Ring Constructions and Generation of the Unbounded Derived Module Category

Given the unbounded derived module category of a ring $A$, we consider the triangulated subcategory closed under arbitrary coproducts generated by injective modules. Similarly we also look at the triangulated subcategory closed under arbitrary products cogenerated by projective modules. For a ring construction $f(A)$, we ask whether $A$ being generated by its injective modules implies $f(A)$ is also generated by its injective modules, and vice versa. Similarly we ask the question with projective modules and cogeneration. In this paper we show when these statements are true for ring constructions including recollements, Frobenius extensions and module category equivalences.


Introduction
In this paper we will be concerned with generation of the unbounded derived module category via localising and colocalising subcategories.A localising subcategory is a triangulated subcategory closed under arbitrary coproducts.Similarly a colocalising subcategory is a triangulated subcategory closed under arbitrary products.If the (co)localising subcategory generated by a class of cochain complexes is the entire unbounded derived module category, then we say these cochain complexes (co)generate the ring.
It is well known that the derived module category of a ring is generated by its projective modules, for one proof see [Ric18, Proposition 2.2], and cogenerated by its injective modules.Here we consider the 'opposite' question.In particular, is a ring A generated by its injective modules?If this is true, we say 'injectives generate for A'.Similarly is A cogenerated by its projective modules?If this is true, we say 'projectives cogenerate for A'.This approach was first mentioned by Keller [Kel01] in a talk where he pointed out an algebra satisfying 'injectives generate' would also satisfy some of the homological conjectures, including the Nunke condition.Rickard furthered this idea and proved 'injectives generate' and 'projectives cogenerate' both imply the big finitistic dimension conjecture, [Ric18, Theorem 4.2, Proposition 5.2].
The big finitistic dimension conjecture is a generalisation of the little finitistic dimension conjecture first stated by Bass in 1960, [Bas60].The little finitistic dimension conjecture states that if A is a finite dimensional algebra over a field then findim(A) < ∞, where findim(A) := sup{proj.dim(MA )|M A ∈ mod-A and proj.dim(MA ) < ∞}.
If the conjecture holds then many other homological conjectures follow including the generalised Nakayama conjecture and Nunke condition.The big finitistic dimension is defined similarly and considers the projective dimension of all A-modules (not necessarily finitely generated) with finite projective dimension.Rickard showed that for A a finite dimensional algebra over a field, if injectives generate for A then A satisfies the big finitistic dimension conjecture and hence also the little finitistic dimension conjecture, [Ric18, Theorem 4.2].Furthermore, if projectives cogenerate for A op then A satisfies the big finitistic dimension conjecture, [Ric18,Proposition 5.2].
The relationship between rings and various, usually more complicated, ring constructions has long been exploited to show they satisfy similar properties.In this paper we consider the relationship between rings and various ring constructions with regards to both 'injectives generate' and 'projectives cogenerate' properties.It is known that for A a finite dimensional algebra over a field, if injectives generate for A then projectives cogenerate for A op , [Ric18, Proposition 5.1].However the converse statement remains unproved so throughout we state results for both injectives generate and projectives cogenerate.
We start by recalling some definitions and well known results about localising and colocalising subcategories in Section 2. In particular we focus on the interaction of these triangulated subcategories with triangle functors.
In Section 3 we focus on our first ring construction, namely the tensor product algebra.This straightforward example will showcase the techniques used in the rest of the paper to prove 'injectives generate' and 'projectives cogenerate' statements.
In Section 4 we show that equivalences at the module category level preserve the properties 'injectives generate' and 'projectives cogenerate'.In particular we prove separable equivalence preserves both properties.
In Section 5 we consider Frobenius extensions.These well known extensions, first defined by Kasch [Kas54], cover many standard examples of ring constructions including strongly G-graded rings for G a finite group and excellent extensions.A natural next step is to consider generalisations of excellent extensions in Section 6.In particular we focus on finite normalising extensions and almost excellent extensions defined by Xue [Xue96].
Finally in Section 7 we look at recollements, defined by Beȋlinson, Bernstein and Deligne [BBD82].We prove generation and cogeneration results for bounded above, bounded below and bounded recollements of both rings and finite dimensional algebras.An example of a ring construction which gives rise to a recollement is the triangular matrix ring.We show that for any rings B and C, and any (C, B)bimodule, if injectives generate (resp.projectives cogenerate) for B and C then injectives generate (resp.projectives cogenerate) for their corresponding triangular matrix ring.Hence to prove injectives generate for all finite dimensional algebras over a field it suffices to consider quiver algebras such that the associated quiver has a directed path between any two vertices.

Acknowledgement
I would like to thank my supervisor Jeremy Rickard for his guidance and many useful discussions.

Preliminaries
In this section we fix some notation and discuss some results relating to localising and colocalising subcategories of triangulated categories which will be used heavily in the rest of the paper.Throughout this paper A, B and C will be unital rings and we consider right modules unless otherwise stated.We will denote the collection of finitely generated A-modules as mod-A and the collection of all A-modules (not necessarily finitely generated) as Mod-A.Furthermore, the collection of indecomposable injective A-modules will be denoted as Inj-A and similarly the collection of indecomposable projective A-modules denoted as Proj-A.The unbounded derived module category of A will be denoted D (Mod-A) and for * ∈ {−, +, b}, D * (Mod-A) will denote the bounded above, bounded below and bounded derived module categories respectively.All complexes of A-modules will be cochain complexes.A triangle functor will be a functor between derived categories which preserves the triangulated structure.
There are many ways to generate the unbounded derived category of a ring, here we focus on generation via localising and colocalising subcategories.First we recall their definitions.

