Let p be a prime. Put k = kbar = \overline{\mathbb F}_p (algebraic
closure of F_p) and O = W(k) (Witt ring of k). Then O is a complete
discrete valuation ring with uniformizer p, residue field k, and
fraction field K_0 of characteristic 0. It is the "absolutely
unramified" complete DVR with residue field k.
Sigma : O \to O denotes the Witt Frobenius (Frobenius lift, with
sigma(p) = p). For later: O^\times = k^\times \times (1 + p O) is the
direct sum of Teichmuller scalars and principal units.
Let G be a finite group. Idempotents in the finite commutative algebra
Z(O G) correspond, by reduction modulo p and unique lifting (O is
henselian), to idempotents in Z(k G).
A block of O G is an algebra O G b for a primitive central idempotent
b in O G. Its reduction k G \bar b (with \bar b the reduction of b)
is a block of k G in the usual sense; the correspondence b <->
\bar b is a bijection between primitive central idempotents.
Defect group, recalled concretely. For a p-subgroup Q of G, the
Brauer map br_Q : (k G)^G \to (k C_G(Q))^G sends sum_g a_g g to
sum_{g in C_G(Q)} a_g g (the terms whose g lies in the centralizer
C_G(Q) survive). A block \bar b of k G has defect group a p-subgroup
D of G that is maximal (with respect to inclusion, hence well defined
up to G-conjugacy) among the p-subgroups Q with br_Q(\bar b) \neq 0.
The defect order is |D|.
The block O G b inherits this defect group from k G \bar b. The
underlying reason: b is the unique lift of \bar b; the two blocks
share a Cartan matrix; and defect is determined by the Cartan matrix
being nonsingular modulo p (standard block theory).
Let R be a commutative ring. Two unital R-algebras A, B are Morita
equivalent (over R) iff their categories of finitely generated left
modules are R-linearly equivalent. Equivalently: there exist bimodules
_B M_A and _A N_B with M, N finitely generated projective on each
side, together with bimodule isomorphisms M \otimes_A N \cong B and
N \otimes_B M \cong A. In block theory one uses a third equivalent
form: there is a progenerator P over A (i.e. finitely generated,
projective, and "generates" the category) with B \cong End_A(P).
"Morita class" is the natural equivalence notion for blocks, because
Morita-equivalent algebras have the same finitely generated module
categories. Donovan's question is a bound on the number of Morita
classes.
Theorem 1.1 (integral Donovan finiteness). Fix a prime p and an
integer M \ge 1, and let O = W(\overline{\mathbb F}_p). Across all
finite groups G and all blocks O G b whose defect group has order
at most M, only finitely many O-linear Morita equivalence classes
of O G b occur.
Notes on the range.
- Every prime p is allowed, including p = 2.
- Every block is allowed, including non-principal blocks.
- Defect groups may be nonabelian.
- G itself has no bound.
- Bounding |D| \le M is equivalent to fixing one defect group D up
to isomorphism, because there are finitely many p-groups of order
at most M.
- "Finitely many Morita classes" means: there exist finitely many
basic O-orders R_1, ..., R_t (basic = the module-category skeleton
of the block) such that every O G b in the stated family is Morita
equivalent to exactly one of the R_i.
Corollary 1.2 (fixed complete DVR). Fix p and M, and fix a complete
discrete valuation ring R of characteristic 0 whose residue field
\ell is algebraically closed of characteristic p. Across all finite
groups G and all blocks R G b whose residue block \ell G \bar b has
defect of order at most M, only finitely many R-linear Morita classes
of R G b occur.
R may be ramified (i.e. W(\ell) \to R is not an isomorphism). No
splitting condition on Frac(R) is needed. The proof is by base change
of the O-result through a fixed embedding O = W(\bar F_p) \to R.
The residue-field version of the same statement — blocks of bounded
defect over k fall into finitely many k-Morita classes, uniformly in
the finite group — is treated in the classical literature on
Donovan's conjecture (Kessar 2004) and in the more recent case
analyses and reductions (Eaton and Livesey 2020). The integral
argument builds on the field case but cannot reduce to it: over O
there are two new phenomena.
Kessar 2004: https://link.springer.com/article/10.1007/s00013-004-4880-8
Eaton Livesey 2020: https://arxiv.org/pdf/2006.11173
Difficulty 1: infinite coefficient sets. The "obvious" descent
argument over k says: finitely many Cartan matrices, finitely many
Cartan invariants, done. Over O the same numerical bounds do not
detect the extra structure of an integral order: the set of
coefficient-Frobenius-fixed points in W(F_q) is infinite, so
point-counting over W(\bar F_p) gives no bound.
Difficulty 2: principal units. Over O, O^\times = k^\times \times (1+pO).
Cohomological obstruction classes often naturally produce elements of
O^\times. The k^\times part is torsion of prime-to-p order, harmless.
The (1+pO) part is a pro-p group and can have non-vanishing
cohomology on a finite p-group. Naive arguments that "the scalar
obstruction is finite because it is a root of unity" can fail.
The fix. In this paper, every specific scalar obstruction produced by
the argument lives in the Teichmuller subgroup
T = [k^\times] = { roots of unity of order prime to p in O },
which is a torsion subgroup of O^\times. For a finite p-group S acting
trivially on T, positive cohomology H^i(S, T) = 0 (Lemma 4.1). So the
obstruction can be removed.
Act I (Section 4, "comparison"). The companion paper produces explicit
integral Morita bimodules and operators between certain Levi and
finite-group blocks (Bonnafe-Rouquier, Section 9 of the companion).
These operators already have strict group actions; what remains is
their scalar error. We prove the scalar errors lie in T, then
H^i(S, T) = 0 kills them. The conclusion (Proposition 4.2): a Sylow
p-subgroup restriction of a certain crossed order is O-linearly
Morita equivalent to a bounded coefficient-Frobenius twist of itself,
with multiplication factors preserved exactly.
Act II (Section 3, "rigidity"). Once we know finitely many Morita
classes of "total orders", we must upgrade to finitely many
labelled/based isomorphism classes: preserving each homogeneous
component with its label, and preserving the identity marking. Two
ingredients:
(i) HH^1_O(A) = 0 for A a product of block orders (Lemma 3.5);
(ii) an O-scheme orbit lemma (Lemma 3.1) that supplies finite
orbits even when the group scheme is not reduced, and applies
for arbitrary finite grading groups, including those divisible
by p.
The key point: no averaging over the grading group is used, because
averaging would fail when p divides |H|.
Act III (Section 6, "kernel"). A crossed order has grading group Q with
bounded [Q : O_{p'}(Q)] but possibly unbounded O_{p'}(Q). We must
remove this kernel. Its twisted group algebra splits into a product of
matrix algebras over O of prime-to-p degree (Clifford-theoretic
corner). Determinant-one choices of implementing matrices sends the
resulting scalar error into T; again T's p-group cohomology vanishes.
Assembly (Section 7). Reduce an arbitrary block to a crossed order via
the imported reduction (Section 2). Act II gives finite based Sylow
restrictions; Act III converts them to finite Morita classes of block
summands. The two corollaries (field case and fixed DVR case) follow.
Section 2 collects the imported block reduction (from the companion
paper plus classical An-Eaton and Kulshammer/Eisele). Section 3 gives
rigidity. Section 4 gives the exact comparison. Section 5 gives finite
based Sylow lists. Section 6 removes the p'-kernel. Section 7
assembles and derives the corollaries.
We recall exactly which imported facts we will use later, phrased so
that the integral argument is self-contained once Section 3 begins.
