On the geometry of lattices and finiteness of Picard groups

Let $(K,\mathcal O, k)$ be a $p$-modular system with $k$ algebraically closed and $\mathcal O$ unramified, and let $\Lambda$ be an $\mathcal O$-order in a separable $K$-algebra. We call a $\Lambda$-lattice $L$ rigid if ${\rm Ext}^1_{\Lambda}(L,L)=0$, in analogy with the definition of rigid modules over a finite-dimensional algebra. By partitioning the $\Lambda$-lattices of a given dimension into"varieties of lattices", we show that there are only finitely many rigid $\Lambda$-lattices $L$ of any given dimension. As a consequence we show that if the first Hochschild cohomology of $\Lambda$ vanishes, then the Picard group and the outer automorphism group of $\Lambda$ are finite. In particular the Picard groups of blocks of finite groups defined over $\mathcal O$ are always finite.


I
Let k =k be an algebraically closed field of characteristic p > 0. Let O = W(k) be the ring of Witt vectors over k, and denote by K the field of fractions of O. In the representation theory of a finite group G over either of these rings, permutation modules, p-permutation modules and endo-permutation modules play a pivotal role. However, to even define permutation modules, one needs to know a group basis of the group ring, a piece of information which is lost when passing to isomorphic or Morita equivalent algebras. Recovering the information lost when forgetting the group basis, or at least quantifying the loss, is a fundamental problem in modular representation theory. This is, for instance, the problem once faces when trying to bridge the gap between Donovan's and Puig's respective conjectures. There are scant results in this direction, apart from Weiss' seminal theorem [Wei88], which gives a criterion for a lattice to be p-permutation requiring only limited knowledge of a group basis (and in some cases none at all). In the present article we study the following property of lattices over an O-order Λ such that A = K ⊗ O Λ is a separable K-algebra, which of course includes lattices over a finite group algebra OG: Definition 1.1. A Λ-lattice L is called rigid if Ext 1 Λ (L, L) = 0. First and foremost we should point out that permutation lattices over a finite group algebra OG are rigid in this sense. The notion of rigid modules over finite-dimensional k-algebras is widely known and well-studied (see for example [DF74,Dad80]), but unfortunately permutation kG-modules and their ilk usually do not have that property. To see why permutation lattices do, we can use a well-known alternative characterisation of rigidity in terms of endomorphism rings, which works for arbitrary Λ-lattices L. Consider the following long exact sequence obtained by applying Hom Λ (L, −) to the short exact sequence 0 → L → L → L/pL → 0:  After this short digression on permutation lattices let us state our first main result, which is that rigid lattices enjoy the same discreteness property as rigid modules over finite-dimensional algebras.
Theorem A. For every n ∈ N there are at most finitely many isomorphism classes of rigid Λ-lattices of O-rank at most n.
An easy corollary of this is that only finitely many isomorphism classes of OG-lattices of any given character can be images of permutation lattices under Morita or stable equivalences originating from another group or block algebra. Perhaps unsurprisingly, this result is proved using geometric methods. The basic idea of using a variety of modules (or a variety of complexes, as the case may be) has been successfully applied to modules and complexes over finite-dimensional algebras in many different contexts [Dad80,Dad82,DF74,HZS01,ANR13,Rou11]. The idea is typically that one gets a homomorphism from the tangent space of such a variety in a point M into Ext 1 (M, M), whose kernel is the tangent space of the subvariety of points N isomorphic to M.
The way we obtain a variety parametrising all lattices with a given K-span is quite different from how one proceeds for finite-dimensional k-algebras. Our analogue of a variety of modules is a smooth family of Λ-lattices (see Definition 3.1). Despite working over O, these smooth families of lattices are actually parametrised by varieties over k. One obstacle to overcome for this to work is that the reduction x ∈ k n of a point x ∈ O n does not determine the p-valuation of f ( x), where f ∈ K(X 1 , . . ., X n ) is a rational function. This is where the theory of Witt vectors enters, proving that x does determine the valuation of f ( x) after all, provided x is sufficiently generic with respect to f . And for polynomials f the condition of f ( x) generically having at least a certain valuation is in fact a Zariski-closed condition on x. With that out of the way, we can prove the two main ingredients of Theorem A: the fact that each smooth family of lattices contains at most one rigid lattice, up to isomorphism (see Theorem 3.3), and the fact that maximal sublattices of the elements of such a family are again partitioned into smooth families (see Theorem 4.2). We then simply start with a fixed lattice, compute its maximal sublattices, which will fall into finitely many of the aforementioned smooth families, then take maximal sublattices of these families of maximal sublattices, and so on and so forth.