Definition 2.1 ((Co)Localising Subcategory). Let A be a ring and S a collection of objects of D (Mod-A).
• A localising subcategory is a triangulated subcategory of D (Mod-A) closed under arbitrary coproducts.The smallest localising subcategory containing S will be denoted Loc (S).
• A colocalising subcategory is a triangulated subcategory of D (Mod-A) closed under arbitrary products.The smallest colocalising subcategory containing S will be denoted Coloc (S).
There are some well known properties of localising and colocalising subcategories which can be found in [Ric18, Proposition 2.1].An important property we will make use of is that both localising and colocalising subcategories are closed under taking direct summands.Throughout this paper we investigate when a localising subcategory or colocalising subcategory generated by some collection of objects S is in fact the entire unbounded derived module category.If D (Mod-A) = Loc (S) then we say S generates for A and similarly if D (Mod-A) = Coloc (S) then we say S cogenerates for A. It is well known that for any ring A, its unbounded derived category D (Mod-A) is generated by the indecomposable projective A-modules and cogenerated by the indecomposable injective A-modules, see [Ric18, Proposition 2.2].Since a localising subcategory is closed under direct sums and summands, it immediately follows that the regular module A A also generates for A. In fact this is true for any generator of Mod-A and similarly any cogenerator of Mod-A cogenerates for A. Definition 2.2 ((Co)Generator).Let A be a ring and M A an A-module.
• The module M A is a generator for Mod-A if for all A-modules N A there exists an index set I and a surjective A-module homomorphism f : i∈I M A → N A .
• The module M A is a cogenerator for Mod-A if for all A-modules N A there exists an index set I and an injective A-module homomorphism f : Lemma 2.3.Let A be a ring and M A an A-module.
Proof.Since M A is a generator of Mod-A, for every projective A-module P A there exists an index set I such that f : i∈I M A → P A is a surjective A-homomorphism.
As P A is projective, f splits and P A is isomorphic to a direct summand of i∈I M A .Thus all projective A-modules are isomorphic to a direct summand of a direct sum of copies of M A .A localising subcategory is closed under direct sums and summands so all projective A-modules are contained in Loc (M A ) and Loc (M A ) = D (Mod-A).
The second claim follows similarly using the injective A-modules and splitting of monomorphisms.

Functors
Most of the results in this paper rely on using functors which preserve properties of localising and colocalising subcategories.Since the ideas will be mentioned often, we collate them here.ii) If F preserves arbitrary products then the preimage of a colocalising subcategory in D (Mod-B) is a colocalising subcategory in D (Mod-A).
Proof.This is a straightforward exercise of applying the definitions of localising and colocalising subcategories.
ii) Let S cogenerate for A. If F preserves arbitrary products and for all Proof.The first statement follows by Lemma 2.4.In particular, the preimage of Loc (T ) under F is a localising subcategory.Furthermore, the preimage contains S so it also contains Loc (S) = D (Mod-A).The second statement follows similarly.