Definition 2.1. A reduced pair is a pair (H, B) where H is a finite
group and B is a block of O H such that:
(r1) every block of any normal subgroup of H that is covered by B is
H-stable;
(r2) if B covers a nilpotent block of a normal subgroup H_0 <= H,
then H_0 <= Z(H) O_p(H) (so H_0 is "small": torsion central
times p-subgroup forces H_0 into the p-part).
Proposition 2.2 (An-Eaton, integral form). Let G be a finite group and
B a block of O G with defect group D. There exists a reduced pair
(H, B_H) together with an O-linear basic Morita equivalence
B \sim B_H (both sides considered as O-algebras) that sends D to a
defect group of B_H. Constants (bounds on D, on the p-part of H, on
the number of quasisimple components of F^*(H), and on [X : O_{p'}(X)]
below) depend only on p and |D|.
[Source: An-Eaton, Proposition 6.1, whose coefficient conventions
include O = W(k).]
An O-order is a unital O-algebra that is free of finite rank as an
O-module. A basic idempotent of a block order is a sum of one
primitive idempotent for each isomorphism type of indecomposable
projective; its corner is the basic order. Basic orders are unique
up to isomorphism in a Morita class.
Put K_0 = Frac(O) and let sigma be Witt Frobenius. For an O-order A,
the coefficient twist sigma^m A is the order obtained by applying
sigma^m to the structure constants of A; equivalently, it is scalar
transport along sigma^m. For a group block, sigma^m(O G b) is
identified with O G sigma^m(b). We write
mf_O(A)
for the least positive m such that A and sigma^m A are O-linearly
Morita equivalent, whenever such an m exists. Only upper bounds on
mf_O(A) will be used.
If the Cartan matrix of a block is (c_{ij}) (c_{ij} the multiplicity
of the simple S_i in the projective P_j, up to indexing), its basic
order has O-rank sum_{i,j} c_{ij}: the reduction is a split basic
k-algebra of that dimension. An integral block and its reduction
share the same Cartan matrix.
Proposition 2.3 (Cartan and integral finiteness inputs). Fix p and
M. Then:
(i) Cartan sum bound. There is a constant c(p, M) < \infty such
that every finite-group block O G b with defect order at most
M has Cartan sum at most c(p, M).
(ii) EEL finiteness. Fix positive bounds on defect order, Cartan
sum, and Morita-Frobenius number. Then finite-group blocks
over O with these bounds fall into only finitely many
O-linear Morita equivalence classes.
Proof. (i) By the companion field theorem (Section 1.5), the
reductions k G \bar b of such blocks fall into finitely many
k-Morita classes. Their Cartan matrices (up to simultaneous
permutation of rows and columns) consequently occupy a finite
list; take c(p, M) to be the maximum Cartan sum in that list.
Integral blocks and their reductions have the same Cartan matrix
(Section 1.2), which proves (i).
(ii) is the integral finiteness theorem of Eaton-Eisele-Livesey
[EEL2020, Theorem 3.10], applied separately to each of the
finitely many possible defect exponents 0, 1, ..., floor(log_p M).
The coefficient conventions of that theorem include the absolutely
unramified complete DVR W(k) and its coefficient Frobenius (fixing
the uniformizer p), so they apply with O = W(k).
Let (H, B) be a reduced pair with defect order at most M. Put
N = F^*(H) (Fitting subgroup times components, the generalized Fitting
subgroup), X = H/N. By Clifford theory B covers a unique H-stable
block b of O N (uniqueness up to H-conjugacy; in the reduced
situation, we may and do fix one).
Then by companion [Lemma 8.1]:
N = P Z K_1 K_2 ... K_j,
where
- P = O_p(H) is the largest normal p-subgroup of H,
- Z = O_{p'}(H) = O_{p'}(Z(H)) is central of prime-to-p order,
- K_1, ..., K_j are quasisimple components.
Bounds: |P| <= |D| and j <= bounded function of |D| (since each K_i
has p-part of its Schur multiplier bounded by |D|; this is the
companion's contribution). Bound on [X : O_{p'}(X)] follows from the
same argument applied to quotients by stable chains. The bound on
[X : O_{p'}(X)] passes to any subgroup of any extension of X by a
p'-group.
The companion [Lemma 8.2] provides a finite extension
N \trianglelefteq Nbar
with abelian p'-quotient A_0 = Nbar/N. For each x in X, choose a
representative h_x in H of x, with h_1 = 1. Write
h_x h_y = n_{x,y} h_{x y}, n_{x,y} \in N.
Conjugation by h_x descends to an automorphism alpha_x of Nbar, and
the actual relations hold:
(a) alpha_x alpha_y = ad(n_{x,y}) alpha_{x y};
(b) n_{x,y} n_{x y,z} = alpha_x(n_{y,z}) n_{x,y z}.
Eq (a) is conjugation by h_x h_y = n_{x,y} h_{x y} applied to
Nbar; Eq (b) is associativity of h_x h_y h_z. Both identities are
in N (not just modulo inner automorphisms), because the h_x are
actual group elements and the n_{x,y} are actual elements of N.
Every block bbar of O Nbar covering b has bounded defect: bbar
covers b, so any defect group of bbar induces one of b, and A_0 is a
p'-group so contributes no p-part. The bound is |D| (the |A_0| part
contributes nothing to a p-group).
Put Xi = Hom(A_0, k^\times), using the Teichmuller lift of each
character to O. Form Y = Xi \rtimes X. By [Lemma 8.3], there is a
Y-crossed order over O whose identity component is O Nbar and in
which O H is a full idempotent corner.
Explicitly, the dual-recovery idempotent is
e = |Xi|^{-1} sum_{chi in Xi} u_chi.
The characters chi are trivial on every n_{x,y} in N (they factor
through A_0 = Nbar/N), so every homogeneous unit u_x for x in X
conjugates e to itself; the idempotent is preserved and we may take
further full corners on the pure X-degrees. This makes the
construction integral: the averaging uses |Xi|^{-1}, and |Xi| is a
p'-number so the division is legal in O. The same integrality
preserves the "actual" factors n_{x,y} (not just their classes
modulo inner automorphisms).
Take the central summand cut out by bbar and a further full corner.
Result: a crossed order with identity component
B_0 = O Nbar bbar
(a block of the finite group Nbar),
and grading group
Q = Stab_Y(bbar) <= Y.
The index [Q : O_{p'}(Q)] is bounded: it equals the corresponding
index for Y, which is [Y : O_{p'}(Y)] (both Xi and X are bounded in
p-part) up to bounding.
[Companion Lemma 8.5] The basic orders of all identity blocks B_0
occurring above belong to a finite list of O-orders
R_1, ..., R_t,
independent of G and b (only depending on p and M).
Also, integral Picard groups Pic_O(R_i) are finite (Eisele, Theorem B
and Corollary 1.2), so outer automorphism groups Out_O(R_i) are
finite.
Let S <= Q be a p-subgroup. We claim S can be conjugated (by an
element of Xi acting on the crossed order) into the pure subgroup
X <= Y = Xi \rtimes X. This follows by Schur-Zassenhaus: the image of
S in X is a p-group, Xi is a p'-group, Schur-Zassenhaus conjugates
the two p-complements in the preimage. In the crossed order this
conjugation is implemented by an integral homogeneous unit, and it
transports bbar. Therefore for the purpose of bounding Sylow
restrictions, we may assume S <= Q \cap X, and this restriction is
"stable": bbar is S-stable, and the transported identity block stays
in the same family.
In particular, orders of relevant p-subgroups of Q are bounded: every
such intersection with O_{p'}(Q) is trivial, so the p-subgroup injects
into the quotient Q/O_{p'}(Q) whose order is bounded in 2.4.
