On abelian subcategories of triangulated categories

The stable module category of a selfinjective algebra is triangulated, but need not have any nontrivial t-structures, and in particular, full abelian subcategories need not arise as hearts of a t-structure. The purpose of this paper is to investigate full abelian subcategories of triangulated categories whose exact structures are related, and more precisely, to explore relations between invariants of finite-dimensional selfinjective algebras and full abelian subcategories of their stable module categories.

is exact in C.
The main motivation for considering this definition is the abundance of distinguished abelian categories in stable module categories of finite-dimensional selfinjective algebras, and the hope that these may therefore shed light on the invariants of selfinjective algebras in terms of their stable module categories.A conjecture of Auslander-Reiten predicts that for A a finite-dimensional algebra over a field, the stable category mod(A) of finitely generated left A-modules should determine the number of isomorphism classes of nonprojective simple A-modules.If this conjecture were true for blocks of finite group algebras, it would imply some cases of Alperin's weight conjecture.By a result of Martinez-Villa [23], it would suffice to prove the Auslander-Reiten conjecture for selfinjective algebras.If A is selfinjective, then mod(A) is triangulated.The following result recasts the Auslander-Reiten conjecture for selfinjective algebras in terms of maximal distinguished abelian subcategories of mod(A).For D an abelian category, we denote by ℓ(D) the number of isomorphism classes of simple objects, with the convention ℓ(D) = ∞ if D has infinitely many isomorphism classes of simple objects.For A a finite-dimensional algebra over a field, we write ℓ(A) = ℓ(mod(A)); that is, ℓ(A) is the number of isomorphism classes of simple A-modules.
Theorem 1.2.Let A be a finite-dimensional selfinjective algebra over a field such that all simple A-modules are nonprojective.The following hold.
(i) If D is a distinguished abelian subcategory of mod(A) containing all simple A-modules, then the simple A-modules are exactly the simple objects in D. In particular, in that case we have ℓ(A) = ℓ(D).(ii) The stable module category mod(A) has a maximal distinguished abelian subcategory D satisfying ℓ(D) = ℓ(A).
Statement (i) of this theorem is Theorem 3.8, and statement (ii) will be proved in Section 6.Note that mod(A) may have distinguished abelian subcategories D satisfying ℓ(D) = ∞; see Example 9.3.A subcategory D of a triangulated catergory C is called extension closed if for any exact triangle U → V → W → Σ(U ) in C with U , W belonging to D, the object V is isomorphic to an object in D. Hearts of t-structures on a triangulated category C are extension closed distinguished abelian subcategories.In general, distinguished abelian subcategories need not be extensions closed.The following result is a basic construction principle for distinguished abelian subcategories in mod(A), together with a sufficient criterion for detecting certain distinguished abelian subcategories which are not extension closed.
Theorem 1.3.Let A be a finite-dimensional selfinjective algebra over a field and let I be a proper ideal in A. Denote by r(I) the right annihilator of I in A.
(i) The canonical map A → A/I induces an embedding mod(A/I) → mod(A) as a distinguished abelian subcategory in mod(A) if and only if r(I) ⊆ I. (ii) Suppose that r(I) ⊆ I ⊆ J(A) and that A/I is selfinjective.Then the distinguished abelian subcategory mod(A/I) of mod(A) is not extension closed in mod(A).
Statement (i) will be proved as part of Theorem 3.1, itself a consequence of the more general Theorem 2.1, and statement (ii) follows from Proposition 7.8.The first part of this theorem points to the fact that distinguished abelian subcategories of mod(A) tend to come in varieties -see Proposition 2.6.
If a triangulated category C carries a structure of a monoidal category and if D is a distinguished abelian subcategory of C which is also a monoidal subcategory of C, we call D a monoidal distinguished abelian subcategory of C. Stable module categories of finite group algebras provide examples of monoidal distinguished abelian subcategories which do not arise as heart of a t-structure, and which are not extension closed.For G a finite group and p a prime, we denote by O p (G) the smallest normal subgroup of G such that G/O p (G) is a p-group.
Theorem 1.4.Let k be a field of prime characteristic p and G a finite group.Let N be a normal subgroup of G of order divisible by p.
(i) Restriction along the canonical surjection G → G/N induces a full embedding of mod(kG/N ) as a symmetric monoidal distinguished abelian subcategory in mod(kG).(ii) There is no nontrivial t-structure on mod(kG); that is, the heart of any t-structure on mod(kG) is zero.In particular, mod(kG/N ) is not the heart of a t-structure on mod(kG).(iii) If O p (N ) is a proper subgroup of N , then the abelian subcategory mod(kG/N ) of mod(kG) is not extension closed in mod(kG).
See Theorem 4.2, Corollary 2.8, Corollary 2.9, and Proposition 7.11 for more precise statements and proofs.For k a field of prime characteristic p and P a finite p-group, the Auslander-Reiten conjecture is known to hold for kP (cf.[19,Theorem 3.4]) .We use this to classify the distinguished abelian subcategories of mod(kP ) which are equivalent to the module categories of split finitedimensional algebras in Theorem 4. 4.
Section 2 describes some construction principles of distinguished abelian subcategories of stable module categories.Section 3 describes distinguished abelian subcategories of mod(A) whose simple objects are the simple A-modules, where A is a finite-dimensional selfinjective algebra.Section 4 specialises previous results to distinguished abelian subcategories in finite group algebras over a field of prime characteristic p, and includes a proof of the first statement of Theorem 1.4.The section 5 contains some general facts on distinguished abelian subcategories.In particular, it is shown in Proposition 5.1 that the morphism h in Definition 1.1 is unique.Section 6 contains technicalities, needed for the proof of Theorem 1.2, on the interplay between short exact sequences in mod(A) and short exact sequences in a distinguished abelian subcategory D of mod(A).The main result of Section 7 is a criterion on extension closure of distinguished abelian subcategories, needed for the last part of Theorem 1.4.Section 8 relates embeddings of module categories of selfinjective algebras to a result of Cabanes.Section 9 contains examples and further remarks.
Remark 1.5.The present paper, investigating abelian subcategories of triangulated categories in situations where there are no nontrivial t-structures, started out as a speculation about a possible analogue of stability spaces (cf.[6]) for stable module categories of finite-dimensional selfinjective algebras.Another interesting angle to pursue would be connections with abelian quotient categories of triangulated categories, which appear in numerous sources, for instance, in [16], [14], [12], in the context of torsion and mutation pairs in triangulated categories.See also [10], which explores this topic with a particular emphasis on stable module categories of finite-dimensional selfinjective algebras.
Notation 1.6.Throughout this paper, k is a field.Modules are unital left modules, unless stated otherwise, and algebras are nonzero unital associative.Let A be a finite-dimensional k-algebra.We denote by mod(A) the abelian category of finitely generated A-modules.We denote by mod(A) the stable module category of mod(A).That is, the objects of mod(A) are the same as in mod(A), and for any two finitely generated A-modules U , V , the morphism space in mod(A) from U to V is the k-space Hom A (U, V ) = Hom A (U, V )/Hom pr A (U, V ), where Hom pr A (U, V ) is the space of all A-homomorphisms from U to V which factor through a projective A-module.Composition in mod(A) is induced by that in mod(A).We write End A (U ) = Hom A (U, U ) and End pr A (U ) = Hom pr A (U, U ). See [21, §2.13] for more details.The Nakayama functor of A is the functor ν = A ∨ ⊗ A − on mod(A), where A ∨ = Hom k (A, k) is the k-dual of A regarded as an A-A-bimodule.
The algebra A is called selfinjective if A is injective as a left (or right) A-module.Equivalently, A is selfinjective if the classes of finitely generated projective and injective A-modules coincide.By results of Happel [13], if A is selfinjective, then mod(A) is a triangulated category, with shift functor Σ induced by the operator sending an A-module to the cokernel of an injective envelope U → I U , and with exact triangles in mod(A) induced by short exact sequences in mod(A).If A is selfinjective, then the Nakayama functor ν on mod(A) is an equivalence and induces an equivalence on mod(A), and Frobenius algebra, hence selfinjective, and the Nakayama functor is isomorphic to the identity functor on mod(A).Finite group algebras, their blocks, and Iwahori-Hecke algebras are symmetric.See for instance [32, Ch.III], [21, § §2.11, 2.14] and [22,Appendix A.3] for more background.
For I a left ideal in A, its right annihilator r(I) = {a ∈ A | Ia = 0} is a right ideal, and for J a right ideal in A, its left annihilator l(J) = {a ∈ A | aJ = 0} is a left ideal.If I is an ideal in A, then so are r(I) and l(I).By results of Nakayama in [24], [25], if A is selfinjective, then the correspondencs I → r(I) and J → l(J) are inclusion reversing bijections between the sets of left and right ideals in A. These bijections are inverse to each other and restrict to bijections on the set of ideals in A. In particular, for any ideal I in A we have r(l(I)) = I = l(r(I)).If A is a Frobenius algebra, then dim k (I) + dim k (r(I)) = dim k (A), and if A is symmetric, then r(I) = l(I) for any ideal I in A. See [32, Chapter IV, Section 6] for details.
We will make use without further comment of the standard Tensor-Hom adjunction.