Adjoint Functors
Adjoint pairs of functors are particularly rich in the various properties they preserve.
To make the best use of this theory we use homomorphism groups to categorise some properties of cochain complexes.Most of these well known results can be found in [Ric89, Proof of Proposition 8.1], [Koe91, Proof of Theorem 1] and [AHKLY17, Lemma 2.4].
Lemma 2.6.Let A be a ring.Then the following hold: i ) The complex X ∈ D (Mod-A) has homology bounded in degree if and only if for all compact objects C ∈ D (Mod-A) (bounded complexes of finitely generated projective A-modules), ) is non zero for finitely many n ∈ Z.
ii ) The complex I ∈ D (Mod-A) is isomorphic to a bounded complex of injectives if and only if for all complexes with homology bounded in degree X ∈ D (Mod-A), ) is non zero for finitely many n ∈ Z.
iii ) The complex P ∈ D (Mod-A) is isomorphic to a bounded complex of projectives if and only if for all complexes with homology bounded in degree X ∈ D (Mod-A), Hom D(Mod-A) (P [n], X) is non zero for finitely many n ∈ Z.
iv ) The complex X ∈ D (Mod-A) is isomorphic to a bounded below complex if and only if for all bounded complexes of injectives I ∈ D (Mod-A), there exists N ∈ Z such that Hom D(Mod-A) (X, I[n]) = 0 for all n < N .
v ) The complex X ∈ D (Mod-A) is isomorphic to a bounded above complex if and only if for all bounded complexes of projectives P ∈ D (Mod-A), there exists N ∈ Z such that Hom D(Mod-A) (P, X[n]) = 0 for all n > N .
Proof.We only prove (ii) as the others follow similar methods.Firstly suppose I ∈ D (Mod-A) is isomorphic to a bounded complex of injectives and let X ∈ D (Mod-A) be a complex with homology bounded in degree.Consider Hom D(Mod-A) (I, X[n]) for n ∈ Z. Since I is a bounded complex of injectives we can pass to the homotopy category and instead work with Hom K(Mod-A) (I, X[n]).Furthermore, as both X and I are bounded in homology there are only finitely many n ∈ Z such that for some l ∈ Z both H l (X[n]) and (I) l are non zero.In particular there are no homomorphisms from acyclic complexes to bounded complexes of injectives.Hence there are only finitely many n ∈ Z such that Hom K(Mod-A) (X[n], I) can be non zero.Now let us consider the other direction.Let Y ∈ D (Mod-A).Suppose that for all complexes X ∈ D (Mod-A) with homology bounded in degree we have Hom D(Mod-A) (X[n], Y ) is non zero for finitely many n ∈ Z. Clearly the complex given by the module A concentrated in degree 0 has homology bounded in degree.Thus Hom D(Mod-A) (A[n], Y ) is non zero for finitely many n ∈ Z. Hence the homology of Y is also bounded in degree and Y is quasi-isomorphic to a bounded below complex of injectives.Denote this bounded below complex of injectives as I with differentials d i for i ∈ Z.In particular note that I also has homology bounded in degree.Consider the bounded complex given by the module K := i∈Z ker d i concentrated in degree 0. Let n ′ ∈ Z be such that for all N ≥ n ′ , I is exact at degree N then ker d n ′ is a direct summand of I n ′ .Thus the good truncation τ ≤n ′ I is a bounded complex of injectives which is still quasi-isomorphic to Y .These criteria can be applied to adjoint functors to show they will preserve the properties considered in Lemma 2.6.Lemma 2.7.Let A and B be rings.Let (F, G) be an adjoint pair of triangle functors such that F : D (Mod-A) → D (Mod-B) and G : D (Mod-B) → D (Mod-A).Then the following hold: i) If G preserves coproducts then F preserves compact objects.
ii) If F preserves compact complexes then G preserves complexes with homology bounded in degree.
iii) If G preserves bounded complexes of injectives then F preserves both bounded and bounded below complexes.

iv) If F preserves complexes with homology bounded in degree then G preserves both bounded and bounded below complexes of injectives. v) If F preserves bounded complexes of projectives then G preserves both bounded and bounded above complexes. vi) If G preserves complexes with homology bounded in degree then F preserves both bounded and bounded above complexes of projectives.
Proof.These results follow from adjunction and Lemma 2.6.Here we prove (ii) as the others are similar.Suppose F preserves compact objects.Note that, by Lemma 2.6, G preserves complexes with homology bounded in degree if and only if for all X ∈ D b (Mod-B) and C ∈ D (Mod-A) a compact object, we have Hom ) is non zero for finitely many n ∈ Z. Hence G(X) is a complex with homology bounded in degree.

Tensor Product Algebra
The first ring construction we consider is the tensor product of two finite dimensional algebras A and B, over a field k.In particular we prove if injectives generate for the two algebras then injectives generate for their tensor product and similarly for projectives cogenerate.Firstly we recall a description of the injective and projective modules for a tensor product algebra.Lemma 3.1.[Xi00, Lemma 3.1] Let A and B be finite dimensional algebras over a field k.Let M A be an A-module and N B be a B-module.
In particular the structure of these modules is functorial in either argument.For a B-module Y B define the functor Since k is a field, for all Y B and X A the functors F Y and G X are exact.Hence these functors are also triangle functors To show injectives generate for A⊗ k B we note that when Y B and X A are finitely generated both F Y and G X preserve arbitrary coproducts and arbitrary products so we can use Proposition 2.5.Proposition 3.2.Let A and B be finite dimensional algebras over a field k.

ii) If projectives cogenerate for A and B then projectives cogenerate for
Proof.Let injectives generate for A. Let X A be an A-module and J B a finitely generated injective B-module.Then we claim that Now suppose injectives generate for both B and A. Let Y B be a B-module and consider the functor G A .By the previous argument for all injective B-modules J B , The projectives cogenerate statement follows similarly by considering F P for P B a finitely generated projective B-module and then G D(A) where D (A) is the dual of A.
The converse to Proposition 3.2 will be shown as an application of the results about finite normalising extensions considered in Section 6.In particular, the converse statement follows immediately from Proposition 6.4.

Separable Equivalence
It is already known that if two algebras are derived equivalent then injectives generate for one if and only if injectives generate for the other [Ric18,Theorem 3.4].This implies that Morita equivalence also preserves 'injectives generate'.Here we show the result extends to both separable equivalence and stable equivalence of Morita type.We only state the proof for separable equivalence as the other result follows immediately.First we recall the definition of separable equivalence using the idea of separably dividing rings.
Recall localising subcategories are closed under direct summands so B ∈ Loc (Inj-B) and injectives generate for B by Lemma 2.3.Suppose projectives cogenerate for A. Since B N is a finitely generated projective B-module it is finitely presented.Thus − ⊗ B N A preserves arbitrary products.Furthermore its right adjoint Hom A ( B N, −) is exact so − ⊗ B N A preserves projective modules.The result follows from the same proof as above.