Note: the order of the universal-cover central kernels in Nbar, or of
the added direct p-factors C_i used in Section 4, is NOT bounded.
The comparison (Section 4) will descend through those kernels before
any bounded-defect integral criterion is applied. That is the reason
Section 4 works so hard to prove the scalar errors lie in T
(Teichmuller), even on the auxiliary groups.
An ungraded Morita list does not record the homogeneous components of
a crossed order or the chosen identity marking. This section couples
an integral orbit lemma to the vanishing of HH^1 for block orders,
yielding finiteness at the finer level we need.
Recall a crossed order from Section 2.4: A = \bigoplus_{q in Q} R u_q
with u_q u_t = a_{q,t} u_{q t} and u_q r = alpha_q(r) u_q for r in R.
Its identity component is R u_1 = R, and every other component is a
copy of R labelled by q in Q. Two types of data must be recorded:
(lab) which homogeneous components are identified to which copy of
R (the group labels q);
(base) how the identity component is identified with the specific
basic order R.
"Based" means: an isomorphism that is the identity on R.
Lemma 3.1 (integral orbit finiteness). Let J be a smooth affine group
scheme of finite type over O, acting on an affine scheme V of finite
type over O. For x in V(O), let T_x V be the integral tangent module
along the O-section x (kernel of the Jacobian map at x on a
closed-immersion presentation V <= A^r_O). Then the points x for which
the differential Lie(J) \to T_x V is surjective all belong to only
finitely many J(O)-orbits.
Proof. Embed V as a closed subscheme of A^r_O with defining
equations. T_x V is the kernel of the Jacobian map O^r \to O^s (s the
number of equation generators), hence is a saturated submodule of O^r.
Write u = rank_O T_x V. Surjectivity of Lie(J) \to T_x V means its
coordinate matrix has a rank-u minor that is a unit of O. This is
Smith normal form over the DVR O.
After extending to the fraction field K_0 = Frac(O), the same
derivative has rank u = dim_{K_0} T_{x_{K_0}} V_{K_0}. Every
irreducible component of V_{K_0} through x_{K_0} has dimension at
most u. Let V_u be the reduced closure in V of the union of the
K_0-irreducible components of dimension at most u. There are
finitely many such components; each has horizontal closure, whose
special fibre has dimension at most its generic point's dimension.
This is the dimension theorem for finite-type schemes over a DVR
(Stacks, Lemma 33.19.2, Tag 0B2J). Hence dim((V_u)_k) \le u. Since
x factors through V_u (x_{K_0} does, and O \to K_0 is injective),
the rank-u point lives on the same closed subscheme.
Special fibre: let x_0 = x mod p. The unit minor of the orbit
derivative is still nonzero mod p, so the special fibre orbit J(k) x_0
has dimension >= u. Also it lies in (V_u)_k: every element of J(k)
lifts to J(O) (O complete, J smooth); the lift acts on x and its image
in V(O) lies in V_u as above. Combined with dim (V_u)_k \le u, we
get that the special-fibre orbit has dimension exactly u.
A locally closed orbit of dimension u in a scheme of dimension at most
u is open in one of its finitely many irreducible components of
dimension u. Two distinct orbits cannot both be dense open subsets
of the same component. Hence only finitely many special-fibre orbits
of dimension u occur.
Hensel slice: fix one such orbit, representative x_0, and u
coordinates whose orbit derivative at x_0 has nonzero u-minor. Lift
the coordinate values of x_0 to integral a_1, ..., a_u. For any
integral point x \in V(O) reducing to x_0, the map
J \to A^u_O, g \mapsto (u chosen coordinates of g x)
is smooth near the identity: J is smooth and its relative differential
is surjective. Hensel then supplies g \equiv 1 (mod p) with g x
having coordinates exactly a_i. So every integral orbit has a
representative in the slice W = V \cap {u chosen coordinates = a_i}.
For a resulting point y in W, the selected-coordinate differential is
an isomorphism on T_{y_{K_0}} V_{K_0} (dimension u, minor is a unit
because y reduces to x_0). Thus T_{y_{K_0}} W_{K_0} = 0. Points with
zero tangent in a finite-type K_0-scheme are isolated, and a
noetherian scheme has only finitely many isolated points. Each
K_0-isolated point admits at most one integral section (O \to K_0
injective). Finitely many points remain in W. Summing over the
finitely many (u, special-fibre orbit) pairs gives finiteness.
Remark 3.2. The integral tangent T_x V need not surject onto the
special-fibre tangent at x_0. Saturation plus the unit minor is all
that is used. No smoothness assumption on V is needed. Rank u = 0 is
covered (no slice coordinates; tangent is zero; finitely many isolated
integral points).
Lemma 3.3 (Hochschild rigidity). Let A be an O-order Morita
equivalent to a finite direct sum of finite-group block orders. Then
HH^1_O(A) = 0 and Out_O(A) is finite.
Proof. For a finite group G,
HH^1_O(OG) = H^1(G, OG_conj) = \bigoplus_{[g]} H^1(C_G(g), O),
where the conjugation action is used and the last isomorphism is
Shapiro. The module O is trivial for C_G(g) and Hom(C_G(g), O_{add}) = 0
because C_G(g) is finite and O is torsion-free. So HH^1 = 0. Since
HH^1 splits over finite products and is Morita invariant, HH^1_O(A)
= 0. Concretely: every O-linear derivation of A is inner.
A^\times: open subscheme of affine underlying space of A where the
regular-representation determinant is invertible; smooth. Aut_O(A):
affine finite type (impose multiplicative and unital equations on an
invertible O-linear map). A^\times acts by inner post-composition.
At an automorphism g, compose with g^{-1} to identify the tangent
with the O-derivations of A. The derivative of the A^\times action is
exactly the map (inner elements) \to (derivations), whose image is the
subspace of inner derivations. That subspace is all of HH^1_O(A) = 0
in the derivation space, so the orbit derivative is surjective.
Lemma 3.1 bounds the number of A^\times(O) = A^\times-orbits of
automorphisms, i.e. the number of cosets modulo inner automorphisms.
Hence Out_O(A) is finite.
Theorem 3.4 (gradings). Let A be an O-order with HH^1_O(A) = 0.
Let H be any prescribed finite group. Then the H-labelled
O-algebra gradings A = \bigoplus_{h in H} A_h occur in finitely many
orbits under Aut_O(A) --- in fact, under inner automorphisms A^\times.
Proof. A grading is determined by its orthogonal projections
e_h in End_O(A), which satisfy:
e_h e_j = delta_{h,j} e_h, sum_h e_h = 1,
e_h(1_A) = delta_{h,1} 1_A,
e_l( e_g(x) e_h(y) ) = 0 for l \ne g h, for x, y in A.
The last family of equations is imposed only on a finite spanning set
of A (basis), so all together these polynomial conditions on the
e_h cut out an affine finite-type grading scheme, on which A^\times
acts by conjugation.
Compute the integral tangent at a grading. A first-order deformation
of the decomposition is represented by \delta in End_O(A) that is
off-diagonal:
e_h \delta e_h = 0 for every h in H.
(Write each summand as a graph over the original summand; equivalently
differentiate the orthogonal-projection equations.) The deformed
summands are (1 + \varepsilon \delta) A_h over O[\varepsilon]/(\varepsilon^2).
Multiply-linear condition. For x in A_g, y in A_h, linearising the
multiplicative equations around the original grading says that
\delta(x y) - \delta(x) y - x \delta(y)
has zero component in every degree other than gh, and zero gh-component
too. The first term is off degree gh by hypothesis. A summand of
\delta(x) y has degree gh only if the first factor lies in degree g
(by cancellation in H); but that component of \delta(x) is zero
since \delta is off-diagonal. So \delta(x) y contributes nothing in
gh; similarly x \delta(y). Conclusion: the linearised expression
vanishes identically, i.e. \delta is a derivation.