Our second main result is an immediate consequence of Theorem A, but we state it as a theorem nevertheless, since it was the main motivation for writing this paper. The study of Picard groups of blocks of finite group algebras over O was initiated in [BKL19]. While [BKL19] primarily studies the group of Morita self-equivalences induced by bimodules of endopermutation source, this is where the possibility of Picard groups of blocks always being finite was first raised. The theorem we obtain is much more general, and provides further evidence for the speculation put forward in [Eis18] that the Lie algebra of Pic O (Λ) (which is shown to be an algebraic group in that paper) should be related to the first Hochschild cohomology of Λ in some way or form.
Here Pic O (Λ) denotes the group of O-linear self-equivalences of the module category of Λ. This group is commensurable with the outer automorphism group Out O (Λ), as well as Outcent(Λ) and Picent(Λ), and we could just as well have stated Theorem B with Pic O (Λ) replaced by any of these groups. The theorem of course immediately implies the following: Corollary 1.2. Let G be a finite group, and let OGb be a block. Then Pic O (OGb) is finite.
There is also a more elementary formulation of this fact which is reminiscent of the second Zassenhaus conjecture. Corollary 1.2 shows that block algebras over O are in some sense extremely rigid, and it turns their Picard groups into a very interesting finite invariant. This is particularly important since Külshammer [Kül95] showed that Picard groups play an important role in both Donovan's conjecture and the classification of blocks of a given defect group. This theme is explored in [Eat16,EEL19]. In the way of actual computations, [HN04] determines the Picard group of the principal block of A 6 for p = 3, [EL18] determines Picard groups of almost all blocks of abelian 2-defect of rank three, and this will certainly not mark the end of the story. Unfortunately, we have very little to offer to aid the calculation of Picard groups. Still, the following might be useful. Note that T (OGb) denotes the subgroup of Pic O (OGb) of equivalences induced by p-permutation bimodules, which is determined in [BKL19].
Proposition 1.4. Let G be a finite group, let OGb be a block and let P G be one of its defect groups.
( The same is true if we set b = 1 and let P be a Sylow p-subgroup of G. It is also sufficient to prove that each indecomposable summand L of O[P\G] · b is the unique rigid lattice in K ⊗ O L with endomorphism ring End OGb (L). It is however unclear whether one can show such uniqueness in any interesting examples, even though it is an algorithmically decidable question by the results of §4. Nevertheless, Proposition 1.4 highlights the importance of understanding and classifying rigid lattices with a given character.
One last thing to note is that while we focus on applications to block algebras in this article, there are other types of O-orders with vanishing first Hochschild cohomology to which Theorem B applies. Iwahori-Hecke algebras defined over O, for instance, should have this property by [GR97, Theorem 5.2].
Notation and conventions. By ν p : K −→ Z ∪ {∞} we denote the p-adic valuation on K. Modules are right modules by default. All varieties are reduced, and by a "point" we mean a closed point. Λ will always denote an O-order in a separable K-algebra.

P : G
In order to perform generic computations over O, we need to assign p-valuations to polynomials f ∈ O[X 1 , . . . , X n ]. We wish to establish that, for any x ∈ k n , there is a "generic" p-valuation ν p,x ( f ) such that ν p ( f ( x)) = ν p,x ( f ) for all "sufficiently generic" x ∈ O n reducing to x. For example, for n = 2, the polynomial X 1 − X 2 can take any valuation 1 on elements of O 2 reducing to (0, 0) ∈ k 2 , but generically it takes p-valuation one (the non-generic case here would be the values for X 1 and X 2 being congruent modulo p 2 ). In what follows we will define ν p,x and establish its most important properties. We will assume that the reader is familiar with the theory of Witt vectors as laid out in [Ser79, §5- §6].