Frobenius Extensions
It is well known that Frobenius algebras are self injective.In particular it is clear that injectives generate for self-injective algebras.Hence in what follows we consider a generalisation of Frobenius algebras, namely Frobenius extensions.Initially Kasch [Kas54] defined free Frobenius extensions which were further generalised by Nakayama and Tsuzuku [NT59], [NT60].
Definition 5.1 ((Free) Frobenius extension).Let A and B be rings, then A is a ring extension of B, denoted as A/B, if there exists a unital ring homomorphism f : A → B. A ring extension A/B is a (free) Frobenius extension if the following are satisfied: • The module A B is a finitely generated projective (free) B-module.

• The bimodule Hom
Note the second condition in the definition of Frobenius extensions implies that the two functors, − ⊗ B A and Hom B (A, −) are isomorphic.In turn this means − ⊗ B A is both left and right adjoint to Hom A ( B A, −).Such a pair of functors is called a strongly adjoint pair.Let A be a ring and G a finite group.Denote the identity of G as 1 and the identity slice of A as A 1 .Then A/A 1 is a Frobenius extension.This collection of graded rings includes skew group algebras, smash products and crossed products for finite groups.
An excellent extension is a ring extension A/B such that A is right B-projective and the modules A B and B A are free with common basis a 1 , ..., a n ∈ A. Note that A is right B-projective [Pas77] if for all A-modules N A and M A such that N A is a submodule of M A and N B a direct summand of M B we have N A is a direct summand of M A .
For example the matrix ring M n (A) is an excellent extension of A.

• The endomorphism ring theorem. ([Kas54], [NT61, Theorem 22])
Let A/B be a Frobenius extension and denote C := End B (A).Then C/A is also a Frobenius extension.
It is well known that Frobenius algebras have the property that injective and projective modules coincide.A Frobenius extension A/B has a similar property when considering relatively B-injective and relatively B-projective modules.We recall the definition here.
Definition 5.3 (Relatively projective/injective).Let A/B be a ring extension and M A an A-module.
• The module M A is relatively B-projective if the counit map of the adjoint pair • The module M A is relatively B-injective if the unit map of the adjoint pair For a generic ring extension A/B it is clear that any projective A-module is relatively B-projective and similarly any injective A-module is relatively B-injective.However when we are considering a Frobenius extension any projective A-module is also relatively B-injective, and vice versa.In fact this holds true for a slightly weaker extension, namely quasi-Frobenius extensions [Mül64].

Lemma 5.4. [Kad99, Proposition 4.1] Let A/B be a Frobenius extension and M
The other direction follows similarly.
Proposition 5.5.Let A/B be a Frobenius extension.
i) If injectives generate for B then injectives generate for A.
ii) If projectives cogenerate for B then projectives cogenerate for A.
Proof.Since − ⊗ B A and Hom B (A, −) are isomorphic they are both exact.Furthermore, as − ⊗ B A and Hom A ( B A, −) are a strongly adjoint pair of functors they both preserve (co)products, injectives and projectives.
Suppose injectives generate for B. Since Hom B (A, −) preserves both coproducts and injective modules, the image of Hom B ( A A, −) is contained in Loc (Inj-A), by Proposition 2.5.By Lemma 5.4 any projective A-module is relatively B-injective hence A A is a direct summand of Hom B (A, A) ∈ Loc (Inj-A).
The second statement follows similarly.
The original Frobenius extension Kasch [Kas54] defined also required A B to be a free B-module.If A B is free then it is also a generator of Mod-B.With the added assumption that A B is a generator, the converse of Proposition 5.5 also holds.It should be noted that A B is not a generator of Mod-B for all Frobenius extensions; a counterexample is given by Morita [Mor67, Example 7.1].However of all the examples we have given only in the endomorphism ring example is A B not necessarily a generator of Mod-B.

Proposition 5.6. Let A/B be a Frobenius ring extension such that A B is a generator of Mod-B. i) Then injectives generate for A if and only if injectives generate for B. ii) Then projectives cogenerate for A if and only if projectives cogenerate for B.
Proof.Suppose injectives generate for A. Since Hom A ( B A, −) preserves injectives and coproducts its image is contained in Loc (Inj-B).Furthermore as A B is a generator and Since A B is a generator for Mod-B, Hom Z (A, Q/Z) is a cogenerator for Mod-B.Hence the projectives cogenerate statement follows from similar reasoning to the above.

Excellent Extensions
Excellent extensions were first introduced by Passman [Pas77].As stated in Example 5.2, excellent extensions are Frobenius extensions.However there are generalisations of excellent extensions which are not.We focus on two of these, namely finite normalising extensions and almost excellent extensions [Xue96].

Finite Normalising Extensions
The first extension we consider is that of finite normalising extensions.

