SIMPLES IN A COTILTING HEART

. Every cotilting module over a ring R induces a t-structure with a Grothendieck heart in the derived category D (Mod-R ). We determine the simple objects in this heart and their injective envelopes, combining torsion-theoretic aspects with methods from the model theory of modules and Auslander-Reiten theory.


Introduction
The notion of a t-structure τ on a triangulated category T appears in the work of Beilinson, Bernstein, and Deligne [12] as a means to associate to T an abelian category, which then arises as the heart H τ of the t-structure.For example, if the triangulated category is the derived category D(A) of an abelian category A, an appropriately chosen t-structure τ will recover, according to this process, the given abelian category H τ ∼ = A. Other choices of t-structure on D(A) will give rise to hearts that may be derived equivalent to A.
The primary aim of [12] was to introduce the abelian category of perverse sheaves (see [25,Ch 8]) on a topological pseudomanifold X of even dimension, equipped with a stratification with no odd-dimensional strata.The triangulated category was the bounded derived category of sheaves on X and the t-structure was chosen to yield the perverse sheaves on X as the objects of the heart H τ .As these perverse sheaves were seen to have finite length, attention naturally turned to the simple ones, which were determined [12,Theorem 4.3.1]by the intersection homology of the connected strata, appropriately shifted (see also [25,Theorem 8.1.8]).
In the same spirit, our interest in this paper are the simple objects of the heart of the HRS tilt of a torsion pair (T , F) in the category Mod-R of modules over a ring.The HRS tilt of (T , F) is a t-structure τ introduced by Happel, Reiten, and Smalø [23] on the derived category D(Mod-R).The objects of the heart H τ need not all be of finite length, but Happel, Reiten and Smalø showed that H τ contains a torsion pair (F, T [−1]) whose constituent classes are equivalent to those of (T , F), with the roles reversed.When viewed with regard to the torsion pair (F, T [−1]), every simple object of H τ is evidently either torsion or torsionfree.
We provide a torsion theoretic description of the simple objects of H τ using the notion of an almost torsionfree module (Definition 3.1) and its dual, that of an almost torsion module.Every torsionfree module is almost torsionfree, but there may be others, which are necessarily torsion.The dual statement also holds and we characterise in Theorem 3.6 the torsionfree simple objects of H τ to be those of the form T [−1] where T R ∈ T ⊆ Mod-R is an almost torsionfree torsion module, and the torsion simples of the heart to be the objects that correspond to almost torsion torsionfree modules.
If C ∈ Mod-R is a 1-cotilting module, then the cotilting class C = ⊥ 1 C = Cogen(C) is a torsionfree class in Mod-R and we call the heart of the HRS tilt of the torsion pair (Q, C) a cotilting heart.Every cotilting heart is a Grothendieck category [15] whose subcategory Inj(H τ ) of injective objects is known to be equivalent to Prod(C) ⊆ C. As such, the injective Date: March 21, 2024.objects of H τ are torsion, but when Prod(C) ⊆ C ⊆ Mod-R is regarded as a subcategory of Rmodules, it consists of torsionfree modules.As cotilting modules are pure-injective, so are all the objects of Prod(C).Corollary 5.12 makes use of the notion of a neg-isolated indecomposable pure-injective module ( §5.5) from the model theory of modules to characterise the injective envelopes of simple objects of the heart, when they are considered as modules in the definable subcategory C ⊆ Mod-R.It states that they are precisely the neg-isolated indecomposable pure-injectives of C that belong Prod(C).
Among the neg-isolated indecomposable pure-injective modules of a definable subcategory of Mod-R such as C, there is the distinguished class of critical neg-isolated indecomposable pure-injectives U, determined by the property that every monomorphism U → V in Mod-R with V ∈ C is a split monomorphism (Proposition 5. 16).These are the torsionfree modules that correspond to the injective envelopes of torsion simple objects of the heart.It is a general fact about definable subcategories that there exist enough critical neg-isolated indecomposables, in the sense that every torsionfree module F ∈ C may be embedded -not necessarily purelyinto a direct product of critical neg-isolated indecomposables in C. It follows that no 1-cotilting module is superdecomposable.
The characterisations of the torsionfree and torsion simple objects of a cotiliting heart in terms of almost torsionfree and almost torsion modules are categorically dual and seem to give the two kinds of simple object equal status.The question of existence however does not.We call the neg-isolated indecomposable pure-injectives of C that correspond to injective envelopes of torsionfree simple objects of the heart special.In stark contrast to the critical neg-isolated indecomposables, there is a 1-cotilting module C Λ over the Kronecker algebra Λ (Example 6.2) whose cotilting class contains no special neg-isolated indecomposable pure-injectives.In other words, every simple object of the cotilting heart of C Λ is torsion.
All of our characterisations of the simple objects of a cotilting heart may be regarded as part of Auslander-Reiten theory, but only the last makes direct appeal to almost split morphisms in the module category.This final description relies on the approximation theory of the complete cotorsion pair (C, C ⊥ ).The almost split morphisms that appear are left almost split morphisms that enjoy the strong uniqueness property (Definition 2.6).It is included in the following summary of all our results on the torsion simple objects of a cotilting heart.
Theorem A (Theorems 3.6 and 4.2, Proposition 4.1, Corollary 5. 18) The following statements are equivalent for a module N .
(1) N is isomorphic to the injective envelope of a torsion simple S in H τ .
(2) N is a critical neg-isolated module in C.
(3) There exists a short exact sequence in Mod-R, where S is torsionfree, almost torsion, a is a C ⊥ 1 -envelope, and b is a strong left almost split morphism in C.
A strong left almost split morphism is either a monomorphism or epimorphism (Lemma 4.3).Theorem A includes a characterisation of the torsion simple objects of the heart as the torsionfree modules that appear as kernels of strong left almost split morphisms in C, while its dual, the next Theorem B, characterises the torsionfree simple objects as shifts of torsion modules that arise as cokernels of strong left almost split morphisms in C.
Theorem B (Theorems 3.6 and 4.2, Proposition 4.1, Proposition 5. 19) The following statements are equivalent for a module N .
(1) N is isomorphic to the injective envelope of a torsionfree simple S[−1] in H τ .
(2) N is a special neg-isolated module in C.
(3) There exists a short exact sequence 0 / / N a / / N b / / S / / 0 in Mod-R, where S is torsion, almost torsionfree, a is a a strong left almost split morphism in C, and b is a C-cover.
The simple objects in cotilting hearts are crucial to understanding the phenomenon of mutation and to describe the lattice tors-R of torsion classes in the category mod-R of finite dimensional modules over a finite dimensional algebra R. Indeed, the simple objects in the heart H τ correspond to the arrows in the Hasse quiver of tors-R which are incident to the torsion class Q ∩ mod-R, or equivalently, to the irreducible mutations of the cotilting module C, cf.[18,9,5].In a forthcoming paper [4], we will employ Theorems A and B to obtain an explicit description of mutation of cotilting (or more generally, cosilting) modules.This will allow us to interpret mutation as an operation on the Ziegler spectrum of R which will amount to replacing critical neg-isolated summands by special ones, or viceversa.

Background
2.1.Notation.In this section we fix our basic notations and conventions.
Let R be a unital associative ring.We denote the category of right R-modules by Mod-R and the category of left R-modules by R-Mod.The full subcategories of finitely presented modules are denoted mod-R and R-mod respectively.The derived category of Mod-R is denoted D(Mod-R).We abbreviate the Hom-spaces in D(Mod-R) in the following way: for all complexes X, Y .
All subcategories will be strict (i.e.closed under isomorphisms) and, for a full subcategory B, we will use the notation B ∈ B to indicate that B is an object of B.
Let X be a set of objects in an additive category A with products.Then we use the notation Prod(X ) for the set of direct summands of products of copies of objects contained in X .In the case where A is Grothendieck abelian, we will use Cogen(X ) to denote the set of subobjects of objects contained in Prod(X ).We will write Inj(X ) for the class of injective objects in the category A that are contained in X .We will consider the following full perpendicular subcategories determined by a subset I ⊆ {0, 1}: A (M, X) = 0 for all X ∈ X and i ∈ I}.In the case where X = {X}, we will use the notation X ⊥ I for X ⊥ I and Prod(X) for Prod(X ) etc. Furthermore, we will often just write X ⊥ 0 instead of X ⊥ {0} etc.

