64c3814
==== TABLE OF CONTENTS ====
Integral-Donovan-Finiteness-over-Witt-Vectors//01_introduction.tex
Integral-Donovan-Finiteness-over-Witt-Vectors//02_inputs.tex
Integral-Donovan-Finiteness-over-Witt-Vectors//03_rigidity.tex
Integral-Donovan-Finiteness-over-Witt-Vectors//04_comparison.tex
Integral-Donovan-Finiteness-over-Witt-Vectors//05_sylow.tex
Integral-Donovan-Finiteness-over-Witt-Vectors//06_kernel.tex
Integral-Donovan-Finiteness-over-Witt-Vectors//07_assembly.tex
Integral-Donovan-Finiteness-over-Witt-Vectors//main.tex
Integral-Donovan-Finiteness-over-Witt-Vectors//preamble.tex
Integral-Donovan-Finiteness-over-Witt-Vectors//references.bib
===========================
==== BEGIN Integral-Donovan-Finiteness-over-Witt-Vectors//01_introduction.tex ====
\section{Introduction}\label{sec:introduction}
Fix a prime $p$, put $k=\overline{\F}_p$, and let $\cO=W(k)$ be its
ring of Witt vectors. A block of a finite group $G$ over $\cO$ is an
algebra $\cO G b$, where $b$ is a primitive central idempotent.
Its reduction $kG\bar b$ has a defect group, well defined up to
conjugacy, and we use the same defect group for the integral block.
All Morita equivalences in this paper are equivalences of categories
of finitely generated modules which are linear over the specified
coefficient ring.
Fixing the defect group is expected to leave only finitely many
Morita types of blocks. This is Donovan's finiteness problem. Over
$\cO$ it concerns integral module categories, including their lattice
structure, and hence is stronger than the corresponding assertion
over the residue field. Our main theorem is the integral assertion for
the Witt ring $W(k)$.
\begin{theorem}\label{thm:main}
Let $p$ be a prime, let $\cO=W(\overline{\F}_p)$, and let
$M\geq1$ be an integer. As $G$ ranges over all
finite groups and $b$ ranges over all blocks of $\cO G$ whose defect
groups have order at most $M$, the algebras $\cO G b$ belong to only
finitely many $\cO$-linear Morita equivalence classes.
\end{theorem}
The theorem includes every prime, including $2$, and every block,
including nonprincipal blocks. The defect groups may be nonabelian;
the ambient finite groups have no order bound. There are finitely
many groups of order at most $M$, so fixing one defect group or bounding
its order gives equivalent finiteness assertions. In terms of basic
orders, Theorem~\ref{thm:main} says that a finite list of
$\cO$-algebras represents all the indicated Morita classes.
The finiteness conclusion also passes to a fixed complete coefficient
ring with algebraically closed residue field.
\begin{corollary}[Fixed complete coefficient rings]\label{cor:complete-dvr}
Let $p$ be a prime, let $\mathcal R$ be a fixed complete discrete
valuation ring of characteristic zero whose residue field $\ell$ is
algebraically closed of characteristic $p$, and let $M\geq1$ be an
integer. As $G$ ranges over all finite groups and $b$ ranges over
primitive central idempotents of $\mathcal R G$ for which the residue
block $\ell G\bar b$ has a defect group of order at most $M$, the
algebras $\mathcal R G b$ belong to only finitely many
$\mathcal R$-linear Morita equivalence classes.
\end{corollary}
Here $\mathcal R$ is fixed before $G$ and $b$ vary. It may be ramified,
and no splitting hypothesis on its fraction field is required. The
proof at the end of Section~\ref{sec:assembly} uses compatible block
idempotent lifts and scalar extension of the integral Morita
equivalences from Theorem~\ref{thm:main}.
\subsection{Finiteness criteria and extension methods}
The local representation-theoretic questions surrounding Donovan's
conjecture were articulated in Alperin's account of local
representation theory \cite{Alperin1980}. Two different sorts of
control enter finiteness: bounds on the size of a basic algebra, and
bounds on its field of definition. Hiss made the field-of-definition
requirement explicit: bounded defect and Cartan invariants yield Morita
finiteness when the blocks have split forms over a common finite field
\cite[Proposition~5.1]{HissBrauer2000}. Kessar separated these issues
using Morita--Frobenius numbers \cite{KessarRemark2004}.
For an integral order $B$, its Morita--Frobenius number
$\mf_{\cO}(B)$ is the least positive coefficient-Frobenius power
giving a Morita-equivalent order. The integral size and rationality
bounds are combined in the finiteness theorem of
Eaton--Eisele--Livesey \cite[Theorem~3.10]{EEL2020}: bounded defect,
bounded Cartan sum, and bounded integral Morita--Frobenius number imply
integral Morita finiteness. The theorem itself allows arbitrary defect
groups; the Cartan-only quasisimple reduction in that paper has the
additional hypothesis of abelian defect.
Their results establish integral Donovan finiteness for all abelian
$2$-groups \cite[Corollary~4.6]{EEL2020}. For nonabelian defect
groups, Eaton--Eisele--Kessar--Linckelmann--Schaeffer Fry prove
integral Donovan finiteness for quaternion defect groups
\cite[Theorem~1.1]{EEKLS2026}.
Crossed products encode the passage from normal subgroups to their
extensions. K\"ulshammer developed this approach in Clifford theory
and in his reduction of Donovan's conjecture
\cite{KulshammerClifford1990,Kulshammer1995}. Eisele established integral
versions and studied the relevant Picard groups
\cite{EiseleReduction2021,EiseleGeometry2022}. His geometry of
rigid lattices supplies the methodological setting for integral
Picard finiteness \cite[Theorems~A and~B]{EiseleGeometry2022}. We use the preliminary
Fong and nilpotent-block reduction in An--Eaton
\cite[Proposition~6.1]{AnEaton2025}; that proposition applies to
arbitrary defect groups over $\cO$, independently of the extraspecial
hypotheses used in their subsequent results.
For quasisimple groups, Farrell--Kessar give the uniform bound
$\mf_{\cO}(B)\leq4$ \cite[Theorem~1.1]{FarrellKessar2019}.
Their theorem does not itself compare the multiplication factors in
an arbitrary group extension. This distinction matters: controlling
the Morita class of an identity component or its outer automorphisms
does not determine a crossed product. Eisele--Livesey's constructions
of arbitrarily large Morita--Frobenius numbers, in families with
growing defect, give further reason to retain the rationality data
explicitly \cite{EiseleLivesey2022}.
The companion article \emph{Donovan's Conjecture over Algebraically
Closed Fields} \cite{OpenAIFieldDonovan2026} establishes
bounded-defect finiteness over $k$ for all finite groups. It supplies
two inputs here: a uniform Cartan bound, through its field theorem,
and the integral extension and comparison constructions of its
Sections~8 and~9. The latter retain the actual multiplication factors
in the normal-subgroup extensions. Their advertised equivariance
is exact after reduction. We will refine the specified integral
operators to obtain exact equivariance over $\cO$ itself.
\subsection{What must be retained over the Witt ring}
The distinction between the two coefficient rings is substantial.
A coefficient-Frobenius-fixed point over $\cO$ has coordinates in
$W(\F_q)$, which is infinite. Thus the finite-field point count used
in a descent argument over $k$ gives no integral finiteness assertion.
Also, a scalar obstruction over $\cO$ may contain principal units,
whose cohomology on a $p$-group need not vanish. We address these
two issues separately: the first requires rigidity of integral
orders, and the second requires control of the particular scalar
errors produced by the comparison.
The normal-subgroup reductions make every relevant block Morita
equivalent to a block summand of a crossed order with identity
component an integral
block $B_0$ and grading group $Q$. The integral basic orders of $B_0$
belong to a finite list, while $[Q:O_{p'}(Q)]$ is bounded. A crossed
order retains both the automorphisms of $B_0$ and the units that occur
when homogeneous generators are multiplied. Its restriction to a
subgroup is the sum of the homogeneous components labelled by the
elements of that subgroup.
Three successive assertions carry the integral proof. The first is
\emph{exact comparison}. The geometric overlap operators have
discrepancies given by prime-to-$p$ character values, so these
discrepancies lie in the Teichm\"uller subgroup
$\T=[k^\times]\subseteq\cO^\times$. We check this membership through
component products, central quotients and the prescribed inner
factors. Only then do we use the vanishing of positive cohomology
of a finite $p$-group with coefficients in $\T$. The resulting
integral comparison extends to every Sylow restriction, giving a
Morita equivalence with a bounded coefficient-Frobenius twist.
The second assertion is \emph{based finiteness}. Each pure Sylow
restriction is an actual finite-group block of bounded defect; we
construct its group from the given automorphisms and factor elements.
The field companion bounds its Cartan sum, so the integral
criterion of Eaton--Eisele--Livesey gives finitely many integral Morita
classes for these total restriction orders. To preserve their
labelled homogeneous components and the chosen identity component, we
prove an integral orbit lemma. Applied to the scheme of gradings of
a fixed order with vanishing first Hochschild cohomology, it gives
finitely many group-labelled gradings, even when $p$ divides the
grading-group order. The word ``fixed'' refers to the \emph{total}
order; this is different from counting all crossed products of a fixed
identity order. We retain the chosen identification of the identity
component; isomorphisms preserving that identification are called
\emph{based}.
The third assertion removes the possibly unbounded normal $p'$-kernel
of $Q$ while retaining those based restrictions.
Its twisted group algebra splits into matrix algebras over $\cO$ of
degree prime to $p$. Determinant-one choices of the matrices
implementing a Sylow action force the new scalar discrepancy to be
Teichm\"uller-valued. Thus the Sylow restrictions remain in their
controlled based list. Restriction and corestriction control the
principal-unit contribution on the remaining bounded quotient, and
its Teichm\"uller cohomology is finite. A finite-fibre argument for
the central factor systems then gives the integral Morita list.
\subsection{Organization and dependencies}
Section~\ref{sec:inputs} fixes the crossed-product conventions and
states the imported block constructions. The integral rigidity
results in Section~\ref{sec:rigidity} are independent of the comparison
construction. Section~\ref{sec:comparison} proves the exact integral
comparison. These two arguments meet in Section~\ref{sec:sylow}:
the comparison gives finite total Morita types of actual extension
blocks, and rigidity recovers their labelled components and markings.