Definition 6.1 (Finite Normalising Extension). A ring extension
A/B is a finite normalising extension if there exist elements a 1 , a 2 , ..., a n ∈ A such that A = n i=1 a i B and a i B = Ba i for all 1 ≤ i ≤ n.Example 6.2.These examples and more can be found in [RS81].
• Let A be a finite dimensional algebra over a field k.Then A/k is a finite normalising extension by the basis elements of A as a k-vector space.Furthermore, the centre of A contains k so A/Z(A) is also a finite normalising extension.
• Let A and B be algebras over a field k with B finite dimensional.Then A ⊗ k B is a finite normalising extension of B for the same reason as above.
• Let B be a ring and J i be a finite collection of ideals of B such that i∈I J i = 0. Define A := i∈I B/J i , then we have that A/B is a finite normalising extension where a i := (0, 0, ..., 1 B , ..., 0) for 1 B in the i th position.The second statement is proved similarly using the cogenerator Hom Z (A, Q/Z).
The tensor product algebra A ⊗ k B is a finite normalising extension of B so we can apply Lemma 6.4.In particular, this proves the converse statement to Proposition 3.2.So injectives generate for A ⊗ k B if and only if injectives generate for both A and B and similarly for projectives cogenerate.

Almost Excellent Extensions
Almost excellent extensions were defined and studied by Xue [Xue96].

Definition 6.5 (Almost Excellent Extension). Let A/B be a finite normalising extension. Then A is an almost excellent extension of B if the following hold:
• The ring A is right B-projective.
• The module B A is flat and A B is projective.
An almost excellent extension has a stronger property than Frobenius extensions in that any A-module is both relatively B-injective and relatively B-projective.Lemma 6.6.[Xue96] Let A/B be a ring extension such that A is right B-projective.Then every A-module is both relatively B-injective and relatively B-projective.
Proof.Let us denote F := Hom A ( B A, −) and G := Hom B ( A A, −).Since (F, G) is an adjoint pair for all A-modules M A , there exists a unit map η M : M → GF (M ).In particular consider F (η M ) : Furthermore this tells us F (η M ) is monic and F is faithful so η M is also monic.Recall A is right B-projective, M is a submodule of GF (M ) and F (M ) is a direct summand of F GF (M ), so M is isomorphic to a direct summand of GF (M ).Thus M A is isomorphic to a direct summand of Hom B (A, Hom A (A, M )) as A-modules and M A is relatively B-injective.Proposition 6.7.Let A/B be an almost excellent extension.

ii) Then projectives cogenerate for A if and only if projectives cogenerate for B.
Proof.By Lemma 6.4, if injectives generate for A then injectives generate for B. To prove the converse firstly note that, as A B is a finitely generated projective Hom B ( A A, −) preserves coproducts.Moreover, as B A is flat Hom B ( A A, −) also preserves injectives.Furthermore as every A-module is relatively B-injective, every projective A-module certainly is.Thus this is the same proof as Proposition 5.5 for Frobenius extensions.

Recollements
Recollements of triangulated categories were first introduced by Beȋlinson, Bernstein and Deligne [BBD82] to study derived categories of sheaves.Throughout we only consider recollements of derived categories of the rings A, B and C. Definition 7.1 (Recollement).Let A, B and C be rings.A recollement is a diagram of triangle functors as in Figure 1 such that the following hold: ii) (i * , i * ), (i * , i ! ), (j !, j * ) and (j * , j * ) are adjoint pairs.iii) i * , j ! and j * are fully faithful.iv) For every X ∈ D (Mod-A) there exist triangles: We will denote a recollement of the form in Figure 1 as (R).If a recollement (R) exists then the properties of A, B and C are often related.This is useful since it allows one to prove properties about A using, the usually simpler, B and C.This has been exploited by Happel [Hap93, Theorem 2] and Chen and Xi [CX17] to prove various statements about the finitistic dimension conjecture and recollements. .This example of a ring construction is particularly useful since it contains a large class of finite dimensional algebras defined as follows.Let A be a quiver algebra (a path algebra with relations) and denote by Q A the associated quiver with vertices Q 0 .Suppose there exists a subset of vertices E such that there are no paths from vertices of Q 0 \E to vertices of E. Define e := e i ∈E e i .Then eAe and (1−e)A(1−e) are finite dimensional algebras with eA(1 − e) a finitely generated (eAe, (1 − e)A(1 − e))-bimodule.Moreover, (1 − e)Ae is zero as there are no paths from vertices of Q 0 \ E to vertices of E. Thus the generalised matrix form of A is a triangular matrix ring, This class of algebras is equivalent to the class of quiver algebras A which have associated quivers that can be drawn as follows for some e constructed as above, Recollements can also be defined on derived categories with different boundedness conditions.Throughout this section for * ∈ {−, +, b} we denote (R * ) to be a recollement of bounded above, bounded below and bounded derived module categories respectively.We will consider the cases when a recollement (R) restricts to a recollement (R * ).It is also possible to lift from a bounded above or bounded recollement to an unbounded recollement.
ii) The lifted recollement restricts, up to equivalence, to the original recollement.
In this section we will prove many results about the dependence of A, B and C on each other with regards to 'injectives generate' and 'projectives cogenerate' statements.In Theorem 7.4 we collate the most useful results.To prove Theorem 7.4 we require some technical results which we state and prove now.We prove these results by exploiting the fact there are four pairs of adjoint functors in a recollement.Thus we can use the ideas in Section 2 to show these functors preserve many properties.We collate these ideas in Table 1  ii) If the image of i * is contained in Coloc (Proj-A) and projectives cogenerate for C then projectives cogenerate for A.
Proof.Let the image of i * be contained in Loc (Inj-A).Let K ∈ D (Mod-C) be a bounded complex of injectives.Consider the triangle Since j * preserves bounded complexes of injectives, j * (K) ∈ Loc (Inj-A).Hence triangle 3 implies j !j * (j * (K)) ∈ Loc (Inj-A).Recall j * is fully faithful so j !j * j * (K) ∼ = j !(K).Thus j !maps bounded complexes of injectives to Loc (Inj-A).Suppose injectives generate for C. Then j !preserves coproducts and maps injective C-modules to Loc (Inj-A).Hence by Proposition 2.5 the image of j ! is contained in Loc (Inj-A).
Thus the images of both i * and j ! are contained in Loc (Inj-A) so for all X ∈ D (Mod-A) both i * i * (X), j !j * (X) ∈ Loc (Inj-A).So injectives generate for A using the triangle, The second result follows similarly.
Lemma 7.6.Let (R) be a recollement.Proof.We prove the first two statements as the others follow similarly.Firstly, suppose injectives generate for both B and C. Since i * preserves bounded complexes of injectives and coproducts, we apply Proposition 2.5 to show the image of i * is contained in Loc (Inj-A).Hence we can apply Proposition 7.5 and injectives generate for A.
Secondly, let i * preserve bounded complexes of injectives.Then we claim j * also preserves bounded complexes of injectives.Since j * preserves complexes bounded in homology, by Lemma 2.7, j !preserves bounded above complexes and j * preserves bounded below complexes.Furthermore, since i * preserves bounded complexes of injectives, by Lemma 2.7, i * preserves bounded below complexes.Let Z ∈ D (Mod-C) be bounded below and consider the triangle: Since i * , i * and j * all preserve bounded below complexes, by the triangle, j ! also does.Hence j !preserves both bounded above and bounded below complexes.Thus j !preserves complexes bounded in homology and j * preserves bounded complexes of injectives.Hence the statement follows immediately from Lemma 7.6.Proof.Since i * is fully faithful both i * and i ! are essentially surjective.Hence if either the image of i * or the image of i ! is contained in Loc (Inj-B) then D (Mod-B) is contained in Loc (Inj-B) and injectives generate for B. The two statements are sufficient conditions for this to happen using Proposition 2.5.The idea is similar for the second statement.