Torsion pairs and HRS-tilts.
In this subsection we introduce the notion of an HRStilt, due to Happel, Reiten and Smalø.The idea of their work is to produce a t-structure in the derived category D(Mod-R) from a given torsion pair in Mod-R.More details about the construction and properties of this t-structure can be found in [23].
Torsion pairs, first introduced by Dickson [19], will be a central object of study in the latter sections of this article.The following is the definition of a torsion pair in abelian category A. Definition 2.1.A pair of full subcategories (T , F) of A is called a torsion pair if the following conditions hold.
(1) For every T ∈ T and F ∈ F, we have that Hom A (T, F ) = 0.
(2) For every X in A, there exists a short exact sequence where t(X) ∈ T and X/t(X) ∈ F.
We call T the torsion class and F the torsionfree class.If, in addition, the class T is closed under subobjects, then the torsion pair is called hereditary.
We extend the above terminology to objects: the objects T in T are called torsion and the objects F in F are called torsionfree.
The next result shows that such a torsion pair in Mod-R yields a t-structure in D(Mod-R), in the sense of [12].Note that we define our t-structure to consist of two Hom-orthogonal classes; this differs from the original definition by a shift.
We will refer to this t-structure as the HRS-tilt of (T , F).It is shown in [12] that the heart such that X, Y and Z are contained in H τ .For any two objects X and Y in H τ , there are functorial isomorphisms We will make use of the following lemma in Section 3.
(1) Let f : X → Y be a morphism in H τ , and let Z be the cone of f in D(Mod-R).Consider the canonical triangle where K ∈ U τ and W ∈ V τ .Then (  Proof.Recall that the cone of a morphism h in Mod-R has homologies Ker (h) in degree −1, Coker(h) in degree 0, and zero elsewhere.
(2) We know from (1) that Ker Hτ (h) = 0 if and only if the cone of h belongs to V τ .This means Ker (h) = 0 and Coker(h) ∈ F. Similarly, Coker Hτ (h) = 0 if and only if the cone of h belongs to U τ , which means that Coker(h) ∈ T .
(3) The cone of h[−1] belongs to V τ if and only if Ker (h) ∈ F, and it belongs to U τ if and only if Coker(h) = 0 and Ker (h) ∈ T .

2.3.
Cotilting modules and cotorsion pairs.In this paper we will focus on HRS-tilts of torsion pairs induced by cotilting modules.We now introduce these modules and collect together some of their important properties.The definition of a (possibly infinitely generated) cotilting module first appeared in [14], dualising the definition of [16].Definition 2.4.A right R-module C is called a cotilting module if the following three statements hold.
(2) Ext 1 R (C κ , C) = 0 for all cardinals κ. (3) There exists a short exact sequence 0 → C 1 → C 0 → I → 0 where C i ∈ Prod(C) for i = 0, 1 and I is an injective cogenerator of Mod-R.We say that cotilting modules C and In [14,Prop. 1.7], the authors show that Cogen(C) = ⊥ 1 C and, moreover, that this equality characterises cotilting modules.We call this class C := Cogen(C) = ⊥ 1 C the cotilting class associated to C and it follows that τ = (Q, C) := ( ⊥ 0 C, Cogen(C)) is a (faithful) torsion pair.We call the heart of the HRS-tilt of τ the associated cotilting heart.
We know from [15] that a cotilting heart H τ is a Grothendieck category with injective cogenerator C so, in particular, we have Inj(H τ ) = Prod(C).
Remark 2.5.Often the term cotilting module is used for the more general notion of an ncotilting module, which was first defined in [1].In that context, the modules specified in Definition 2.4 are called 1-cotilting modules.Since we will not be considering n-cotilting modules for n > 1, we will use the term cotilting module to refer to a 1-cotilting module.
It was shown in [10] that every cotilting module is pure-injective and every cotilting class is definable (see Sections 5.3 and 5.4 for definitions of these terms).As a consequence, the class C is closed under direct limits, and the cotorsion pair (C, cogenerated by C is a perfect cotorsion pair.In particular, for every module M in Mod-R, there exist special approximation sequences 0 In particular, a is a C-cover and b is a C ⊥ 1 -envelope.Moreover, we have that C ∩ C ⊥ 1 = Prod(C).For more details on covers, envelopes and cotorsion pairs, we refer the reader to [21].
2.4.Injective envelopes of simples and left almost split morphisms.In this section we will prove some preliminary results connecting simple objects in a cotilting heart to left almost split morphisms.Our considerations are inspired by [17].
Definition 2.6.Let X be an additive category.A morphism f : X → Y in X is called a left almost split morphism if it is not a split monomorphism and, for any g : X → Z that is not a split monomorphism, there exists a morphism h : Y → Z such that g = hf .If the morphism h is unique for every such g, then we call f a strong left almost split morphism.
We begin with the following general result about Grothendieck abelian categories.Proof.
(1) Consider the morphism f given by the composition of the quotient E(S) → E(S)/S with the injective envelope E(S)/S → E(E(S)/S).This morphism is not a split monomorphism.Any other morphism g : E(S) → F in Inj(G) that is not a split monomorphism must have a non-trivial kernel K and so K necessarily contains S because S is essential in E(S).It follows that g factors through f as required.
(2) Consider the kernel 0 We will show that K = Ker (f ) is simple.Clearly K = 0 because f is not a split monomorphism.Moreover, every non-zero subobject G ⊂ K coincides with K, because the composition of the quotient E → E/G with the injective envelope E/G → E(E/G) of E/G is not a split monomorphism and thus factors through f .
Let e : K → E(K) be the injective envelope of K. Since k : K → E is a monomorphism and E is injective, there exists a split epimorphism m : E → E(K) such that e = mk.If m is not a monomorphism, then there exists a morphism g : E + → E(K) such that gf = m.This implies that 0 = gf k = mk = e, which is a contradiction.Therefore m is an isomorphism.
(3) By (2), we have an exact sequence 0 → S i → E f → E + where i is the injective envelope of S and S is simple.Consider the short exact sequence We will show that g is a strong left almost split morphism in G.Note that g is not a monomorphism and so cannot be a split monomorphism.Consider a morphism a : E → M that is not a split monomorphism.If ai = 0, then ai must be a monomorphism because S is simple.Then a is a monomorphism because i is an essential monomorphism.This implies that a splits because E is injective, but this is a contradiction.Thus ai = 0 and therefore a factors uniquely through the cokernel E/S ∼ = Im(f ) of i, as required.
(4) Consider f := eg where e : Ẽ → E( Ẽ) is the injective envelope of Ẽ.We will show that f is a left almost split morphism in Inj(G).Firstly, f is not a split monomorphism because otherwise g is a monomorphism and therefore split (since E is injective).Let a : E → E ′ be a morphism in Inj(G) that is not a split monomorphism.As g is a left almost split morphism in G, we have that there exists a morphism b : Ẽ → E ′ such that a = bg.Moreover, since E ′ is injective and e is a monomorphism, we have that there exists a morphism c : E( Ẽ) → E ′ such that a = c(eg) = cf , as required.
Remark 2.8.Following all the notation of Proposition 2.7, assume that the Grothendieck category G = H τ is a cotilting heart with respect to the cotilting torsion pair τ = (Q, C).Then, in the argument for Proposition 2.7(3), the object Im Hτ (f ) is in C because C is a torsion class.Hence g is a strong left almost split morphism in the subcategory C.Moreover, the argument for Proposition 2.7(4) only requires that g is a left almost split morphism in C since the injective objects in Corollary 2.9.Let G be a Grothendieck abelian category and let Inj(G) denote the full subcategory of injective objects in G.The following statements are equivalent for an object E of Inj(G).
(1) E is isomorphic to the injective envelope E(S) of a simple object S in G.
(2) There exists a left almost split morphism f : In that case, the torsion pair (T , F) is cogenerated by its torsionfree injective objects.Define the localisation G/T of G at T to be the category whose objects are the same as the objects X of G, but denoted by X T .The morphisms between two objects are given by the set Hom where X ′ ranges over the subobjects of X such that X/X ′ ∈ T and Y ′ ranges over the subobjects of Y such that Y ′ ∈ T .The work of Gabriel shows that the localisation functor The adjoint property allows us to calculate hom groups in the localisation: if The left adjoint L T is exact and the right adjoint R T : G/T → G is fully faithful.We may therefore identify the localisation category G/T with the full subcategory of G given by the image of R T .We note that G/T is contained in F. Because the right adjoint of an exact functor preserves injective objects, we may regard Inj(G/T ) under this identification as a subcategory of Inj(G); it is precisely the subcategory Inj(G/T ) = Inj(G) ∩ F of torsionfree injective objects.
As the right adjoint R T is left exact, we may identify the entire localisation G/T with the equivalent subcategory Cogen 2 (Inj(G/T )) ⊆ G consisting of the objects in G with a copresentation by torsionfree injectives.For more details on localisation in Grothendieck categories, we refer the reader to [29,Ch. 4].