Distinguished abelian subcategories in stable module categories
The stable module category of a finite-dimensional non-semisimple selfinjective k-algebra A need not have any t-structures (see Proposition 2.7 and Corollary 2.8), but it always has distinguished abelian subcategories, and these tend to come in varieties (see Proposition 2.6).The first result in this section describes those distinguished abelian subcategories of mod(A) which arise as image of a full abelian subcategory of mod(A) equivalent to mod(D), for some other finite-dimensional k-algebra D.
Theorem 2.1.Let A be a finite-dimensional selfinjective k-algebra, and let D be a finite-dimensional k-algebra.Let Y be a finitely generated A-D-bimodule.The functor Y ⊗ D − is a full exact embedding of mod(D) into mod(A) and induces an embedding of mod(D) as a distinguished abelian subcategory of mod(A) if and only if the following conditions hold. (1) Y is projective as a right D-module, and (3) the Tensor-Hom adjunction unit maps isomorphisms, for all finitely generated D-modules V .
We state some parts of the proof of Theorem 2.1 as separate lemmas in slightly greater generality.
Lemma 2.2.Let A be a finite-dimensional selfinjective k-algebra.Let D be a full abelian subcategory of mod(A) such that Hom pr A (U, V ) = {0} for all A-modules U , V in D. Then the image of D in mod(A) is a distinguished abelian subcategory of mod(A), which as an abelian category, is equivalent to D.
Proof.The fact that D is a full subcategory of mod(A), together with the hypothesis Hom pr A (U, V ) = {0} for all U , V in D, implies that the image of D in mod(A) is a full subcategory of mod(A) which is equivalent to D. By the assumptions on D, exact sequences in D remain exact in mod(A).Since distinguished triangles in mod(A) are induced by short exact sequences in mod(A), it follows that the image of D in mod(A) is a distinguished abelian subcategory.
The following observation is well-known (see the papers [1] and [26] on static and adstatic modules).We include a short proof for convenience.
Proof.The functor Y ⊗ D − is a full embedding if and only if for any two finitely generated Dmodules U , V , the map is an isomorphism.By considering the case U = D, one sees that this is the case if and only if the adjunction map V → Hom A (Y, Y ⊗ D V ) itself is an isomorphism, whence the result.
Proof of Theorem 2.1.By Lemma 2.4, the functor Y ⊗ D − : mod(D) → mod(A) is a full embedding, if and only if the condition (3) holds.This embedding is exact if and only if Y is flat as a right D-module.Since Y is finitely generated, this is equivalent with requiring condition (2).It follows from the Lemmas 2.2 and 2.3 that the composition with the canonical functor mod(A) → mod(A) yields an embedding of mod(D) as a distinguished abelian subcategory in mod(A) if and only if (1) holds as well.This concludes the proof of Theorem 2.1.
If both A and D are selfinjective, then Theorem 2.1 yields the following result.Proof.If (i) holds, then (ii) holds by Theorem 2.1.Suppose that (ii) holds.Note that the hypotheses imply that D = End A (Y ) op is selfinjective and that Y is projective as a right D-module.
Thus the conditions ( 1) and ( 2) in Theorem 2.1 are satisfied.We need to show that condition (3) in that Theorem holds as well.That is, given a finitely generated D-module V , we need to show that the adjunction map Thus this is the case for V any free D-module of finite rank.In general, since D is selfinjective, V is isomorphic to a submodule of a free D-module of finite rank.Thus there is an exact sequence of D-modules of the form for some positive integers n and m.By the hypotheses in (ii), the functor Y ⊗ D − is exact, and hence we have an exact sequence of A-modules of the form Since the functor Hom A (Y, −) is left exact, this yields an exact sequence of D-modules of the form By naturality of the adjunction maps, we get a commutative diagram of D-modules with exact rows where α, β, γ are the adjunction maps.By the above remarks, β and γ are isomorphisms.The exactness of the rows implies that α is an isomorphism as well.This shows that condition (3) in Theorem 2.1 holds as well, and hence the result follows from Theorem 2.1.
Using a Theorem of Cabanes [8,Theorem 2] one can identify the image of the functor Y ⊗ D − in Theorem 2.5 more precisely; see Section 8.
If A is a Frobenius algebra over an algebraically closed field, then the ideals I containing their right annihilators form subvarieties of certain Grassmannians.Proposition 2.6.Let A be a finite-dimensional Frobenius algebra over k.Suppose that k is algebraically closed.The set of proper ideals I in A satisfying r(I) ⊆ I is a projective variety whose connected components are subvarieties of the Grassmannians Gr(n, A), Proof.If A is a Frobenius algebra, then dim k (A) = dim k (I)+dim k (r(I)).Since r(I) ⊆ I, it follows that dim k (A) ≤ 2dim k (I).Thus the ideals satisfying r(I) ⊆ I satisfy dim k (A) 2 ≤ dim k (I).In each dimension, they form subvarieties of the Grassmannians, since being an ideal with an annihilator of a fixed dimension is obviously a polynomial condition (obtained by fixing a k-basis of A).
If A is symmetric, then A is selfinjective, but none of the distinguished abelian subcategories constructed above arises as the heart of a t-structure.More precisely, we have the following result.
Proposition 2.7.Let A be a finite-dimensional selfinjective k-algebra.Denote by ν the Nakayama functor on mod(A).Then mod(A) has no nontrivial t-structure which is stable under ν; that is, the heart of any ν-stable t-structure on mod(A) is zero.
Corollary 2.8.Let A be a finite-dimensional symmetric k-algebra.Then mod(A) has no nontrivial t-structure; that is, the heart of any t-structure on mod(A) is zero.
Proof.The Nakayama functor of a symmetric algebra is isomorphic to the identity functor, and hence the statement is a special case of Proposition 2.7.
Since finite group algebras are symmetric, this implies in particular the second statement of Theorem 1.4: Corollary 2.9.Let G be a finite group.Then mod(kG) has no nontrivial t-structure; that is, the heart of any t-structure on mod(kG) is zero.
Remark 2.10.With the notation of Theorem 2.1, not every embedding of mod(D) as a distinguished abelian subcategory of mod(A) lifts in general to a full embedding mod(D) → mod(A).Suppose that Y ⊗ D − : mod(D) → mod(A) is a full exact embedding and induces an embedding mod(D) → mod(A) as distinguished abelian subcategory.Let M be an A-A-bimodule inducing a stable equivalence of Morita type on A. Then the functor M ⊗ A Y ⊗ D − : mod(D) → mod(A) is exact but no longer necessarily full.It induces still an embedding of mod(D) as a distinguished abelian subcategory, because the functor M ⊗ A − induces a triangulated equivalence on mod(A), hence permutes distinguished abelian subcategories.It is not clear whether an embedding mod(D) → mod(A) as a distinguished abelian subcategory is necessarily induced by tensoring with a suitable A-D-bimodule.

Simple modules in distinguished abelian subcategories
We consider in this section distinguished abelian subcategories of mod(A) whose simple objects are simple A-modules, where A is a finite-dimensional selfinjective k-algebra.The first theorem is essentially a special case of Theorem 2.1, and it implies Theorem 1.3 (i), describing those distinguished abelian subcategories which arise from quotients of A. Any full abelian subcategory of a distinguished abelian subcategory of a triangulated category is clearly again a distinguished abelian subcategory.In particular, if the canonical functor mod(A/I) → mod(A) is an embedding as a distinguished abelian subcategory, then so is the canonical functor mod(A/J) → mod(A) for any ideal J which contains I, because this factors through the embedding mod(A/J) → mod(A/I) induced by the canonical surjection A/I → A/J.
Every ideal which squares to zero gives rise to a distinguished abelian subcategory in the stable module category of a selfinjective algebra.Corollary 3.2.Let A be a finite-dimensional selfinjective k-algebra, and let J be an ideal in A such that J 2 = {0}.Set I = l(J).Then the canonical surjection A → A/I induces an embedding mod(A/I) → mod(A) of mod(A/I) as a distinguished abelian subcategory in mod(A).
Proof.We have r(I) = r(l(J)) = J, hence r(I) 2 = {0} by the assumptions.The result follows from the equivalence of the statements (i) and (v) in Theorem 3.1.
If A is a finite-dimensional Hopf algebra, then mod(A) is a monoidal abelian category, and A is selfinjective, by a result of Larson and Sweedler [18].Given two A-modules U , V , if one of U , V is projective, then so is U ⊗ k V (see e. g. [5,Proposition 3.1.5]).Thus mod(A) is a monoidal triangulated category.If I is a proper Hopf ideal in A, then A/I is a Hopf algebra, the canonical surjection A → A/I is a homomorphism of Hopf algebras, and hence induces a full embedding of monoidal categories mod(A/I) → mod(A).Thus Theorem 3.1 implies immediately the following observation.
Corollary 3.3.Let A be a finite-dimensional Hopf algebra over k and let I be a proper Hopf ideal in A containing its right annihilator r(I).Then the composition of canonical functors mod(A/I) → mod(A) → mod(A) is an embedding of mod(A/I) as a monoidal distinguished abelian subcategory in the monoidal triangulated category mod(A).Some of the implications in Theorem 3.1 hold in slightly greater generality.Lemma 3.4.Let A be a finite-dimensional k-algebra, and let J be a proper left ideal in A. We have End pr A (A/J) = {0} if and only if r(J) ⊆ J.
Proof.Note first that if β : A/J → A is an A-homomorphism, then β • π is an A-endomorphism of A with kernel containing J, hence induced by right multiplication with an element y ∈ r(J).
Conversely, right multiplication with an element y ∈ r(J) factors through π.Let α : A/J → A/J be an endomorphism of A/J as a left A-module such that α factors through a projective A-module.Then α factors through the canonical surjection π : A → A/J; that is, there is an A-homomorphism β : A/J → A such that α = π • β.By the above, the endomorphism β • π of A is induced by right multiplication with an element y ∈ r(J).Since π is surjective, we have α = 0 if and only if α  Proof.If r(I) ⊆ I, then taking left annihilators yields I = l(r(I)) ⊇ l(I), so (i) implies (ii).A similar argument shows that (ii) implies (i).Since I • r(I) = 0, it follows that if r(I) ⊆ I, then r(I) 2 = 0. Thus (i) implies (iii).A similar argument shows that (ii) implies (iv).If r(I) 2 = 0, then r(I) ⊆ l(r(I)) = I, so (iii) implies (i), and a similar argument shows that (iv) implies (ii).
Proof of Theorem 3.1.We are going to prove Theorem 3.1 as a special case of Theorem 2.1.Set Y = A/I, regarded as an A-A/I-bimodule.We have Clearly Y is projective as a right A/I-module.Given an A/I-module V , the adjunction unit V → Hom A (A/I, A/I ⊗ A/I V ) is trivially an isomorphism.Thus the A-A/I-bimodule satisfies the conditions ( 2) and (3) in Theorem 2.1.Therefore the composition of functors mod(A/I) → mod(A) → mod(A) is an embedding of mod(A/I) as a distinguished abelian subcategory if and only if (1) holds, that is, if and only if End pr A (A/I) = {0}.This proves the equivalence of (i) and (iii).It follows from Lemma 3.4 that the statements (ii) and (iii) are equivalent.The implication (iii) ⇒ (iv) follows from Lemma 3.5.The implication (iv) ⇒ (i) follows from Lemma 2.2.The equivalence of (ii) and (v) holds by Lemma 3.6.
Remark 3.7.Lemma 3.6 implies that working with left or right modules yields equivalent statements.To illustrate this point, by Theorem 3.1, we have r(I) ⊆ I if and only if we have a full embedding mod(A/I) → mod(A).There is an obvious right module analogue which states that l(I) ⊆ I if and only if we have a full embedding mod((A/I) op ) → mod(A op ).Thus Lemma 3.6 implies that we have a full embedding mod(A/I) → mod(A) if and only if we have a full embedding mod((A/I) op ) → mod(A op ).In other words, the full distinguished abelian subcategories in mod(A) and mod(A op ) constructed in Theorem 3.1 and its right module analogue correspond bijectively to each other.Theorem 3.8.Let A be a finite-dimensional selfinjective k-algebra such that all simple A-modules are nonprojective.Let D be a distinguished abelian subcategory of mod(A).Suppose that D contains all simple A-modules.The simple A-modules are exactly all simple objects in D. In particular, we have ℓ(D) = ℓ(A).
Proof.Let U be an indecomposable nonprojective A-module belonging to D, and let S be a simple A-module.Let ψ : U → S be an A-homomorphism, and denote by ψ the image of ψ in Hom A (U, S).Note that U , S are both nonzero objects D, by the assumptions.
We are going to show first that if ψ is not an isomorphism in mod(A), then ψ is not a monomorphism in D. If ψ is zero, there is nothing to show.If ψ is nonzero, then ψ is surjective because S is a simple A-module.Since ψ is not an isomorphism we have ker(ψ) is nonzero.Let T be a simple A-submodule of ker(ψ).The inclusion T → U is an injective A-homomorphism, hence its image in mod(A) is a nonzero morphism in mod(A).By construction, the composition T → U → S is zero in mod(A).Since also T belongs to D, it follows that ψ is not a monomorphism in D.
This argument shows that S is a simple object of D. Indeed, if not, there would have to be a monomorphism U → S in D which is not an isomorphism.But by the first paragraph, any such monomorphism is inducd by an isomorphism in mod(A), so is an isomorphism in D as well.This argument also shows that D contains no other simple objects.Indeed, let U be an indecomposable nonprojective A-module which is a simple object in D. Consider a surjective Ahomomorphism ψ : U → S onto some simple A-module S. Then S belongs to D, and the image ψ is a monomorphism in D because U is simple in D. But then ψ is an isomorphism by the first argument.Thus the simple A-modules are exactly the simple objects in D, whence the result.Corollary 3.9.Let A be a finite-dimensional selfinjective k-algebra such that all simple A-modules are nonprojective.Let I be an ideal such that r(I) ⊆ I ⊆ J(A).Let D be a distinguished abelian subcategory of mod(A) containing mod(A/I).Then the simple A-modules are exactly the simple objects in D.
Proof.The hypothesis r(I) ⊆ I implies, by Theorem 3.1, that mod(A/I) is a distinguished abelian subcategory of mod(A).The hypothesis I ⊆ J(A) implies that mod(A/I) contains all simple A-modules.The result follows from Theorem 3.8.
Removing the reference to simple A-modules yields the following statement.For the proof, we will need the following elementary observation, which is a sufficient criterion for an epimorphism in the category of k-algebras to be an isomorphism.Lemma 3.12.Let D be a finite-dimensional k-algebra and A a unital subalgebra of D. Suppose that the restriction functor Res D A : mod(D) → mod(A) is a full embedding which sends every simple D-module to a simple A-module.Then A = D.
Proof.We first show that J(A) = A ∩ J(D).Since simple D-modules restrict to simple A-modules, it follows that J(A) annihilates every simple D-module, and hence J(A) ⊆ A∩J(D).Since A∩J(D) is a nilpotent ideal in A, we have the other inclusion as well, whence the equality J(A) = A ∩ J(D).Thus the inclusion A ⊆ D induces an injective algebra homomorphism A/J(A) → D/J(D).Since A/J(A) is semisimple, every A/J(A)-module is injective, and hence A/J(A) is isomorphic to a direct summand of D/J(A) as a right A/J(A)-module.Thus, for any simple A-module S, the D/J(D)-module D/J(D) ⊗ A/J(A) S is nonzero.Regarded as a left D-module, this is a quotient of D⊗ A S. In particular D⊗ A S is nonzero.Let T be a simple quotient of D⊗ A S and let D⊗ A S → T be a nonzero D-homomorphism.The standard adjunction yields a nonzero A-module homomorphism S → Res D A (T ).Since S and Res D A (T ) are both simple A-modules, it follows that S ∼ = Res D A (T ).