Section~\ref{sec:kernel} removes the large normal $p'$-kernel, and
Section~\ref{sec:assembly} proves Theorem~\ref{thm:main}.
The field theorem is used before the integral theorem, in the Cartan
input and the specified companion constructions. The reduction
corollary records that the resulting finite integral list also
represents all field basic algebras; it is not a premise of the argument.
The fixed-coefficient-ring corollary is then proved by scalar extension.
==== END Integral-Donovan-Finiteness-over-Witt-Vectors//01_introduction.tex ====
==== BEGIN Integral-Donovan-Finiteness-over-Witt-Vectors//02_inputs.tex ====
\section{Integral orders and the imported block constructions}\label{sec:inputs}
The proof requires numerical bounds for actual blocks and a precise
description of the crossed orders produced by reduction. We state
these inputs separately. The distinction will matter when the
numerical criterion is applied: first we identify a restriction as a
group block, and then we use its Cartan and Frobenius bounds.
\subsection{Orders, Frobenius and the numerical criterion}
An $\cO$-order means a unital $\cO$-algebra free of finite rank as an
$\cO$-module. For a general order, a block summand means its direct
factor at a primitive central idempotent. Put $K_0=\Frac(\cO)$ and
let $\sigma$ be Witt
Frobenius. For an order $A$, the coefficient twist $\sigma^m A$ is
the order obtained by applying $\sigma^m$ to its structure constants;
equivalently it is scalar transport along $\sigma^m$. For a group
block we identify it with $\cO G\sigma^m(b)$. We write
$\mf_{\cO}(A)$ for the least positive $m$ for which $A$ and
$\sigma^m A$ are $\cO$-linearly Morita equivalent, whenever such an
$m$ exists. Only upper bounds for this number will be used.
Every finite $\cO$-algebra is semiperfect. A basic idempotent of a
block order is a sum of one primitive idempotent for every isomorphism
type of indecomposable projective. It is full, and its corner is the
basic order. Basic orders are unique up to isomorphism within a
Morita class. If the Cartan matrix of a block is $(c_{ij})$, its
basic order has $\cO$-rank $\sum_{i,j}c_{ij}$: reduction is a split
basic $k$-algebra, whose dimension is that sum.
\begin{proposition}[Cartan and integral finiteness inputs]\label{prop:criteria}
The following assertions will be used.
\begin{enumerate}
\item For every $p$ and $M$, the Cartan sums of all finite-group
blocks over $\cO$ with defect order at most $M$ are bounded by a
constant $c(p,M)$.
\item For fixed positive bounds on the defect order, Cartan sum,
and integral Morita--Frobenius number, finite-group blocks over
$\cO$ have only finitely many $\cO$-linear Morita equivalence
classes.
\end{enumerate}
\end{proposition}
\begin{proof}
By \cite[Theorem~1.1]{OpenAIFieldDonovan2026}, the reductions of the
blocks in (i) have finitely many $k$-Morita classes. Their Cartan
matrices, up to simultaneous permutation, consequently form a finite
list. Integral blocks and their reductions have the same Cartan
matrix, which proves (i).
Assertion (ii) is \cite[Theorem~3.10]{EEL2020}, applied to the
finitely many possible defect exponents. The coefficient conventions
of that theorem include the absolutely unramified complete discrete
valuation ring $W(k)$ and its coefficient Frobenius, which fixes the
uniformizer $p$.
Its defect-zero case can also be separated: a defect-zero block over
$W(k)$ is a matrix algebra over $W(k)$ and has basic order $\cO$.
Indeed, its reduction is a full matrix algebra over $k$; lift its
matrix units over the complete ring $\cO$, whose resulting primitive
corner has rank one.
\end{proof}
\subsection{Multiplication factors and identity markings}
\begin{definition}[Crossed orders and based isomorphisms]
Let $R$ be an $\cO$-order and $Q$ a finite group. A normalized
crossed system consists of $\alpha_q\in\Aut_{\cO}(R)$ and
$a_{q,t}\in R^\times$ such that
\begin{align}
\alpha_q\alpha_t&=\ad(a_{q,t})\alpha_{qt},
\label{eq:action-law}\\
a_{q,t}a_{qt,v}&=\alpha_q(a_{t,v})a_{q,tv},
\label{eq:factor-law}
\end{align}
with $\alpha_1=\id$ and $a_{1,q}=a_{q,1}=1$.
Its crossed order is $A=\bigoplus_{q\in Q}Ru_q$, with
\[
u_qr=\alpha_q(r)u_q,\qquad
u_qu_t=a_{q,t}u_{qt},\qquad u_1=1.
\]
A $Q$-labelled graded isomorphism preserves each homogeneous
component with its label. When the identity component is identified
with a fixed $R$, such an isomorphism is \emph{based} if it is the
identity on $R$.
\end{definition}
Changing each $u_q$ to $t_qu_q$, with $t_q\in R^\times$ and
$t_1=1$, gives exactly the based changes of crossed systems. The
induced homomorphism $Q\to\Out_{\cO}(R)$ is the outer action.
It remembers the actions only modulo inner automorphisms. Both the
units implementing their products in Equation~\eqref{eq:action-law}
and the compatibility in Equation~\eqref{eq:factor-law} are needed to
recover the order. For $S\leq Q$ the \emph{restriction to $S$} is
the crossed suborder $A_S=\bigoplus_{s\in S}Ru_s$, with the same
identity marking and the restricted factors.
\subsection{Reduction and dual recovery}
We call a pair $(H,B)$ reduced if $B$ is a block of $\cO H$ such
that every block of a normal subgroup covered by $B$ is $H$-stable,
and
\begin{equation}\label{eq:reduced}
B\text{ covers a nilpotent block of }H_0\trianglelefteq H
\quad\Longrightarrow\quad H_0\leq Z(H)O_p(H).
\end{equation}
An--Eaton's preliminary reduction
\cite[Proposition~6.1]{AnEaton2025} replaces an arbitrary block by a
reduced pair through an $\cO$-linear basic Morita equivalence,
preserving the defect group. Their coefficient conventions include
$\cO=W(k)$. We apply this integral reduction directly.
The next proposition collects the group structure, actual factor
elements and integral full corners supplied by the field companion.
Constants in it depend only on $p,M$.
\begin{proposition}[Controlled integral extensions]\label{prop:structure}
Let $(H,B)$ be a reduced pair with defect order at most $M$. Put
$N=F^*(H)$, $X=H/N$, and let $b$ be the block of $\cO N$ covered by
$B$. Then the following constructions and bounds hold.
\begin{enumerate}
\item The group $N$ is a central product
\[
N=PZK_1\cdots K_j,\qquad P=O_p(H),\quad Z=O_{p'}(H)\leq Z(H),
\]
where the $K_i$ are quasisimple components, $|P|$ and $j$ are
bounded, and $[X:O_{p'}(X)]$ is bounded. The latter index is
bounded also for every subgroup of every extension of $X$ by a
$p'$-group.
\item There is an extension $N\trianglelefteq\bar N$ with abelian
$p'$-quotient $A_0=\bar N/N$. For representatives $h_x$ of $x\in X$,
normalized by $h_1=1$, write $h_xh_y=n_{x,y}h_{xy}$ with
$n_{x,y}\in N$. Conjugation extends to automorphisms $\alpha_x$ of
$\bar N$ satisfying the actual identities
\begin{align}
\alpha_x\alpha_y&=\ad(n_{x,y})\alpha_{xy},
\label{eq:actual-actions}\\
n_{x,y}n_{xy,z}&=\alpha_x(n_{y,z})n_{x,yz}.
\label{eq:actual-factors}
\end{align}
Every block $\bar b$ of $\cO\bar N$ covering $b$ has bounded defect.
\item Put $\Xi=\Hom(A_0,k^\times)$, using Teichm\"uller lifts of
the characters over $\cO$, and $Y=\Xi\rtimes X$. There is a
$Y$-crossed order with identity component $\cO\bar N$ in which
$\cO H$ is a full idempotent corner. After taking an appropriate
central summand and a further full corner, this is a crossed order
with identity component
\[
B_0=\cO\bar N\bar b
\]
and grading group $Q=\Stab_Y(\bar b)$. In the pure $X$-degrees the
actions and factors are those in (ii). The index
$[Q:O_{p'}(Q)]$ is bounded.
\item The basic orders of all these identity blocks $B_0$ belong
to a finite list of integral orders $R_1,\ldots,R_t$. Their integral
Picard groups, and in particular their outer automorphism groups,
are finite.
\end{enumerate}
\end{proposition}
\begin{proof}[Source of the assertions]
For a reduced pair, the structural conclusions follow from
\cite[Lemma~8.1]{OpenAIFieldDonovan2026}. Its opening Morita
reduction is stated over $k$; we use only its conclusions about the
already reduced pair, obtained here by the integral An--Eaton
reduction. The compatible partial extension and its actual factors
are \cite[Lemma~8.2]{OpenAIFieldDonovan2026}. The integral dual
recovery construction and full corners are
\cite[Lemma~8.3]{OpenAIFieldDonovan2026}.
For clarity, the dual-recovery idempotent is
$|\Xi|^{-1}\sum_{\chi\in\Xi}u_\chi$. The characters are trivial
on every $n_{x,y}\in N$, so the pure $X$-units preserve this
idempotent and retain exactly these factors. This explains why
the construction is integral and why it retains the multiplication
data needed later. Finally (iv) is the integral assertion of
\cite[Lemma~8.5]{OpenAIFieldDonovan2026}, using the trivial-subgroup
case of its Proposition~9.1. The finite Picard assertion is also
\cite[Theorem~B and Corollary~1.2]{EiseleGeometry2022}.
\end{proof}
Every $p$-subgroup $S$ of $Y$ can be conjugated by an element of
$\Xi$ into the pure subgroup $X$. Indeed, its image in $X$ is a
$p$-group, and Schur--Zassenhaus conjugates the two complements to
$\Xi$ in the inverse image of that group. In the crossed order this
conjugation is implemented by an integral homogeneous unit; it also
transports $\bar b$. We may therefore work with pure $p$-subgroups,
provided we keep the transported identity block in the same family.