Simple objects in the heart
In this section we consider the simple objects in the heart H τ of the HRS-tilt of a torsion pair τ = (T , F) in Mod-R.Since (F, T [−1]) is a torsion pair in H τ , it follows that any simple object S in H τ is either of the form S = F for some F in F or S = T [−1] for some T in T .In other words, the simple objects in H τ correspond to certain modules in Mod-R.The aim of this section is to identify these modules.We remark that our results remain valid when replacing Mod-R by an arbitrary abelian category.
Any torsionfree module is trivially almost torsionfree and any torsion module is trivially almost torsion.The condition (ATF1) implies that if an almost torsionfree object is not torsionfree, then it must be torsion.Similarly, any almost torsion object is either torsion or torsionfree.We will consider the non-trivial cases: the objects contained in T that are almost torsionfree and the objects contained in F that are almost torsion.These objects are also known as torsion, almost torsionfree and torsionfree, almost torsion respectively.Example 3.3.Suppose that the torsion pair τ = (T , F) in Mod-R is hereditary.If T is a torsion, almost torsionfree module, then (ATF1) implies that T is simple.Conversely, if T ∈ T is simple, then (ATF1) is clearly satisfied, and (ATF2) follows from the hereditary property of T .
Next, we show that the torsionfree, almost torsion modules are precisely the modules in Cogen 2 (Inj(F)) ∼ = Mod-R/T which become simple in the localisation.To see that a torsionfree, almost torsion module F belongs to Cogen 2 (Inj(F)), take the injective envelope of F, which is torsionfree by the hereditary property.Condition (AT2) implies that Ω −1 (F ) too is torsionfree; if we take its injective envelope, we get a copresentation of F = F T by torsionfree injective modules.Conversely, any module F in Cogen 2 (Inj(F)) satisfies condition (AT2).For, suppose w.l.o.g. that there is a short exact sequence 0 → F → A → B → 0 with A in F and B = 0 in T .Then we have a commutative diagram with exact rows where Ω −1 (F ) ∈ F and thus h = 0.But then the upper row is split exact, a contradiction.Now it is easy to see that a module in Cogen 2 (Inj(F)), regarded as an object of the localisation, contains no proper subobjects, and must therefore be simple, if and only if it satisfies condition (AT1).Finally, observe that the torsionfree, almost torsion module F is uniform, for if M 1 ∩ M 2 = 0 are two nonzero submodules of F, then, by (AT1), the direct sum F/M 1 ⊕ F/M 2 is a torsion module.As F embeds in a canonical way into this direct sum, the hereditary property would give the contradiction that F was also torsion.We conclude that E(F ) is indecomposable and, because Ω −1 (F ) also belongs to F, the injective envelope a : The following proposition is essentially a rephrasing of [38,Lem. 2.3].Proposition 3.4.Let τ = (T , F) be a torsion pair in Mod-R.
(1) The following statements are equivalent for a non-zero module T .
(a) T is almost torsionfree.
(2) The following statements are equivalent for a non-zero module F .
(a) F is almost torsion.(b) For every exact sequence Proof.We will prove (1), the argument for (2) is completely dual.
(1) [(a)⇒(b)] Assume T is almost torsionfree and consider an arbitrary exact sequence 0 → X → Y g → T with X in F. The case where g is an epimorphism is covered by the dual of [38,Lem. 2.3] (noting that the argument does not require T to be in T ).It remains to consider the case where g is not an epimorphism.Then Im(g) is in F by (ATF1), so Y is in F because F is closed under extensions.
[(b)⇒(a)] Suppose T satisfies (1)(b).By the dual of [38,Lem. 2.3] (noting again that the argument does not require T to be in T ), it suffices to show that every proper subobject of T is in F. But this follows immediately if we consider the exact sequence 0 → 0 → Y → T .Remark 3.5.Almost torsionfree and almost torsion modules are closely related to the minimal (co)extending modules over finite-dimensional algebras introduced in [9], and also the brick labelling given in [8] for functorially finite torsion pairs and in [18] for general torsion pairs.The precise connections between these concepts are made clear in [38].
If τ = (Q, C) is a cotilting torsion pair, then C is closed under direct limits, hence all torsion, almost torsionfree modules are finitely generated.On the other hand, there may be torsionfree, almost torsion modules which are not finitely generated, as Example 6.1 will show.For the only-if part, we start by considering a proper submodule U of F .Then, since F = S is simple in H τ , the map h : U → F gives rise to an epimorphism h : Conversely, we show that (AT1) and (AT2) imply that S = F is simple.To this end, we claim that every morphism 0 = f : S → A in H τ is a monomorphism.Since f factors through the torsion part of A with respect to the torsion pair (F, T [−1]), we can assume that A = C for some C ∈ F. Then f is a morphism in Mod-R, and f is a monomorphism (by (AT1)) with cokernel in F (by (AT2)).But then it follows from Lemma 2.3 that Ker Hτ (f ) = 0, and the claim is proven.
(1) Assume that T, T ′ are both torsion, almost torsionfree.If g : f is an isomorphism.
In particular, we have shown that the torsion, almost torsionfree modules and the torsionfree, almost torsion modules are bricks (i.e., their endomorphism rings are division rings).

Injective envelopes in a cotilting heart
In this section we will consider the case where our torsion pair τ = (Q, C) is a cotilting torsion pair.We know from [15] that the associated cotilting heart H τ is a Grothendieck category and so, in particular, has enough injectives.Next we relate the injective envelopes of simple objects in H τ to the special approximation sequences induced by the perfect cotorsion pair (C, (1) Let M ∈ Q and consider a short exact sequence 0 be the corresponding triangle in D(Mod-R).The following statements are equivalent.
Then there is g ∈ End Hτ (Y ) yielding a commutative diagram whose rows are given by triangles It follows that b = b • g and hence g is an isomorphism by the minimality of b.As g is an isomorphism, we conclude that h is an isomorphism as desired.
Since injective envelopes are unique up to isomorphism, there exists an isomorphism h : X → X ′ and a commutative diagram: where the induced morphism f must also be an isomosrphism.It follows that b is a C-cover of M in Mod-R. ( We have seen that a ′ is an injective envelope of M in H τ .Since injective envelopes are unique up to isomorphism, there exists an isomorphism h :   (1) There is a left almost split morphism f : M → M in M that is not a monomorphism.
(2) There is a left almost split morphism g : M → M in M that is an epimorphism.Moreover, a strong left almost split morphism is either a monomorphism or an epimorphism.
Proof.[(2) ⇒ (1)] is trivially true.To prove [(1) ⇒ (2)], observe that, if f : M → M is a left almost split morphism in M that is not a monomorphism, then g : M → Im(f ) is a left almost split morphism in M that is an epimorphism because Im(f ) ∈ M by assumption.
The final statements follows immediately because strong left almost split morphisms are unique up to isomorphism.In some cases, the heart H τ turns out to be locally finitely generated and in this case we have the converse of the second part of Corollary 4.4.
Corollary 4.5.Suppose H τ is locally finitely generated. ( Then C is a cotilting module that is equivalent to C and, moreover, C is isomorphic to a direct summand of every other cotilting module that is equivalent to C. Proof.We have already seen that every N ∈ N C arises as a direct summand of a cotilting module D that is equivalent to C. The rest of the corollary follows from the corresponding statements for locally finitely generated Grothendieck categories (see, for example, [27, Prop.3.17 and Cor.3.18]).Note that the special C ⊥ 1 -envelope of N ∈N C N becomes the injective envelope of N ∈N C N in the heart by Proposition 4.1.Example 4.6.There are some important cases where we know that the heart H τ is locally finitely generated and so we may apply Corollary 4.5.
( and the finitely presented objects in H τ are precisely the objects which belong to the bounded derived category D b (mod-R).

Neg-isolated modules
Let R be a ring and consider the category (R-mod, Ab) of additive functors from the category R-mod of finitely presented left R-modules to the category Ab of abelian groups.This functor category is a locally coherent Grothendieck category.We will use the notation (M, −) := Hom R (M, −) for the representable objects in (mod-R, Ab), M ∈ mod-R, and we will write [F, G] to denote the set Hom (R-mod,Ab) (F, G) of natural transformations from F to G.