This shows that Res D
A induces a bijection between the isomorphism classes of simple D-modules and simple A-modules.Since Res D A is a full embedding, we also have End D (T ) = End A (T ).Thus the simple modules for D and A which correspond to each other through the bijection induced by Res D A have the same dimensions and isomorphic endomorphism rings.The Artin-Wedderburn Theorem implies that A/J(A) ∼ = D/J(D), and hence D = A + J(D).
We show next that every maximal A-submodule of D is in fact a maximal D-submodule.Indeed, let M be a maximal A-submodule of D. Then S = D/M is a simple A-module.By the previous argument, there is a simple D-module T and an Taking the intersection of all maximal submodules of D as an A-module yields J(A)D = J(D).
It follows that The converse of Lemma 3.12 holds trivially.One cannot drop in this Lemma the hypothesis that Res D A sends simple modules to simple modules.Consider the subalgebra A of upper triangular matrices in D = M 2 (k).The restriction from D to A of the unique (up to isomorphism) simple D-module is the unique (up to isomorphism) projective indecomposable A-module of dimension 2. The converse in Lemma 3.13 need not hold; the issue is that the functor Hom A (Y, −) need not be right adjoint to the functor induced by Y ⊗ D −.The Tensor-Hom adjunction induces a natural transformation between the induced bifunctors at the level of the stable category mod(A) (cf.Lemma 9.11 and Remark 9.12), but this need not be an isomorphism.Proof of Theorem 3.11.Since Y ⊗ D − induces a full embedding mod(D) → mod(A), it follows from Lemma 3.13 that for any finitely generated D-modules V , we have an isomorphism

One verifies easily that Res
be a short exact sequence of nonzero D-modules.This sequence is isomorphic to the short exact sequence Since Y ⊗ D − induces a full embedding mod(D) → mod(A), this yields an exact sequence and applying the left exact functor Hom A (Y, −) yields an exact sequence Thus we have a commutative exact diagram of the form where the vertical maps are the canonical surjections.Arguing by induction, the left and right vertical maps are isomorphisms.Thus the top right horizontal map is surjective, and comparing dimensions implies that the middle vertical map is an isomorphism as well.This shows that Example 3.15.Let A be a finite-dimensional selfinjective k-algebra.Suppose that the simple A-modules are non-projective.Then J(A) contains its right annihilator soc(A) in A. Thus the subcategory of all semisimple A-modules, which is equivalent to mod(A/J(A)), is a distinguished abelian subcategory of mod(A).We have ℓ(A) = ℓ(A/J(A)), so for trivial reasons, mod(A) has distinguished abelian subcategories D whose number of isomorphism classes ℓ(D) of simple objects in D is equal to the number ℓ(A) of isomorphism classes of simple A-modules.
Example 3.16.Let A be a finite-dimensional selfinjective k-algebra.Suppose that soc 2 (A) ⊆ J(A) 2 .Since soc 2 (A) is the right annihilator of J(A) 2 , it follows from Theorem 3.1 that the composition of canonical functors mod(A/J(A) 2 ) → mod(A) → mod(A) is an embedding of mod(A/J(A) 2 ) as a distinguished abelian subcategory in mod(A).If A is indecomposable as an algebra, then so is A/J(A) 2 , and both have the same quiver.Therefore, in this situation, mod(A) has a connected distinguished abelian subcategory D = mod(A/J(A) 2 ) satisfying ℓ(D) = ℓ(A) and Ext 1 D (S, T ) ∼ = Ext 1 C (S, T ), for any two simple objects S, T in D. Remark 3.17.The property soc 2 (A) ⊆ J(A) 2 in the previous Example is not invariant under stable equivalences, in fact, not even under derived equivalences.For instance, the Brauer tree algebra of a star with four edges has this property, but the Brauer tree algebra of a line with four edges (and no exceptional vertex) does not.

Distinguished abelian subcategories for finite group algebras
We describe special cases of the situation arising in Theorem 3.1 involving finite group algebras.We use without further comments standard properties of finite p-group algebras in prime characteristic p; see e. Let G be a finite group.The module category mod(kG) of the finite group algebra kG over a field k is a symmetric monoidal category with respect to the tensor product − ⊗ k − of kGmodules over k.It is well-known that if U , V are finitely generated kG-modules with at least one of U , V projective, then U ⊗ k V is projective as well.Therefore the tensor product over k induces a commutative monoidal structure on the triangulated category mod(kG).If N is a normal subgroup of G, then the canonical surjection G → G/N induces an embedding of symmetric monoidal categories mod(kG/N ) → mod(kG).The following result implies the first statement in Theorem 1.4.Alternatively, one can also show this using a special case of Higman's criterion.Let U , V be kG/N -modules.When regarded as kG-modules, the elements of N act as identity on U , V .Thus any k-linear map τ : U → V is a kN -homomorphism, and then this expression is zero in k.It follows from the special case [21, Proposition 2.13.11] of Higman's criterion that Hom pr kG (U, V ) = {0}.Equivalently, we have Hom kG (U, V ) ∼ = Hom kG (U, V ) = Hom kG/N (U, V ).This shows that if p divides |N |, then mod(kG/N ) can indeed be identified canonically with a full subcategory of mod(kG).Note that kG/N is a projective kG/N -module.Thus every kG/N -endomorphism of kG/N is equal to Tr G/N 1 (σ) for some linear endomorphism σ of kG/N .Equivalently, every kG-endomorphism of kG/N is of the form , which shows that τ factors through a projective kG-module.Equivalently, the canonical map End kG/N (kG/N ) → End kG (kG/N ) is zero.This shows that if |N | is coprime to p, then the canonical functor mod(kG/N ) → mod(kG) is not an embedding.Remark 4.3.Let k be a field of prime characteristic p.
(1) Let G be a finite group having a nontrivial normal p-subgroup Q.It is well-known that the kernel I of the canonical algebra homomorpism kG → kG/Q is contained in the radical J(kG) and hence that ℓ(kG) = ℓ(kG/Q).Thus Theorem 4.2 illustrates Theorem 3.8, constructing explicitly the distinguished abelian subcategory mod(kG/Q) of mod(kG) whose number of isomorphism classes of simple objects is equal to that of mod(kG).(2) Theorem 4.2 implies that if P is a nontrivial finite p-group, then any cyclic subgroup of Z(P ) yields a distinguished abelian subcategory of mod(kP ).But then so does any shifted cyclic subgroup of Z(P ), suggesting that distinguished abelian subcategories should form varieties which are related to cohomology support varieties.
Combining Theorem 3.11, a result of J. F. Carlson [9, Theorem 1], and [19, Theorem 3.4] yields the following classification of those distinguished abelian subcategories of the stable module category of a finite p-group algebra in prime characteristic p which are equivalent to module categories of finite-dimensional split k-algebras.If P is a finite p-group, then a finiteley generated kP inverse equivalences on mod(kP ).In particular, V ⊗ k − sends in that case any distinguished abelian subcategory D of mod(kP ) to a distinguished abelian subcategory, denoted V ⊗ k D, of mod(kP ).
Theorem 4.4.Let p be a prime, P a nontrivial finite p-group and k a field of characteristic p.Let D be a finite-dimensional split basic k-algebra such that there is an embedding Φ : mod(D) → mod(kP ) as distinguished abelian subcategory of mod(kP ).
(i) We have ℓ(D) = 1; that is, D is split local.
(ii) Let V is an indecomposable kP -module corresponding to a simple D-module under the functor Φ.Then V is an endotrivial kP -module. .By Lemma 4.5, the category D has a unique isomorphism class of simple objects, whence (i).Let V be an indecomposable kP -module such that V is simple as an object in D. Then in particular End kP (V ) ∼ = k.A result of J. F. Carlson [9, Theorem 1] implies that V is endotrivial, which shows (ii).The exact functor V ⊗ k − on mod(kP ) induces an equivalence on mod(kP ), with inverse induced by the functor V * ⊗ k −.Thus after replacing D by the image of D under the functor V * ⊗ k −, we may (and do) assume that the trivial kP -module k belongs to D, and is the -up to isomorphism unique -simple object of D. In other words, with the notation and hypotheses of statement (iii), the kP -D-bimodule Y is projective as a right D-module and the functor Y ⊗ D − sends a simple D-module to the trivial kP -module.Thus the hypotheses of Theorem 3.11 are satisfied, implying statement (iii).