Ladders of Recollements
Let us fix rings A, B and C. A ladder of recollements is a collection of finitely or infinitely many rows of triangle functors between D (Mod-A), D (Mod-B) and D (Mod-C), of the form given in Figure 2, such that any three consecutive rows form a recollement.This definition is taken from [AHKLY17, Section 3].The height of a ladder is the number of distinct recollements it contains.i) The recollement (R) can be extended down one step if and only if j * (equivalently i ! ) has a right adjoint.This occurs exactly when j * (equivalently i * ) preserves compact objects.
ii) The recollement (R) can be extended up one step if and only if j !(equivalently i * ) has a left adjoint.If A is a finite dimensional algebra over a field this occurs exactly when j !(equivalently i * ) preserves bounded complexes of finitely generated modules.
If the recollement (R) can be extended one step down then we have a recollement (R ↓ ) as in Figure 3.

Bounded Above Recollements
In this section we consider the case of a recollement which restricts to a bounded above recollement.In particular we use a classification by [AHKLY17].
ii) The functor i * preserves bounded complexes of projectives.
If A is a finite dimensional algebra over a field then both conditions are equivalent to: iii) The functor i * preserves compact objects.Proof.Since (R − ) is a recollement of bounded above derived categories i * preserves bounded complexes of projectives [AHKLY17, Proposition 4.11].Hence we apply Proposition 7.7 to get (i).Furthermore, if A is a finite dimensional algebra over a field then i * preserves compact objects.Then the recollement also extends down by one and we apply Proposition 7.11.

Bounded Below Recollements
Similarly to the last section we consider bounded below recollements.First we prove an analogous statement to Proposition 7.15 about the conditions under which a recollement (R) restricts to a recollement (R + ), Figure 5. i) The recollement (R) restricts to a bounded below recollement (R + ), see Figure 5.
ii) The functor i * preserves bounded complexes of injectives.
If A is a finite dimensional algebra over a field then both conditions are equivalent to: iii) The functor j !preserves bounded complexes of finitely generated modules.
Proof.Let i * preserve bounded complexes of injectives.Then by the proof of Proposition 7.7 all six functors preserve bounded below complexes.Hence the recollement (R) restricts to a bounded below recollement (R + ).
For the converse statement, suppose (R) restricts to a bounded below recollement (R + ), that is all six functors preserve bounded below complexes.Since i * preserves complexes with homology bounded in degree by Lemma 2.7, i * preserves bounded above complexes.Hence i * preserves both bounded above and bounded below complexes.Thus i * preserves complexes with homology bounded in degree and by Lemma 2.7, i * preserves bounded complexes of injectives.
Finally, let A be a finite dimensional algebra and that (R) restricts to (R + ).Let X ∈ D b (mod-C) be a bounded complex of finitely generated A-modules.Since A is a finite dimensional algebra over a field, j !(X) is a bounded above complex of finitely generated modules by [AHKLY17, Lemma 2.10 (b)].Suppose (R) restricts to a bounded below recollement (R + ).Then j !preserves bounded below complexes so j !(X) is bounded below.Hence we can truncate j !(X) from below and j !(X) is quasi-isomorphic to a bounded complex of finitely generated A-modules.Thus by Proposition 7.9, (R + ) extends one row upwards.
The converse follows immediately from Proposition 7.9.
We can use these results to get an analogous statement to Proposition 7.15 about bounded below recollements.Proposition 7.17.Let (R) be a recollement that restricts to a bounded below recollement (R + ).Then the following hold: Proof.The proof is dual to the proof of Proposition 7.15.