5.1.
Finitely generated subfunctors of I.In this section, we study the subfunctors of the most important object of (mod-R, Ab), the forgetful functor I : mod-R → Ab.There is a natural isomorphism I ι / / (R, −) between I and the functor represented by R R .For M ∈ mod-R, the M -component is given by the morphism M is actually a morphism of R-modules which induces an isomorphism of pointed R-modules (M, m) / / ((R, M ), f m ) for each m ∈ M .So if φ ⊆ I is a subfunctor of the forgetful functor, then m ∈ φ(M ) if and only if f m ∈ φ(R, M ).In this way we observe that the rule φ → φ(R, −) is a bijective correspondence between the subfunctors of the forgetful functor and those of (R, −).
Associated to a pointed finitely presented module (M, m), is the finitely generated subfunctor Im (f m , −) ⊆ (R, −) induced by the natural transformation (f m , −) : (M, −) / / (R, −).It corresponds to the subfunctor H M,m of the forgetful functor I which takes N ∈ mod-R to the finite matrix subgroup (or pp-definable subgroup) H M,m (N ) = {h(m) | h ∈ Hom R (M, N )}.On the other hand, if φ ⊆ (R, −) is a finitely generated subfunctor, then there is a natural transformation η : (M, −) → (R, −) with image Im η = φ.By Yoneda's Lemma, there exists an R-linear morphism f m : R / / M such that η = (f m , −).Thus, every finitely generated subfunctor of I arises from a pointed finitely presented module (M, m) in this way.
The finitely presented objects of (mod-R, Ab) admit a canonical extension to Mod-R that respects direct limits.As all representable functors and their finitely generated subfunctors are finitely presented, this pertains to the finitely generated subfunctors φ ⊆ I of the forgetful functor.
Proposition 5.1.Let (M, m) be a pointed finitely presented module and φ = H M,m ⊆ I. Then (M, m) is a free realisation of φ, in the sense that m ∈ φ(M ), and whenever (N, n) is a pointed module with n ∈ φ(N ), then there exists a morphism h : (M, m) → (N, n) of pointed modules.
Proof.The case when (N, n) is finitely presented is clear by definition of H M,m .For the general case, use the fact that φ respects direct limits, so there exists a pointed finitely presented module (N ′ , n ′ ) and a morphism (N ′ , n ′ ) → (N, n) of pointed modules with the property that n ′ ∈ φ(N ′ ).Now use again the definition of φ.

5.2.
The pp-type of a pointed module.The finitely generated subobjects of I form a modular lattice.We will now use techniques from the model theory of modules [44,30,31] to investigate this lattice.The model theoretic approach allows us to represent the finitely generated subfunctors of I by formulas φ(x) in a certain first-order language.The formula endows the corresponding subfunctor φ ⊆ I with semantic content, so that if (M, m) is a pointed right R-module, we may evaluate the statement φ(m) as true in M, denoted by M |= φ(m), or not.In other words, the M -component of the inclusion φ ⊆ I consists of the "solutions" in M to the formula φ(x).We refer to the formulas φ(x) as pp-formulas, see [31, §12.2] for details.
The pp-type, denoted by pp(N, n), of a pointed module (N, n) is the collection of ppformulas φ(x), for which N |= φ(n).Equivalently, we can think of pp(N, n) as the collection of finitely generated subfunctors φ ⊆ I for which n ∈ φ(N ).As such, the pp-type pp(N, n) may be regarded as a filter Φ in the lattice of finitely generated subfunctors of the forgetful functor I ∈ (mod-R, Ab); we say that (N, n) is a realisation of Φ.The Completeness Theorem of first-order logic ensures that every filter Φ arises as the pp-type Φ = pp(N, n) of some pointed module.The functorial property of pp-formulas ensures that if The general question thus arises of when an inclusion pp(M, m) ⊆ pp(N, n) of pp-types is induced by a morphism of pointed modules.If M is finitely presented, it is easy to see that pp(M, m) is the principal filter generated by the subfunctor H M,m ⊆ I. Proposition 5.1 therefore implies that every inclusion pp(M, m) ⊆ pp(N, n) is induced by a morphism h : (M, m) → (N, n).Next we consider a condition on the module (N, n) that ensures the existence of a morphism of pointed modules.which is called coYoneda embedding.It allows us to consider the exact structure of the functor category inside the module category.This is known as the pure exact structure in Mod-R.
Definition 5.2.A short exact sequence 0 In this case, we refer to f as a pure monomorphism and to g as a pure epimorphism.
It is natural to consider the modules that are injective with respect to the pure exact structure.Definition 5.3.A module N is called pure-injective if every pure exact sequence of the form (5.2) is a split exact sequence.
Clearly any module that becomes an injective object under the coYoneda embedding is pureinjective and, in fact, all injective objects in (R-mod, Ab) arise in this way.That is, the coYoneda embedding restricts to an equivalence cY : Pinj(R) ∼ / / Inj(R-mod, Ab) where Pinj(R) denotes the full subcategory of pure-injective objects in Mod-R and Inj(R-mod, Ab) denotes the full subcategory of injective objects in (R-mod, Ab).Furthermore, if M R is an Rmodule with pure-injective envelope ι : M → PE(M ), then the corresponding monomorphism A pure monomorphism f : M → N may also be characterised [31, Proposition 2.1.6]in terms of pp-formulas, which is equivalent to the condition that pp(M, m) = pp(N, f (m)) for every m ∈ M .So if Φ is a filter of finitely generated subfunctors of the forgetful functor with realisation Φ = pp(N, n), then the pure-injective envelope Φ = pp(PE(N ), n) too is a realisation of Φ.