In particular, the orders of the relevant $p$-subgroups of $Q$ are
bounded: their intersection with $O_{p'}(Q)$ is trivial, so they
inject into the quotient whose order was bounded in (iii).
There is no bound here on the orders of the central kernels in the
universal-cover presentations, or on the extra central $p$-factors
used in the partial regular extensions. The comparison in
Section~\ref{sec:comparison} descends through those kernels before
any bounded-defect integral criterion is applied.
\subsection{The companion's comparison data}
The comparison convention in
\cite[Definition~8.4 and Proposition~9.1]{OpenAIFieldDonovan2026}
uses an integral Morita bimodule but asserts exact covariance and
factor identities on its reduction. We use the particular integral
operators constructed there. Their ingredients have four distinct
roles. Lemmas~9.2--9.4 realize the genuine field, graph and inner
relations, construct an invariant torus and character, and choose a
common parabolic--Levi pair with the specified inner overlap.
Lemma~9.5 gives an integral Levi Morita bimodule with strict geometric
actions and matching central actions. Lemma~9.6 gives a uniformly
bounded coefficient-Frobenius return and its prime-to-$p$ character
twist. Finally, Lemma~9.7 computes the integral scalar overlap of
the chosen operators.
Section~\ref{sec:comparison} recalls these constructions with their
exact hypotheses and shows that their scalar errors stay in $\T$.
The subsequent normalization over $\cO$ is proved here. Thus the
imported comparison supplies operators and overlap identities, while
Proposition~\ref{prop:comparison} supplies the exact integral factor
law required by the later extension argument.
==== END Integral-Donovan-Finiteness-over-Witt-Vectors//02_inputs.tex ====
==== BEGIN Integral-Donovan-Finiteness-over-Witt-Vectors//03_rigidity.tex ====
\section{Integral orbits and gradings of a fixed order}\label{sec:rigidity}
An ungraded Morita list does not specify the components of a crossed
order. This section gives the additional rigidity needed to recover
both their group labels and the identity marking. There are three
steps: an orbit lemma over $\cO$, a tangent calculation for gradings
of a fixed total order, and a rank argument which passes from Morita
classes to such fixed orders. The orbit method is related to the
geometry of rigid lattices and Picard groups in
\cite{EiseleGeometry2022}; the grading argument below works for every
finite grading group.
\subsection{Integral orbit finiteness}
\begin{lemma}[Integral orbit finiteness]\label{lem:orbit}
Let $J$ be a smooth affine group scheme of finite type over $\cO$,
acting on an affine scheme $V$ of finite type over $\cO$. For
$x\in V(\cO)$ let $T_xV$ denote the integral tangent module along
the section $x$. The points for which the orbit derivative
\[
\Lie(J)\longrightarrow T_xV
\]
is surjective belong to only finitely many $J(\cO)$-orbits.
\end{lemma}
\begin{proof}
We bound the special-fibre orbits and then cut each one by a fixed
integral slice. The latter has only finitely many possible generic
points arising from the sections under consideration.
\emph{The integral tangent and horizontal components.}
Embed $V$ as a closed subscheme of $\mathbb A^r_{\cO}$, with
finitely many defining equations. The module $T_xV$ is the kernel
of their Jacobian map on $\cO^r$. It is therefore saturated in
$\cO^r$. Write $u=\rank_{\cO}T_xV$. Surjectivity of the orbit
derivative and saturation imply that its coordinate matrix has a
unit minor of size $u$. This follows, for example, from Smith
normal form over the discrete valuation ring $\cO$.
After extension to $K_0$, this derivative has rank $u$, which is
also $\dim_{K_0}T_{x_{K_0}}V_{K_0}$. Every irreducible component
of $V_{K_0}$ through $x_{K_0}$ consequently has dimension at most
$u$. Let $V_u$ be the reduced closure in $V$ of the union of the
generic-fibre irreducible components of dimension at most $u$.
There are finitely many such components. Each of their horizontal
closures has special fibre of dimension at most its generic-fibre
dimension. This is the dimension theorem for an irreducible
finite-type scheme over a valuation ring
\cite[Lemma~33.19.2, Tag~0B2J]{StacksProject}. Thus
\begin{equation}\label{eq:fibre-dimension}
\dim (V_u)_k\leq u.
\end{equation}
The section $x$ factors through $V_u$, because its generic point
does and $\cO\hookrightarrow K_0$ is injective.
\emph{The special-fibre orbits.}
Let $x_0$ be the reduction of $x$. The unit minor remains nonzero
modulo $p$, so the special-fibre orbit of $x_0$ has dimension at
least $u$. This orbit lies in $(V_u)_k$. One way to verify that
last assertion is to lift every element of $J(k)$ to $J(\cO)$,
using smoothness and completeness, and then apply it to the section
$x$. The transformed generic point still lies on components of
dimension at most $u$. Equation~\eqref{eq:fibre-dimension} now
shows that the orbit has dimension exactly $u$.
A locally closed orbit of dimension $u$ in a scheme of dimension
at most $u$ contains an open subset of an irreducible component of
dimension $u$. Two distinct orbits cannot both contain dense open
subsets of the same component. There are therefore finitely many
possible special-fibre orbits for this $u$.
\emph{A fixed Hensel slice.}
Fix one such orbit and a representative $x_0$. By lifting elements
of $J(k)$, move all integral points under consideration so that
their reduction equals $x_0$. Choose $u$ coordinate functions
whose orbit derivative has a nonzero $u$-minor at $x_0$, and fix
integral lifts $a_1,\ldots,a_u$ of their values there. For each
such integral point $x$, the map
\[
J\longrightarrow\mathbb A^u_{\cO},\qquad
g\longmapsto\text{the selected coordinates of }g x
\]
is smooth near the identity: $J$ is smooth and its relative
differential there is surjective. Hensel lifting supplies
$g\equiv1\pmod p$ for which the selected coordinates of $gx$
are exactly the $a_i$. Thus every integral orbit has a
representative in the fixed affine slice
\[
W=V\cap\{\text{selected coordinates }=a_1,\ldots,a_u\}.
\]
For any resulting point $y$, the selected-coordinate differential
is an isomorphism on $T_{y_{K_0}}V_{K_0}$: that space has dimension
$u$, and the same minor is a unit since $y$ reduces to $x_0$.
It follows that $T_{y_{K_0}}W_{K_0}=0$. Such a point is isolated
in the finite-type $K_0$-scheme $W_{K_0}$. A noetherian scheme
has only finitely many isolated points. Each generic point gives
at most one integral section, again by the injection
$\cO\hookrightarrow K_0$. Hence there are finitely many
representatives in this slice. Taking the finite union over the
special-fibre orbits and the possible ranks $0\leq u\leq r$
proves the lemma.
\end{proof}
The integral tangent module in this proof need not reduce to the
whole tangent space at $x_0$. Saturation and the unit minor are
what the proof uses. In particular, no smoothness assumption on
$V$, nor on an automorphism scheme occurring as $V$, is needed.
The rank-zero case is included by using no slice coordinates. The
argument uses smoothness of $J$ for lifting and for its differential;
it does not require $J$ to be connected.
\subsection{Block rigidity and labelled gradings}
\begin{lemma}[Hochschild rigidity and outer automorphisms]
\label{lem:HH}
Let $A$ be an $\cO$-order Morita equivalent to a finite direct sum
of finite-group block orders. Then $\HH^1_{\cO}(A)=0$, and
$\Out_{\cO}(A)$ is finite.
\end{lemma}
\begin{proof}
For a finite group $G$, the conjugation permutation lattice and
Shapiro's lemma give
\[
\HH^1_{\cO}(\cO G)
\cong H^1(G,\cO G_{\mathrm{conj}})
\cong\bigoplus_{[g]}H^1(C_G(g),\cO)=0.
\]
The action on the last coefficient module is trivial, and
$\Hom(C_G(g),\cO_{\mathrm{add}})=0$ because $C_G(g)$ is finite
and $\cO$ is torsion-free. Hochschild cohomology splits over
finite direct products and is Morita invariant, proving the
first assertion. Thus every $\cO$-linear derivation of $A$ is
inner.
The unit group scheme $A^\times$ is an open subscheme of the
affine space underlying $A$, defined by invertibility of the
regular-representation determinant. It is smooth. The
automorphisms of $A$ form an affine scheme of finite type: impose
the multiplicative and unital equations on an invertible linear
map. Let $A^\times$ act by postcomposition with inner
automorphisms. At an automorphism, compose with its inverse to
identify the tangent module with the derivations of $A$.
The orbit derivative then consists of the inner derivations,
so it is surjective. Lemma~\ref{lem:orbit} says exactly that
there are finitely many cosets modulo inner automorphisms.
\end{proof}
\begin{theorem}[Gradings of a fixed integral order]\label{thm:gradings}
Let $A$ be an $\cO$-order with $\HH^1_{\cO}(A)=0$, and let $H$
be any fixed finite group. There are only finitely many
$H$-labelled gradings of $A$ up to $\cO$-algebra automorphism.
In fact there are only finitely many orbits under inner
automorphisms.
\end{theorem}
\begin{proof}
We apply Lemma~\ref{lem:orbit} to the space of decompositions
compatible with multiplication. The key point is that the tangent
calculation uses cancellation in $H$, without averaging over $H$.
A grading $A=\bigoplus_{h\in H}A_h$ is specified by its
orthogonal projections $e_h\in\End_{\cO}(A)$. These satisfy
\begin{align*}
e_he_j&=\delta_{h,j}e_h,& \sum_{h\in H}e_h&=1,\\
e_h(1_A)&=\delta_{h,1}1_A,&
e_l(e_g(x)e_h(y))&=0\quad(l\ne gh).
\end{align*}
The last equations need only be imposed on a fixed finite basis
of $A$. They define an affine finite-type grading scheme, on
which $A^\times$ acts by conjugation.
We compute its integral tangent module at a grading. A
first-order deformation of the direct-sum decomposition is
uniquely represented by an off-diagonal $\cO$-linear map
$\delta:A\to A$, where
\[
e_h\delta e_h=0\quad(h\in H).