Basic properties of distinguished abelian subcategories
We show that a short exact sequence in a distinguished abelian subcategory D of a triangulated category C determines a unique exact triangle in C; that is, we show that the morphism h in Definition 1.1 is unique.If not stated otherwise, the shift functor of a triangulated category is denoted by Σ.
is an exact triangle in C.
Proof.The existence of h is clear by definition; we need to show the uniqueness.Let h, h ′ : Z → Σ(X) be morphisms in C such that the triangles are exact.The pair of identity morphisms (Id X , Id Y ) can be completed to a morphism of triangles (Id X , Id Y , a).That is, there is a morphism a : Z → Z satisfying a • g = g and h ′ • a = h.Since Z belongs to D and since D is a full subcategory of C, it follows that a is a morphism in the abelian category D. Since g is an epimorphism in D, this forces a = Id Z , whence h ′ = h.
induced by composition with g and precomposition with Σ(f ) are surjective.
Proof.Since D is a distinguished abelian subcategory in C, there is a morphism h : Z → Σ(X) such that the triangle in C is exact.Applying the functor Hom C (W, −) yields a long exact sequence of the form The right map is induced by composing with the monomorphism f in D. Thus if W belongs to D, then the right map is injective.But then the map in the middle is zero, so the left map is surjective.
Since Σ is an equivalence, it follows that the map Hom C (Σ(W ), Y ) / / Hom C (Σ(W ), Z) is surjective.Similarly, we have a long exact sequence The right map is induced by precomposing with the epimorphism g in D. Thus if W belongs to D, then the right map is injective, hence the map in the middle is zero, and therefore the left map is surjective.This concludes the proof.
Unlike hearts of t-structures, distinguished abelian subcategories need not be disjoint from their shifts -they may contain periodic objects.The following consequence of Proposition 5.3 shows that if D is a distinguished abelian subcategory in a triangulated category C, then D ∩ Σ(D) is a subcategory of the additive category proj(D) generated by the projective objects in D. induced by precomposition with g is injective.
Proof.Since D is a distinguished abelian subcategory in C, there is a morphism h : Z → Σ(X) such that the triangle in C is exact.Applying the functor Hom C (W, −) yields a long exact sequence of the form If W is projective in D, then g * is surjective, hence h * is zero.This implies that Σ(f ) * is injective, proving (i).A dual argument, applying the functor Hom C (−, W ), and using the fact that Σ is an equivalence, shows (ii).
A category D is called split if every morphism f : X → Y in D is split; that is, if there exists a morphism g : Y → X such that f = f • g • f .If D is an abelian category, an easy verification shows that D is split if and only if every monomorphism (resp.epimorphism) in D is split.All epimorphisms and monomorphisms in a triangulated category are split.Proof.Suppose that the inclusion functor D ⊆ C has a left adjoint Φ.That is, for any object U in D and any object X in C we have a natural isomorphism Hom D (Φ(X), U ) ∼ = Hom C (X, U ). Thus any monomorphism U → U ′ in D induces an injective map Hom C (X, U ) → Hom C (X, U ′ ).This shows that the morphism U → U ′ is a monomorphism in C, hence split in C. Since D is a full subcategory of C it follows that the monomorphism U → U ′ is split in D, and hence D is split.A similar argument shows that if the inclusion functor D ⊆ C has a right adjoint, then every epimorphism in D is split, whence the result.
If D is a finite-dimensional k-algebra, then D is semisimple if and only if mod(D) is split.Thus Proposition 5.6 has the following immediate consequence.
Corollary 5.7.Let A be a finite-dimensional selfinjective k-algebra, D a finite-dimensional kalgebra, and Φ : mod(D) → mod(A) a full embedding of mod(D) as a distinguished abelian subcategory in mod(A).If Φ has a left adjoint or a right adjoint, then D is semisimple.This Corollary implies in particular that even if Φ is induced by tensoring with a suitable A-Dbimodule, the Tensor-Hom adjunction does not in general yield a right adjoint to Φ; see Lemma 9.11 and Remark 9.12 for some more comments.
Remark 5.8.By a result of Balmer and Schlichting [4, Theorem 1.5], the idempotent completion Ĉ of a triangulated category C is triangulated in such a way that the canonical embedding C → Ĉ is an exact functor.Since this embedding is full, it follows that a distinguished abelian subcategory D of C remains a distinguished abelian subcategory of Ĉ.Since any abelian category is idempotent split, it follows that the indecomposable objects in D remain indecomposable in Ĉ.
Example 5.9.Let k be a field of characteristic 2 and let P be a finite 2-group of order at least 4. Let Z be a central subgroup of order 2 of P .Set Y = kP/Z as left kP -module.Then End pr kP (Y ) = {0} and End kP (Y ) ∼ = (kP/Z) op .Clearly Y is a progenerator of the distinguished abelian subcategory mod(kP/Z) of mod(kP ), obtained from the restriction functor Φ given by the canonical surjection kP → kP/Z.The functor Φ is trivially isomorphic to Y ⊗ kP/Z −.We have Σ(Y ) ∼ = Y , where Σ is the shift functor in mod(kP ).In other words, as a kP -module, Y has period 1.Therefore, Y is also a progenerator of the distinguished abelian subcategory Σ(mod(kP/Z)).The subcategories mod(kP/Z) and Σ(mod(kP/Z)) are different; in fact, their intersection is add(Y ) because of Proposition 5.4.In particular, the embedding Σ • Φ : mod(kP/Z) → mod(kP ) is not induced by the functor Y ⊗ kP/Z −.It is, though, still induced by tensoring with a bimodule, namely the kP -kP/Z-bimodule Σ P ×P/Z (Y ).This is because we have composed the embedding Φ with the self-equivalence Σ on mod(kP ), which is a stable equivalence of Morita type, hence induced by tensoring with a suitable bimodule.One should expect that composing Φ with a stable equivalence on mod(kP ) which is not of Morita type would yield embeddings mod(kP/Z) → mod(kP ) as distinguished abelian subcategories which are not induced by tensoring with any bimodule.

Exact sequences in distinguished abelian subcategories
Any exact triangle X → Y → Z → Σ(X) in a triangulated category C such that X, Y , Z belong to the heart A of a t-structure is in fact induced by a short exact sequence 0 → X → Y → Z → 0 in A. In an arbitrary distinguished abelian subcategory, this need not be the case.The case where a nonzero object W and its shift Σ(W ) belong to a distinguished abelian subcategory D of C yields an exact triangle which is not induced by an exact sequence in D. As noted in Corollary 5.4, in that situation W is injective in D and Σ(W ) is projective in D. The following result shows that these are essentially the only exact triangles with three terms in D which can arise besides those induced by short exact sequences in D. As before, we denote the shift functor in a triangulated category by Σ.
Proposition 6.1.Let D be a distinguished abelian subcategory in a triangulated category C, and let be an exact triangle in C such that X, Y , Z belong to D. Then this triangle is isomorphic to a direct sum of two exact triangles of the form where X ′ , Z ′ , W , Σ(W ) are in D, and where the sequence Proof.Since D is abelian, the morphism g has a kernel f ′ : X ′ → Y in D. Then in particular g • f ′ = 0, and hence there is a morphism v : Since D is distinguished, this can be completed to an exact triangle in C with a morphism h ′ : Z ′ → Σ(X ′ ).The morphisms v and w yield morphisms of triangles is an automorphism of the third triangle, and hence so is its inverse.After replacing b by (b • a) −1 • b, we therefore may choose b in such a way that b • a = Id Z ′ .It follows that the first triangle is a direct summand of the second, and that it has a complement isomorphic to where W is the complement of X ′ in X determined by ker(w).The last statement on W (resp. Σ(W )) being injective (resp.projective) in D follows from Corollary 5.4.Proof.Since T is closed under powers of Σ, it follows from Corollary 5.4 that all objects in T are projective and injective in D. Let f : X → Y be a morphism in T .Complete f to an exact triangle in T (or equivalently, in C).Since T is contained in D, it follows that (possibly after replacing Z by an isomorphic object) Z belongs to D, and hence the morphism g belongs to D. This triangle is the direct sum of two triangles as in Proposition 6.1.All terms in these two triangles are in T , hence projective and injective in D. The first of the two triangles is induced by a short exact sequence in D, and therefore split.The second of the two triangles is trivially split.The result follows.
Remark 6.3.With the notation of Corollary 6.2, suppose that D is equivalent to mod(D) for some finite-dimensional symmetric k-algebra D and that T is a thick subcategory of C which is contained in D. Then T is generated, as an additive category, by indecomposable projective (or equivalently, injective) objects in D. By the assumption on D, each projective indecomposable object U in D has an endomorphism with image the socle of U .This endomorphism is split, hence an isomorphism, and thus U is simple.Therefore, in this situation, T consists of projective semisimple objects in D which are permuted by Σ.
Proposition 6.4.Let C be a triangulated category and let D, D ′ be distinguished abelian subcategories of C such that D ⊆ D ′ .Then D is an abelian subcategory of D ′ ; that is, the inclusion functor / / 0 be a short exact sequence in D. We need to show that this sequence remains exact in D ′ .Since D is a distinguished abelian subcategory of C, it follows that there is a morphism h : Z → Σ(X) in D such that the triangle is exact in D ′ .Since D is full in C, hence in D ′ and since any abelian category is idempotent complete, it follows that X ′ , Z ′ , W belong to D (up to isomorphism).Since D is a full subcategory of C, hence of D ′ , it follows that the morphisms f ′ , g ′ belong to D. Thus f is the direct sum in D of f ′ and the zero morphism W → 0. But f is also a monomorphism in D, and hence W = 0.The result follows.Proposition 6.5.Let C be an essentially small triangulated category.Every distinguished abelian subcategory of C is contained in a maximal distinguished abelian subcategory of C, with respect to the inclusion of subcategories.
Proof.We may assume that C is small, so that the distinguished abelian subcategories form a set.Let T be a totally ordered set of distinguished abelian subcategories of C, where the order is by inclusion.In view of Zorn's Lmma, we need to show that T has an upper bound.We claim that E = ∪ D∈T D is such an upper bound.We need to show that E is a distinguished abelian subcategory.By construction, E is a full subcategory of C. We show next that E is an abelian category.Let f : X → Y be a morphism in E. Then there is D ∈ T containing X, Y , and since D is a full subcategory of C, it follows that f is a morphism in D. Thus f has a kernel a : W → X in D. We are going to show that a is a kernel of f in E. Let g : Z → X a morphism in E such that f • g = 0. We need to show that g factors uniquely through a.Since T is totally ordered, there is D ′ ∈ T such that D ⊆ D ′ and such that g is a morphism in D ′ .By Proposition 6.4, the morphism a remains a kernel of f as a morphism in D ′ .Thus there is a unique morphism h : Z → W in D ′ such that a • h = g.We need to show that h is unique in E with this property.Let j : Z → W be a morphism in E such that a • j = g.Then j belongs to a category D ′′ ∈ T , which we may choose such that D ′ ⊆ D ′′ .Again by Proposition 6.4, the morphism a remains a monomorphism in D ′′ .Since a • j = g = a • h, it follows that j = h.This shows that the kernel of f in any subcategory D ∈ T containing f is the kernel of f in E. A similar argument shows that the cokernel of f in any subcategory D ∈ T containing f is the cokernel of f in E. This implies also that the canonical map coker(ker(f )) → ker(coker(f )) in E is an isomorphism, since it is an isomorphism in any subcategory D ∈ T containing the morphism f .By the above arguments, any short exact sequence in E is a short exact sequence in D for some D ∈ T , hence can be completed to an exact triangle in C.This shows that E is a distinguished abelian subcategory in C. Thus T has an upper bound in the set of distinguished abelian subcategories of C. Zorn's Lemma implies the result.
Proof of Theorem 1.2.Statement (i) is Theorem 3.8.For statement (ii), let A be a finite-dimensional selfinjective algebra over a field k such that all simple A-modules are nonprojective.Then J(A) contains its annihilator soc(A).Thus mod(A/J(A)) is a distinguished abelian subcategory of mod(A) containing all simple A-modules such that ℓ(mod(A/J(A))) = ℓ(A).By Proposition 6.5 there is a maximal distinguished abelian subcategory in C which contains mod(A/J(A)).By Corollary 3.9 the simple A-modules are exactly the simple objects in D. Thus ℓ(D) = ℓ(A), whence the result.
The next two Propositions are tools for passing between short exact sequences in mod(A), for some finite-dimensional selfinjective algebra A, and short exact sequences in a distinguished abelian subcategory of the stable category mod(A).Proposition 6.6.Let A be a finite-dimensional selfinjective algebra over a field k, and let D be a distinguished abelian subcategory of mod(A).Let be a short exact sequence in D. Suppose that X, Y , Z have no nonzero projective direct summands as A-modules.The following hold.
(i) There is a finitely generated projective A-module Q and a short exact sequence of A-modules such that f and g are the images of a and b in mod(A), respectively.(ii) In addition, if X or Z is simple as an A-module, then Q = {0} in the first statement.
Proof.Since D is a distinguished abelian subcategory of mod(A), it follows that the given exact sequence in D gives rise to an exact triangle in mod(A), for some morphism h.By the construction of exact triangles in mod(A), this exact triangle is induced by a short exact sequence of A-modules of the form for some finitely generated projective A-modules P , Q, R, such that f and g are the images of a and b in mod(A).Since a is injective, and since the A-module P is projective, hence also injective, it follows that a(P ) ∼ = P splits off the middle term Y ⊕ Q.Since Y has no nonzero projective summand, it follows that we may assume P = {0}.A similar argument shows that we may assume R = {0}, whence the first statement.
For the second statement, assume first that Z is simple as an A-module.Write b = r s , where , it follows that r = 0, hence r is surjective as Z is simple.Since Q is projective as an A-module, it follows that s factors through by the first statement, and so also Q = {0}.Assume next that X is simple.Writing a = (u, v) : X → Y ⊕ Q, we have that u = 0, so u is injective.Thus a(X) is not contained in the summand Q, hence intersects this summand trivially since X is simple.Thus b sends Q to a submodule of Z isomorphic to Q. Since Q is also injective as an A-module, it follows that Q is isomorphic to a direct summand of Z, hence zero by the first statement.This proves the second statement.
Proposition 6.7.Let A be a finite-dimensional selfinjective algebra over a field k, and let D be a distinguished abelian subcategory of mod(A).Let be a short exact sequence of A-modules such that Q is a projective A-module, and such that the A-modules X, Y , Z belong to D. Suppose that as an object in D, X has no nonzero injective direct summand, or that as an object in D, Z has no nonzero projective direct sumand.Then the sequence where f = a and g = b are the images in mod(A) of a and b, respectively.
Proof.By the construction of mod(A) as a triangulated category, the given short exact sequence of A-modules determines an exact triangle in mod(A) of the form for some morphism h in mod(A).By Proposition 6.1, this triangle is isomorphic to a direct sum of two exact triangles of the form such that the sequence Thus f is a monomorphism in D if and only if W = {0}, which is equivalent to Σ(W ) = {0}, hence to g being an epimorphism in D. The result follows.
Remark 6.8.Any commutative rectangle in a triangulated category C In general, c is not uniquely determined by (a, b).If, however, the two exact triangles are determined by short exact sequences in a distinguished abelian subcategory D of C, then a and c are both determined by b alone.Indeed, D is a full subcategory of C, so a, b, c all are morphisms in D, and since f ′ is a monomorphism and g an epimorphism in D, it follows that b determines both a and c.In particular, any endomorphism (a, b, c) of the exact triangle is determined by the endomorphism b of Y , or equivalently, the algebra homomorphism from the endomorphism algebra of this triangle to End C (Y ) sending (a, b, c) to b is injective.This is a necessary criterion for an exact triangle to have the property that its components belong to a distinguished abelian subcategory of C. Remark 6.9.Remark 6.8 can be rephrased as stating that the inclusion functor of a distinguished abelian subcategory D of a triangulated category C sends morphisms of exact sequences in D to morphisms of exact triangles in C. Indeed, if is a commutative exact diagram in D, then c is uniquely determined by b, and hence the diagram in C is commutative, where h, h ′ are the unique morphisms such that the rows are exact triangles.
In a similar vein, given two composable monomorphisms describing the third isomorphism theorem can be extended uniquely to an octahedral diagram in C of the form The kernel and cokernel of a morphism in a distinguished abelian subcategory are related with the third term of the exact triangle determined by that morphism, via the octahedron in C associated with an epi-mono factorisation of the morphism.Proposition 6.10.Let D be a distinguished abelian subcategory in a triangulated category C. Every morphism f : X → Y in D gives rise to an octahedron in C of the form where C is an object in D, u an epimorphism in D, v a monomorphism in D, and where ker(f ) and coker(f ) denote the kernel and cokernel of f in D, respectively.
Proof.Since D is abelian, we have a canonical isomorphism C = coker(ker(f )) ∼ = ker(coker(f )) in D. Since D is a distinguished abelian subcategory, the obvious short exact sequences Turning the first of these two exact triangles yields an exact triangle Thus an octahedron associated with the factorisation has the form as stated.
Remark 6.11.With the notation of Proposition 6.10, the kernel and cokernel of f and the factorisation of f via C are unique up to unique isomorphism.Once fixed, they determine the morphisms in the top horizontal and left vertical exact triangle uniquely, by Lemma 5.1.