Definition 2.1. Let f ∈ O[X 1 , . . . , X n ] be a polynomial. We define ν p ( f ) as the minimal We call ν p,x ( f ) the generic valuation of f at x. We may extend this generic valuation to K(X 1 . . . , X n ) in the obvious way. One should note though that if f is a rational function rather than a polynomial, then ν p ( f ( x)) could be either bigger or smaller than ν p,x ( f ), depending on the choice of x ∈ O n reducing to x. Proposition 2.2. Assume the situation of Definition 2.1.
(1) Note that the right hand side cannot be bigger than the left hand side. On the other reduces to a non-zero polynomial in k[Z 1 , . . . , Z n ], and we can certainly find values for the Z i for which the polynomial does not vanish. This shows that the right hand side cannot be smaller than the left hand side either.
(2) By the first part we know that p −ν p,x ( f ) f ( x 1 + pZ 1 , . . . , x n + pZ n ) reduces to a non-zero polynomial g ∈ k[Z 1 , . . ., Z n ], and ν p ( f ( x + p · z)) = ν p,x ( f ) if and only if g(z) 0, where z is the reduction of z modulo p. Hence, what we are looking for is a z ∈F n p which avoids the zero locus of a polynomial defined over k. Since A n (F p ) is Zariski-dense in A n (k) such a z exists trivially. Corollary 2.3. Let E be the ring of Witt vectors over an algebraically closed field k ′ containing k, and let f ∈ O[X 1 , . . . , X n ] be a polynomial. For x ∈ k n the value of ν p,x ( f ) is independent of whether we consider f as a polynomial over O or over E.
Proof. This follows from the first part of Proposition 2.2.
While the theory of Witt vectors is obviously a very important theoretical ingredient of the present article, Proposition 2.4 below is actually the only place where the polynomials defining addition and multiplication on Witt vectors enter directly. And even there we only make use of their existence, rather than of their explicit form.
Fix an algebraically closed k ′ ⊇ k, and set E = W(k ′ ). Let x ∈ A n (k ′ ), and let x ∈ E n be the element reducing to it given by Teichmüller representatives (i.e. the only non-zero component of the Witt vector x i is x i ). As E is the ring of Witt vectors over k ′ and f is defined over k, we get polynomials f i ∈ k[X 1 , . . . , X n , Z 1, j , . . . , Z n, j | 0 j i] (where i ∈ Z 0 ) such that f ( x + p z) (for arbitrary z ∈ O n ) is given by the Witt vector whose i-th component is the evaluation of f i at x and the first i +1 components of the Witt vectors z 1 , . . ., z n . The polynomials f i do not depend on k ′ . Now ν p,x ( f ) r if and only if, for all 0 i < r, x is a zero of all coefficients of f i as a polynomial in k[X 1 , . . ., X n ][Z 1, j , . . ., Z n, j | 0 j i]. Hence we can define V ν p,− ( f ) r as the zero locus of these coefficients, which are elements of k[X 1 , . . ., X n ].

S
In this section we introduce the structure we use to endow isomorphism classes of lattices with the structure of a variety over k. We call this structure a smooth family of Λ-lattices. Its definition is actually quite straight-forward, but at this stage it is not yet obvious that we can in fact parametrise all lattices of a given dimension by finitely many such families.