5.4.
Definable subcategories of modules.We are interested in localisating the functor category (R-mod, Ab) at hereditary torsion classes associated to a particular kind of category of modules.Definition 5.5.A full subcategory D of Mod-R is called definable if it is closed under products, pure submodules and directed colimits.
A definable subcategory D ⊆ Mod-R is closed under pure-injective envelopes, so that its image under the coYoneda embedding cY(D) = D ⊗ − ⊆ (R-mod, Ab) is closed under injective envelopes in (R-mod, Ab).It follows that the torsion pair (T D , Cogen(D ⊗ −)) in (R-mod, Ab) cogenerated by D ⊗ −, is hereditary.Notation 5.6.We will denote the localisation (R-mod, Ab)/T D by (R-mod, Ab) D and the corresponding localisation functor by We will denote the Hom-spaces between two objects F, G in (R-mod, Ab) D by [F, G] D .
If D ⊆ Mod-R is a definable subcategory, then the hereditary torsion pair (T D , Cogen(D ⊗−)) in (R-mod, Ab) is of finite type in the sense of the following definition.We refer the reader to [24] for more details on the theory surrounding hereditary torsion pairs of finite type in locally coherent Grothendieck categories.Definition 5.7.A torsion pair (T , F) (not necessarily hereditary) in a Grothendieck category G is said to be of finite type if the torsionfree class F is closed under directed limits in G.
The following theorem will be very important in what follows.For details, we refer to [31, §12.3].
Theorem 5.8.The rule D → (T D , F D ) is a bijective correspondence between the collection of definable subcategories D ⊆ Mod-R and hereditary torsion pairs in (R-mod, Ab) of finite type, with inverse given by (T , Consider the functor Mod-R → (R-mod, Ab) D given by the composition of the functor (5.1) with the localisation functor (−) D .Since the injective objects in (R-mod, Ab) D coincide with the injective objects in (R-mod, Ab) that are contained in the torsionfree class, this functor restricts to an equivalence of categories where Pinj(D) denotes the full subcategory of pure-injective objects in D and Inj ((R-mod, Ab) D ) denotes the full subcategory of injective objects in (R-mod, Ab) D .
Remark 5.9.In the proofs below we will use the following observation several times.If D is a definable subcategory of (R-mod, Ab), then, for any M ∈ Pinj(D) and any R-module L, we have that This follows directly from the fact that (M ⊗ −) is injective and torsionfree with respect to the torsion pair (T D , F D ) induced by D in (R-mod, Ab).In particular, we have Proof.Notice that, since N is an indecomposable pure-injective module, the endomorphism ring of N is then local.This implies that every endomorphism of pointed modules (N, n) → (N, n) with n ∈ N being nonzero is an automorphism.The equivalence [(1) ⇔ (2)] follows directly from Corollary 2.9 and the discussion following Notation 5.6.
[(2) ⇒ (3)] Let n ∈ N be nonzero and let f : N → N + be a left almost split morphism in Pinj(D).Set n + = f (n) and consider the D-filters Φ = pp(N, n) and Φ + = pp(N + , n + ).Clearly Φ + ⊇ Φ, and equality would imply by Remark 5.4 that there is a map of pointed modules (N + , n + ) → (N, n).Because f is not a split monomorphism, we get that Φ + ⊃ Φ is strictly larger.On the other hand, if Ψ ⊃ Φ is any strictly larger D-filter with realisation Ψ = (U, u) where U in Pinj(D), then there is a morphism g : (N, n) → (U, u) which is not a split monomorphism.Then g factors through f, and pp(U, u) ⊇ pp(N + , n + ), that is, Ψ ⊇ Φ + .
Consider the pushout of (N, n) with (M, m), postcomposed with a D-approximation (which exists e.g. by [31,Proposition 3.4.39])as in the middle row of (R, 1) Clearly, pp(N + , n + ) contains both Φ and ψ, so it must contain Φ + .Then the composition a K • p : (N, n) → (N + , n + ) cannot be a split monomorphism.In fact, it must be a left almost split morphism in D. For, suppose that we are given a morphism g : N → U in D that is not a split monomorphism.Then there can't be a morphism of pointed modules (U, u) → (N, n), and we infer from Remark 5.4 that pp(U, u) ⊃ Φ is strictly larger.Thus ψ ∈ pp(U, u).By Proposition 5.1 there exist a morphism (M, m) → (U, u) from the free realisation of ψ and a factorisation through the pushout.As U ∈ D, this map from the pushout then factorises through its D-approximation, as required.
[( 4) ⇒ (2)] Suppose there exists a left almost split morphism h : N → N in D and let e : N → P E( N ) be the pure-injective envelope of N .We will show that g := eh is a left almost split morphism in Pinj(D).Let u : N → U be a morphism in Pinj(D) that is not a split monomorphism.Then there exists some v : N → U such that u = vh.Using that e is a pure monomorphism and that U is pure-injective, we have that there exists f : P E( N ) → U such that f g = f eh = vh = u as desired.
The theorem above is a relative version of results in [31, §5.3.5] for the case D = Mod-R.Indeed, in that case condition (3) means precisely that Φ = pp(N, n) is a neg-isolated pptype.This suggests the following terminology.Definition 5.11.Let D be a definable subcategory of Mod-R.A pure-injective module N in D is called neg-isolated in D if it satisfies the equivalent conditions of Theorem 5.10.
We have seen that left almost split morphisms in a cotilting class are intimately related to the injective envelopes of simple objects in the cotilting heart.Since cotilting classes C are always definable subcategories, it is natural to ask how the results of Section 4 are related to the neg-isolated modules in C. Corollary 5.12.Let C be a cotilting module with torsion pair τ = (Q, C).The R-modules that become injective envelopes of simple objects in H τ are precisely the neg-isolated modules in C which lie in Prod(C).
Proof.This follows immediately from Theorem 5.10 and Corollary 4.4.5.6.Critical modules.Let (Q, C) be a cotilting torsion pair in Mod-R.The aim of this section is to investigate the neg-isolated modules in C that are domains of left almost split morphisms in C that are epimorphisms.In Lemma 4.3 we saw that these coincide with the domains of left almost split morphisms in C that are not monomorphisms; these are the critical neg-isolated modules.
Definition 5.13.Let D be a definable subcategory of Mod-R.We call a neg-isolated module N in D critical in D if there exists an morphism h : N → N + that is a left almost split morphism in Pinj(D) such that h is not a monomorphism.
When the definable subcategory is a torsionfree class, we have the following alternative characterisation of critical modules showing that they are exactly the neg-isolated modules such that the associated strong left almost split morphism is an epimorphism.Proposition 5.14.Let (T , F) be a torsion pair in Mod-R such that F is a definable subcategory.The following statements are equivalent for a module N in F.
(1) N is a critical neg-isolated module in F.
(2) There exists a left almost split morphism f : N → N in F that is an epimorphism.In particular, f is a strong left almost split morphism.
Proof.[(1) ⇒ (2)] Let h : N → N + be a left almost split morphism in Pinj(F) that is not a monomorphism.Note that Im(h) is contained in F because F is closed under submodules.We will show that h : N → Im(h) is a left almost split morphism in F. Since h is not a monomorphism, it follows that h is not a split monomorphism.Suppose u : N → U is a morphism in F that is not a split monomorphism and consider eu : N → P E(U ) where e is the pure-injective envelope of U .Then eu can't be a split monomorphism.Since h is left almost split in Pinj(F) and P E(U ) lies in F by [31,Thm. 3.4.8],there exists a morphism f : N + → P E(U ) such that eu = f h.Let k : Ker (h) → N be the kernel of h.Then euk = f hk = 0 and, moreover, uk = 0 because e is a monomorphism.Using that h is the cokernel of k, we conclude that there exists a unique morphism g : Im(h) → U such that u = g h.
[(2) ⇒ (1)] Let h : N → N be a left almost split morphism in F that is an epimorphism.Then N must belong to Pinj(F), since otherwise the pure-injective envelope e : N → P E(N ) would factor through h and h would be an isomorphism.In the proof of Theorem 5.10, we showed that eh : N → N → P E( N ) is a left almost split morphism in Pinj(F).Since h is an epimorphism, it is clear that eh is not a monomorphism.We saw in Theorem 5.10 that neg-isolated modules in definable subcategories are in bijection with injective envelopes of simple objects in the corresponding localisation of the functor category.In the next lemma we identify which injective envelopes of simple objects give rise to critical modules.Lemma 5.15.Let D be a definable subcategory of Mod-R and suppose N is neg-isolated in D. The following statements are equivalent.
(1) N is critical in D. ( Proof.Since N is neg-isolated in D, we have the following set up according to Section 5.3.There exists a left almost split morphism h : N → N + in Pinj(D) and, moreover, the morphism By Proposition 2.7, we see that the kernel of (h ⊗ −) D is isomorphic to the monomorphism i : [(1)⇒( 2)] Suppose that Ker (h) = 0. Consider an element k ∈ Ker (h) and consider it as the morphism k : R → N that takes 1 → k.This yields a non-zero morphism and it follows from Remark 5.9 that (k ⊗ −) D = 0.Moreover, we have that (h⊗ −)•(k ⊗ −) = (h(k)⊗ −) = 0 and so (h⊗ −) D •(k ⊗ −) D = 0. Therefore, the morphism (k ⊗ −) D factors through the kernel of (h ⊗ −) D .Thus we have a factorisation i • g = (k ⊗ −) D and, in particular, we have a non-zero morphism g : (R ⊗ R −) D → S.
[(2)⇒(1)] Now suppose that there exists a non-zero morphism f : Then, by Remark 5.9, there exists a non-zero morphism g : (R ⊗ R −) → (N ⊗ R −) such that g D = i • f .Since the coYoneda embedding given in Section 5.3 is fully faithful, we can find a non-zero element n ∈ N that determines a morphism n : R → N such that 1 → n and (n ⊗ −) = g.By definition, we have that (n ⊗ −) D = i • f and so we have By Remark 5.9, this implies that (h(n) ⊗ −) = 0 and hence 0 = n ∈ Ker (h).This characterisation of critical modules in a definable subcategory D allows us to show in the next proposition that they cogenerate D.Moreover, it follows from this that the critical modules in a definable subcategory D are related to the split injective modules in Proposition 5.16.Let D be a definable subcategory of Mod-R and let D 0 denote the set of critical modules in D.
(2) The split injective modules in D are contained in the set Prod(D 0 ).
(3) A module L belongs to D 0 if and only if it is neg-isolated and split injective in D. Proof.
(1) The definable subcategory D is closed under pure-injective envelopes and so it suffices to show that Pinj(D) ⊆ Cogen(D 0 ).Let N ∈ Pinj(D) and let n ∈ N and consider is the canonical quotient morphism and j is a monomorphism.Observe that the functor (R ⊗ −) ∼ = Hom R (R, −) is finitely presented, and so is its localisation (R ⊗ −) D by [24,Prop. 2.15].Then K must be contained in a maximal subobject M , which induces an epimorphism r The isomorphisms in Remark 5.9 yield that h ∼ = (h n ⊗ −) D for some h n : N → L n .The commutativity of the above diagram yields that 0 Applying this argument to every element of N , yields a monomorphism N → n∈N L n with components equal to h n .
(3) Let m : L → N be a monomorphism in Mod-R with N ∈ D and L ∈ D 0 .If m is not split then, the composition em is not split, where e : N → PE(N ) is the pure-injective envelope of N .But then em must factor through the left almost split morphism h : L → L + in Pinj(D).But this is not possible because h is not a monomorphism.
Conversely, if L is neg-isolated and split injective in D, then there exists a left almost split morphism h : L → L + in Pinj(D), and h cannot be a monomorphism.Proposition 5.17.Now suppose (Q, C) is a cotilting torsion pair and let C 0 denote the set of critical modules in C. We fix an injective cogenerator I of Mod-R with a special C-cover Proof.First we show that C 0 is split injective.Consider a monomorphism h : C 0 → C ∈ C. Since C is cogenerated by C 0 and I is injective, there are a cardinal α and maps e : C ֒→ C α 0 and f : C α 0 →I such that f eh = g.As g is a C-cover, there is also f ′ : C α 0 →C 0 such that f = gf ′ .Now the right minimality of g yields that h is a split monomorphism.
The Proposition above shows in particular that every cotilting class contains critical negisolated modules and torsionfree, almost torsion modules.In contrast, we will see in Example 6.2 that torsion, almost torsionfree modules need not exist.Notice moreover that in general the class of split injectives in C is properly contained in Prod(C 0 ), cf.Example 6.3.where h must be non-zero.Indeed, if h = 0, then e factors through g and f is a split monomorphism, a contradiction.Since T ∈ Q, we conclude that Z is not in C = Q ⊥ 0 .
[(4)⇒(1)] Let g : N → N + be as in (4).Note that, by definition, the module N is neg-isolated in C. Set Z := Cokerg and consider the special / / 0 where g = ip is the canonical factorisation of g through Im g.Then kp must be a split monomorphism.Indeed, if kp is not a split monomorphism, then there exists a morphism l : N + → X such that kp = lg = lip.Since p is an epimorphism, it follows that k = li.A standard argument shows that the bottom sequence splits, which is not possible because Z would then be isomorphic to a summand of Y ∈ C. We have therefore shown that N is contained in Prod(C) because it is isomorphic to a direct summand of X ∈ Prod(C).Moreover, since kp = lg is a monomorphism, so is g and hence N is not critical.(1) Every critical neg-isolated module in C is a direct summand of C 0 .
(2) Every special neg-isolated module in C is a direct summand of C 1 .
Proof.To prove the corollary we will use the following general property of neg-isolated modules (see [30,Prop. 9.29]): if a neg-isolated module N in a definable subcategory D is a direct summand of i∈I M i where M i ∈ D for all i ∈ I, then N is a direct summand of M i for some i ∈ I.
is a coproduct factor of C ⊗ R −, as required.