\]
The deformed summands are $(1+\varepsilon\delta)A_h$ over
$\cO[\varepsilon]/(\varepsilon^2)$. This description follows
by differentiating the equations for orthogonal idempotent
projections, or by writing the summands as graphs over the
original summands. It uses no division by $|H|$.
For $x\in A_g$ and $y\in A_h$, the linearized multiplicative
equations say that
\begin{equation}\label{eq:grading-derivative}
\delta(xy)-\delta(x)y-x\delta(y)
\end{equation}
has zero component in every degree other than $gh$. It also has
zero $gh$-component. The first term is off degree $gh$ by
definition. A summand of $\delta(x)y$ can have degree $gh$ only
if its first factor has degree $g$, by cancellation in $H$; that
component of $\delta(x)$ is zero. The same reasoning applies
to $x\delta(y)$. Hence Equation~\eqref{eq:grading-derivative}
vanishes, and $\delta$ is a derivation.
The hypothesis gives $\delta=[a,-]$ for some $a\in A$.
Conjugation by $1+\varepsilon a$ produces the specified tangent
to the grading. Thus the orbit derivative from the smooth
group $A^\times$ is surjective onto every integral tangent
module. Lemma~\ref{lem:orbit} proves the assertion.
\end{proof}
\begin{remark}\label{rem:fixed-total}
The total order is fixed in Theorem~\ref{thm:gradings}. Finiteness
of all crossed orders on a fixed identity order is a different
assertion. For example, for odd $p$ the based $C_p$-crossed orders
\[
\cO[t]/(t^p-a),\qquad a\in\cO^\times,
\]
have classes parametrized by $\cO^\times/(\cO^\times)^p$.
The principal-unit quotient is infinite: the $p$-adic logarithm
identifies $(1+p\cO)/(1+p\cO)^p$ with
$p\cO/p^2\cO\cong k$. Their total orders vary.
The crossed-product finiteness theorem in the published
\cite[Corollary~4.9]{EiseleReduction2021} assumes a $p'$-grading
group. Theorem~\ref{thm:gradings} uses a different hypothesis and
has just been proved also for groups divisible by $p$.
\end{remark}
\subsection{From Morita classes to based graded types}
For a crossed order on a fixed identity order, fixing the grading
group also fixes the total rank. This converts a finite Morita list
into a finite isomorphism list, to which the grading theorem applies.
\begin{proposition}[Retaining the identity marking]\label{prop:marked}
Fix a basic block order $R$ and a finite group $H$. Let
$\mathcal A$ be a collection of $H$-crossed orders on $R$, each
Morita equivalent to a finite-group block. If the total orders
in $\mathcal A$ have finitely many $\cO$-Morita classes, then
$\mathcal A$ has finitely many based graded isomorphism classes.
\end{proposition}
\begin{proof}
\emph{The total order.}
Every order in the collection has rank $|H|\rank_{\cO}R$.
Choose a basic representative in each of its finitely many
Morita classes. Any order in that class is an endomorphism
order of a projective generator
$\bigoplus_i P_i^{m_i}$, with positive integer multiplicities
$m_i$ and the $P_i$ ranging over the finitely many indecomposable
projectives. Its endomorphism order contains
$M_{m_i}(\End(P_i))$ as a direct $\cO$-module summand, so its
rank bounds $m_i^2$. There are consequently only finitely many
ungraded isomorphism types in the collection.
\emph{The grading and the marking.}
Lemma~\ref{lem:HH} and Theorem~\ref{thm:gradings} give finitely
many labelled $H$-gradings on every such total order. Fix one
graded representative with identity component isomorphic to
$R$. Two identifications of that component with $R$ differ by
an element of $\Aut_{\cO}(R)$. Inner changes extend to graded
automorphisms of the total order by conjugation with a unit in
degree one. Since $\Out_{\cO}(R)$ is finite by
Lemma~\ref{lem:HH}, there are finitely many remaining markings.
These are precisely the based graded types in the assertion.
\end{proof}
==== END Integral-Donovan-Finiteness-over-Witt-Vectors//03_rigidity.tex ====
==== BEGIN Integral-Donovan-Finiteness-over-Witt-Vectors//04_comparison.tex ====
\section{Teichm\"uller overlaps and integral Sylow comparison}
\label{sec:comparison}
The rationality bound for a Sylow restriction must compare its
multiplication factors as well as its identity block. We obtain it
from the explicit operators in Section~9 of
\cite{OpenAIFieldDonovan2026}. Proposition~9.1 there asserts exact
equivariance after reduction; here we prove exact equivariance over
$\cO$ by following the scalar errors through the entire integral
construction. Their membership in the Teichm\"uller group is the
point that permits the final normalization.
\subsection{The scalar coefficient group}
Write
\[
\T=[k^\times]\subseteq\cO^\times
\]
for the Teichm\"uller subgroup. Since $k$ is the algebraic closure
of a finite field, $\T$ consists precisely of the roots of unity
in $\cO$ of order prime to $p$. It is preserved by Witt Frobenius,
which acts on it by $p$th powering.
\begin{lemma}\label{lem:teich-cohomology}
If $S$ is a finite $p$-group acting trivially on $\T$, then
$H^i(S,\T)=0$ for every $i>0$.
\end{lemma}
\begin{proof}
Positive-degree cohomology of a finite group is killed by its
order. Raising to $|S|$ is an automorphism of $\T$, since $\T$
has no $p$-torsion and has unique $p$-power roots. This map also
induces an automorphism on cohomology. It is simultaneously the
zero map there, so the cohomology vanishes.
\end{proof}
\subsection{Exact covariance with the prescribed factors}
\begin{proposition}[Exact integral comparison]\label{prop:comparison}
Fix $p,M$ and the data of Proposition~\ref{prop:structure}.
Let $S\leq Q$ be a $p$-subgroup contained in the pure subgroup
$X\leq Y$, so that $S$ stabilizes $B_0=\cO\bar N\bar b$.
Put $a_{s,t}=n_{s,t}\bar b$. There is a positive integer $m$,
bounded in terms of $p,M$, and a
$(\sigma^m B_0,B_0)$-Morita bimodule $\mathcal M$ with invertible
$\cO$-linear maps $J_s:\mathcal M\to\mathcal M$, normalized by
$J_1=\id$, such that
\begin{align}
J_s(avb)&=\alpha'_s(a)J_s(v)\alpha_s(b),
\label{eq:comparison-covariance}\\
J_sJ_t(v)&=a'_{s,t}J_{st}(v)a_{s,t}^{-1}.
\label{eq:comparison-exact}
\end{align}
Here $a\in\sigma^m B_0$, $b\in B_0$,
$\alpha'_s=\sigma^m(\alpha_s)$, and
$a'_{s,t}=\sigma^m(a_{s,t})$. The assertion includes $S=1$.
\end{proposition}
\subsection{The imported component operators}
We first specify the integral operators that will prove
Proposition~\ref{prop:comparison}. Fix its data and pure subgroup
$S$. The companion's construction temporarily replaces the identity
group $\bar N$ by a central cover. More precisely,
\cite[Lemma~8.2]{OpenAIFieldDonovan2026} supplies
\[
\bar N=\bigl(P\times Z\times\prod_i V_i\bigr)/D_0,
\qquad J_i=V_i\times C_i.
\]
Here $D_0$ is the original central kernel, each $V_i$ enlarges the
universal cover of a component, and $C_i$ is an added direct central
$p$-group. On the cross-characteristic Lie components outside the
finite exceptional list, $J_i$ is the full fixed-point group of a
connected reductive group with simply
connected derived subgroup; on the remaining components put $C_i=1$.
There is no bound on $|D_0|$ or $|C_i|$.
The operators below are constructed on these auxiliary groups; their
descent will return us to the bounded-defect block $B_0$.
Consider one $S$-orbit of these Lie components.
By \cite[Lemmas~9.2--9.4]{OpenAIFieldDonovan2026}, the product of
its groups $J_i$ has a realization $J=\mathbf J^F$ with simply
connected algebraic derived subgroup and Steinberg endomorphism $F$.
The chosen representatives
$h_s$, together with actual inner operations, generate an operation
group $I$, represented by algebraic automorphisms commuting with $F$.
The source chooses a rational Levi $L=\mathbf L^F$,
a parabolic with Levi $\mathbf L$, and a subgroup $A_1$ of $I$
preserving this pair, such that
\begin{equation}\label{eq:I-decomposition}
I=\Inn(J)A_1,\qquad \Inn(J)\cap A_1=\Inn(L),
\end{equation}
where $A_1$ is the particular subgroup of the cited construction.
The Levi is $F$-stable; its parabolic need not be. The field
operations are represented by algebraic permutations with their
actual relations, including the diagram twist at the cyclic wrap
in the ordinary twisted realization. The exceptional graph-isogeny
realization is the separate
construction of that source. In particular,
the representatives of $S$ still have the specified inner factors
$n_{s,t}$, lifted to the central-product cover.
Let $b_J$ be the lifted block and $b_L$ its Levi correspondent.
Set $A_J=\cO Jb_J$ and $B_L=\cO Lb_L$. The integral
Bonnaf\'e--Rouquier $(A_J,B_L)$-Morita bimodule $U$ has a strict
$A_1$-action: its operators satisfy the group law exactly, and the
operator for $\ad(\ell)$ is $u\mapsto\ell u\ell^{-1}$ for
$\ell\in L$. Every central element of $J$ acts equally from
the two sides of $U$
\cite[Lemma~9.5]{OpenAIFieldDonovan2026}. Existence of the
equivalence uses the full-dual-centralizer hypothesis in
\cite[Section~11.4, Theorem~B$'$]{BR2003}. Geometric comparison and
extension methods are developed further in
\cite[Sections~6--7]{BDR2017}. The strict action used here is the
companion's action by actual automorphisms of the fixed common
parabolic--Levi pair; it is not a choice of comparison maps between
different parabolics.
By \cite[Lemmas~9.3 and~9.6]{OpenAIFieldDonovan2026}, a bounded
positive exponent $m$ and an $A_1$-invariant linear character
$\lambda_m$ of $L$ of $p'$-order satisfy
\[
\sigma^m(b_L)=\lambda_m b_L.
\]
The bound is uniform in classical rank and field size, including
type~A where the diagonal index need not be bounded. The exponent
likewise does not depend on the orders of the
added central tori or the original covering kernels.