such that all of the following hold: ( is well-defined, and the isomorphism type of the corresponding E ⊗ O Λ-lattice only depends on x. A few remarks are in order. First off, the polynomial f ensures that the rational functions occurring as entries of images of ∆ become elements of O (or E) upon specialisation at an admissible x. However, Definition 3.1 does not state anywhere that this is necessarily the only function of f , and it is not clear whether the smooth families of Λ-lattices we construct later would remain such families if we were to just replace f by the product of all denominators occurring in the image of ∆. A second thing to note is that the reason we are considering extensions E of O is that we need those to specialise at "generic points" in the proof of Theorem 3.3 below. In applications we actually only require the property that specialisation at points in Z(k) ∩ U(k) is well-defined. The last thing we should note is that the requirement that Z ∩ U be smooth and irreducible only serves to get uniqueness in Corollary 3.4 and make the proof of Theorem 3.3 slightly nicer. As such, its inclusion in the definition is really a matter of preference. Proof. Assume that we have an x ∈ Z(k) ∩ U(k) such that L(x) is rigid. Let m be the ideal of elements of k[Z] vanishing at x. As Z is smooth at x, the completion of the local ring k , which factors through k[Z], and whose composition with the evaluation map k[[T 1 , . . . , T r ]] ։ k is the same as evaluation at x (the n spare variables will come in handy later on). Define w = (ϕ(X 1 ), . . ., ϕ(X n )) ∈ k[[T 1 , . . ., T r ]] n . Then w lies in Z(k ((T 1 , . . . , T r ))), and w(0, . . . , 0) = x . Here w(0, . . ., 0) denotes the evaluation of w at (T 1 , . . ., T r ) = (0, . . ., 0). Now note that the localisation O[[T 1 , . . . , T r ]] pO[[T 1 ,...,T r ]] is a discrete valuation ring with residue field k((T 1 , . . . , T r )), and therefore is a discrete valuation ring with residue field We can find a w ∈ O[[T 1 , . . ., T r ]] n such that w 1 , . . ., w n are algebraically independent over K and w reduces modulo p to w (which implies that w(0, . . ., 0) reduces to x). To do this simply choose an element w ′ ∈ O[[T 1 , . . . , T dim Z ]] n reducing to w and then define w by w i = w ′ i + p · T dim Z+i (for each 1 i n) to ensure algebraic independence. By working over E instead of O and using Proposition 2.2 (2) we can modify w by an element of p · W(F p ) n to ensure that, in addition to the above properties, ν p,x ( f ) = ν p ( f ( w(0, . . . , 0))). Note that our construction of w actually ensures that ν p,w (g) = ν p (g( w)) for all g ∈ O[X 1 , . . ., X n ] (where we view w as an element of E n ), since w is polynomial in the spare variables T dim Z+i , and therefore ν p,w (g) = ν p (g( w ′ 1 + pT dim Z+1 , . . . , w ′ n + pT dim Z+n )) = ν p (g( w)) (using Proposition 2.2 and the fact that ν p is independent of whether we regard the T dim Z+i as indeterminates or as elements of the valuation ring E).
We now know that E ⊗ O L(x) is isomorphic to L(w), as these are the modules afforded by the two representations we just showed are conjugate. In elementary terms, this means that the linear system of equations has p-valuation zero (i.e. some coefficient has p-valuation zero). But then one can easily find z ∈ O d (or even W(F p ) d ) such that the p-valuation of det(M 1 · z 1 + . . . + M d · z d ) is zero. Then M = M 1 · z 1 + . . . + M d · z d is an element of GL m (E) with entries that have preimages in K(X 1 , . . . , X n ) such that M −1 · ∆ w (λ) · M = ∆ w(0,...,0) (λ) for all λ ∈ Λ. Now recall that ∆ w (λ) is obtained from ∆(λ) by entry-wise application of ϕ (again, extended to fields of fractions). Let M ′ be a preimage under ϕ of M, that is, M ′ has entries in K(X 1 , . . ., X n ). Then M ′−1 · ∆(λ) · M ′ must be a preimage under ϕ of ∆ w(0,...,0) (λ) (for any λ ∈ Λ). But since ϕ is injective by construction, we get which is now an equation entirely in K(X 1 , . . ., X n ) = frac(O[X 1 , . . ., X n ]). Now let y be another point in U(k) ∩ Z(k), and let y ∈ O n be an element reducing to y such that ∆ y is defined. Since ∆ y is obtained from ∆ by substituting X i = y i , equation (14) implies that L(y) L(x) provided M ′ | (X 1 ,...,X n )= y ∈ GL m (O). Note that all g ∈ O[X 1 , . . ., X n ] which occur as numerators or denominators of either M ′ or M ′−1 satisfy ν p,w (g) = ν p (g( w)), and substituting (X 1 , . . . , X n ) = w in M ′ and M ′−1 gives back M and M −1 by definition. By Proposition 2.4 there are closed subvarieties V ν p,− (g) ν p,w (g) and V ν p,− (g) ν p,w (g)+1 of A n defined over k such that w lies in V ν p,− (g) ν p,w (g) (k ′ ) but not in V ν p,− (g) ν p,w (g)+1 (k ′ ). Since w was chosen as a generic point for Z, any subvariety of A n defined over k contains w if and only if it contains Z. It follows that (V ν p,− (g) ν p,w (g) \ V ν p,− (g) ν p,w (g)+1 ) ∩ Z is an open subvariety of Z, whose k-rational points are by definition those y for which ν p,y (g) = ν p,w (g).