Extension closed distinguished abelian subcategories
As mentioned in the introduction, unlike hearts of t-structures, distinguished abelian subcategories in a triangulated category need not be extension closed -see Proposition 7.8 below.We develop criteria for a distinguished abelian subcategory D to be extension closed in a triangulated category C. Since we will compare the shift functor on C to the shift operator on D, we specify in this section shift functors of triangulated categories.
Let (C, Σ) be a k-linear triangulated category and let D be a distinguished abelian subcategory of C such that D ∼ = mod(D) for some finite-dimensional k-algebra D. This hypothesis ensures that D has enough injective objects.We are going to compare Ext n C (U, V ) = Hom C (U, Σ n (V )) and Ext n D (U, V ), where n ≥ 0. For n = 0 these two spaces are equal since D is full in C. We investigate the case n = 1.In order to calculate Ext 1 D , we will make use of the usual shift operator Σ D on D, defined as follows.For each object U in D choose a (minimal) injective envelope ι U : U → I U in D, and set Σ D (U ) = coker(ι U ).That is, we have a short exact sequence in D of the form Let (C, Σ) be a k-linear triangulated category and let D be a distinguished abelian subcategory of C such that D ∼ = mod(D) for some finite-dimensional k-algebra D. For each object U in D and each nonnegative integer n, define a morphism in C inductively as follows.We set σ 0,U = Id U , assuming implicitly that Σ 0 D (resp.Σ 0 ) is the identity operator (resp.identity functor) on D (resp.C).We define as the unique morphism such that the triangle is exact.For n ≥ 2, we define Let (C, Σ) be a k-linear triangulated category and let D be a distinguished abelian subcategory of C such that D ∼ = mod(D) for some finite-dimensional k-algebra D. For any two objects U , V in D, the morphism with notation chosen such that I 0 = I V , and Im(δ 0 ) = ker(δ 1 ) = Σ D (V ).By definition, Ext 1 D (U, V ) is the degree 1 cohomology of the cochain complex obtained from applying Hom D (U, −) to the above injective resolution of V , of the form ) is contravariantly functorial in U .In order to show functoriality in V , let ψ : V → W be a morphism in D. Then ψ extends to a morphism I V → I W , and hence there is a commutative diagram of exact triangles The morphism τ depends on the choice of an extension of ψ to Proof.The hypotheses and Theorem 7.2 imply that Ext 1 D (U, V ) = 0 for any two finitely generated D-modules U , V , whence the result.
See Dugas [10] for torsion pairs and mutation in stable module categories of selfinjective algebras, as well as the references therein.The next result is a criterion when the canonical maps Ext C (U, V ) for all objects U , V in D. Proof.We adjust the notation slightly in order to ensure that the terms in exact triangles in the proof below appear in alphabetical order; that is, we consider Ext Since D is extension closed, it follows that V can be chosen to belong to D (possibly after replacing V by an isomorphic object).The morphism U → V belongs then to D, and although it need not be a monomorphism in D, it is a direct sum of a monomorphism and a zero morphism, by Proposition 6.1.Thus the morphism ι U : U → I U extends to a morphism V → I U , and hence there exists a morphism of exact triangles This shows that ψ is the image of the class represented by ϕ under the map Ext 1 D (W, U ) → Ext 1 C (W, U ) induced by composition with σ 1,U , and so the latter map is surjective.Suppose conversely that the map Then there is ϕ ∈ Hom D (W, Σ D (U )) such that the rectangle This in turn can be completed to morphism of exact triangles Completing the previous short exact sequence to an exact triangle and applying the functor Hom C (U, −) yields a long exact sequence of Ext C -spaces.Since Ext 1 D (U, Y ) = {0} by the hypotheses, this long exact sequence yields in particular an exact sequence The first three terms coincide with the first three nonzero terms in the previous 4-term exact sequence because D is a full subcategory of C, and hence the same is true for the fourth terms.By construction, the isomorphism Ext 1 D (U, V ) ∼ = Ext 1 C (U, V ) arising in this way is induced by σ 1,V .The result follows from Theorem 7.4.subcategory of mod(A) by Theorem 3.1.If also I ⊆ J(A), then Ext 1 A (A/I, A/I) = {0} by (i).Since A/I is selfinjective, it follows from Theorem 7.5 that mod(A/I) is not extension closed in mod(A) Remark 7.9.Let A be a finite-dimensional k-algebra, and let I be a nonzero proper ideal in A such that A/I is selfinjective.Then Ext kG (kG/N, kG/N ) is nonzero.It follows from Theorem 7.5 that mod(kG/N ) is not extension closed in mod(kG).
Corollary 7.12.Let k be a field of prime characteristic p, let G be a finite group, and let Q be a nontrivial normal p-subgroup of G. Then the canonical image of mod(kG/Q) in mod(kG) is a distinguished abelian subcategory which is not extension closed.
Proof.Since O p (Q) is trivial but Q is not, this is a special case of Proposition 7.11.
Remark 7.13.A result in [11] gives a sufficient criterion when an extension closed exact subcategory D of a triangulated category C has the property that Ext 1 D and Ext 1 C are isomorphic as bifunctors on D.

Selfinjective distinguished abelian subcategories
Let A be a finite-dimensional k-algebra and Y a finitely generated A-module.We denote by mod Y (A) the full k-linear subcateory of mod(A) of all A-modules which are isomorphic to Im(ϕ) for some ϕ ∈ End A (Y m ) and some positive integer m.We denote by add(Y ) the full additive subcategory of mod(A) of modules which are isomorphic to finite direct sums of direct summands of Y .Clearly mod Y (A) contains add(Y ).By a result of Cabanes [8,Theorem 2], if E = End A (Y ) is selfinjective, then the canonical functor Hom A (Y, −) : mod(A) → mod(E op ) restricts to a klinear equivalence mod Y (A) ∼ = mod(E op ).We use the results and methods from Cabanes [8] to identify in a similar vein the distinguished abelian subcategories constructed earlier in Theorem 2.5.We denote by mod Y (A) the image of mod Y (A) in mod(A).If Y has no nonzero projective direct summand, then no module in mod Y (A) has a nontrivial projective summand, and hence the canonical functor mod Y (A) → mod Y (A) induces a bijection on isomorphism classes of objects.As mentioned earlier, not every distinguished abelian subcategory of mod(A) is of the form as described in Theorem 8.1, since any selfequivalence of mod(A) as a triangulated category induces a permutation on distinguished abelian subcategories which need not preserve the distinguished abelian subcategories of the form as described in Theorem 8.1.Note that the hypothesis End pr A (Y ) = {0} implies that Y has no nonzero projective direct summand.Therefore, the second statement of Theorem 8.1 implies that the isomorphism classes of indecomposable summands of the A-module Y are determined by mod Y (A).Remark 8.3.Corollary 8.2 does not imply that a distinguished abelian subcategory is necessarily determined by a progenerator -the same object in C could be a progenerator of several distinguished abelian subcategories.This Corollary only asserts that the distinguished abelian subcategories obtained as in Theorem 8.1 are determined by their progenerators.See the Example 5.9.
We use the following notation from [8].For any k-algebra A, any two A-modules Y , U , and and this map is obviously surjective.To show that this map is also injective, consider the diagram where the vertical maps are induced by the inclusions M ⊆ Hom A (Y, Y n ) and M • Y ⊆ Y n , and where the bottom horizontal map is the obvious evaluation map.A trivial verification shows that this diagram commutes.Since Hom A (Y, Y n ) is a free D-module of rank n, it follows that the bottom horizontal map is an isomorphism.Since Y is projective as a right D-module by the assumptions, it follows that the left vertical map is injective.This implies that Ψ is injective, whence (i).Since E, hence D, is self-injective, we may assume that the D-module M in (ii) is a submodule of Hom A (Y, Y n ) for some positive integer n.Applying the functor Hom A (Y, −) to the isomorphism Ψ yields an isomorphism By [8, Lemma 5], the right side in this isomorphism is equal to M (this is an equality of subsets of Hom A (Y, Y n )).The inverse of this isomorphism is the map described in (ii).For (iii), observe first that the map Φ is surjective, since U is in mod Y (A), hence a quotient of a finite direct sum of copies of Y .For the injectivity, again since U is in mod Y (A), hence isomorphic to a submodule of Y n for some positive integer n, it follows that there is a commutative diagram of the form where the right vertical map is injective.The left vertical map is then injective, too, since Y is projective as an E op -module, and the bottom horizontal map is an isomorphism.This shows that Φ is injective, whence (iii).One can prove (iii) also by applying (i) and (ii) with U = M • Y .Statement (iv) follows from (ii) and (iii).Statement (v) is an immediate consequence of (iv).Statement (vi) follows from Lemma 3.5.
Proof of Theorem 8.1.In the situation of Theorem 8.1, Cabanes' linear equivalence from [8, Theorem 2] is an equivalence of abelian categories, by Proposition 8.4 (iv).By construction, mod Y (A) is the image in mod(A) of mod Y (A).Thus we need to show that the inclusion mod Y (A) ⊆ mod(A) composed with the canonical functor mod(A) → mod(A) is still a full embedding.By Proposition 8.4 (vi) we have Hom pr A (U, V ) = {0} for any two A-modules U , V in mod Y (A).This implies that the canonical functor mod Y (A) → mod(A) is a full embedding.Since exact triangles in mod(A) are induced by short exact sequences in mod(A), it follows that the image mod Y (A) of mod Y (A) in mod(A) is a distinguished abelian subcategory in mod(A).Thus Theorem 8.1 follows from Proposition 8.4.