Use primes for coefficient-Frobenius images. The corresponding
isomorphism is
\[
f:B_L\longrightarrow B_L',\qquad
g b_L\longmapsto\lambda_m(g)^{-1}g\sigma^m(b_L).
\]
Let $T_{\lambda}$ be the invertible $(B_L',B_L)$-bimodule with
underlying left module $B_L'$ and right action through $f$.
Writing $U^\vee$ for a $(B_L,A_J)$-Morita inverse of $U$, the
component comparison bimodule is
\begin{equation}\label{eq:comparison-bimodule}
\mathcal M_J=
\sigma^m(U)\otimes_{B_L'}T_{\lambda}\otimes_{B_L}U^\vee.
\end{equation}
All three factors have strict $A_1$-actions: geometric action and
its dual on the outside, and the action on the group basis on
the middle factor. Denote their tensor action by $D_a$.
For $g\in J$ let $E_g(v)=gvg^{-1}$ be the actual inner
operator on $\mathcal M_J$. The construction gives
\begin{align}
E_gE_h&=E_{gh},& D_aE_gD_a^{-1}&=E_{a(g)},
\label{eq:strict-operators}\\
D_{\ad(\ell)}&=\lambda_m(\ell)^{-1}E_\ell
&&(\ell\in L).
\label{eq:overlap}
\end{align}
The last equality is the explicit overlap calculation in the
proof of \cite[Lemma~9.7]{OpenAIFieldDonovan2026}: on the middle
factor, right multiplication uses
$f(\ell^{-1})=\lambda_m(\ell)\ell^{-1}$. It is an equality of
integral operators, not merely of their reductions.
\subsection{Proof of the exact comparison}
\begin{proof}[Proof of Proposition~\ref{prop:comparison}]
The imported operators retain the actual inner factors. We show
first that all their presentation ambiguities lie in $\T$, then
assemble and descend the component comparisons. Only after this
descent will we remove the resulting scalar cocycle.
\emph{Teichm\"uller-valued presentation ambiguities.}
Every scalar in Equation~\eqref{eq:overlap} belongs to $\T$,
because $\lambda_m$ has $p'$-order. To represent an operation
$i\in I$, choose $i=\ad(g)a$ using
Equation~\eqref{eq:I-decomposition} and take $E_gD_a$. Changing
this presentation is a change through $\Inn(L)$, and hence
introduces only a value of $\lambda_m$, apart from the ambiguity
of a central element in the choice of $g$.
That central ambiguity is also Teichm\"uller-valued. If
$c\in Z(J)$ is a $p$-element, then $c\in L$ and
Equation~\eqref{eq:overlap} applied to it says
$E_c=1$: the inner operation is the identity and
$\lambda_m(c)=1$. If $c$ has $p'$-order, it acts on each block
side by the value of its central character. The operator $E_c$
is the ratio of these two $p'$-roots of unity, and lies in $\T$.
An arbitrary central element is a product of its $p$- and
$p'$-parts. Thus any two presentations give operators differing
by $\T$, rather than by an arbitrary unit of $\cO$.
Multiplication of chosen operators uses only
Equation~\eqref{eq:strict-operators} and a change of presentation.
Its scalar discrepancy is consequently in $\T$. Applying this
to the relation
$\alpha_s\alpha_t=\ad(n_{s,t})\alpha_{st}$ shows, with the
\emph{actual} inner factor retained, that the resulting transport
has the form
\[
D_sD_t(v)=c_J(s,t)n'_{s,t}D_{st}(v)n_{s,t}^{-1},
\qquad c_J(s,t)\in\T.
\]
\emph{Products, component permutations and the remaining factors.}
We now use the remaining constructions in the proof of
\cite[Proposition~9.1]{OpenAIFieldDonovan2026}.
There are boundedly many component orbits. Taking a common
bounded multiple of their exponents is legitimate by tensoring
successive Frobenius twists of the comparison bimodules.
Frobenius preserves $\T$, and tensor products multiply the scalar
discrepancies. This operation therefore preserves their membership
in $\T$.
For the finite list of exceptional, sporadic, and relevant
defining-characteristic components, a common Frobenius period
allows the identity bimodule with strict group transports.
The unbounded alternating-cover family also has a uniformly
bounded coefficient period, as proved in that proposition, and
uses the same strict transports. The bounded $p$-group factor
$P$ uses its identity bimodule. The central $p'$-factor $Z$
has a rank-one character algebra, so its left/right discrepancy
is a ratio of $p'$-character values and belongs to $\T$.
Permutation of factors uses the ordinary flips of bimodules and
satisfies its group relations strictly. These are tensors of
ordinary Morita bimodules, so no graded symmetry sign enters
this step, including when $p=2$.
\emph{Descent through the original central kernels.}
It remains to pass from $P\times Z\times\prod_i J_i$ to
$\bar N$: quotient by the added factors $C_i$ and the original
kernel $D_0$. The established construction
identifies the left and right actions of every central $p$-element
on the comparison. Quotienting by $c-1$ on the two sides
therefore gives a Morita bimodule over the corresponding quotient
blocks: tensor the inverse equivalence as well and use equality
of the central actions in the two inverse identities. The
transport maps preserve these ideals and descend. The
$p'$-part of the quotient kernel acts trivially on both sides in
the chosen averaging sector. Thus this descent introduces no
new scalar factor. Differences between lifts of the prescribed
$n_{s,t}$ vanish in the quotient.
In particular, the Morita equivalence and its transport maps now
live on the original identity blocks, where the defect bound of
Proposition~\ref{prop:structure}(ii) applies.
We have obtained an integral $(\sigma^m B_0,B_0)$-Morita
bimodule $\mathcal M$ with $\cO$-linear covariance maps $D_s$
satisfying
\begin{equation}\label{eq:projective-comparison}
D_sD_t(v)=c(s,t)a'_{s,t}D_{st}(v)a_{s,t}^{-1},
\qquad c(s,t)\in\T.
\end{equation}
The exponent remains bounded in terms of $p,M$; the orders of
the central kernels have not been bounded or used in a finiteness
criterion.
\emph{Removal of the scalar error.}
Normalize $D_1=\id$. Compare $(D_sD_t)D_v$ and
$D_s(D_tD_v)$. Covariance and
Equation~\eqref{eq:actual-factors} on the two block sides cancel
all the non-scalar terms and leave
\[
c(s,t)c(st,v)=c(t,v)c(s,tv).
\]
Because the maps are $\cO$-linear, the scalar action is trivial.
Thus $c$ is a normalized $2$-cocycle in $Z^2(S,\T)$.
Lemma~\ref{lem:teich-cohomology} makes it a coboundary.
Rescaling each $D_s$ by a normalized $\T$-valued $1$-cochain
gives $J_s$ and Equation~\eqref{eq:comparison-exact}, without
changing the covariance equation or the exponent $m$.
\end{proof}
The construction proves the scalar restriction before it uses
cohomology. General $\cO^\times$-valued projective transports would
not suffice, since their principal-unit errors need not vanish on
$S$. Here Equation~\eqref{eq:comparison-bimodule}, its overlap
characters and its central descent give the more precise coefficient
group $\T$ at every stage.
==== END Integral-Donovan-Finiteness-over-Witt-Vectors//04_comparison.tex ====
==== BEGIN Integral-Donovan-Finiteness-over-Witt-Vectors//05_sylow.tex ====
\section{Finite based lists on \texorpdfstring{$p$}{p}-subgroups}\label{sec:sylow}
Exact comparison and fixed-total-order rigidity now meet. We first
extend the comparison to a Morita equivalence of crossed orders. We
then realize the relevant orders as actual blocks and apply the
numerical finiteness criterion. Finally we recover the labelled
components and the identity markings using
Proposition~\ref{prop:marked}.
\subsection{Extending an exact comparison}
The following is the one-term crossed-order version of the
diagonal extension argument of Marcus
\cite[Theorem~3.4]{Marcus1996}, also recorded in
\cite[Lemma~10.2.8]{Rouquier1998}. We give its algebraic
proof to retain the specified multiplication factors.
\begin{lemma}[Extension of a comparison bimodule]\label{lem:induced}
Let $B,B'$ be $\cO$-orders with normalized crossed systems
$(\alpha_s,a_{s,t})$ and $(\alpha'_s,a'_{s,t})$ for a finite group
$S$, and let $A,A'$ be their crossed orders. Suppose an
$(B',B)$-Morita bimodule $\mathcal M$ has invertible
$\cO$-linear maps $J_s$ satisfying
Equations~\eqref{eq:comparison-covariance} and
\eqref{eq:comparison-exact}. Then $A$ and $A'$ are
$\cO$-linearly Morita equivalent.
\end{lemma}
\begin{proof}
The induced right module carries the required left crossed action
precisely because the comparison has the exact factor identity.
Put $P=\mathcal M\otimes_B A$, a right $A$-module. The left
$B'$-action is the original action on $\mathcal M$. Define
right $A$-linear operators
\[
L_s(v\otimes a)=J_s(v)\otimes u_s a.
\]
They are well defined on the balanced tensor product: covariance
and $u_sb=\alpha_s(b)u_s$ identify the images of
$vb\otimes a$ and $v\otimes ba$. They are invertible, and
covariance gives $L_s b'=\alpha'_s(b')L_s$.
The exact factor identity gives
\begin{align*}
L_sL_t(v\otimes a)
&=a'_{s,t}J_{st}(v)a_{s,t}^{-1}\otimes
a_{s,t}u_{st}a\\
&=a'_{s,t}L_{st}(v\otimes a).
\end{align*}
Thus these operators define a left $A'$-action.
The right $B$-module $\mathcal M$ is a finitely generated
projective generator. Inducing its projective summand and
generator identities to $A$ shows that $P$ is a projective
generator as a right $A$-module. It remains to identify its
endomorphism order with $A'$.
As a right $B$-module, $P$ is the direct sum of the components
$\mathcal M\otimes_B Bu_s$. Induction--restriction adjunction
gives the following isomorphisms of $\cO$-modules:
\[
\End_A(P)\cong\Hom_B(\mathcal M,P)
\cong\bigoplus_{s\in S}
\Hom_B(\mathcal M,\mathcal M\otimes_B Bu_s).