We conclude that there is an open subvariety V of U such that for any y ∈ V(k) there is a y for which M ′ | (X 1 ,...,X n )= y and M ′−1 | (X 1 ,...,X n )= y lie in O m×m (in fact, the valuation of each entry is the same as that of the corresponding one of M). Hence L(y) L(x) for all y ∈ V(k), which completes the proof. Proof. We assume Z to be irreducible, which means that any two non-empty Zariski-open subsets have non-trivial intersection. In particular, if the family contains two rigid lattices, then their respective sets of points parametrising lattices isomorphic to them have non-trivial intersection. This implies that any two rigid lattices in the family must be isomorphic.

A
In this section we show how to parametrise all Λ-lattices in a given finite-dimensional K ⊗ O Λmodule V, up to isomorphism. The idea is taken from the classical algorithm that yields all such lattices when k is a finite field. Namely, one can find all maximal sublattices of a given Λlattice L by considering kernels of all non-zero homomorphisms from L into a simple module. For finite k, there are only finitely many such homomorphisms, and therefore only finitely many maximal sublattices. One can then proceed to compute the maximal sublattices of each of these finitely many maximal sublattices of L, and so on. That is, we can compute the sublattices of L layer by layer. In practice one would check isomorphism to see when to stop, but for our theoretical application it suffices to stop after a certain number of layers. The point of this section is that we can perform this algorithm as is for smooth families of Λ-lattices defined over our algebraically closed field k. We should point out that what we describe in this section is in fact an algorithm, although an effective implementation would still require an isomorphism test.
Lemma 4.1. Let M ∈ K(X 1 , . . . , X n ) m×r (for n, m, r ∈ N) be a matrix, let Z ⊆ A n be a closed subvariety and let U ⊆ A n be an open subvariety such that for any x ∈ Z(k) ∩ U(k) we have ν p,x (M i j ) 0 for all i, j. Then there exist (1) Z 1 . . ., Z d ⊆ A n closed and U 1 . . ., U d ⊆ A n open (for some d ∈ N), (2) matrices N 1 , . . . , N d with entries in K(X 1 , . . ., X n ), Proof. To compute the kernel in (16) we would first need to perform Gaussian elimination on the matrix (M| (X 1 ,...,X n )= x | id m×m ) considered as an element of k m×(r+m) . The idea now is to perform a series of manipulations on (M | id m×m ) ∈ K(X 1 , . . ., X n ) m×(r+m) in such a way that specialising at x and reducing modulo p turns these manipulations into the steps of Gaussian elimination. When performing Gaussian elimination, we need to do the following for the i-th column of the matrix, where i ranges from 1 to m + r: (1) Find the smallest 1 j m such that the ( j, i ′ )-entry of the matrix is zero for all i ′ < i and the ( j, i)-entry is non-zero. If no such j exists, we skip the next three steps and proceed immediately to the next column. Also find the smallest 1 j ′ j such that the ( j ′ , i ′ )-entry of the matrix is zero for all i ′ < i.
(2) Divide the j-th row by the ( j, i)-entry, thus creating a "1" in the ( j, i)-position.
(3) For every 1 j ′′ j m subtract the j-th row multiplied by the ( j ′′ , i)-entry from the j ′′ -th row, thus creating a "0" in the ( j ′′ , i)-position. (4) Swap the j-th and the j ′ -th row. The second, third and fourth step can be performed over K(X 1 , . . . , X n ) just as they can be over k, and it does not matter whether we specialise at x and reduce modulo p before or after we perform the step. The only issue is therefore the first step: checking whether an element of K(X 1 , . . . , X n ) becomes zero upon specialisation.
To redress this, let us modify Gaussian elimination as follows: we start by defining f as the product of all non-zero numerators and denominators occurring in entries of M. This ensures that M (X 1 ,...,X n )= x actually has entries in O whenever ν p ( f ( x)) = ν p,x ( f ). We then start the process of elimination. Whenever we need to check whether an entry g h (for non-zero g, h ∈ O[X 1 , . . . , X n ]) will become zero upon specialisation and reduction modulo p, we cut Z ∩ U up into for i, j ∈ Z 0 , where the V's are as in Proposition 2.4. While these are technically infinitely many varieties, only finitely many of them are non-empty, since the V ν p,− (g) i 's and V ν p,− (h) j 's form descending chains of closed subvarieties of A n which, for non-zero g and h, can only become stationary at ∅. We then perform the rest of the algorithm for each of the finitely many non-trivial X i, j separately. In the branch belonging to X i, j : (1) X i, j effectively takes the role of Z ∩ U for the remainder of the algorithm.