The hypothesis HH^1_O(A) = 0 gives \delta = [a, -] for some a in A.
Conjugation by 1 + \varepsilon a produces exactly this tangent.
So the orbit derivative at every grading point is surjective onto
every integral tangent module. Lemma 3.1 applies and gives finiteness.
Remark 3.5 (total order is fixed). The theorem counts gradings on a
fixed total order A. Count of all crossed orders on a fixed identity
order R is different and can be infinite. For odd p the based
C_p-crossed orders
O[t] / (t^p - a), a in O^\times,
have classes parametrized by O^\times/(O^\times)^p (infinite: the
p-adic logarithm identifies (1+pO)/(1+pO)^p with pO/p^2O \cong k).
The total orders here vary. Theorem 3.4 uses a different hypothesis
and covers grading groups of order divisible by p.
Proposition 3.6 (retaining the marking). Fix a basic block order R
and a finite group H. Let A be a collection of H-crossed orders on R,
each Morita equivalent to a finite-group block. If the total orders in
A have only finitely many O-Morita classes, then A has only finitely
many based graded isomorphism classes.
Proof. Every order in A has O-rank |H| rank_O(R). Finitely many of
them lie in each Morita class: choose a basic representative, any
other order is End_A(P) for a progenerator P = \bigoplus_i P_i^{m_i}
with positive m_i and P_i ranging over the finitely many indecomposable
projectives of the basic representative. End_A(P) contains
M_{m_i}(End(P_i)) as a direct O-module summand, so its rank
bounds m_i^2. Finitely many (m_i) for each tuple i.
For each ungraded total order, apply Theorem 3.4 with H = the grading
group. This is allowed because each total order is Morita equivalent
to a block product, hence HH^1_O vanishes (Lemma 3.3). We get
finitely many labelled H-gradings.
Fixing one graded representative with identity component R, two
identifications of that component with R differ by Aut_O(R). Inner
changes extend to graded automorphisms of the total order (conjugation
by a degree-one unit). Since Out_O(R) is finite (Lemma 3.3), finitely
many markings remain. Together with the finitely many gradings, this
gives finitely many based graded types.
We must strengthen the Sylow restriction of the crossed order A
\cdot alpha_{stv}(b), moving the right-hand factors through, and then
using exactness of D_{_,_} as bimodule maps on both sides):
LHS(w) = c(s,t) c(st,v) a'_{s,t} a'_{st,v} D_{stv}(w)
alpha_{stv}(alpha_v^{-1}(a_{s,t}^{-1})) a_{st,v}^{-1};
RHS(w) = c(t,v) c(s,tv) alpha'_s(a'_{t,v}) a'_{s,tv} D_{stv}(w)
a_{s,tv}^{-1} alpha_s(a_{t,v}^{-1}).
Since D_{stv} : M \to M is invertible, its image is all of M, so as w
varies the value u = D_{stv}(w) ranges over all of M. Both sides are
linear in u (left action and right action of B_0 on M). Comparing
left-side and right-side factors separately, and using the actual
factor law (Section 2.3 (b))
a_{s,t} a_{st,v} = alpha_s(a_{t,v}) a_{s,tv}
and its sigma^m-tilde version
a'_{s,t} a'_{st,v} = alpha'_s(a'_{t,v}) a'_{s,tv},
the non-scalar factors cancel, leaving:
c(s,t) c(st,v) = c(t,v) c(s,tv).
Thus c is a normalized 2-cocycle in Z^2(S, T) with trivial action.
By Lemma 4.1, c is a coboundary: c = \partial(e) for some
normalized 1-cochain e : S \to T. Rescaling D_s by e_s,
J_s = e_s D_s,
gives:
J_s J_t = e_s e_t D_s D_t = e_s e_t c(s,t) a'_{s,t} D_{st} a_{s,t}^{-1}
= (e_s e_t c(s,t)/e_{st}) a'_{s,t} J_{st} a_{s,t}^{-1}
= a'_{s,t} J_{st} a_{s,t}^{-1},
using e_s e_t c(s,t) = e_{st} since c = \partial e. Covariance of J_s
follows directly from covariance of D_s and the scalar nature of e_s
(scalar acts trivially under cov). Normalize J_1 = id by choosing
e_1 = 1 (D_1 = id already).
This completes the proof of Proposition 4.2.
If c were only in O^\times, the cocycle would not automatically be
a coboundary, because principal units 1 + pO can have non-trivial
H^2(S, 1+pO) on a finite p-group. The extra precision of the
argument is that (i) the specific lambda_m has p'-order (giving
scalars in T), (ii) the overlap identity produces T-scalars only,
(iii) the descent through D_0 and C_i introduces no additional
non-T scalar. In particular, no step asserts vanishing of
principal-unit cohomology on a p-group: the very reason that
integral comparison is more subtle than the field case.
The comparison of Section 4 produces a Morita equivalence of crossed
orders; combined with EEL's numerical criterion, it gives finitely
many Morita classes for Sylow restricted orders. Fixed-total-order
rigidity (Section 3) then upgrades to based graded types.
Lemma 5.1 (Marcus extension, integral form). Let B, B' be O-orders
with normalized crossed systems (\alpha_s, a_{s,t}) and
(\alpha'_s, a'_{s,t}) for a finite group S, and A, A' the associated
crossed orders (A = \bigoplus_s B u_s, A' = \bigoplus_s B' v_s).
Suppose an O-linear (B', B)-Morita bimodule M has invertible
O-linear maps J_s: M \to M, with J_1 = id, satisfying:
(cov) J_s(a v b) = \alpha'_s(a) J_s(v) \alpha_s(b),
(exact) J_s J_t(v) = a'_{s,t} J_{st}(v) a_{s,t}^{-1},
for all a in B', b in B, v in M, s, t in S. Then A and A' are
O-linearly Morita equivalent.
Proof. Put P = M \otimes_B A (tensor over B); P is a right A-module.
The left B'-action is from M on the outer tensor factor. Define
left-A'-action operators, on simple tensors:
L_s(v \otimes a) = J_s(v) \otimes u_s a, s in S.
These are well defined on the balanced tensor: identity
v \cdot b \otimes a = v \otimes b \cdot a uses u_s b = \alpha_s(b) u_s
combined with (cov). Explicitly, for v in M, b in B, a in A:
L_s(v b \otimes a) = J_s(v b) \otimes u_s a
= J_s(v) \alpha_s(b) \otimes u_s a
= J_s(v) \otimes \alpha_s(b) u_s a
= J_s(v) \otimes u_s (b a)
= L_s(v \otimes b a),
so the map is well defined. Similarly
L_s(b' \cdot (v \otimes a)) = L_s(b' v \otimes a)
= J_s(b' v) \otimes u_s a
= \alpha'_s(b') J_s(v) \otimes u_s a
= \alpha'_s(b') \cdot L_s(v \otimes a),
so L_s is B'-semilinear with respect to (cov), and
L_s L_t(v \otimes a) = L_s(J_t(v) \otimes u_t a)
= J_s J_t(v) \otimes u_s u_t a
= a'_{s,t} J_{st}(v) a_{s,t}^{-1} \otimes a_{s,t} u_{st} a
= a'_{s,t} J_{st}(v) \otimes u_{st} a
= a'_{s,t} L_{st}(v \otimes a),
so L_s L_t = a'_{s,t} L_{st}. Hence L_s extend to a left A'-action
on P.