Bounded Recollements
Finally we consider the case of a recollement (R) which restricts to a bounded recollement (R b ), Figure 6.Since all the functors must preserve complexes bounded in homology the middle functors i * and j * must also preserve bounded complexes of injectives and projectives.

Lemma 2. 4 .
Let A and B be rings and let F : D (Mod-A) → D (Mod-B) be a triangle functor.i) If F preserves arbitrary coproducts then the preimage of a localising subcategory in D (Mod-B) is a localising subcategory in D (Mod-A).

Proposition 2. 5 .
Let A and B be rings and F : D (Mod-A) → D (Mod-B) be a triangle functor.Let S and T be collections of objects in D (Mod-A) and D (Mod-B) respectively.i) Let S generate for A. If F preserves arbitrary coproducts and for all Definition 4.1 (Separably dividing rings.).Let A and B be rings.Then B separably divides A if there exist bimodules A M B and B N A such that: i) The modules A M , M B , B N and N A are all finitely generated projectives.ii) There exists a bimodule B Y B such that B N ⊗ A M B ∼ = B ⊕ B Y B as B-bimodules.Theorem 4.2.Let A separably divide B. i) If injectives generate for A then injectives generate for B. ii) If projectives cogenerate for A then projectives cogenerate for B. Proof.Consider the adjoint functors − ⊗ B N A and Hom A ( B N, −).Since both B N and N A are projective, − ⊗ B N A and Hom A ( B N, −) are exact.As Hom A ( B N, −) has an exact left adjoint it preserves injective modules.Furthermore, the module N A is a finitely generated projective so Hom A ( B N, −) also preserves coproducts.Let injectives generate for A. Since Hom A ( B N, −) preserves injective modules and coproducts its image is contained in Loc (Inj-B) by Proposition 2.5.By the tensor-hom adjunction Hom B (N ⊗ A M , B ) ∼ = Hom A (N, Hom B (M, B)) as Bmodules and so Hom A

Example 5. 2 .
There are many familiar examples of Frobenius extensions.• Strongly G-graded rings for a finite group G. [BF93, Example B].

Lemma 6. 3 .
[Sou87, Corollary 4],[Sha92, Proposition 2.1] Let A/B be a finite normalising extension and N B a B-module.Then the following hold:i) If N ⊗ B A = 0 then N B = 0. ii) If Hom B (A, N ) = 0 then N B = 0.Hence both − ⊗ B A and Hom B ( A A, −) are faithful.In particular it follows from adjunction that the restriction functor, Hom A ( B A, −) preserves both generators and cogenerators.It should be noted that Kitamura independently proved the result that Hom A ( B A, −) preserves generators for a generalised extension of finite normalised extensions, [Kit81, Proposition 1.3].Lemma 6.4.Let A/B be a finite normalising extension.• If B A is flat and injectives generate for A then injectives generate for B. • If A B is projective and projectives cogenerate for A then projectives cogenerate for B. Proof.Since B A is flat, − ⊗ B A is exact and so Hom A ( B A, −) preserves injectives.Hence the image of Hom A ( B A, −) is contained in Loc (Inj-B).In particular A A is a generator for Mod-A thus Hom A ( B A, A) is a generator for Mod-B.Hence injectives generate for B by Proposition 2.5.

Example 7. 2 .
One example of a recollement can be defined using triangular matrix rings, [AHKLY17, Example 3.4].Let B and C be rings and C M B a finitely generated (C, B)-bimodule.Define A := C C M B 0 B to be the triangular matrix ring.Then A, B and C define a recollement (R).The functors of (R) are defined by using idempotents of A. Let e 1 := 1 0 0 0 and e 2 := 1 − e 1 .Then j !:= − ⊗ L C e 1 A and i * := −⊗ L B e 2 A. This follows from the work on stratified recollements in [AHKLY17]

Theorem 7. 4 .
Let (R) be an unbounded recollement.i) Let injectives generate for both B and C. If one of the following conditions holds then injectives generate for A. a) The recollement (R) is in a ladder of height greater than or equal to 2. [Proposition 7.11] b) The recollement (R) restricts to a bounded below recollement (R + ).[Proposition 7.17] c) The recollement (R) restricts to a bounded above recollement (R − ) and A is a finite dimensional algebra over a field.[Proposition 7.15] ii) Let projectives cogenerate for both B and C. If one of the following conditions holds then projectives cogenerate for A. a) The recollement (R) is in a ladder of height greater than or equal to 2. [Proposition 7.11] b) The recollement (R) restricts to a bounded above recollement (R − ).[Proposition 7.15] c) The recollement (R) restricts to a bounded below recollement (R + ) and A is a finite dimensional algebra over a field.[Proposition 7.17]

Lemma 7. 8 .
Let (R) be a recollement.i) Let injectives generate for A. Then injectives generate for B if one of the following two conditions holds: (a) The functor i !preserves coproducts.(b) For any bounded complex of injectives I ∈ D (Mod-A), we have i * (I) ∈ Loc (Inj-B).ii)Let projectives cogenerate for A. Then projectives cogenerate for B if one of the following two conditions holds:(a) The functor i * preserves products.(b)For any bounded complex of projectives P ∈ D (Mod-A), we have i !(P ) ∈ Coloc (Proj-B).

