-infinite-little-finitistic-dimension//references.bib ====