P is a projective generator of Mod-A: M is a progenerator on the B-
side, and inducing M by - \otimes_B A preserves f.g. projectivity and
generation (standard Morita context). Finally, compute End_A(P) as a
right B-module. As right B-modules,
P = M \otimes_B A = \bigoplus_{s in S} (M \otimes_B B u_s) = \bigoplus_s M u_s,
a direct sum of copies of M. Then by induction-restriction:
End_A(P) = \bigoplus_{s in S} Hom_B(M, M u_s).
Explicitly, a B-linear f: M \to M u_s extends uniquely to a right
A-linear endomorphism F of P by F(v \otimes a) = f(v) a. Composing
with L_s^{-1}: M \to M u_s \to M yields a B-linear map M \to M, so an
element of End_B(M) = B'. Conversely, any b' in B' gives the
degree-one operator F = b' L_s^{-1} on M u_s. The A-linearity and the
L_s L_t = a'_{s,t} L_{st} relation identify End_A(P) with the crossed
order A'. More precisely, the resulting bimodule A' \to End_A(P) has
image exactly \bigoplus_s B' L_s, and the L_s L_t = a'_{s,t} L_{st}
relation shows this equals A'. Hence A and A' are Morita equivalent.
Proposition 5.2. Keep the data of Proposition 4.2. Let S <= Q \cap X
be a pure p-subgroup. Then the restriction A_S = \bigoplus_{s in S}
R u_s of the crossed order (with identity B_0) is, up to the
identification B_0 = O \bar N \bar b, exactly a block of a finite
group. Its defect group D_S has order bounded in p, M; its integral
Morita-Frobenius number mf_O(A_S) is bounded in p, M; and the
collection A_S, over all pure p-subgroups S occuring, has only
finitely many O-Morita classes.
Proof. Construction of the group. Use the actual data: automorphisms
alpha_s of Nbar, and elements n_{s,t} in N. On the underlying finite
set Nbar \times S define
(g,s)(h,t) = (g alpha_s(h) n_{s,t}, st).
Associativity: use (a) alpha_s alpha_t = ad(n_{s,t}) alpha_{st} and
(b) n_{s,t} n_{st,v} = alpha_s(n_{t,v}) n_{s,tv} from Section 2.3.
The identity is (1,1) and the inverse of (g,s) exists by
normalization. This is a finite group Gamma_S with normal subgroup
Nbar and quotient S (via (g,s) \mapsto s).
Since the crossed order A_S is generated by u_s (s in S) with
u_s u_t = a_{s,t} u_{st} and u_s b u_s^{-1} = alpha_s(b) for b in B_0,
mapping u_s (1,s) and Nbar (n,1) gives an isomorphism between the
crossed order A_S and O Gamma_S \bar b: both have the same
O-module decomposition \bigoplus_s B_0 u_s and the same multiplication
u_s u_t = a_{s,t} u_{st}. Therefore A_S = O Gamma_S \bar b is itself
the block of O Gamma_S containing \bar b.
The block \bar b (an idempotent in O Nbar) is stable under S. A
stable block over a p'-free normal subgroup Nbar inside a p-extension
Gamma_S extends uniquely to a block of O Gamma_S covering it (Fong-
Reynolds, special case of p-power quotient). Since \bar b itself
occurs inside O Gamma_S as an idempotent covering \bar b, it IS that
covering block. So A_S = O Gamma_S \bar b is the unique block of
O Gamma_S covering bbar. In particular, A_S is a block of O Gamma_S,
not just a summand of one.
Defect. Let D_S be a defect group of A_S. Normal block theory (e.g.
standard references on blocks over DVRs) gives that D_S \cap Nbar is
a defect group of bbar in Nbar, after conjugation; and D_S maps onto
S. Hence
|D_S| \le |S| |D_S \cap Nbar| \le |S| \cdot (\text{bound on
defect of bbar}).
|S| is bounded (Section 2.6), and the defect of bbar is bounded by
Section 2.3. In particular |D_S| is bounded in p, M. Call the bound
M_1 = M_1(p, M).
MF number. By Proposition 4.2, with this same actual factor data,
there is a (sigma^m B_0, B_0)-Morita bimodule M with operators J_s
satisfying the exact comparison. Lemma 5.1 then gives a Morita
equivalence between A_S and sigma^m(A_S), with m bounded in p, M.
Hence mf_O(A_S) \le m, uniformly.
Finite Morita list. Apply Proposition 2.3 (i): the Cartan sum of any
finite-group block with defect order at most M_1 is bounded by
c(p, M_1). Apply Proposition 2.3 (ii) (EEL Theorem 3.10): with
bounded defect, bounded Cartan sum, and bounded MF number, only
finitely many O-linear Morita classes occur.
Finally, the abstract p-groups S of order bounded as above are
themselves finite in number, so the union over all possible S is
finite.
Theorem 5.3 (finite based list on p-subgroups). Compress the crossed
orders of Proposition 4.2 by a basic idempotent e of the identity
block B_0, and identify the resulting identity order R = e B_0 e with
one of the finitely many basic orders R_i from Section 2.5. Then for
every possible abstract p-group S, the restrictions to S of these
crossed orders have only finitely many based graded isomorphism
classes. The identity-order identifications can be chosen arbitrarily
over O.
Proof. Step 1 (pure subgroups, fixed identity R). Suppose S is pure.
Let e be a basic idempotent of B_0. e is also full in A_S, since e
is full in B_0 = identity component and A_S contains B_0 as direct
summand. We want a crossed order on R = e B_0 e that is Morita
equivalent to (indeed a basic corner of) A_S.
Construction. For every homogeneous unit u_s of A_S, conjugation by
u_s sends the basic idempotent u_s e u_s^{-1} to a basic idempotent
of B_0. Two basic idempotents of a semiperfect O-order are conjugate
in the order: both represent a projective module containing one copy
of every simple / indecomposable-projective type. Choose a unit
c_s in B_0^\times with c_s (u_s e u_s^{-1}) c_s^{-1} = e. Replace u_s
by c_s u_s: since c_s is in the identity component,
c_s u_s (c_s u_s)^{-1} fixes e. The new homogeneous units (still
called u_s) fix e.
Then
C_S := e A_S e = \bigoplus_{s in S} (e B_0 e) u_s
is a crossed order on R = e B_0 e with grading S. The O-rank of
C_S is:
rank_O(C_S) = |S| \cdot rank_O(R). (*)
Step 2 (Morita classes). By Proposition 5.2, A_S ranges over finitely
many O-Morita classes. Since C_S is the corner e A_S e of A_S at the
full idempotent e, C_S is Morita equivalent to A_S (Morita's theorem
on e A e for full idempotent e). Hence C_S also falls into finitely
many Morita classes.
Step 3 (based graded types). Each C_S is Morita equivalent to the
actual block A_S = O Gamma_S \bar b, so HH^1_O(C_S) = 0 (Lemma 3.3).
Apply Proposition 3.6 with H = S: fixed total order C_S, and the
finitely many Morita classes just found, produce finitely many based
graded isomorphism classes. The rank formula (*) is exactly the
explicit bound used in the proof of Proposition 3.6 to pass from
Morita classes to ungraded iso classes of total orders.
Step 4 (arbitrary S). A p-subgroup S of Q can be conjugated by an
element of Xi (Schur-Zassenhaus, Section 2.6) into a pure subgroup
S' of X. In the crossed order, this conjugation is implemented by
an integral homogeneous unit u_chi (for chi in Xi). u_chi conjugates
A_S to A_{S'}, transports the identity block bbar to itself, and
transports the full basic corner accordingly. The transported
identity order is still in the finite list {R_i} (Section 2.5),
because conjugation by Xi acts trivially on B_0 (it acts on the
p'-factor A_0 = Nbar/N, and the identity block is fixed).