Figure 5 :
Figure 5: Recollement of bounded below derived categories (R + ) i) If injectives generate for B and C then injectives generate for A. ii) If injectives generate for A then injectives generate for C.Moreover, if A is a finite dimensional algebra over a field then the following hold: iii) If projectives cogenerate for A then projectives cogenerate for B. iv) If projectives cogenerate for B and C projectives cogenerate for A.

DFigure 6 :
Figure 6: Recollement of bounded derived categories (R b ) Definition 4.3 (Separable Equivalence).Let A and B be rings.Then A and B are separably equivalent if A separably divides B and B separably divides A. Let G be a group and H a Sylow p-subgroup of G. Let k be a field of characteristic p. Then the group algebras kG and kH are separably equivalent using the bimodules kG kG kH and kH kG kG ; this was stated by Linckelmann [Lin11].Let A and B be separably equivalent rings.i) Injectives generate for A if and only if injectives generate for B. ii) Projectives cogenerate for A if and only if projectives cogenerate for B. Proof.Since A and B are separably equivalent, A separably divides B and B separably divides A.
preserves bounded complexes of projectives and projectives cogenerate for A then projectives cogenerate for C.Proof.Suppose injectives generate for A. Since j * preserves bounded complexes of injectives and coproducts, its image is contained in Loc (Inj-C).Furthermore j * is essentially surjective as it is right adjoint to j ! which is fully faithful.Thus the image of j * contains D (Mod-C) and D (Mod-C) is contained in Loc (Inj-C).Hence injectives generate for C.The proof of the second statement is similar.If projectives cogenerate for both B and C then projectives cogenerate for A.
i) If j * preserves bounded complexes of injectives and injectives generate for A then injectives generate for C. ii) If j * (b) If projectives cogenerate for A then projectives cogenerate for C.
3: Recollement of derived categories extended one step down (R ↓ ) The triangular matrix ring defines a recollement (R) as seen in Example 7.2.We claim this recollement extends down by one step.Recall i * := In particular note that e 2 A A is a finitely generated projective A-module so i != Hom A (e 2 A, −) is exact and preserves coproducts.Hence i * preserves compact objects by Lemma 2.7.Thus we can apply Proposition 7.9 to show (R) extends down one row.The bottom recollement of the ladder is a recollement as in (R) but with the positions of B and C swapped.Hence in this bottom recollement j * acts as i * in the recollement (R).Moreover, j * preserves bounded complexes of injectives.Thus we apply Proposition 7.7 to prove injectives generate for A if injectives generate for B and C. Furthermore, we can apply this to the class of algebras defined in Example 7.2.This implies that injectives generate for all finite dimensional algebras over fields if and only if injectives generate for all quiver algebras with associated quivers such that for any ordered pair of vertices (e i , e j ) there exists a non zero directed path from e i to e j .Let (R) be a recollement in a ladder of height ≥ 3.i) Then injectives generate for A if and only if injectives generate for both B and C.ii) Then projectives cogenerate for A if and only if projectives cogenerate for both B and C.Proof.If the recollement is in a ladder of height greater than 3 then there are at least two distinct ladders of recollements of height 2. One with B on the left as in (R ↓ ) and another with B and C swapped.Hence we can apply Proposition 7.11 to both (R ↓ ) and the swapped version of (R ↓ ) to get the desired result.
Proposition 7.11.Let (R) be the top recollement in a ladder of height 2. With notation as in (R ↓ ) the following hold: i) If injectives generate for A then injectives generate for B. ii) If injectives generate for both B and C then injectives generate for A. iii) If projectives cogenerate for A then projectives cogenerate for C. iv) If projectives cogenerate for both B and C then projectives cogenerate for A.Proof.Since (R) extends down one row i ! has a right adjoint and so preserves coproducts.Hence we apply Lemma 7.8 to show injectives generate for B if injectives generate for A.
Moreover, if A is a finite dimensional algebra over a field then the following hold: i) Injectives generate for A if and only if injectives generate for both B and C.ii) Projectives cogenerate for A if and only if projectives cogenerate for both B andC.Proof.Since (R b ) is a recollement of bounded derived categories both i * and i !preserve bounded complexes.Hence i * preserves both bounded complexes of injectives and bounded complexes of projectives.Thus the results follow immediately from Proposition 7.17 and Proposition 7.15.
Then the following hold: i) If injectives generate for both B and C then injectives generate for A. ii) If injectives generate for A then injectives generate for C. iii) If projectives cogenerate for both B and C then projectives cogenerate for A. iv) If projectives cogenerate for A then projectives cogenerate for C.