\]
Explicitly, a $B$-linear map $f$ extends uniquely to the
$A$-endomorphism $v\otimes a\mapsto f(v)a$. For a map in
the degree-$s$ summand, composing its extension with $L_s^{-1}$
gives a degree-one endomorphism, hence an element of
$\End_B(\mathcal M)=B'$. Therefore
\[
\End_A(P)=\bigoplus_{s\in S}B'L_s.
\]
The relations already checked identify this order with $A'$.
The projective-generator characterization of Morita equivalence
now proves the lemma.
\end{proof}
\subsection{The actual extension blocks}
\begin{proposition}[Sylow restrictions are block orders]
\label{prop:sylow-block}
For the data of Proposition~\ref{prop:structure}, let
$S\leq Q\cap X$ be a pure $p$-subgroup. The restriction of the
crossed order to $S$, with identity component $B_0$, is a block
of a finite group. Its defect order is bounded in terms of
$p,M$, and its integral Morita--Frobenius number is bounded in
the same parameters. All these restriction blocks have
finitely many integral Morita classes.
\end{proposition}
\begin{proof}
\emph{Constructing the group and its block.}
Use the actual automorphisms $\alpha_s$ of $\bar N$ and actual
elements $n_{s,t}\in N$ supplied by
Proposition~\ref{prop:structure}(ii). On the finite set
$\bar N\times S$ define
\begin{equation}\label{eq:extension-group}
(g,s)(h,t)=\bigl(g\alpha_s(h)n_{s,t},st\bigr).
\end{equation}
Equations~\eqref{eq:actual-actions} and
\eqref{eq:actual-factors} give associativity, and their
normalizations give the identity and inverses. Hence this is
a finite group $\Gamma_S$ with normal subgroup $\bar N$ and
quotient $S$. Sending the crossed generator $u_s$ to $(1,s)$
identifies the restriction of the unprojected crossed order with
$\cO\Gamma_S$.
The block $\bar b$ is $S$-stable. A stable block has a unique
covering block in an extension of $p$-power index. Its idempotent
is $\bar b$, so our restriction is exactly
\[
A_S=\cO\Gamma_S\bar b.
\]
\emph{Bounding its defect and coefficient period.}
If $D$ is a defect group of this block, normal block theory gives
a defect group $D\cap\bar N$ of $\bar b$, after conjugacy.
Moreover $D$ maps into $S$, and in this stable $p$-extension it
maps onto $S$. In particular,
\[
|D|\leq |S|\,|D\cap\bar N|.
\]
Both quantities on the right are bounded by
Proposition~\ref{prop:structure}. Notice that the bound is
applied to the group after the central descent in
Section~\ref{sec:comparison}.
Proposition~\ref{prop:comparison}, applied to these same actual
factors, and Lemma~\ref{lem:induced} give a Morita equivalence
between $A_S$ and $\sigma^m A_S$, with $m$ uniformly bounded.
Thus $\mf_{\cO}(A_S)\leq m$. We now have an actual block with both
required local bounds. Write $M_1(p,M)$ for the defect-order bound
just obtained. Proposition~\ref{prop:criteria}(i), applied with
$M_1(p,M)$, bounds its Cartan sum. Part
(ii) of that proposition gives the claimed finite integral
Morita list. There are only finitely many abstract groups $S$
of the possible bounded orders.
\end{proof}
\subsection{Full basic corners and arbitrary markings}
Passing to a basic identity component is the integral crossed-product
corner construction of \cite[Proposition~4.15 and Corollary~4.16]{EiseleReduction2021}.
We recall it below to retain the homogeneous units and their labels.
\begin{theorem}[Based finiteness on $p$-subgroups]
\label{thm:based-sylow}
Compress the crossed orders of Proposition~\ref{prop:structure}
by a basic idempotent in the identity block, and identify the
resulting identity order with a representative $R_i$ of its
finite list. For every possible abstract $p$-group $S$, their
restrictions to $S$ have finitely many based graded isomorphism
classes. The identity-order identifications may be chosen
arbitrarily over $\cO$.
\end{theorem}
\begin{proof}
\emph{Pure subgroups and a fixed identity order.}
First suppose $S$ is pure. Let $e$ be a basic idempotent of
$B_0$. For every homogeneous automorphism, its translate of
$e$ is conjugate to $e$ by a unit of $B_0$: both idempotents
represent a projective module containing one copy of every
indecomposable projective type. Correcting each homogeneous
unit by such a unit of $B_0$ makes it commute with $e$.
Consequently
\[
C_S=eA_Se
\]
is a crossed order on $R=eB_0e$, and
\begin{equation}\label{eq:corner-rank}
\rank_{\cO}C_S=|S|\rank_{\cO}R.
\end{equation}
The idempotent $e$ is full in $A_S$, since it is already full
in $B_0$.
Proposition~\ref{prop:sylow-block} gives finitely many Morita
classes for these total orders. For fixed $R$ and $S$,
Proposition~\ref{prop:marked} now gives finitely many based
graded types. The proof of that proposition applies here
because each $C_S$ is Morita equivalent to the actual block
$A_S$. In particular, its first Hochschild cohomology
vanishes. Equation~\eqref{eq:corner-rank} also displays
explicitly the rank bound used to pass from Morita classes to
isomorphism classes of total orders.
\emph{Transport from arbitrary subgroups.}
For an arbitrary $p$-subgroup of $Q$, conjugate by a character
in $\Xi$ to make it pure, as explained after
Proposition~\ref{prop:structure}. Conjugation is implemented
by an integral homogeneous unit and transports the identity
block and its full basic corner. The transported identity
order is still in the same finite list. Arbitrary integral
markings are already included in
Proposition~\ref{prop:marked}; inner differences are absorbed
by conjugation in degree one, and outer differences form a
finite set. Transporting back proves the assertion for the
original subgroup and all its markings.
\end{proof}
The information retained by this theorem is stronger than a Morita
list: an isomorphism fixes the specified copy of $R$ and every group
label. This is exactly what the kernel argument will need when it
compares central factor systems after correcting the matrix actions.
The integral finiteness came from actual block orders and rigidity,
without a point count over $W(\F_q)$.
==== END Integral-Donovan-Finiteness-over-Witt-Vectors//05_sylow.tex ====
==== BEGIN Integral-Donovan-Finiteness-over-Witt-Vectors//06_kernel.tex ====
\section{Removing an unbounded prime-to-\texorpdfstring{$p$}{p} kernel}\label{sec:kernel}
We now turn the finite based lists on $p$-subgroups into a finite
Morita list for block summands of the whole crossed order. Its
grading group may have unbounded order. The bounded quantity is
its index over its largest normal $p'$-subgroup.
The proof refines the field argument of
\cite[Lemma~8.7]{OpenAIFieldDonovan2026} through the
Clifford-theoretic matrix corners of
\cite{KulshammerClifford1990,Kulshammer1995}. Over $\cO$, the
principal-unit argument uses unique prime-to-$p$ divisibility.
Removing a matrix factor creates a second scalar error; determinant
normalization places it in $\T$, so the original based Sylow
restriction is recovered exactly. After these two steps, the
remaining crossed systems have bounded grading groups and finite
fibres under restriction.
\subsection{Central units and the restriction kernel}
The principal-unit argument follows
\cite[Lemma~4.4]{EiseleReduction2021}. Sylow restriction will control
the part for which the grading group may have $p$-torsion.
\begin{lemma}[Units of the centre]\label{lem:center-units}
Let $R$ be an indecomposable $\cO$-order and put $Z_R=Z(R)$.
Then $Z_R$ is local with residue field $k$, and
\begin{equation}\label{eq:center-units}
Z_R^\times=\T\times U_R,
\qquad U_R=1+\rad Z_R.
\end{equation}
This decomposition is invariant under every $\cO$-algebra
automorphism of $R$. If $n$ is prime to $p$, the map
$u\mapsto u^n$ is an automorphism of $U_R$.
\end{lemma}
\begin{proof}
The finite commutative $\cO$-algebra $Z_R$ is a product of
complete local algebras, by henselianity of $\cO$. More than
one factor would give a nontrivial central idempotent of $R$.
It is therefore local. Its residue field is a finite extension
of the algebraically closed field $k$, hence is $k$.
The residue map on units is split by the scalar Teichm\"uller
units, and its kernel is $U_R$, proving
Equation~\eqref{eq:center-units}. Both factors are canonical
and are preserved by the stated automorphisms; these act
trivially on $\T$.
For $u\in U_R$, the polynomial $X^n-u$ has the root $1$ modulo
the maximal ideal, with derivative $n$ a unit. Hensel's lemma
gives a unique root in $U_R$. This proves the last assertion.
\end{proof}
\begin{samepage}
\begin{lemma}[Finite kernel of Sylow restriction]
\label{lem:restriction}
Let a finite group $Q$ act by $\cO$-algebra automorphisms on
$Z_R$, and let $S$ be a Sylow $p$-subgroup of $Q$. Then
\[
\res:H^2(Q,Z_R^\times)\longrightarrow H^2(S,Z_R^\times)
\]
has finite kernel.
\end{lemma}
\end{samepage}
\begin{proof}
The two factors in Equation~\eqref{eq:center-units} play different
roles: restriction is injective on the principal-unit part, while
the entire Teichm\"uller cohomology group is finite. On $U_R$ the composite
$\cor\circ\res$ is multiplication by $[Q:S]$. This is an
automorphism on $U_R$, by Lemma~\ref{lem:center-units}, and
therefore on its cohomology. Restriction is consequently
injective on $H^2(Q,U_R)$.
On $\T$, the action is trivial and $H^2(Q,\T)$ is finite.
Indeed, let $d=|Q|$. The $d$th-power map on $\T$ is onto
and has the finite kernel $\mu_d(\cO)\cap\T$. The exact
sequence in cohomology and the annihilation of positive
cohomology by $d$ show that $H^2(Q,\T)$ is a quotient of
\[
H^2(Q,\mu_d(\cO)\cap\T),
\]
which is finite because both the group and the coefficient
module are finite. Combining the two factors proves the lemma.