(2) We replace our current f by f · g · h. Note that the definition of X i, j ensures that ν p,x (g) = i and ν p,x (h) = j for all x ∈ X i, j (k), and the replacement of f ensures that ν p (g( x)) = ν p,x (g) and ν p (h( x)) = ν p,x (h) for any admissible x. Therefore the valuation of g/h specialised at x is i − j.
(3) If i − j > 0 we replace the entry g/h in the matrix by 0. If i − j = 0 we henceforth treat g/h as invertible. Obviously our algorithm will branch at at most finitely many junctures, giving us finitely many matrices N ′ 1 , . . ., N ′ d ∈ K(X 1 , . . ., X n ) m×(r+m) in row echelon form, together with varieties Z i , U i and polynomials f i as in the statement of the lemma.
Let us fix 1 i d. Take the last m entries of those rows of N ′ i for which the first r entries are zero. By definition of N ′ i , this is a matrix in row echelon form such that if we specialise at an admissible x and reduce modulo p, we get a basis of the kernel of M| (X 1 ,...,X n )= x considered as an element in k m×r . If we want N i that reduce to a basis of (16) then we can stop here. If we want the N i to give an O-basis of (15) we simply append a row p · e i (where e i is the i-th standard basis vector) for each i indexing a column which does not contain a pivot.
By construction, for every x ∈ Z i (k) ∩ U i (k), every non-zero element of Hom Λ (L(x), S) is obtained as the specialisation of N i at ( x, y) ∈ O n+d(i) such that x reduces to x, y reduces to a certain Conversely, every such x and y yield a non-zero element of Hom Λ (L(x), S). We now plug the N i together with Z i ∩ U i into Lemma 4.1 to get varieties Z i, j ∩ U i, j , matrices K i, j ∈ K(X 1 , . . . , X n , Y 1 , . . . , Y d(i) ) rank L×rank L and polynomials f i, j ∈ O[X 1 , . . ., X n , Y 1 , . . . , Y d(i) ] such that the K i, j specialise to the kernels of the non-zero elements of Hom Λ (L(x), S). That is, the rows of the K i, j specialise to bases of the maximal sublattices of L(x) with quotient S. Now we simply define (20) ∆ i, j : Λ −→ K(X 1 , . . ., X n Y 1 , . . ., Y d(i) ) rank L×rank L : λ → K i, j · ∆(λ) · K −1 i, j . Then the various (∆ i, j , Z i, j ∩ U i, j , f · f i · f i, j ) are almost the smooth families of Λ-lattices we are looking for. The only remaining issue is that the Z i, j are neither irreducible nor smooth in general. This can however easily be rectified by simply decomposing each Z i, j into irreducible components, removing the singular loci, then splitting the singular loci into irreducible components, and so forth. This divides the (∆ i, j , Z i, j ∩ U i, j , f · f i · f i, j ) into finitely many smooth families of Λ-lattices M 1 (−), . . ., M d (−) with properties as required. Proof. Fix some full Λ-lattice L 0 V. It is well-known that every full Λ-lattice in V is isomorphic to a lattice L such that L 0 · (Γ : Λ) 2 L L 0 , where (Γ : Λ) denotes the biggest two-sided Γ-ideal contained in Λ, Γ being a maximal order containing Λ. Therefore there is an upper bound l ∈ N on the composition length of L/L 0 which depends only on V and Λ. Now we can just apply Theorem 4.2 to L 0 (viewed as a constant smooth family of Λ-lattices) and the different simple modules of Λ, such as to obtain finitely many smooth families of Λ-lattices containing all maximal sublattices of L. Then we apply Theorem 4.2 to these families to obtain finitely many smooth families of Λ-lattices containing all sublattices of L 0 with quotient of length two. We can repeat this process until we have parametrised all sublattices of L 0 with quotient of length at most l, which, up to isomorphism, are all full Λ-lattices in V.