Proof of Corollary 8.2. Note that the hypothesis End pr
A (Y ) = {0} implies that Y has no nonzero projective direct summand; similarly for Y ′ .Thus the linear subcategories add(Y ) and add(Y ′ ) of mod(A) are equal if and only if their images in mod(A) are equal.This equality is clearly equivalent to mod Y (A) = mod Y ′ (A), whence the result.

Examples and further remarks
The following example illustrates that Theorem 2.5 and Theorem 8.1 cover some cases not covered by Theorem 3.1.Example 9.2.Let A be a split finite-dimensional selfinjective k-algebra.Let n be a positive integer and let {X i | 1 ≤ i ≤ n} be a set of A-modules which are pairwise orthogonal in mod(A); that is, End A (X i ) ∼ = k and Hom A (X i , X j ) = {0}, where 1 ≤ i, j ≤ n, i = j.Set Y = ⊕ n i=1 X i .Then End A (Y ) is a commutative split semisimple k-algebra, and the image of add(Y ) is a semisimple distinguished abelian subcategory of mod(A), equivalent to mod(End A (Y )).See for instance [28], [29], [31], [17], [10] for more details on orthogonal sets of modules in mod(A).
Example 9.3.Let A be a finite-dimensional selfinjective k-algebra.If A has two nonisomorphic simple modules S, T such that dim k (Ext 1 A (S, T )) ≥ 2 and if k is infinite, then A has infinitely many pairwise non-isomorphic uniserial modules of length 2 with composition factors S and T , from top to bottom.The A-endomorphism algebra of any such module is 1-dimensional, and there is no nonzero A-homomorphism between any two non-isomorphic uniserial modules with these composition factors.Therefore the full additive subcategory of mod(A) generated by these modules is a semisimple distinguished abelian subcategory with infinitely many isomorphism classes of simple objects.
The next example shows that the hypothesis on D to contain all simple A-modules in the statement of Theorem 3.8 is necessary.
Example 9.4.Suppose that k is an algebraically closed field of characteristic 5. Consider the algebra A = kD 10 ∼ = k(C 5 ⋊ C 2 ).This is a Nakayama algebra with two nonisomorphic simple modules S, T and uniserial projective indecomposable modules of length 5. Let U be a uniserial module of length 2 with composition factors S and T .Then End A (U ) = End A (U ) ∼ = k.Thus the finite direct sums of modules isomorphic to U form a semisimple distinguished abelian subcategory D of mod(A) in which U is up to isomorphism the unique simple object.We have soc 2 (A) = r(J(A) 2 ) ⊆ J(A) 2 .Thus mod(A/J(A) 2 ) is a distinguished abelian subcategory of mod(A) containing the simple A-modules S and T and all uniserial modules of length 2. In particular, mod(A/J(A) 2 ) contains the subcategory D, but the simple object U in D does not remain simple in mod(A/J(A) 2 ).
Let A be a symmetric k-algebra and I a proper ideal in A. By a result of Nakayama [24,Theorem 13], the quotient algebra A/I is symmetric if and only if I = ann(z) for some z ∈ Z(A).In that case, if s is a symmetrising form on A, then z • s has kernel I and induces a symmetrising form on A/I.The following Proposition shows that the elements z ∈ Z(A) satisfying z 2 = 0 parametrise the symmetric quotients A/I of A satisfying End pr A (A/I) = {0} through the correspondence z → ann(z).The Tensor-Hom adjunction induces a natural transformation of bifunctors at the level of stable categories, but this need not be an isomorphism (cf.Proposition 5.6).

Lemma 2 . 3 .
Let A be a finite-dimensional k-algebra and Y a finitely generated A-module such that End pr A (Y ) = {0}.Set D = End A (Y ) op .The following hold.(i) Let m, n be positive integers and let U , V be quotients of the A-modules Y m , Y n , respectively.Then Hom pr A (U, V ) = {0}.(ii) For any two finitely generated D-modules M , N we have Hom pr A (Y ⊗ D M, Y ⊗ D N ) = {0}.Proof.With the assumptions in (i), there are surjective A-homomorphisms α : Y m → U and β : Y n → V .Let ψ : U → V be an A-homomorphism which factors through a projective A-module P .Let γ : U → P and δ : P

Lemma 2 . 4 .
Let A, D be finite-dimensional k-algebras, and let Y be a finitely generated A-Dbimodule.The functor Y ⊗ D − : mod(D) → mod(A) is a full k-linear embedding if and only if the adjunction unit

Theorem 2 . 5 .
Let A be a finite-dimensional selfinjective k-algebra.Let Y be a finitely generated A-module.Suppose that End A (Y ) is selfinjective.Set D = End A (Y ) op , and regard Y as an A-Dbimodule.The following are equivalent.(i) The functor Y ⊗ D − : mod(D) → mod(A) is a full exact embedding and induces an embedding of mod(D) as a distinguished abelian subcategory in mod(A).(ii) We have End pr A (Y ) = {0}, and Y is projective as an End A (Y )-module.

Remark 2 . 11 .
Let A be a finite-dimensional selfinjective k-algebra, D a finite-dimensional kalgebra, and Y an A-D-bimodule which is finitely generated projective as a right D-module.Then the functor mod(D) → mod(A) induced by Y ⊗ D − extends to a functor of triangulated categories D b (mod(D)) → mod(A).Indeed, since Y is finitely generated projective as a right D-module, it follows that Y ⊗ D − induces a functor D b (mod(D)) → D b (mod(A)).Composed with the canonical functor D b (mod(A)) → mod(A) from [30, Theorem 2.1] or [7, Theorem 4.4.1],this yields a functor D b (mod(D)) → mod(A) as stated.

Theorem 3 . 1 .
Let A be a finite-dimensional selfinjective k-algebra and let I be a proper ideal in A. The following statements are equivalent.(i) The composition of canonical functors mod(A/I) → mod(A) → mod(A) is an embedding of mod(A/I) as a distinguished abelian subcategory in mod(A).(ii) The ideal I contains its right annihilator r(I).(iii) We have End pr A (A/I) = {0}.(iv) For any two finitely generated A/I-modules U , V , we have Hom pr A (U, V ) = {0}.(v) We have r(I) 2 = {0}.

Lemma 3 . 5 .
Let A be a finite-dimensional k-algebra and let I be a proper ideal in A. Suppose that r(I) ⊆ I. Then for any two A/I-modules U , V we have Hom pr A (U, V ) = {0} .Proof.Set Y = A/I, regarded as an A-A/I-bimodule.Then End pr A (Y ) = {0} by Lemma 3.4, and we have End A (Y ) ∼ = (A/I) op , hence D = End A (Y ) op ∼ = A/I.Using this isomorphism, if U is an A/I-module, then Y ⊗ A/I U = A/I ⊗ A/I U ∼ = U , regarded as an A-module via the canonical surjection A → A/I.The result follows from Lemma 2.3.Lemma 3.6.Let A be a finite-dimensional selfinjective k-algebra, and let I be an ideal in A. The following are equivalent.(i) We have r(I) ⊆ I. (ii) We have l(I) ⊆ I. (iii) We have r(I) 2 = 0. (iv) We have l(I) 2 = 0.

Corollary 3 . 10 .
Let A be a finite-dimensional selfinjective k-algebra such that all simple Amodules are nonprojective.Then mod(A) has a semisimple distinguished abelian subcategory D such that ℓ(D) is finite and such that for any distinguished abelian subcategory D ′ of mod(A) containing D we have ℓ(D ′ ) = ℓ(D).Proof.Let D be the full subcategory of mod(A) consisting of all semisimple modules in mod(A).The result follows from Theorem 3.8.The distinguished abelian subcategories of mod(A) of the form mod(A/I) in Theorem 3.1 have the property that the simple objects in mod(A/I) remain simple in mod(A).The next result explores the question under what circumstances a distinguished abelian subcategory of mod(A) whose simple objects correspond to simple A-modules is of the form mod(A/I) for some ideal I in A. Theorem 3.11.Let A be a finite-dimensional selfinjective k-algebra, and let D be a finitedimensional k-algebra.Let Y be a finitely generated A-D-bimodule such that Y is projective as a right D-module and such that Y has no nonzero projective direct summand as a left A-module.Suppose that for any simple D-module T the A-module Y ⊗ D T is simple and that the functor Y ⊗ D − induces a full embedding of mod(D) as a distinguished abelian subcategory of mod(A).Let I be the annihilator in A of Y as a left A-module.Then r(I) ⊆ I, and Y ⊗ D − induces an equivalence mod(D) ∼ = mod(A/I).
D A is a full embedding.(The inclusion A → D is thus an epimorphism in the category of rings; see Stenström [33, Chapter XI, Proposition 1.2].)Lemma 3.13.Let A be a finite-dimensional selfinjective algebra over a field k, let D be a finitedimensional k-algebra, and let Y be a finitely generated A-D-bimodule.Suppose that the functor Y ⊗ D − induces a full embedding mod(D) → mod(A).Then, for any finitely generated D-module V , the map V → Hom A (Y, Y ⊗ D V ) induced by the adjunction unit is an isomorphism of D-modules.In particular, the functor Hom A (Y, −) : mod(A) → mod(D) is a left inverse of the embedding mod(D) → mod(A) induced by Y ⊗ D −. Proof.By the assumptions, for any two finitely generated D-modules U , V the map Hom D (U, V ) → Hom A (Y ⊗ D U, Y ⊗ D V ) induced by the functor Y ⊗ D − is an isomorphism.Specialising this isomorphism to U = D and combining it with the canonical isomorphism V ∼ = Hom D (D, V ) yields the result.