Arbitrary integral markings: inner differences are absorbed by
conjugation in degree one (Section 3.5), and outer differences form
a finite set (Out_O(R) finite, Lemma 3.3). Transporting back gives
finiteness for the original S and all its markings.
The point of Theorem 5.3: the information retained is not just a
Morita list but full labeled-component + marking data. This is
precisely what Section 6 will need to control central factor systems
after the p'-kernel is removed.
The grading group Q of a crossed order has bounded [Q : O_{p'}(Q)]
but unbounded O_{p'}(Q). We must remove that kernel. The method
refines the field argument (companion, Lemma 8.7) via Kulshammer's
Clifford matrix corners, with two scalar-normalization tricks that
only make sense over O: unique prime-to-p divisibility on the
principal-unit factor, and determinant-one matrices to force the
second scalar error into T.
Lemma 6.1 (units of the centre). Let R be an indecomposable
O-order with centre Z_R = Z(R). Put U_R = 1 + rad(Z_R). Then:
(a) Z_R is a local commutative O-algebra with residue field k;
(b) Z_R^\times = T \times U_R;
(c) the decomposition is canonical (preserved by every
O-algebra automorphism of R);
(d) if n is coprime to p, the n-power map u \mapsto u^n is an
automorphism of U_R.
Proof. Z_R is a finite commutative O-algebra. Since O is henselian,
it is a finite product of complete local O-algebras. More than one
factor would give a nontrivial central idempotent of R (an element
of Z_R reducing to 1 mod some factor; it splits R as a product).
So Z_R is local. Its residue field is a finite extension of k and
k is algebraically closed, so residue field = k.
The residue map Z_R^\times \to k^\times has kernel 1 + rad(Z_R) =
U_R. The Teichmuller scalars [k^\times] = T hit k^\times bijectively,
so Z_R^\times = T \times U_R, canonical given the k-algebra structure.
Both factors are preserved by O-algebra automorphisms (scalars map
to themselves in T, principal units in U_R).
For u in U_R, the polynomial X^n - u has root 1 mod p (u \equiv 1
mod rad) and derivative n\cdotu^{(n-1)/... } = n at that point, a
unit in O since (n, p) = 1. Hensel's lemma produces a unique root
in U_R. So u \mapsto u^n is a bijection U_R \to U_R.
Lemma 6.2 (finite kernel of Sylow restriction). Let a finite group
Q act by O-algebra automorphisms on Z_R, and let S \le Q be a Sylow
p-subgroup. Then
res : H^2(Q, Z_R^\times) \to H^2(S, Z_R^\times)
has finite kernel.
Proof. By Lemma 6.1, Z_R^\times = T \times U_R with trivial action on
T (automorphisms of R preserve T and act trivially on it; T is
[k^\times] and inner automorphisms, being conjugation, act trivially
on the canonical scalar subring).
Part U_R. Compute \cor \circ \res on H^2(Q, U_R): this is
multiplication by [Q : S]. By Lemma 6.1 (d), [Q : S] is coprime to p,
so multiplication by [Q : S] is an automorphism of U_R and hence an
automorphism of H^2(Q, U_R) (applied degree by degree). Hence \res
has trivial kernel on U_R-cohomology: if \res(x) = 0, then
[Q : S] x = \cor(\res(x)) = 0, and [Q : S] is invertible, so x = 0.
Part T. Q acts trivially on T, so H^2(Q, T) is the ordinary group
cohomology with trivial T coefficients. Let d = |Q|. The d-th
power map T \to T is surjective and has finite kernel
\mu_d(O) \cap T = { roots of unity of order dividing d in T }
(a finite group). From the short exact sequence
0 -> \mu_d \cap T -> T \xrightarrow{d} T -> 0,
and the annihilation of H^i(Q, T) by d for i > 0, we get that
H^2(Q, T) is a quotient of H^0(Q, \mu_d \cap T)- twisted, i.e. a
quotient of H^2(Q, \mu_d \cap T) which is finite (finite group Q,
finite coefficient module, standard finiteness). Hence H^2(Q, T)
is finite, and \res restricted to T has finite kernel (its target
is finite, so the kernel is at worst H^2(Q, T) which is finite).
Combining: \ker(\res) is a subgroup of \ker(\res_{|U_R}) \times
H^2(Q, T), which is trivial \times finite = finite.
Theorem 6.3 (integral kernel removal). Fix an indecomposable basic
O-order R Morita equivalent to a finite-group block, and a finite
subgroup F of Out_O(R). Consider the class A of O-crossed orders on
R by finite groups Q with:
(i) [Q : O_{p'}(Q)] bounded (by a constant C independent of Q);
(ii) every outer action Q \to Out_O(R) has image contained in F;
(iii) for each abstract p-group S, the restrictions of orders in
A to S have finitely many based graded isomorphism classes.
Then the block summands of orders in A have only finitely many
O-linear Morita equivalence classes.
Proof. Write A = \bigoplus_q R u_q for a typical order in A. Put
K = O_{p'}(Q) \cap \ker(Q \to Out_O(R)).
Then K is a normal p'-subgroup of Q, and
[Q : K] \le [Q : O_{p'}(Q)] \cdot |\mathcal F| \le C |\mathcal F|,
so [Q : K] is bounded.
Step 1 (K-units become central). Since K acts trivially modulo inner
automorphisms, we may change the K-degree units by inner automorphisms
of R so that u_k centralizes R for every k in K. Then the u_k generate
a subalgebra E = \bigoplus_k R u_k inside A that commutes with R; in
fact E = \bigoplus_k Z_R u_k = Z_R-twisted version. More usefully, E
is a crossed order on R with grading K and trivial action, i.e., a
twisted group algebra R \otimes_O O_\beta K for a 2-cocycle \beta \in Z^2(K, Z_R^\times)
with trivial K-action.
Step 2 (kill the principal-unit cocycle). By Lemma 6.1,
Z_R^\times = T \times U_R. On U_R, |K| is invertible, so H^i(K, U_R) = 0 for
i > 0 (restriction-corestriction, as in Lemma 6.2 U_R-part). After
a change of the u_k by central U_R-valued scalars, \beta takes values in T:
v_k v_l = \beta(k,l) v_{kl}, \beta(k,l) \in T, for the rescaled u_k
(still denoted u_k, or v_k for the rescaled version).
Step 3 (matrix decomposition of E). The finite subgroup of T
generated by the values \beta(k,l) is p'-torsion, so it defines a finite
p'-central extension of K; its O-twisted group algebra E = \oplus_k R v_k
enlarges to R \otimes_O O_\beta K, and O_\beta K is a finite p'-group algebra over
O. A finite p'-group algebra over a complete DVR is split semisimple;
its primitive central idempotents lift over O, and the matrix units
lift, so
O_\beta K \cong \prod_j M_{n_j}(O),
with every n_j prime to p (ordinary character degrees of a finite
p'-group, dividing the group order).
Step 4 (matrix corners and bounded quotient). Let e_j be the identity
of one matrix factor in this decomposition. Q acts on the set {e_j};
the orbit sum \sum_{q \in Q-orbit(e_j)} q e_j q^{-1} is a central
idempotent of A, giving a central summand of A. In that orbit
summand, e_j is full (its homogeneous conjugates sum to the orbit
idempotent). Let Q_j^pre = Stab_Q(e_j). Since K fixes e_j,
Q_j^pre \supseteq K. Set Q_j = Q_j^pre / K; then |Q_j| \le [Q : K], bounded.