There is at least one situation in which we get a feasible algorithm to determine the smooth families in Theorem 4.3: Remark 4.4. In case V is simple and Z(Λ) maps surjectively onto a maximal order E ⊂ End K ⊗ O Λ (V ), the sublattices of L 0 not contained in L 0 ·rad(E) are mutually non-isomorphic and contain a representative for each isomorphism class of Λ-lattices in V. If we keep track of the bases of sublattices (which we can do as we get them out of Lemma 4.1) then we can throw away lattices contained in L 0 · rad(E), as containment can be verified by multiplying the basis by π −1 and checking whether the p-valuations of all entries are non-negative. Here π denotes a generator of rad(E). Proposition 2.4 tells us that we can then simply slice up the Z ∩ U's such as to remove the lattices contained in L 0 · rad(E).

P
It should be fairly clear by now how Corollary 3.4 and Theorem 4.3 imply Theorem A, and Theorem B is an immediate consequence of that. We still include proofs for completeness' sake.
Proof of Theorem A. As K ⊗ O Λ is assumed to be separable, there are only finitely many isomorphism classes of K ⊗ O Λ-modules V of dimension n. By Theorem 4.3 there are, for each such V , finitely many smooth families of Λ-lattices such that each full Λ-lattice in V is contained in one of these families. Hence every Λ-lattice of rank n is contained in one of finitely many smooth families of Λ-lattices, and by Corollary 3.4 each such family can contain at most one isomorphism class of rigid lattices.
Proof of Theorem B. We can assume without loss of generality that Λ is basic. Then every element of Pic O (Λ) is represented by a Λ-Λ-bimodule Λ α , where α ∈ Aut O (Λ). Define Λ e = Λ op ⊗ O Λ. Then Λ e is again an O-order in a separable K-algebra, and we can view elements of Pic O (Λ) as Λ e -lattices. Note that if α ∈ Aut O (Λ), then id ⊗α ∈ Aut O (Λ e ), and Λ α (that is, the Λ-Λ-bimodule Λ twisted by α on the right) is the same as Λ id ⊗α (that is, the right Λ e -module Λ twisted by id ⊗α). Hence Ext 1 Λ e (Λ α , Λ α ) Ext 1 Λ e (Λ, Λ) = HH 1 (Λ), and the latter is zero by assumption. That is, the elements of Pic O (Λ) are rigid Λ e -lattices of O-rank equal to the O-rank of Λ. By Theorem A there are only finitely many such Λ e -lattices, up to isomorphism. Proposition 1.4 is actually just the combination of Theorem B and Weiss' criterion.
Proof of Proposition 1.4. Consider (OGb) e as a block of O(G op × G). Let M be an (OGb) emodule representing an element of Pic(OGb). By Weiss' criterion (see [Wei88], and [MSZ18] for a version allowing O as a coefficient ring) our M has trivial source if and only if the lattice of P-fixed points taken on the left To finish, let us briefly mention the following nice consequence of Theorems A and B, even though it is implied by [Thé95, Theorem (38.6)], a theorem due to Puig.
Proposition 5.1. Let {OG i b i } i∈I (for some index set I) be a family of block algebras defined over O with fixed defect group P, each of which Morita equivalent to some fixed O-algebra A by means of some fixed OG i b i -A-bimodule M i . If there is a bound, independent of i, on the dimension of L ⊗ OG i b i M i for indecomposable p-permutation OG i b i -modules L, then the OG i b i split into finitely many equivalence classes with respect to splendid Morita equivalence.
Proof. Note that by assumption P is a subgroup of G i for every i ∈ I. Equivalently, one could also assume that there is a fixed embedding P ֒→ G i for each i, but we are going to take the former point of view. By Theorem A there are only finitely many rigid A-lattices of rank smaller than the given bound on images of indecomposable p-permutation modules. Hence I splits up into finitely many sets I 1 , . . ., I d such that I = d j=1 I j and, for every i ∈ I j , the A-module  [Sco90]) this implies that the M i ⊗ A M ∨ i ′ for i, i ′ in a fixed I j are splendid up to restriction along an automorphism of P, of which there are only a finite number. That is, each I j splits into finitely many subsets such that the blocks parametrised by any one of the subsets are pair-wise source algebra equivalent by means of the M i ⊗ A M ∨ i ′ . Acknowledgements. I would like to thank Radha Kessar and Markus Linckelmann for pointing the finiteness problem for Picard groups out to me, and for helpful comments on a first version of this preprint.