Lemma 3 . 14 .
Let A, D be finite-dimensional k-algebras and let Φ : mod(D) → mod(A) be a full exact embedding sending simple D-modules to simple A-modules.Then Φ is isomorphic to a functor of the form Y ⊗ D − : mod(D) → mod(A), where Y is an A-D-bimodule which is a progenerator as a right D-module.Moreover, if I is the annihilator in A of Y as a left A-module, then Φ factors through an equivalence Ψ : mod(D) ∼ = mod(A/I) and the inclusion functor mod(A/I) → mod(A).Proof.The first statement is a special case of the Eilenberg-Watts Theorem: since Φ is a full exact embedding, it is induced by tensoring with an A-D-bimodule Y which is flat as a right D-module.Since this is a functor between finite-dimensional module categories, preserving simple modules, it follows that Y is a progenerator as a right D-module.Thus D is Morita equivalent to D ′ = End D op (Y ), via the functor from mod(D) to mod(D ′ ) induced by Y ⊗ D −, with Y here regarded as a D ′ -D-bimodule.The action of A on Y induces an algebra homomorphism A → D ′ .Let I be the annihilator of Y in A. Then the algebra homomorphism A/I → D ′ induced by the action of A on Y is injective.The functor Φ is the composition of the Morita equivalence Y ⊗ D − : mod(D) → mod(D ′ ) followed by the restriction functor along the injective algebra homomorphism A/I → D ′ .By the assumptions, Φ preserves simple modules.Since Y ⊗ D − induces an equivalence mod(D) ∼ = mod(D ′ ) it induces in particular a bijection between isomorphism classes of simple D-modules and simple D ′ -modules.It follows that simple D ′ -modules restrict to simple A/I-modules.Lemma 3.12 implies that A/I ∼ = D ′ .Thus Φ factors through an equivalence mod(D) → mod(A/I) as stated.
for all finitely generated D-modules V .Applied to V = D this implies that End pr A (Y ) = {0}.By the first paragraph, this also implies that the canonical map V → Hom A (Y, Y ⊗ D V ) is an isomorphism for all V .By Theorem 2.1, the functor Y ⊗ D − induces a full embedding mod(D) → mod(A), and by the assumptions, this embedding sends simple D-modules to simple A-modules.Since I is the annihilator in A of Y , it follows from Lemma 3.14, that the full embedding Y ⊗ D − : mod(D) → mod(A) factors through an equivalence mod(D) ∼ = mod(A/I).By the assumptions, the functor Y ⊗ D − induces a full embedding mod(D) → mod(A).Thus the inclusion mod(A/I) → mod(A) induces a full embedding mod(A/I) → mod(A) as distinguished abelian subcategory.The inclusion r(I) ⊆ I follows from Theorem 3.1, whence the result.
g. [21, Section 1.11].Theorem 4.1.Let k be a field of prime characteristic p and A a finite-dimensional selfinjective k-algebra.Suppose that Z(A) × has a nontrivial finite p-subgroup Z such that A is projective as a kZ-module.Set I = I(kZ) • A, where I(kZ) is the augmentation ideal of kZ.Then I contains its right annihilator in A. In particular, restriction along the canonical surjection A → A/I induces a full embedding mod(A/I) → mod(A) of mod(A/I) as a distinguished abelian subcategory in mod(A).Proof.The right annihilator of I(kZ) in kZ is the 1-dimensional ideal soc(kZ) = ( z∈Z z) • kZ, and we have soc(kZ) ⊆ I(kZ).Since A is a free left or right kZ-module, an easy argument shows that the right annihilator of I(kZ) • A = A • I(kZ) is therefore soc(kZ) • A = A • soc(kZ), which is contained in I(kZ) • A. Thus the statement is the special case of Theorem 3.1 with I = I(kZ) • A.

Theorem 4 . 2 .
Let k be a field of prime characteristic p and G a finite group.Let N be a normal subgroup of G. Restriction along the canonical surjection G → G/N induces a full embedding mod(kG/N ) → mod(kG) of mod(kG/N ) as a symmetric monoidal distinguished abelian subcategory in mod(kG) if and only if p divides the order of N .Proof.The fact that the functor mod(kG/N ) → mod(kG) is a functor of symmetric monoidal categories is obvious (see the remarks preceding the Theorem).We need to show that this induces an embedding as a distinguished abelian subcategory in mod(kG) if and only if |N | is divisible by p.The kernel of the canonical algebra homomorphism kG → kG/N is equal to I = kG • I(kN ), where I(kN ) is the augmentation ideal of kN .Arguing as in the previous proof, the right annihilator of I(kN ) in kN is the 1-dimensional ideal ( y∈N y)kN .This is contained in I(kN ) if and only if p divides |N |.Indeed, if p divides |N |, then y∈N y = y∈N (y − 1) ∈ I(kN ).If p does not divide |N |, then ( y∈N y)kN is a complement of I(kN ) in kN .Since kG is free as a right kN -module of rank |G : N |, it follows that the right annihilator of I is equal to kG • ( y∈N y).Therefore, if p divides |N |, then the right anihilator of I is contained in I by the previous argument.The result follows in that case from Theorem 3.1.If |N | is prime to p, then kG/N is a projective kG-module, so End kG (kG/N ) vanishes, and in particular, the canonical functor mod(kG/N ) → mod(kG) is not an embedding.
(iii) If Φ is induced by a functor Y ⊗ D − for some finitely generated kP -D-bimodule Y which is projective as a right D-module, then there is an ideal I of kP containing its right annihilator in kP such that D ∼ = (kP )/I and such that Φ induces an equivalence between mod(D) and the distinguished abelian subcategory V ⊗ k mod((kP )/I).The first statement of Theorem 4.4 holds slightly more generally, based on the following observation which is a consequence of the proof of[19, Theorem 3.4].Slightly extending standard terminology from finite-dimensional algebras, if C is a k-linear triangulated category, then a distinguished abelian subcategtory D of C is called split if for every object X in D which is simple as an object of D we have End D (X) ∼ = k.Lemma 4.5.Let p be a prime, P a nontrivial finite p-group, and suppose that char(k) = p.Let D be a split distinguished abelian subcategory of mod(kP ) such that D has a simple object.Then ℓ(D) = 1.Proof.By the assumptions on D we have ℓ(D) ≥ 1 (we include here by convention the case where D has infinitely many isomorphism classes of simple objects).Arguing by contradiction, suppose that ℓ(D) ≥ 2. Thus D has two nonisomorphic simple objects S, T .Since D is a full subcategory of mod(kP ), the objects S, T remain nonisomorphic in mod(kP ).Since D is split, we have End kP (S) ∼ = k ∼ = End kP (T ), and we have Hom kP (S, T ) = {0} = Hom kP (T, S).It is shown in the proof of [19, Theorem 3.4] that this is not possible.Proof of Theorem 4.4.Denote by D the distinguished abelian subcategory of mod(kP ) obtained from taking the closure under isomorphisms in mod(kP ) of the image of the embedding Φ : mod(D) → mod(kP )

Proposition 5 . 1 .
Let C be a triangulated category and let D be a distinguished abelian subcategory of C. Let0 / / X f / / Y g / / Z / / 0be a short exact sequence in D. There is a unique morphism

Remark 5 . 2 .Proposition 5 . 3 .
The definition of a distinguished abelian subcategory D of a triangulated category C does not require D to be closed under isomorphisms in C. One easily checks that the closure of D under isomorphisms in C is again a distinguished abelian subcategory of C which is equivalent to D as an abelian category.Let C be a triangulated category and let D be a distinguished abelian subcategory of C. Let 0 / / X f / / Y g / / Z / / 0 be a short exact sequence in D and let W be an object in C. If W belongs to D, then the maps Hom C (Σ(W ), Y ) / / Hom C (Σ(W ), Z)

Corollary 5 . 4 .
Let C be a triangulated category and let D be a distinguished abelian subcategory of C. Let W be an object in D such that Σ(W ) is an object in D. Then W is injective in D and Σ(W ) is projective in D. Proof.By Proposition 5.3, if g : Y → Z is an epimorphism in D, then every morphism Σ(W ) → Z lifts through g.Since Σ(W ) belongs to D, it follows that Σ(W ) is projective in D. Similarly, by Proposition 5.3 (applied with Σ(W ) instead of W ), if f : X → Y is a monomorphism in D, then every morphism X → W factors through f , and hence W is injective in D. Proposition 5.5.Let C be a triangulated category and let D be a distinguished abelian subcategory of C. Let 0 / / X f / / Y g / / Z / / 0 be a short exact sequence in D and let W be an object in C. (i) If W is a projective object in D, then the map Hom C (W, Σ(X)) → Hom C (W, Σ(Y )) induced by composition with Σ(f ) is injective.(ii) If W is an injective object in D, then the map Hom C (Z, Σ(W )) → Hom C (Y, Σ(W ))

Proposition 5 . 6 .
Let C be a triangulated category and D a distinguished abelian subcategory.If the inclusion functor D ⊆ C has a left adjoint or a right adjoint as an additive functor, then D is split.

Corollary 6 . 2 .
Let C be a triangulated category, D a distinguished abelian subcategory of C, and T a thick subcategory of C. Suppose that T ⊆ D. Then all objects in T are projective and injective in D, and T is split.
Hom D (U, I 0 ) / / Hom D (U, I 1 ) / / Hom D (U, I 2 ) / / • • • where the first two maps are induced by composing with δ 0 and δ 1 , respectively.A morphism ϕ : U → I 1 is in the kernel of the second map if and only if it factors through ker(δ 1 ) = Im(δ 0 ) = Σ D (V ).Thus the kernel of the map Hom D (U, I 1 ) / / Hom D (U, I 2 ) can be identified with Hom D (U, Σ D (V )), and hence Ext 1 D (U, V ) is the cokernel of the map Hom D (U, I 0 ) / / Hom D (U, Σ D (V )) induced by composition with δ 0 .Applying the functor Hom C (U, −) to the exact triangle Taking the quotient of the middle term by the image of the left term yields a monomorphism Ext D (U, V ) → Hom C (U, Σ(V )) = Ext 1 C (U, V ) as stated.We need to show the naturality.Since this map is defined by applying the functor Hom C (U, −) to the above diagram, and since the Yoneda embedding U → Hom C (U, −) is contravariantly functorial in U , it follows immediately that the map Ext D τ ′ are two morphisms making the above diagram commutative, then σ 1,W • (τ − τ ′ ) = 0, and hence τ − τ ′ factors through the morphism I W → Σ D (V ) in the diagram.Thus applying Hom D (U, −) to τ − τ ′ induces the zero map Ext 1 D (U, V ) → Ext 1 D (U, W ), showing the functoriality in V .This proves the result.The following immediate consequence of Theorem 7.2 shows that D-mutation pairs (cf.[14, Definition 2.5]), with D a distinguished abelian subcategory equivalent to the module category of a finite-dimensional algebra, arise only if D is semisimple.Corollary 7.3.Let (C, Σ) be a k-linear triangulated category and let D be a distinguished abelian subcategory of C such that D ∼ = mod(D) for some finite-dimensional k-algebra D. Suppose that Ext 1 C (X, Y ) = 0 for all X, Y in D. Then the k-algebra D is semisimple.

1 DTheorem 7 . 4 .
(U, V ) → Ext 1 C (U, V ) in the previous Theorem yield an isomorphism of bifunctors on D. Let (C, Σ) be a k-linear triangulated category and let D be a distinguished abelian subcategory of C such that D ∼ = mod(D) for some finite-dimensional k-algebra D. The category D is extension closed in C if and only if the morphisms σ 1,V induce isomorphisms Ext 1 D (U, V ) ∼ = Ext 1

1 C
(W, U ) instead of Ext 1 C (U, V ).Suppose first that D is extension closed in C. Let U , W be objects in D, and let ψ : W → Σ(U ) be a morphism in C; that is, ψ ∈ Ext 1 C (W, U ). Itsuffices to show that there exists a morphism ϕ : W → Σ D (U ) in C (which is then automatically in D as D is full, thus representing an element in Ext 1 D (W, U )) such that ψ = σ 1,U • ϕ.Complete ψ to an exact triangle in C of the form and this forces V ′ ∼ = V in C.This completes the proof.If I is an injective module over a finite-dimensional k-algebra D, then Ext 1 D (U, I) = {0} for any D-module U .Therefore, if a triangulated category C has an extension closed distinguished abelian subcategory equivalent to mod(D), then Ext 1 C (U, I) must also vanish thanks to the previous Theorem (where we identify U , I to their images in C).It turns out This yields the following characterisation of extension closed distinguished abelian subcategories which are equivalent to mod(D).Theorem 7.5.Let (C, Σ) be a k-linear triangulated category and let D be a distinguished abelian subcategory of C such that D ∼ = mod(D) for some finite-dimensional k-algebra D. Then D is extension closed if and only if for any two objectsU , Y in D such that Y is injective in D we have Ext 1 C (U, Y ) = {0}.Proof.Suppose that D is extension closed.Let U , Y be objects in D such that Y is injective in D. Then Ext 1 D (U, Y ) = {0}.Theorem 7.4 implies Ext 1 C (U, Y ) = {0}.Conversely, suppose that Ext 1 C (U, Y ) = {0} for any two objects U , Y in D such that Y is injective in D. By Theorem 7.4, it suffices to show that there is an isomorphism Ext 1 D (U, V ) ∼ = Ext 1 C (U, V ) inducedby σ 1,V , for any two objects U , V in D. Let U , V be objects in D. Consider a short exact sequence in D of the form 0 / / V ιV / / Y / / Σ D (V ) / / 0 for some injective object Y in D. Applying Hom D (U, −) yields a long exact sequence of Ext Dspaces.Since Ext 1 D (U, Y ) = {0}, this long exact sequence yields in particular an exact 4-term sequence 0 / / Hom D (U, V ) / / Hom D (U, Y ) / / Hom D (U, Σ D (V )) / / Ext 1 D (U, V ) / / 0