The e_j-corner, with grading coarsened by K, is crossed over
R \otimes_O M_n(O) (n = n_j) by the group Q_j, i.e. the corner is a
Q_j-crossed order:
e_j A e_j = \bigoplus_{q \in Q_j} (R \otimes M_n(O)) e_j u_q
where we pick one unit e_j u_q for each q in Q_j. Every O-algebra
automorphism of M_n(O) is inner, so the conjugation action of e_j u_q
on M_n(O) is by a unique-up-to-scalar matrix.
Step 5 (Sylow restriction is preserved exactly). Let S \le Q_j be a
Sylow p-subgroup. By Schur-Zassenhaus, in the preimage of S in
Q_j^pre there is a p-subgroup \tilde S \le Q_j^pre mapping isomorphically
onto S. Use the original units u_s for s \in \tilde S, with original factors
a_{s,t} \in R^\times. Conjugation by these units on E_j = M_n(O) is honest
(a_{s,t} centralizes M_n(O)).
Choose implementing matrices c_s \in GL_n(O), with c_1 = 1 and
det(c_s) = 1. (Since p \nmid n, every unit of O has an n-th root:
take a Teichmuller root and apply Hensel to the principal-unit part.
Multiplying c_s by a scalar adjusts determinant without changing the
automorphism.) Because the conjugation action on M_n(O) is a group
action, c_s c_t c_{st}^{-1} = \gamma(s,t) I_n for some \gamma(s,t) \in O^\times.
Taking determinants: \gamma(s,t)^n = 1 (since det(c_s) = 1). Since n is
prime to p, \gamma(s,t) is a root of unity of p'-order, so \gamma(s,t) \in T.
Now consider the corrected units w_s = c_s^{-1} e_j u_s in the corner
e_j A e_j. Compute:
w_s w_t = c_s^{-1} e_j u_s c_t^{-1} e_j u_t.
Since u_s conjugates M_n(O) via conj(c_s) and R via \alpha_s (they do not
mix), u_s c_t^{-1} u_s^{-1} = c_s c_t^{-1} c_s^{-1} (as an element of
M_n(O) \subseteq corner). So
w_s w_t = c_s^{-1} (u_s c_t^{-1} u_s^{-1}) u_s u_t
= c_s^{-1} (c_s c_t^{-1} c_s^{-1}) a_{s,t} u_{st}
= c_t^{-1} c_s^{-1} a_{s,t} u_{st}.
From c_s c_t c_{st}^{-1} = \gamma(s,t) I, we have c_t^{-1} c_s^{-1} =
\gamma(s,t)^{-1} c_{st}^{-1}, so
w_s w_t = \gamma(s,t)^{-1} a_{s,t} c_{st}^{-1} u_{st} = \gamma(s,t)^{-1} a_{s,t} w_{st}.
\gamma is a normalized T-valued 2-cocycle on S (associativity; trivial
action). Lemma 4.1 gives \gamma = \partial(\varepsilon) for some 1-cochain \varepsilon : S -> T;
rescaling w_s by \varepsilon_s gives units w'_s with
w'_s w'_t = a_{s,t} w'_{st}.
These w'_s provide a based graded isomorphism between the Sylow
restriction of the corner crossed order C_j and the Sylow restriction
of the original A_S (up to based restriction to \tilde S, transported back
via Schur-Zassenhaus). Therefore the finiteness hypothesis (iii)
gives finite based Sylow S-restrictions for the family {C_j}.
Step 6 (bounded quotient gives finitely many). The quotient Q_j is
bounded, so only finitely many abstract groups Q_j and outer
homomorphisms Q_j -> Out_O(R) occur. Fix one such quotient group and
outer image, and fix representatives \alpha_q \in Aut_O(R) of its outer
classes. Changing units makes every crossed system with this outer
action use exactly these \alpha_q (inner changes only rescale). The
remaining choices form a torsor under
Z^2(Q_j, Z_R^\times) / B^2(Q_j, Z_R^\times) = H^2(Q_j, Z_R^\times).
(Indeed, if two factor systems both realize the same \alpha_q, their
quotient is a 2-cocycle; and a change of units that preserves each
\alpha_q is central. See Section 6.3.)
Two crossed systems C_j, C_j' in this torsor are based-isomorphic
iff their class in H^2(Q_j, Z_R^\times) is trivial after a based
identification; the quotient H^2(Q_j, Z_R^\times) has finitely many
restrictions to Sylow subgroups (finiteness of H^2(S, T) and
injectivity on H^2(Q_j, U_R), Lemma 6.2), so the restriction map
has finite fibres on H^2(Q_j, Z_R^\times). Therefore there are
finitely many based crossed orders C_j.
Step 7 (finiteness of block summands). Each C_j has only finitely
many block summands (finitely many primitive central idempotents in
a finite-dimensional semisimple-over-K_0 algebra; central idempotents
lift uniquely). Every block summand of the original A is Morita
equivalent to one of these block summands: the central summands of A
correspond to Q-orbits on {e_j}, and each orbit corner is
M_n(O) \otimes C_j (matrix algebra over C_j), so its blocks are matrix
algebras over blocks of C_j.
Therefore the block summands of all orders in A fall into finitely
many O-Morita classes.
The identification of crossed system classes with H^2(Q_j, Z_R^\times)
needs one line of justification. Fix the \alpha_q \in Aut_O(R). If
a = (a_{q,t}) and a' = (a'_{q,t}) are both normalized 2-cocycles
satisfying the compatibility (factor-law), then a' / a is a 2-cocycle
in Z^2(Q_j, Z_R^\times): its defining crossed-product relations imply
the cocycle condition on Z_R^\times, and the action on the centre is
trivial (automorphisms of R act trivially on its canonical centre).
Changing homogeneous units that fix every \alpha_q is central (an element
of R^\times that is fixed by every \alpha_q, hence in Z_R^\times), so it changes
a by an element of B^2(Q_j, Z_R^\times).
Fix p and M. We show there is a finite list of O-orders representing
all O G b (across all finite G and blocks b with defect order <= M).
Step A (reduction to a reduced pair). For each such (G, b), the
An-Eaton Proposition 2.2 gives a reduced pair (H, B_H) with
B_H \sim b as O-algebras (Morita equivalent) and with defect group
of B_H conjugate to that of b. So B_H has defect order <= M.
It suffices to bound the Morita classes of the B_H.
Step B (crossed order). Apply Section 2.4 (from the companion).
We obtain a crossed order on an identity block B_0, with grading
group Q and outer action through Out_O(B_0), such that B_H is a
block summand of a full corner (Section 2.4). The bounds are:
- basic orders of B_0 lie in the finite list R_1, ..., R_t
(Section 2.5);
- Out_O(B_0) is finite (Section 2.5);
- [Q : O_{p'}(Q)] is bounded (Section 2.4).
Step C (based basic identity order). Compress by a basic idempotent
e of B_0 (Section 5.3, proof of Theorem 5.3): the resulting
homogeneous units induce a crossed order C on R = e B_0 e = one of
the R_i, by the same Q, with the same outer action on R (the
finitely many outer automorphisms of R, still in the finite list).
Step D (apply kernel removal). Theorem 5.3 provides the finiteness of
hypothesis (iii) of Theorem 6.3 (finite based Sylow restrictions).
The bounds (i), (ii) of Theorem 6.3 also hold by Steps B, C.
Fix one i in {1, ..., t} and one finite subgroup F_i <= Out_O(R_i)
containing all outer images of all Q acting on R_i. Then Theorem 6.3
applied to R_i and F_i gives finitely many Morita classes of
block summands.