\end{proof}
\subsection{The integral extension theorem}
\begin{theorem}[Integral kernel removal]\label{thm:kernel}
Fix an indecomposable basic $\cO$-order $R$ Morita equivalent
to a finite-group block, and a finite subgroup
$\mathcal F\leq\Out_{\cO}(R)$. Consider crossed orders on
$R$ by finite groups $Q$ with the following properties:
\begin{enumerate}
\item the indices $[Q:O_{p'}(Q)]$ are bounded;
\item every outer action has image in $\mathcal F$;
\item for each possible abstract $p$-group, the restrictions
to subgroups of that type have finitely many based graded
isomorphism classes.
\end{enumerate}
Then the block summands of these crossed orders have only
finitely many $\cO$-linear Morita equivalence classes.
\end{theorem}
\begin{proof}
Write $A=\bigoplus_{q\in Q}Ru_q$ for one crossed order and set
\begin{equation}\label{eq:large-kernel}
K=O_{p'}(Q)\cap\ker(Q\longrightarrow\mathcal F).
\end{equation}
It is a normal $p'$-subgroup, and
$[Q:K]\leq[Q:O_{p'}(Q)]|\mathcal F|$ is bounded.
We isolate its twisted group algebra, pass to matrix corners,
verify their Sylow restrictions, and finally count central factor
systems on the bounded quotient.
\emph{Scalar factors on $K$.}
Since $K$ acts trivially modulo inner automorphisms, change the
$K$-degree units so that they centralize $R$. Their factors
belong to $Z_R^\times$ and form an ordinary $2$-cocycle for
the trivial $K$-action. Multiplication by $|K|$ is invertible
on $U_R$, so $H^i(K,U_R)=0$ for $i>0$. Removing the
$U_R$-component by a central change of units gives units $v_k$
with
\[
v_kv_l=\beta(k,l)v_{kl},\qquad \beta(k,l)\in\T.
\]
Their $\cO$-span is a twisted group algebra
$E=\cO_\beta K$, and the restriction to $K$ is
$R\otimes_{\cO}E$.
Every homogeneous unit of $A$ normalizes $E$. To verify this,
write
\[
u_qv_ku_q^{-1}=z_kv_{qkq^{-1}},\qquad z_k\in Z_R^\times.
\]
The coefficient is central because both sides centralize $R$.
Comparing the products for $k,l\in K$ shows that their
$U_R$-components define a homomorphism $K\to U_R$: the
original factors, their conjugates, and the factors in the
new $K$-degrees all lie in $\T$. Such a homomorphism is
trivial, since multiplication by $|K|$ is invertible on
$U_R$. Hence every $z_k$ belongs to $\T$, proving
normalization of $E$.
\emph{The matrix factors of $E$.}
The finitely many values of $\beta$ generate a finite
$p'$-subgroup of $\T$. The usual central extension defined
by $\beta$ is therefore a finite $p'$-group, and $E$ is its
character summand over $\cO$. A finite $p'$-group algebra
over $\cO$ is a product of full matrix algebras: its reduction
is split semisimple, and the matrix units lift over the
complete ring $\cO$. Each lifted primitive corner has rank
one and is $\cO$. We obtain
\begin{equation}\label{eq:matrix-E}
E\cong\prod_j M_{n_j}(\cO).
\end{equation}
Every $n_j$ is prime to $p$. Indeed these are ordinary
irreducible character degrees in the indicated character
sector of a finite $p'$-group; each such degree divides the
group order.
Let $e_j$ denote the identity of one matrix factor. The
$Q$-orbits on these idempotents give central summands of
$A$. In each orbit summand a single $e_j$ is full, because
its homogeneous conjugates sum to that orbit idempotent.
Let $Q_j^{\mathrm{pre}}$ be its stabilizer. It contains
$K$, and the $e_j$-corner, with its grading coarsened by
$K$, is crossed over
\[
R\otimes_{\cO}M_n(\cO),\qquad n=n_j,
\]
by the group $Q_j=Q_j^{\mathrm{pre}}/K$. The order of $Q_j$
is bounded by $[Q:K]$.
In each $Q_j$-degree choose a unit $e_ju_q$ inherited from
an original $Q$-degree. Its conjugation preserves both $R$
and $M_n(\cO)$, by the normalization of $E$ proved above.
Every $\cO$-algebra automorphism of
$M_n(\cO)$ is inner. For example, its images of the standard
matrix idempotents split $\cO^n$ into free rank-one summands;
choosing compatible basis vectors for the matrix units
constructs an implementing matrix. Correct the homogeneous
units by these matrices so that they centralize the matrix
factor. Their multiplication factors then lie in its
centralizer inside $R\otimes_{\cO}M_n(\cO)$, namely $R$.
The orbit corner is consequently
\[
M_n(\cO)\otimes_{\cO}C_j,
\]
where $C_j$ is a $Q_j$-crossed order on $R$ with the same
outer action on $R$. Equivalently, $C_j$ is the centralizer
of the matrix factor in the orbit corner.
This passage replaces the unbounded group $Q$ by the bounded group
$Q_j$. It has not yet supplied a finite list: the matrix corrections
may change the multiplication factors of a restriction. We next
recover those factors. The matrix size need not be bounded; its
prime-to-$p$ property is what the recovery uses.
\emph{Preservation of the based Sylow restriction.}
Let $S$ be a Sylow $p$-subgroup of $Q_j$. Schur--Zassenhaus
in its inverse image gives a $p$-subgroup
$\widetilde S\leq Q_j^{\mathrm{pre}}$ mapping isomorphically
onto $S$. Use the original units $u_s$ in these degrees,
with original factors $a_{s,t}\in R^\times$. Their
conjugations on $E_j=M_n(\cO)$ form an honest group action,
since $a_{s,t}$ centralizes $E_j$.
Choose implementing matrices $c_s\in\mathrm{GL}_n(\cO)$,
with $c_1=1$. We may take $\det c_s=1$. In fact, since
$p\nmid n$, every unit of $\cO$ has an $n$th root: take a
Teichm\"uller root of its residue and then use Hensel's
lemma on the principal-unit part. Multiplication by a
scalar thus normalizes the determinant without changing
the implemented automorphism.
Because the automorphisms form a group action, there is a
scalar $\gamma(s,t)\in\cO^\times$ such that
\[
c_sc_tc_{st}^{-1}=\gamma(s,t)I_n.
\]
Taking determinants gives $\gamma(s,t)^n=1$. Thus
$\gamma(s,t)\in\T$, since $n$ is prime to $p$.
Associativity shows that $\gamma$ is a normalized
$\T$-valued $2$-cocycle on $S$.
For the corrected units $w_s=c_s^{-1}e_ju_s$, their
actions on $R$ are unchanged. The formula for their
products is
\begin{equation}\label{eq:corrected-matrix-factors}
w_sw_t=\gamma(s,t)^{-1}a_{s,t}w_{st}.
\end{equation}
For example, this follows by using
$u_sc_tu_s^{-1}=c_sc_tc_s^{-1}$ in the product on the
left. Lemma~\ref{lem:teich-cohomology} removes $\gamma$
by scalar rescaling. Therefore the restriction of $C_j$
to $S$ is based-isomorphic to the original restriction to
$\widetilde S$. The corrected units generate the corresponding
homogeneous $R$-components of the centralizer $C_j$. For a fixed
original unit, any two implementing matrices differ by a scalar
unit; the subsequent scalar rescaling also changes only the
generator of its component. Thus these choices change the
presentation, not the based graded order. Hypothesis (iii) supplies a finite list
for the restrictions now obtained.
\emph{Crossed systems on a bounded quotient.}
There are finitely many possible abstract groups $Q_j$
and outer homomorphisms $Q_j\to\mathcal F$. Fix one of
each and choose representatives
$\alpha_q\in\Aut_{\cO}(R)$ of its outer classes.
Changing homogeneous units makes every crossed system
with this outer action use these same automorphisms.
If any factors satisfying the crossed identities exist,
their classes under central changes of units form a
torsor under
\begin{equation}\label{eq:factor-torsor}
H^2(Q_j,Z_R^\times).
\end{equation}
Indeed, the ratio of two factor systems with these fixed
actions is central by Equation~\eqref{eq:action-law}.
Equation~\eqref{eq:factor-law} says that the ratio is a
$2$-cocycle. A homogeneous-unit change preserving every
$\alpha_q$ must itself be central, and changes the ratio
by a coboundary. The action on the centre is a genuine
group action because inner automorphisms act trivially
there.
On a Sylow $p$-subgroup, the based finiteness just proved
gives finitely many restricted torsor classes. To see
that the fixed representatives cause no extra ambiguity,
observe that a based isomorphism between systems with
the same automorphisms must change their units centrally.
Lemma~\ref{lem:restriction} gives finite fibres for the
restriction map from Equation~\eqref{eq:factor-torsor}: a nonempty
fibre is a coset of its finite kernel.
Thus there are finitely many based crossed orders $C_j$.
Each $C_j$ has only finitely many block summands. The
orbit corners above are matrix algebras over them, and
the selected $e_j$ was full in the corresponding orbit
summand of $A$. Hence every block summand of $A$ is
Morita equivalent to one of this finite list of block
summands of the $C_j$. Neither the number of matrix
factors in Equation~\eqref{eq:matrix-E} nor their sizes
had to be bounded. This proves the theorem.
\end{proof}
The two scalar normalizations in this proof have different reasons.
On $K$, its prime-to-$p$ order makes principal-unit cohomology vanish.
On the lifted Sylow subgroup, the determinant first forces the error
into $\T$, and only its $\T$-cohomology is used. Neither step asserts
vanishing of principal-unit cohomology on a $p$-group.
==== END Integral-Donovan-Finiteness-over-Witt-Vectors//06_kernel.tex ====
==== BEGIN Integral-Donovan-Finiteness-over-Witt-Vectors//07_assembly.tex ====
\section{Proof of integral Donovan finiteness}\label{sec:assembly}
Fix $p$ and $M$. The previous sections supply three inputs for the
last step: finitely many integral basic identity orders, finite based
restrictions on their $p$-subgroups, and the integral kernel-removal
theorem. We check these inputs for an arbitrary finite-group block.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Let $B$ be a block of $\cO G$ with defect order at most $M$.