1 A
(A/I, A/I) = {0} if and only if Ext 1 A (U, A/I) = {0} for every finitely generated A/I-module.Indeed, if Ext 1 A (A/I, A/I) = {0}, then Ext 1 A (Y, Y ) = {0} for any finitely generated projective A/I-module Y .Applying the functor Hom A (−, U ) to a short exact sequence of the form0 / / V / / Y / / U / / 0for some free A/I-module Y of finite rank yields a long exact sequence starting0 / / Hom A/I (U, Y ) / / Hom A/I (Y, Y ) / / Hom A/I (V, Y ) / / Ext 1 A (U, Y ) / / 0 .Since Y is also injective as an A/I-module, the mapHom A/I (Y, Y ) → Hom A/I (V, Y ) is surjective.Thus Ext 1 A (U, Y ) = 0, and hence Ext 1 A (U, A/I) = {0}.The converse is obvious.We will need the following well known fact; we sketch a proof for convenience.Lemma 7.10.Let H be a finite group.Denote by I(kH) the augmentation ideal of kH, regarded as left kH-module, and by Hom(H, k) the k-vector space of group homomorphisms from H to the additive group ofg k.We have a canonical k-linear isomorphism Hom kH (I(kH), k) ∼ = Hom(H, k) sending a kH-homomorphism α : I(kH) → k to the group homomorphism β : H → k defined by β(y) = α(y − 1) for all y ∈ H. Proof.We check first that β is a group homomorphism.Let y, z ∈ H. Then yz − 1 = y(z − 1) + (y − 1), hence β(yz) = yα(z − 1) + α(y − 1) = β(y) + β(z), where we use that y acts as identity on k.The map α → β is therefore well-defined, k-linear, and injective.For the surjectivity, let β : H → k be a group homomorphism.We need to show that the k-linear map α : I(kH) → k sending y − 1 to β(y) is a kH-homomorphism.Let y, z ∈ H.We have α(z(y − 1)) = α(zy − 1 − (z − 1)) = β(zy) − β(z) = β(zyz −1 ) = β(y) = α(y − 1) = zα(y − 1), whence the result.The following result proves the last statement in Theorem 1.4.Proposition 7.11.Let k be a field of prime characteristic p, let G be a finite group, and let N be a normal subgroup of order divisible by p in G.If O p (N ) is a proper subgroup of N , then the canonical image of mod(kG/N ) in mod(kG) is a distinguished abelian subcategory which is not extension closed.Proof.The fact that mod(kG/N ) embeds as a distinguished abelian subcategory into mod(kG) follows from Theorem 4.2.The kernel of the canonical algebra homomorphism kG → kG/N is equal to I = kG • I(kN ), where I(kN ) is the augmentation ideal of kN .Since kG is free as a right kN -module, we have I ∼ = kG ⊗ kN I(kN ) as left kG-modules.Thus Hom kG (I, kG/I) ∼ = Hom kG (kG ⊗ kN I(kN ), kG/N ) ∼ = Hom kN (I(kN ), kG/N ) , where the last isomorphism uses Frobenius reciprocity.Since N acts trivially on kG/N on the left, it follows that the space Hom kN (I(kN ), kG/N ) is nonzero if and only if Hom kN (I(kN ), k) is nonzero.By Lemma 7.10 his space is equal to the space Hom(N, k) of group homomorphisms from N to the additive group k.Since k has characteristic p, this implies that Hom k (N, k) is trivial if and only if O p (N ) = N .Thus if O p (N ) is a proper subgroup of N , then by Lemma 7.7, the space Ext 1

Theorem 8 . 1 .
Let A be a finite-dimensional self-injective k-algebra.Let Y be a finitely generated A-module such that the algebra E = End A (Y ) is selfinjective.Suppose that End pr A (Y ) = {0} and that that Y is projective as an E-module.Set D = E op .(i) The functor Y ⊗ D − : mod(D) → mod(A) induces a full embedding Φ Y : mod(D) → mod(A) of mod(D) as a distinguished abelian subcategory in mod(A), and moreover Φ Y induces an equivalence of abelian categories mod(D) ∼ = mod Y (A).(ii) The A-module Y , regarded as an object in the abelian category mod Y (A), is a progenerator of mod Y (A).(iii) The canonical functor mod(A) → mod(A) induces an isomorphism of abelian categories mod Y (A) ∼ = mod Y (A).

Corollary 8 . 2 .
Let A be a finite-dimensional self-injective k-algebra.Let Y , Y ′ be finitely generated A-modules which both satisfy the hypotheses on Y in Theorem 8.1.Then mod Y (A) = mod Y ′ (A) if and only if add(Y ) = add(Y ′ ) in mod(A).
spanned by the sum of the images of the A-homomorphisms in M .Setting E = End A (Y ), we consider Y as an A-E op -bimodule in the obvious way.The following Proposition collects the technicalities for the proof of Theorem 8.1.Proposition 8.4.Let A be a finite-dimensional self-injective k-algebra and Y a finitely generated A-module such that E = End A (Y ) is selfinjective.Suppose that Y is projective as an E-module.Set D = E op .(i)Let n be a positive integer and let M be anE op -submodule of Hom A (Y, Y n ).The canonical A-homomorphism Ψ : Y ⊗ D M → M • Ysending y ⊗ µ to µ(y), where y ∈ Y and µ ∈ M , is an isomorphism.In particular, Y ⊗ D M belongs to mod Y (A).(ii) Let M be a finitely generated D-module.The canonical D-homomorphism M → Hom A (Y, Y ⊗ D M ) sending m ∈ M to the map y → y ⊗ m for y ∈ Y is an isomorphism.(iii) Let U be an A-module contained in mod Y (A).The canonical evaluation map Φ : Y ⊗ D Hom A (Y, U ) → U sending y ⊗ η to η(y), where y ∈ Y and η ∈ Hom A (Y, U ), is an isomorphism.(iv) The category mod Y (A) is an abelian subcategory of mod(A), and the functor Hom A (Y, −) : mod(A) → mod(D) restricts to an equivalence of abelian categories mod Y (A) ∼ = mod(D) with an inverse induced by the functor Y ⊗ D − : mod(D) → mod(A).(v) The equivalence mod Y (A) ∼ = mod(D) in (iv) sends Y to the regular D-module D. In particular, Y is a progenerator of the category mod Y

Example 9 . 1 .
Let A be a finite-dimensional selfinjective k-algebra.Let Y be a nonprojective uniserial A-module of length 2 with two non-isomorphic simple composition factors S and T .Then End A (Y ) ∼ = End A (Y ) ∼ = k, and hence mod Y (A) = add(Y ) is abelian semisimple, but is not the category of a quotient of A. Indeed, such a quotient algebra would have to be semisimple, but mod Y (A) contains no simple A-module, because neither the simple quotient S of Y nor the simple submodule T of Y are contained in mod Y (A).There are trivial examples of distinguished abelian subcategories beyond those constructed in Theorem 2.5 and Theorem 8.1.

Proposition 9 . 5 .Remark 9 . 9 .
Let A be a symmetric k-algebra and I a proper ideal in A such that I = ann(z) for some z ∈ Z(A).We have End pr A (A/I) = {0} if and only if z 2 = 0.homomorphism Z(C) → Z(D).If in addition D a finite-dimensional k-algebra, then this yields finite-dimensional k-algebra quotients of Z(C).Finally, if C = mod(A) for some finite-dimensional selfinjective k-algebra A, then the canonical isomorphism Z(A) ∼ = Z(mod(A)) induces an algebra homomorphism Z(A) → Z(mod(A)), where Z(A) is the stable center of A. Thus restriction to a distinguished abelian subcategory of mod(A) which is equivalent to mod(D) for some finitedimensional k-algebra D yields a k-algebra homomorphism Z(A) → Z(D).Such a homomorphism is in general neither injective nor surjective.Let C be an essentially small triangulated category, and let D be a distinguished abelian subcategory of C. The Grothendieck group a(D) of D is the abelian group generated by the isomorphism classes [X] of objects X in D subject to the relations [X] − [Y ] + [Z] for any short exact sequence 0 → X → Y → Z → 0 in D. The Grothendieck group a(C) of C is the abelian group generated by the isomorphism classes [X] of objects X in C subject to the relations [X] − [Y ] + [Z] for any exact triangle X → Y → Z → Σ(X) in C. Since short exact sequences in D can be completed to exact triangles in D, it follows that the inclusion D → C induces a canonical group homomorphism a(D) → a(C).In general, this group homomorphism need not be injective or surjective.If C is monoidal and D a monoidal distinguished abelian subcategory such that tensor products with objects in C and D preserve exact triangles in C and short exact sequences in D, respectively, then the canonical map a(D) → a(C) is a ring homomorphism.If C = mod(A) for some finite-dimensional selfinjective k-algebra A and D contains all simple A-modules, then the canonical group homomorphism a(D) → a(C) is surjective.Note that if the Cartan matrix of A is nonsingular, then a(C) is finite, while if D ∼ = mod(D) for some finite-dimensional k-algebra, then a(D) is the free abelian group on the set of isomorphism classes of simple D-modules.Remark 9.10.Let A be a finite-dimensional self-injective algebra over a field k and let I be an ideal in A which contains its right annihilator r(I).Then the distiguished abelian subcategory mod(A/I) is functorially finite in mod(A) (cf.[2, §3], [3]).Indeed, let U be an A-module.Then U/IU and the annihilator U I of I in U are in mod(A/I).The canonical map U → U/IU , regarded as a morphism in mod(A), is a left approximation of U , and the inclusion U I → U , again regarded as a morphism in mod(A), is a right approximation of U .(Of course, mod(A/I) is also functorially finite in mod(A), by the same argument.)

Lemma 9 . 11 .
Let A be a finite-dimensional selfinjective k-algebra, and let D be a finite-dimensional k-algebra.Let Y be a finitely generated A-D-bimodule, U a finitely generated A-module and V a finitely generated D-module.The Tensor-Hom adjunctionΨ : Hom A (Y ⊗ D V, U ) ∼ = Hom D (V, Hom A (Y, U )) sends Hom pr A (Y ⊗ D V, U ) to Hom D (V, Hom pr A (Y, U ))and induces a natural map Ψ : Hom A (Y ⊗ D V, U ) → Hom D (V, Hom A (Y, U )) .

Proof.
We need to show that Ψ sends Hom pr A (Y ⊗ D V, U ) to Hom D (V, Hom pr A (Y, U )). Let π : P → U be a projective cover of U .Then any A-homomorphism ending at U which factors through a projective A-module factors through π.Let ϕ : Y ⊗ D V → U be an A-homomorphism which factors through π.That is, there is an A-homomorphism α : Y ⊗ D V → P such that ϕ = π • α.
Let C be a triangulated category with shift functor Σ.A distinguished abelian subcategory of C is a full additive subcategory D of C which is abelian, such that for any short exact sequence f / / Y g / / Z / / 0 in D there exists a morphism h : Z → Σ(X) in C such that the triangle or equivalently, if and only if Im(β • π) ⊆ ker(π) = J.Since the image of β • π is Ay, it follows that α = 0 if and only if y ∈ J.The result follows.
is a quotient of Y and since Y has no nonzero projective summand as an A-module, it follows that the simple A-module Y ⊗ D V is nonprojective and hence that Hom pr A