Step E (finite union). The crossed order C of Step C is in the family
of Theorem 6.3. Its block summands include B_H (up to full corner
and basic corner, both Morita equivalences). Since the family has
finitely many Morita classes of block summands, and only finitely
many i occur, the union over i is finite. This list is independent
of G and b (it depends only on p, M).
Step F (back to O G b). The original O G b is Morita equivalent to
B_H (Step A), which is Morita equivalent to a full corner of B_0
(Section 2.4), which is Morita equivalent to the basic corner C
(Step C). So O G b lies in the finite Morita list of Step E.
This proves Theorem 1.1.
Corollary 7.1. For every p, M, there is a finite list of integral
basic orders R'_1, ..., R'_v such that every basic algebra of a block
of k G with defect order <= M is isomorphic to k \otimes_O R'_j for
some j. In particular, blocks over k of bounded defect have finitely
many k-linear Morita classes.
Proof. Let B be a block of k G with defect order <= M. By
Hensel-lifting, B lifts uniquely to a block O G b (Section 1.2), and
b has the same defect group. Lift a basic idempotent \bar e of B to
an idempotent e of O G b (unique lift in a finite-dimensional algebra
by nilpotence of the radical mod p). The corner e (O G b) e is an
integral basic order, and e (O G b) e \otimes_O k = \bar e B \bar e
(the original basic algebra).
By Theorem 1.1, e (O G b) e falls into the finite Morita list of
integral basic orders, say Morita equivalent to R'_j. Then \bar e B \bar e
is Morita equivalent to k \otimes_O R'_j. So k \otimes_O R'_j
represents the field block. Finiteness of k-linear Morita classes
follows.
The residue-field finiteness conclusion was also derived directly by
the field companion (used in Proposition 2.3 for the Cartan bound);
Corollary 7.1 identifies a finite list of integral basic orders whose
reductions represent the same field algebras. The stronger statement
needed the explicit integral constructions throughout; it has not
required lifting an arbitrary field Morita equivalence.
Fix p, M, and a complete DVR \mathcal R of characteristic 0 whose
residue field \ell is algebraically closed of characteristic p. Set
k = \overline{\mathbb F}_p and O = W(k).
Step 1 (fixed coefficient embedding). Choose an embedding
\iota : k \hookrightarrow \ell. Since \ell is perfect, W(\ell) is a
Cohen ring at \ell (a complete DVR with residue field \ell). The
Cohen structure theorem (Stacks, Theorem 10.160.8, Tag 032A) gives a
local map W(\ell) \to \mathcal R that induces the identity on \ell.
Injectivity: the DVR W(\ell) has maximal ideal (p), and \mathcal R
has characteristic 0, so any nonzero ideal of W(\ell) (containing
some power of p) cannot be killed by the map. Composing with the
Witt-vector map W(\iota) : W(k) \to W(\ell) (injective on Witt
coordinates, since \iota is injective and W is exact for perfect
residue fields) gives a fixed local embedding
O = W(k) \xrightarrow{W(\iota)} W(\ell) \hookrightarrow \mathcal R,
whose residue map is \iota.
No finiteness of \mathcal R over O is used.
Step 2 (idempotent and defect correspondence). For every finite group
G, scalar extension gives Z(\ell G) = \ell \otimes_k Z(k G). Each
factor of the finite commutative algebra Z(k G) is local with residue
field k (Section 6.1, Lemma 6.1 (a)). Extending to \ell, the extended
factor is local with residue field \ell: the radical ideal survives
nilpotent, and the residue field goes from k to \ell. So primitive
central idempotents \bar b of \ell G are in bijection with \bar b_0 of
k G, via \bar b = 1 \otimes \bar b_0.
For a p-subgroup Q \le G, restriction of coefficients C_G(Q) gives
Br^{\ell}_Q(\bar b) = 1 \otimes Br^k_Q(\bar b_0).
Faithfulness of field extension (both sides embed into a fixed
algebraic closure of Frac(R)) shows the Brauer images vanish
simultaneously. So the defect groups of \bar b and \bar b_0 coincide.
Similarly, for the coefficient pairs (\Lambda, F) = (\mathcal O, k)
or (\mathcal R, \ell), the class sums give a \Lambda-basis of Z(\Lambda G)
whose reductions form an F-basis of Z(F G). The algebra Z(\Lambda G) is
a finite free complete commutative \Lambda-algebra with residue
algebra Z(F G). Idempotents lift uniquely by henselianity (the
polynomial x^2 - x has nonvanishing derivative 2x - 1 at an idempotent
mod p, including p = 2 where 2x - 1 \equiv 1). Primitivity is
preserved (central decompositions lift, and an idempotent reducing
to 0 is 0).
Step 3 (idempotent lifting over the embedding). Let b be a block
idempotent of \mathcal R G, and \bar b its reduction. Let \bar b_0
in k G be as in Step 2, and let b_0 in O G be the unique lift of
\bar b_0. Push b_0 forward to \mathcal R G along the fixed O -> \mathcal R;
its image reduces to \bar b in \ell G; uniqueness of lift makes it
equal to b. Hence
\mathcal R G b \cong \mathcal R \otimes_O (O G b_0)
as \mathcal R-algebras, and b_0 has the same defect group as b.
Step 4 (base change of Morita contexts). Let A, B be finite O-algebras
with an O-linear Morita equivalence; write it as inverse Morita
bimodules _B P_A and _A Q_B with
P \otimes_A Q \cong B, Q \otimes_B P \cong A,
and compatible Morita-context maps. After base change along O ->
\mathcal R, define A_R = \mathcal R \otimes_O A, B_R, P_R, Q_R.
Base change of the tensor decompositions gives
P_R \otimes_{A_R} Q_R \cong \mathcal R \otimes_O (P \otimes_A Q) \cong B_R,
Q_R \otimes_{B_R} P_R \cong \mathcal R \otimes_O (Q \otimes_B P) \cong A_R.
Base change preserves finite projectivity and the generator property.
So A_R and B_R are \mathcal R-linearly Morita equivalent.
Step 5 (finish). By Theorem 1.1, blocks O G b_0 with defect order <= M
fall into finitely many O-linear Morita classes, represented by a
finite list \mathcal B_1, \ldots, \mathcal B_s of O-orders. The base
changed list \mathcal R \otimes_O \mathcal B_1, \ldots,
\mathcal R \otimes_O \mathcal B_s is therefore a finite set of
\mathcal R-orders; by Step 4, every block \mathcal R G b with defect
order <= M is \mathcal R-linearly Morita equivalent to one of them.
This proves Corollary 1.2.
The proof shows three things independently:
(i) The scalar obstruction produced by the companion's integral
comparison operators is Teichmuller-valued (Proposition 4.2); only
such scalars need to be killed by H^2(S, T) = 0 for a p-group S.
(ii) The rigidity necessary to go from an ungraded Morita list to
labelled-and-based gradings uses HH^1 vanishing plus a non-averaging
scheme-orbit lemma; it works for arbitrary finite grading groups,
including p-groups (Sections 3.3--3.5).
(iii) The unbounded p'-kernel of the grading group is removed by
Clifford matrix corners whose determinant-one implementation forces
the new scalar error into T; again T's cohomology finishes
(Section 6).
Only three numerical inputs are used, all from the companion field
paper and from EEL:
- bounded Cartan sum in bounded defect (Proposition 2.3 (i)),
- EEL's integral finiteness: bounded defect + bounded Cartan sum
+ bounded mf_O imply finitely many Morita classes (Proposition 2.3 (ii)),
- the companion field theorem over k.
The integral structure appears entirely through T-valued scalar
errors and through the fixed-total-order rigidity; the point-counting
and Frobenius-fixed-point arguments used over k are not used over O.