\emph{Reduction and the identity order.}
The integral An--Eaton reduction gives a reduced pair
$(H,B_H)$ with $B_H$ $\cO$-Morita equivalent to $B$ and with
the same defect group. Apply
Proposition~\ref{prop:structure}. Its integral dual recovery
realizes $B_H$, up to full corners, as a block summand of a
crossed order on an identity block $B_0$, with grading group
$Q$ satisfying a uniform bound on $[Q:O_{p'}(Q)]$.
Compress by a basic idempotent of $B_0$. As in the proof of
Theorem~\ref{thm:based-sylow}, correction of the homogeneous
units makes the resulting full corner a crossed order on
one of the finite list $R_1,\ldots,R_t$. Its outer action
lies in the finite group $\Out_{\cO}(R_i)$, by
Lemma~\ref{lem:HH} or Proposition~\ref{prop:structure}(iv).
\emph{The restrictions and the kernel.}
Theorem~\ref{thm:based-sylow} supplies finitely many based
restrictions on every possible $p$-subgroup. Its proof used the exact
integral comparison on the actual extension blocks, followed by
fixed-total-order rigidity; hence these are the based restrictions
required in the kernel theorem.
All three hypotheses of Theorem~\ref{thm:kernel} now hold for this
family over $R_i$: bounded $[Q:O_{p'}(Q)]$, finite outer-action
image, and finite based $p$-subgroup restrictions. Their block
summands therefore have finitely many integral Morita classes.
\emph{The finite union.}
Taking
the finite union over $i$ gives a list independent of $G$
and $b$. The full-corner equivalences and the preliminary
integral reduction place the original $B$ in that list.
This proves the theorem.
\end{proof}
\begin{corollary}\label{cor:reduction}
For every $p,M$, there is a finite list of integral basic
orders $R_1',\ldots,R_v'$ such that the basic algebra of
every block of $kG$ of defect order at most $M$ is
isomorphic to $k\otimes_{\cO}R_i'$ for some $i$.
In particular the blocks over $k$ of bounded defect have
finitely many $k$-linear Morita classes.
\end{corollary}
\begin{proof}
Every block idempotent over $k$ has its unique central
idempotent lift to $\cO G$. Lift a basic idempotent in
that block. Its corner is an integral basic order, and
reduction gives the original basic algebra. Theorem~\ref{thm:main}
makes the integral basic orders a finite list, proving
both assertions.
\end{proof}
The residue-field finiteness in Corollary~\ref{cor:reduction} was
already established by the field companion and supplied the Cartan
input in Proposition~\ref{prop:criteria}. The corollary now identifies
a finite list of integral basic orders whose reductions represent
those field algebras. The proof of that stronger conclusion has used
the explicit integral constructions throughout; it has not required
lifting an arbitrary field Morita equivalence.
\subsection{Scalar extension to a fixed complete coefficient ring}
\begin{proof}[Proof of Corollary~\ref{cor:complete-dvr}]
Keep $k=\overline{\F}_p$ and $\cO=W(k)$, and fix $\mathcal R$ and
its residue field $\ell$ as in the corollary. Choose an embedding
$\iota:k\hookrightarrow\ell$. Since $\ell$ is perfect, $W(\ell)$ is
a Cohen ring. The coefficient-ring theorem
\cite[Theorem~10.160.8, Tag~032A]{StacksCohen} gives a local map
$W(\ell)\to\mathcal R$ inducing the identity on $\ell$. It is
injective: any nonzero ideal of the DVR $W(\ell)$ contains a power of
$p$, whereas $\mathcal R$ has characteristic zero. Composing with the
Witt-vector map $W(\iota)$, which is injective on Witt coordinates,
gives a fixed local embedding
\[
\cO=W(k)\xrightarrow{W(\iota)}W(\ell)\hookrightarrow\mathcal R
\]
whose residue map is $\iota$. Only this coefficient embedding is
used; no finiteness of $\mathcal R$ over $\cO$ is needed.
We recall the residue-field argument of
\cite[Section~2.1]{OpenAIAlperinWeight2026}. For every finite group
$G$, scalar extension identifies
$Z(\ell G)=\ell\otimes_k Z(kG)$. Each local factor of the finite
commutative algebra $Z(kG)$ has nilpotent radical and residue field
$k$. After extending to $\ell$, the extended radical is a nilpotent
ideal with quotient $\ell$, so the factor remains local.
Consequently every primitive central idempotent $\bar b$ of $\ell G$
is uniquely of the form $1\otimes\bar b_0$ for a primitive central
idempotent $\bar b_0$ of $kG$. For every $p$-subgroup $Q\leq G$,
restriction of coefficients to $C_G(Q)$ gives
\[
\operatorname{Br}^{\ell}_Q(\bar b)
=1\otimes\operatorname{Br}^{k}_Q(\bar b_0).
\]
Faithfulness of field extension shows that these Brauer images are
nonzero for the same $Q$. The characterization of defect groups as
the maximal $p$-subgroups with nonzero Brauer image therefore gives the
same defect groups for the two residue blocks.
For either coefficient pair $(\Lambda,F)=(\cO,k)$ or
$(\mathcal R,\ell)$, the class sums give a $\Lambda$-basis of
$Z(\Lambda G)$ whose reductions give an $F$-basis of $Z(FG)$.
Thus $Z(\Lambda G)$ is a finite free complete commutative
$\Lambda$-algebra with residue algebra exactly $Z(FG)$. Idempotents
in this residue algebra lift uniquely: the derivative $2x-1$ of
$x^2-x$ is a unit at an idempotent modulo the maximal ideal of
$\Lambda$, so the complete-ring Hensel argument applies, including
when $p=2$. Primitivity is preserved, since central decompositions lift and
an idempotent reducing to zero is zero.
Now let $b$ be a block idempotent of $\mathcal R G$, and let $b_0$ be
the unique central idempotent of $\cO G$ lifting the corresponding
$\bar b_0$. The image of $b_0$ in $\mathcal R G$ is a central
idempotent reducing to $\bar b$, so uniqueness of the central lift
makes it equal to $b$. In particular
\begin{equation}\label{eq:complete-dvr-block-base-change}
\mathcal R G b\cong\mathcal R\otimes_{\cO}(\cO G b_0)
\end{equation}
as $\mathcal R$-algebras, and $b_0$ has the same defect group as $b$.
Finally, scalar extension preserves the specified linear Morita
equivalences. Indeed, for finite $\cO$-algebras $A,B$, a
$\cO$-linear Morita equivalence is represented by inverse
$\cO$-central Morita bimodules ${}_B P_A$ and ${}_A Q_B$ with
\[
P\otimes_A Q\cong B,\qquad Q\otimes_B P\cong A.
\]
Write $A_{\mathcal R}=\mathcal R\otimes_{\cO}A$ and similarly for
$B,P,Q$. Base change of the bimodules and their context maps gives
\[
\begin{aligned}
P_{\mathcal R}\otimes_{A_{\mathcal R}}Q_{\mathcal R}
&\cong\mathcal R\otimes_{\cO}(P\otimes_A Q)
\cong B_{\mathcal R},\\
Q_{\mathcal R}\otimes_{B_{\mathcal R}}P_{\mathcal R}
&\cong\mathcal R\otimes_{\cO}(Q\otimes_B P)
\cong A_{\mathcal R}.
\end{aligned}
\]
The Morita-context identities are preserved, as are finite
projectivity and the generator property. These bimodules therefore
give an $\mathcal R$-linear Morita equivalence on finitely generated
modules.
Choose the finite list of $\cO$-block representatives of defect order
at most $M$ from Theorem~\ref{thm:main}. The block $\cO G b_0$ in
\eqref{eq:complete-dvr-block-base-change} has that defect bound, so it
is $\cO$-linearly Morita equivalent to one representative. The
base-changed context puts $\mathcal R G b$ in the Morita class of its
scalar extension. These scalar extensions form a finite list over the
fixed ring $\mathcal R$, proving the corollary.
\end{proof}
==== END Integral-Donovan-Finiteness-over-Witt-Vectors//07_assembly.tex ====
==== BEGIN Integral-Donovan-Finiteness-over-Witt-Vectors//main.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{Integral Donovan Finiteness over Witt Vectors}
\author{OpenAI}
\date{September 25, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove integral Donovan finiteness: for every prime $p$ and positive
integer $M$, the blocks of all finite groups with defect groups of order
at most $M$ have only finitely many Morita equivalence classes over
$W(\overline{\mathbb F}_p)$. The defect groups need not be abelian,
and the result includes $p=2$. The same bounded-defect finiteness holds
over each fixed complete discrete valuation ring of characteristic zero
with algebraically closed residue field of characteristic $p$,
including ramified rings.
\end{abstract}
\noindent\small\textbf{Article identifier:}
\nolinkurl{Integral-Donovan-Finiteness-over-Witt-Vectors-September-25-2026}
\normalsize
\tableofcontents
\input{sections/01_introduction}
\input{sections/02_inputs}
\input{sections/03_rigidity}
\input{sections/04_comparison}
\input{sections/05_sylow}
\input{sections/06_kernel}
\input{sections/07_assembly}
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END Integral-Donovan-Finiteness-over-Witt-Vectors//main.tex ====
==== BEGIN Integral-Donovan-Finiteness-over-Witt-Vectors//preamble.tex ====
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\hypersetup{pdftitle={Integral Donovan Finiteness over Witt Vectors},pdfauthor={OpenAI}}
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\newtheorem{definition}[theorem]{Definition}
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==== END Integral-Donovan-Finiteness-over-Witt-Vectors//preamble.tex ====
==== BEGIN Integral-Donovan-Finiteness-over-Witt-Vectors//references.bib ====
@misc{OpenAIFieldDonovan2026,
author = {{OpenAI}},
title = {{Donovan's Conjecture over Algebraically Closed Fields}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026/main.pdf}{OAI:Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026}},
year = {2026}
}
@misc{OpenAIAlperinWeight2026,
author = {{OpenAI}},
title = {{The Blockwise Alperin Weight Conjecture}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/paper.pdf}{OAI:The-Blockwise-Alperin-Weight-Conjecture-September-23-2026}},
year = {2026}
}
@misc{EEKLS2026,
author = {Eaton, Charles W. and Eisele, Florian and Kessar, Radha and
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==== END Integral-Donovan-Finiteness-over-Witt-Vectors//references.bib ====