Examples
In this section, we discuss some examples that illustrate our results.We also study the special case of cotilting modules induced by ring epimorphisms, where we establish some interesting properties of special and critical neg-isolated modules.
6.1.The Kronecker algebra.Let Λ be the Kronecker algebra, i.e. the path algebra of the quiver • − → − → • over an algebraically closed field k.It is well known that the category of finite dimensional indecomposable modules admits a canonical trisection (p, t, q), where p and q denote the families given by the preprojective and preinjective modules, respectively, and t = x∈X t x is the tubular family formed by the regular modules and indexed over the projective line X = P 1 (k).Given a simple regular module S, we denote by S ∞ and S −∞ the Prüfer and the adic module on the corresponding ray and coray, respectively.Further, we denote by G the generic module.Recall from [34] that End R G is a division ring, and G is the unique (up to isomorphism) indecomposable module which has infinite length over Λ, but finite length over its endomorphism ring.Moreover, G cogenerates the class F = t ⊥ 0 of all torsionfree modules, it generates the class D = ⊥ 0 t of all divisible modules, and the intersection F ∩ D = AddG is equivalent to the category of all modules over a simple artinian ring Q which is obtained as universal localization of Λ at t and is Morita equivalent to End R G.
A complete description of the hearts of all cotilting Λ-modules is given in [32].Let us focus on two special cases.The torsion, almost torsionfree modules coincide with the simple regular modules, while G is the only torsionfree, almost torsion module.We refer to [32] for the first statement and prove the second for the reader's convenience.We have shown that G is almost torsion.For the uniqueness, observe that any other torsionfree, almost torsion module X ∈ F is cogenerated by G, hence Hom Λ (X, G) = 0, and X ∼ = G by Corollary 3.7.
It follows that G is simple injective in the heart H τ .Moreover, every simple regular module S gives rise to a short exact sequence as in Theorem B, and to a minimal injective coresolution We infer that G is the only critical neg-isolated module in F, and the special neg-isolated modules coincide with the adic modules.These are the neg-isolated modules in F which belong to Prod(C).Observe that also the modules in p are neg-isolated in F, see Theorem 5.10.
Finally, let us remark that H τ is not hereditary.Indeed, it is shown in [42, 5.2] that the heart of a torsion pair is hereditary only if the torsion pair splits.But if S x denotes the simple regular in the tube t x , then by [35,Prop. 5] there is a non-split exact sequence with x∈X S x ∈ T and G (α) ∈ F. This shows that (T , F) is not a split torsion pair.
We have just seen that torsionfree, almost torsion modules may be infinite dimensional, while this is not possible for torsion, almost torsionfree modules according to Remark 3.5.The next example, however, exhibits a cotilting module without torsion, almost torsionfree modules.Example 6.2.Consider now the torsion pair τ = (Q, C) in Mod-Λ generated by the set q.It is cogenerated by the cotilting module The heart H τ is locally coherent and hereditary, and it is equivalent to the category of quasicoherent sheaves over X.In particular, all simple objects in H τ are torsion.In other words, there are no torsion, almost torsionfree modules, and the torsionfree, almost torsion modules coincide with the simple regular modules.For details, we refer again to [32].
Every simple regular module S gives rise to a short exact sequence as in Theorem A, which can also be regarded as the minimal injective coresolution of the simple torsion object S in H τ .The critical neg-isolated modules in C thus coincide with the Prüfer modules, and there are no special neg-isolated modules.
The last example also shows that in general the class ProdC 0 in Proposition 5.16 does not coincide with the class of split injectives in C. Example 6.3.Let τ = (Q, C) be as in Example 6.2.Given an injective cogenerator I of Mod-Λ, there is a short exact sequence where g is a C-cover, C 0 is a direct sum of Prüfer modules and C 1 is a direct sum of copies of G, see [33,Thm. 7.1].Observe that C 1 is a direct summand of a product of copies of C 0 by [35,Prop. 3], but it is not split injective in C because the sequence (6.1) is not split.

6.2.
Hereditary torsion pairs.In this section, we assume that τ = (Q, C) is a hereditary cotilting torsion pair with cotilting module C. Equivalently, the torsionfree class C is closed under injective envelopes.If V ∈ C is split injective, then its injective envelope V → E(V ) is a monomorphism in C, and must therefore be an isomorphism.We conclude that the subcategory of split injective objects of C is given by the category Inj(Mod-R) ∩ C of torsionfree injective modules, and that if C 0 = E C denotes the set of critical modules in C as in Proposition 5.16, then Prod(C 0 ) = C ∩ Inj(Mod-R).
Given a module M , we denote by C(M ) a C-cover of M and by K(M ) its kernel, If C is a cotilting module cogenerating C, then K(M ) is an object of Prod(C) which is uniquely determined by M , up to isomorphism.If F ∈ C is a simple torsionfree module, then its injective envelope E(F ) is also torsionfree and so serves as its own C-cover, but if Q ∈ Q is a torsion simple module, then its injective envelope E(Q) is not torsionfree so its C-cover is given by the epimorphism in In Example 3.3, we elaborated on the equivalence of Theorem 4.2(2) for a hereditary torsion pair, showing that the torsionfree, almost torsion modules correspond to the simple objects of the localisation Mod-R/Q, and that their injective envelopes are the critical neg-isolated indecomposable pure-injectives.The following elaborates on the equivalence given by Theorem 4.2(1).Theorem 6.4.Suppose that (Q, C) is a hereditary cotilting torsion pair and Q R ∈ Q is a torsion simple module.The C-cover of Q is given by the pullback of (6.2) along its injective envelope Proof.As in the argument used in the proof of Proposition 5.17, the module C(E(Q)) must be split injective and therefore, by the hereditary property, injective.Moreover, any indecomposable summand of C(E(Q)) which does not intersect K(E(Q)) must be isomorphic to E(Q), which is impossible as Q ∈ Q.We conclude that the kernel morphism of c is an injective envelope.This also implies that the embedding of F into C(E(Q)) is an injective envelope.Furthermore, it follows that K(E(Q)) is indecomposable.For, suppose that K(E(Q)) = K 1 ⊕ K 2 were a proper decomposition.Neither of the summands can be injective, since they are contained in the kernel of a C-cover.The injective envelope of K(E(Q)) as well as its cosyzygy would then be decomposable, contradicting the fact that E(Q) is not.
Because F is torsionfree and K(E(Q)) belongs to C ⊥ 1 , b : F → Q is a special C-precover.As such, it contains the C-cover C(Q) → Q as a direct summand, whose kernel would be a direct summand of K(E(Q)).But K(E(Q)) is indecomposable, and Q is not torsionfree, so the C-cover of Q must contain K(E(Q)), and properly so.As Q is simple, so we see that b : The last statement follows from Theorem 4.2(1).Theorem 6.4 allows us to infer that the module PE(⊕ N C N ) of Corollary 5.22 that arises as a summand of any cotilting module C for C is itself cotilting.Because it is a summand of a cotilting module, it certainly satisfies the first two conditions of being one.To verify the third, consider the injective cogenerator I = Sim(Mod-R) E(S) of Mod-R, where the index set runs over the set of all simple modules.It may be decomposed as where the index set has been partitioned into the simple torsion and simple torsionfree modules, respectively.Take the special C-precover of I given by the product of the respective C-covers, Because every K(Q) is neg-isolated, the kernel belongs to Prod(PE(⊕ N C N )).On the other hand, the middle term is injective and therefore belongs to C ∩ Inj(Mod-R) = Prod(C 0 ).
is a 1-cotilting module for C that is isomorphic to a direct summand of C.

6.3.
Commutative noetherian rings.Let now R be a commutative noetherian ring.It is shown in [6] that the cotilting torsion pairs are precisely the hereditary torsion pairs in Mod-R with R being torsionfree.They are parametrized by the subsets P ⊂ Spec(R) that are closed Moreover, if X is an indecomposable module in Prod(C), then it is pure-injective and therefore has a local endomorphism ring.This shows that π i ι i must be an isomorphism for i = 0 or i = 1, that is, X lies in Prod(C 0 ) or Prod(C 1 ).Finally, X can't lie in both, because Prod(C 0 ) ⊂ ⊥ 0 C 1 by conditions (M1) and (M2).This proves statement (2).
For statement (3), observe that the C-cover in (6.3) can't be an isomorphism, so C 1 = 0 and C 1 = ∅.The existence of torsion, almost torsionfree modules then follows from Theorem B. Proposition 6.11.[39] Let assumptions and notation be as above, and suppose that R is right artinian.Assume further that C is a proper subcategory of Mod-R, and let W be the class of all modules with a finite filtration by torsion, almost torsionfree modules.Then the class of all direct limits of modules in W coincides with the left perpendicular category ⊥ 0,1 C 0 of the module C 0 in (6.3).Theorem 6.12.Let R be a right artinian ring and let C be a minimal cotilting module with associated ring epimorphism λ : R → S.
(1) Every indecomposable summand of a product of special neg-isolated modules in C is special neg-isolated in C. (2) S is a right coherent ring.
Proof.(1) Suppose M is an indecomposable module in Prod(M C ) which is not special.Recall that Y = ⊥ 0,1 C 1 is the essential image of λ * .By [3,Proposition 4.15] there is an exact sequence where M ′′ belongs to Q and Hom R (η, Y ) is an isomorphism for every module Y ∈ Y.In particular, Hom R (η, C 0 ) is an isomorphism.Furthermore, Ext 1 R (M ⊗ R S, C 0 ) = 0 because M ⊗ R S ∈ Y ⊆ ⊥ 1 C 1 = ⊥ 1 C and C 0 ∈ Prod(C).It follows that Ext 1 R (M ′′ , C 0 ) = 0, and of course also Hom R (M ′′ , C 0 ) = 0.But then M ′′ lies in the left perpendicular category of C 0 , and by Proposition 6.11 it is a direct limit of modules in W, the class of all modules with a finite filtration by torsion, almost torsionfree modules.
Consider now W ∈ W. We have Hom R (W, M ⊗ R S) = 0 because W is torsion and M ⊗ R S is torsionfree.Moreover, Ext 1 R (W, M ) = 0, because M is an indecomposable, non-special module in Prod(C), thus becomes indecomposable injective in H τ and satisfies Ext 1 R (S, M ) ∼ = Hom Hτ (S[−1], M ) = 0 for all torsion, almost torsionfree modules S. We conclude that Hom R (W, M ′′ ) = 0.
In conclusion, we have shown that M ′′ = 0 and M ∼ = M ⊗ R S belongs to Y. Bu then M ∈ ⊥ 0 C 1 , which contradicts the assumption M ∈ Prod(M C ) = Prod(C 1 ).
(2) By Example 4.6, the heart H τ is a locally coherent Grothendieck category.Then we know from [24], [26] that the (isoclasses of) indecomposable injective objects form a topological space Spec(H τ ), where a basis of open subsets is given by the collection Moreover, there is a one-one-correspondence between the open subsets of Spec(H τ ) and the hereditary torsion pairs of finite type in H τ , which maps a torsion pair (S, R) to the set of indecomposable injectives which are not contained in R. Let

Theorem 3 . 6 .
Let τ = (T , F) be a torsion pair in Mod-R.The simple objects S in the heart H τ of the HRS-tilt of (T , F) are precisely those of the form S = T [−1] with T torsion, almost torsionfree and S = F with F torsionfree, almost torsion.Proof.Using the canonical exact sequence 0 → F → S → T [−1] → 0 in H τ with F ∈ F and T ∈ T , we see that a simple object S is either of the form S = F or S = T [−1].Let us show that an object of the form S = F with F ∈ F is simple if and only if F is almost torsion.The other case is proven dually.

Theorem 4 . 2 .( 1 )
Let τ = (Q, C) be a cotilting torsion pair with associated cotilting module C. Consider a short exact sequence 0 → L a → M b → N → 0 in Mod-R.The following statements are equivalent.(a) The module N is torsion, almost torsionfree and the morphism b is a special C-cover of N in Mod-R.(b) The module L is in Prod(C) and the morphism a is a strong left almost split morphism in C. (2) The following statements are equivalent.
(a) The module L is torsionfree, almost torsion and the morphism a is a special C ⊥ 1envelope of L in Mod-R.(b) The module M is in Prod(C) and the morphism b is a strong left almost split morphism in C. Proof.(1)[(a)⇒(b)] By Theorem 3.6, the object N is simple in the heart H τ and by Proposition 4.1, we have that c[−1] : N [−1] → L is an injective envelope, where L → M → N c→ L[1] is the completion of the exact sequence to a triangle in D(Mod-R).In particular, this means that L is injective in H τ and hence L ∈ Prod(C).It follows from Lemma 2.3, that 0→ N [−1] c[−1] → L a → M → 0 is a short exact sequence in H τ .Then Remark 2.8 tells us that a is a strong left almost split morphism in C.[(b)⇒(a)] It follows from our assumptions that L is injective in H τ .By Proposition 2.7 and Remark 2.8, the kernel S := Ker Hτ (a) is simple and the inclusion 0 → S c → L is an injective envelope.Moreover, since strong left almost split morphisms starting at an object are unique up to isomorphism, a is an epimorphism in H τ .By Lemma 2.3, we have that N is in Q and also that N [−1] ∼ = S.By Theorem 3.6 we have that N is almost torsionfree.Finally, since c is an injective envelope, it follows from Proposition 4.1 that b is a C-cover of N .(2)[(a)⇒(b)]By Theorem 3.6, we have that L is simple in H τ and by Proposition 4.1 the morphism a is an injective envelope in H τ .In particular, we have that M is contained in Prod(C) and the sequence 0→ L a → M b → N → 0 is exact in H τ .By Remark 2.8 and Proposition 2.7(1) we have that b is a strong left almost split morphism in C. [(b)⇒(a)] Since M is in C and C is closed under submodules, we have that L is also in C. Therefore 0 → L a → M b → N → 0 is a short exact sequence in H τ .By our assumption, we have that b is a strong left almost split morphism in C and so, by Remark 2.8 and Proposition 2.7(2), we have that L is simple and a is an injective envelope.By Theorem 3.6 and Proposition 4.1, we have shown that condition (a) holds.Next we show that the strong left almost split morphisms arising in Theorem 4.2 are the only strong left almost split morphisms in C with domain contained in Prod(C).

Lemma 4 . 3 .
Let M be a full subcategory of Mod-R that is closed under subobjects.Then the following are equivalent for a module M in M.

DefineN
C := {N ∈ Prod(C) | ∃N → N a (strong) left almost split morphism in C}.Note that, by Proposition 2.7 and the subsequent corollaries, the set N C does not depend on whether we choose to include the word strong or not.The previous lemma shows that N C is a disjoint union N C = M C ⊔ E C where M C := {L ∈ Prod(C) | ∃L → L a strong left almost split monomorphism in C} and E C := {M ∈ Prod(C) | ∃M → M a strong left almost split epimorphism in C}.Note that M C consists of the modules L ∈ Prod(C) arising in Theorem 4.2(1) and E C consists of the modules M ∈ Prod(C) arising in Theorem 4.2(2).

Corollary 4 . 4 .
The following statements hold for a module N in Prod(C).

( 1 )
N ∈ N C if and only if N is the injective envelope of a simple object in H τ .In this case N is isomorphic to an indecomposable direct summand of any cotilting module that is equivalent to C. (2) N ∈ M C if and only if N is the injective envelope of T [−1] in H τ where T is a torsion, almost torsionfree module with respect to τ .In this case T is the cokernel of the strong left almost split monomorphism N → N in C. (3) N ∈ E C if and only if N is the injective envelope of F in H τ where F is a torsionfree, almost torsion module with respect to τ .In this case F is the kernel of the strong left almost split epimorphism N → N in C. Proof.(1) The first statement follows immediately from Corollary 2.10.The latter statement follows from the fact that the injective envelope of a simple object in a Grothendieck category is indecomposable and the fact that the injective envelopes of simple objects arise as direct summands of any injective cogenerator, up to isomorphism.The statements (2) and (3) follow directly from Theorem 4.2 and Proposition 4.1.

5. 5 .
Neg-isolated pure-injective modules.Next we consider the pure-injective objects in a given definable subcategory that correspond to injective envelopes of simple objects in the Grothendieck category (R-mod, Ab) D .They will be characterised by a condition on pp-types.A filter Φ in the lattice of finitely generated subfunctors of the forgetful functor in (mod-R, Ab) will be called a D-filter if it admits a realisation pp(D, d) = Φ with D in D. Recall that we can always choose D to belong to the full subcategory Pinj(D) of pure-injective objects in D.Theorem 5.10.Let D be a definable subcategory in Mod-R.The following statements are equivalent for an indecomposable pure-injective module N in D.(1) (N ⊗ R −) D is the injective envelope of a simple object in (R-mod, Ab) D .(2)There exists a left almost split morphism N → N + in Pinj(D).(3)If n ∈ N is a nonzero element and Φ = pp(N, n) is the associated pp-type, then there exists a D-filter Φ + ⊃ Φ which properly contains Φ such that whenever a D-filter Ψ ⊃ Φ properly contains Φ, then Ψ ⊇ Φ + .(4) N is the source of an almost split morphism in D.

Corollary 5 . 18 . 4 )/
The set of critical neg-isolated modules in C coincides with the set E C of modules M ∈ Prod(C) admitting a strong left almost split epimorphism M → M in C.Proof.The statement follows immediately from Proposition 5.14 and Proposition 5.17 since Prod(C 0 ) ⊆ Prod(C).5.7.Special modules.Let τ = (Q, C) be a cotilting torsion pair with cotilting module C. We saw in Corollary 4.4 that the injective envelopes of simple objects in the heart H τ are exactly the objects in the set N C := {N ∈ Prod(C) | ∃N → N a (strong) left almost split morphism in C}.By Corollary 5.12 the elements of N C are the neg-isolated modules in C which belong to Prod(C).Moreover, we showed in Lemma 4.3 that N C = E C ⊔ M C where E C := {M ∈ Prod(C) | ∃M → M a strong left almost split epimorphism in C} and M C := {L ∈ Prod(C) | ∃L → L a strong left almost split monomorphism in C}.By Corollary 5.18 we have that the elements of E C are the critical neg-isolated modules in C. In this section we will identify which of the non-critical neg-isolated modules in C are contained in M C .In other words, we wish to determine the non-critical neg-isolated modules in C which are contained in Prod(C).Proposition 5.19.Let τ = (Q, C) be a cotilting torsion pair.The following statements are equivalent for a module N in C. (1) N is contained in Prod(C) and is a non-critical neg-isolated module in C. (2) N is contained in the set M C of modules in Prod(C) that admit a strong left almost split monomorphism f : N → N in C. (3) There exists a strong left almost split monomorphism f : N → N in C such that the cokernel of f is torsion, almost torsionfree.(There exists a left almost split morphism g : N → N + in Pinj(C) such that the cokernel of g is not contained in C. Proof.[(1)⇒(2)] By Theorem 5.10 and Proposition 5.18, if N ∈ Prod(C) is non-critical negisolated, then N is contained in N C \ E C = M C .[(2)⇒(3)] This follows from Theorem 4.2(1).[(3)⇒(4)] Let f : N → N be as in (3) and let T := Cokerf .By the proof of Theorem 5.10, we have that the composition g := ef is a left almost split morphism in Pinj(C) where e : N → N + := PE( N ) is the pure-injective envelope of N .We show that Z := Cokerg is not contained in C. Thus we have a commutative diagram: / N + / / Z / / 0

Definition 5 . 20 .
Let τ = (Q, C) be a cotilting torsion pair.A neg-isolated module N in C is called special if it satisfies the equivalent conditions of Proposition 5.19.Corollary 5.21.Let I be an injective cogenerator of Mod-R with a special C-cover (5.5) 0 → C 1 → C 0 g → I → 0.
Since C 0 ⊕ C 1 is a cotilting module equivalent to C, it follows from Corollary 4.4(1) that every N ∈ N C is a direct summand of C 0 ⊕ C 1 .By Proposition 5.16(3), the neg-isolated summands of C 0 are exactly the critical ones and so the special neg-isolated modules are all direct summands of C 1 by [30, Prop.9.29].Corollary 5.22.If τ = (Q, C) is a cotilting torsion pair with cotilting module C, then PE(⊕ N C N ) is isomorphic to a direct summand of C. Proof.The argument used in the first part of the proof of Corollary 5.21 implies that every N ∈ N C arises as a direct summand of C. Equivalently, the functor N ⊗ R −, regarded as an object in the localised functor category (R-mod, Ab) C , is a coproduct factor of the injective object C⊗ R −.Since N is neg-isolated, the socle soc(N ⊗ R −) is a simple object in (R-mod, Ab) C .It follows that N ∈N C soc(N ⊗ R −) ⊆ soc(C ⊗ R −) and therefore that the injective envelope
(AT1) Let g : G → B be a proper epimorphism, and 0 → B ′ → B → B → 0 the canonical exact sequence with B ′ ∈ T and B ∈ F. Then B lies in F ∩ D = AddG , so G g → B → B is a morphism in AddG and is therefore zero or a split monomorphism.It follows that B = 0 and B ∈ T .(AT2) Let 0 → G → B → C → 0 be an exact sequence with B ∈ F. By applying the functor Hom Λ (S, −) given by a simple regular module S, we obtain an exact sequence Hom Λ (S, B) → Hom Λ (S, C) → Ext 1 Λ (S, G) ∼ = DHom Λ (G, S) where the first and third term are zero.Hence C ∈ F.

Corollary 6 . 5 .
If (Q, C) is a hereditary cotilting torsion pair with cotilting module C, then the module PE(⊕ N C N ) = PE((⊕ E C M ) ⊕ (⊕ M C L)), given more explicitly by PE( us now consider the set Ω = {S[−1] | S torsion, almost torsionfree} of all simple torsionfree objects in H τ .It generates a hereditary torsion pair (S, R) of finite type in H τ , which is associated to the open setO = {E ∈ Spec(H) | Hom H (M, E) = 0 for some M ∈ Ω} = M Cin Spec(H τ ).Moreover, S is a localizing subcategory of H τ , and the corresponding quotient category H τ /S is again a locally coherent Grothendieck category whose spectrum is formed by the indecomposable injective objects in R, that is, by the complement O c of O, cf.[24,  Thm.2.16 and Prop.3.6].
If f : E → E + is a left almost split morphism in Inj(G), then the kernel Ker (f ) is simple and the canonical embedding Ker (f ) → E is the injective envelope of Ker (f ).(3)If f : E → E + is a left almost split morphism in Inj(G), then the canonical epimorphism g : E → Im(f ) is a strong left almost split morphism in G.(4) If g : E → Ẽ is a left almost split morphism in G with E in Inj(G), and e : Ẽ → E( Ẽ) is the injective envelope of Ẽ, then f := eg is a left almost split morphism in Inj(G).
Corollary 2.10.Let τ = (Q, C) be a cotilting torsion pair in Mod-R and let Inj(H τ ) denote the full subcategory of injective objects in H τ .The following statements are equivalent for an object E of Inj(H τ ).(1) E is isomorphic to the injective envelope E(S) of a simple object S in H τ .(2)There exists a left almost split morphism f : E → E + in Inj(H τ ).Localisation in abelian Grothendieck categories.A torsion pair (T , F) in an abelian category is hereditary if the torsion class T is closed under subobjects.If the abelian category is Grothendieck, this is equivalent to the torsionfree class being closed under injective envelopes.
) There exists a strong left almost split morphism g : E → Ẽ in G. (4) There exists a left almost split morphism g : E → Ẽ in G.