aa r X i v : . [ m a t h . R T ] F e b AN INTRODUCTION TO THE LATTICE OF TORSION CLASSES
HUGH THOMAS
Abstract.
In this expository note, I present some of the key features of thelattice of torsion classes of a finite-dimensional algebra, focussing in partic-ular on its complete semidistributivity and consequences thereof. This is in-tended to serve as an introduction to recent work by Barnard–Carroll–Zhu andDemonet–Iyama–Reading–Reiten–Thomas.
Let A be a finite-dimensional algebra over a field k . We write mod A for thecategory of finite-dimensional left A -modules. There is a class of subcategories of mod A which are called torsion classes. The torsion classes, ordered by inclusion,form a poset which we denote tors A . This poset is in fact a lattice, and its lattice-theoretic properties have recently been the focus of some attention, as in [Ja, GM,BCZ, DI+, AP].In this note I will present some of the interesting features of these lattices. Theproofs in this note are self-contained except for the final section, where we presentwithout proof an application of these ideas to the study of finite semidistributivelattices from [RST]. This note is intended as a gentle introduction to the subject.No results in this note are new. The presentation is, of course, novel in somerespects, and I hope that it is helpful as an introduction to the subject.Let me now quickly summarize the contents of this note. Terms which are unde-fined here will be introduced later where they logically fit. In addition to presentingthe easy explanation that tors A is a lattice, I will prove the result of Barnard, Car-roll, and Zhu [BCZ] that the completely join irreducible elements of tors A are inbijection with the bricks of A . I will show that tors A is completely semidistribu-tive. I will not take the most direct route to this result, but rather spend sometime developing independently properties of tors A and corresponding properties ofsemidistributive lattices, in an attempt to illuminate how semidistributivity givesus a helpful perspective through which to view the combinatorics of tors A . I willshow that tors A is weakly atomic. We will then see how an algebra quotient inducesa lattice quotient map between the corresponding lattices of torsion classes, andstudy this lattice quotient. In the final section, I will present (without proof) aconstruction of finite semidistributive lattices developed in [RST], and inspired bythe study of lattices of torsion classes.1. Definition of torsion classes
For the elementary material in this section and the two following, a furtherreference is [ASS, Chapter VI].A torsion class in mod A is a subcategory T of mod A which is • closed under quotients (i.e., Y ∈ T and Y ։ Z implies Z ∈ T ). Mathematics Subject Classification.
Primary: 16G20, Secondary: 16S90, 06B99.
Key words and phrases.
Torsion classes, semidistributive lattices, lattice congruences. [10] [21] [32] · · · [11][22] ... [23] [12] [01] · · · preprojective component tubes preinjective component
Figure 1.
The AR quiver of the path algebra of the Kronecker quiver. • closed under extensions (i.e., X, Z ∈ T and 0 → X → Y → Z → Y ∈ T .I should clarify that for me a subcategory is always full, closed under direct sums,direct summands, and isomorphisms. In other words, a subcategory of mod A canbe specified as the direct sums of copies of some subset of the indecomposablemodules of A .We write tors A for the set of torsion classes of mod A , and we think of it as aposet ordered by inclusion. Example 1.1 (Type A ) . Our quiver Q is ← , and the algebra is the pathalgebra A = kQ . The category mod A has three indecomposable objects S , P , S ,which I denote by their dimension vectors as [10] , [11] , and [01] , respectively.The torsion classes are as follows, where the angle brackets denote additive hull. h [10] , [11] , [01] ih [11] , [01] ih [01] i h [10] i Example 1.2 (Type A n ) . For an example in type A n , where Q is ← · · · ← n ,see [Kr] . Example 1.3 (Kronecker quiver) . Let k be algebraically closed. Let Q be the quiver and let A = kQ .The AR quiver is displayed in Figure 1, where I write [ ab ] for an indecomposablemodule with dimension vector ( a, b ) .The tubes are indexed by points in P ( k ) = k S {∞} ; they each look the same.The torsion classes consist of the additive hull of each of the following sets: • any final part of the preinjective component, • all preinjectives and a subset of the tubes, • all preinjectives, all tubes, and a final part of the preprojectives, • S = [10] . N INTRODUCTION TO THE LATTICE OF TORSION CLASSES 3 h [01] ih [12] , [01] i ... IhI , R − i · · · hI , R i · · · hI , R ihI , S x ∈ P ( k ) R x i· · · · · ·· · · · · ·· · ·· · · · · · [01] [12] Boolean lattice... h ind mod A \ { [10] , [21] }ih ind mod A \ { [10] }i mod A h [10] i [10][01] [10] [21] Figure 2.
The lattice of torsion classes for A the path algebra ofthe Kronecker quiver. The labels on the edges should be ignoredfor now; they are the brick labelling ˆ γ defined in Section 11. An image of the lattice of torsion classes is displayed in Figure 2. There, I denotes the preinjective component, and R x denote the tube corresponding to x ∈ P ( k ) . I write ind mod A for the set of indecomposable A -modules.The interval between I and hI , S x ∈ P ( k ) R x i is isomorphic to the Boolean latticeof all subsets of P ( k ) , ordered by inclusion. Specifying a torsion class
In general, how can we specify a torsion class? For C a subcategory of mod A ,define T ( C ) to be the subcategory whose modules are filtered by quotients of objectsfrom C . That is to say M ∈ T ( C ) if and only if M admits a filtration 0 = M ⊂ M ⊂ · · · ⊂ M r = M with M i /M i − a quotient of an object of C for all i . Proposition 2.1.
For C an arbitrary subcategory, T ( C ) is the smallest torsion classcontaining all the objects from C . HUGH THOMAS
Proof.
Suppose that M ∈ T ( C ), so that we have a filtration 0 = M ⊂ M ⊂· · · ⊂ M r = M with M i /M i − a quotient of an object of C for all i . Considersome quotient of M , say N = M/L . Then define N i = ( M i + L ) /L , which formsa filtration of N . We see that N i /N i − is a quotient of M i /M i − , and therefore aquotient of an object of C . This shows that N ∈ T ( C ).Suppose next that we have two modules M and N , both in T ( C ), and an exten-sion 0 → M → E → N → . Now E has a two-step filtration 0 ⊂ M ⊂ E , with E/M ≃ N , and we can refinethe two steps of the filtration to filtrations of M and N with subquotients beingquotients of C , since we know such filtrations exist. This shows that E ∈ T ( C ).It follows that T ( C ) satisfies the two defining properties, and is therefore a torsionclass. T ( C ) is the smallest torsion class containing C because any element of T ( C ) is aniterated extension of quotients of C , which must be in any torsion class containing C . (cid:3) We now consider a second way to specify a torsion class. For C a subcategory of mod A , define ⊥ C = { X ∈ mod A | Hom(
X, Y ) = 0 for all Y ∈ C} . Proposition 2.2.
For C an arbitrary subcategory, ⊥ C is a torsion class.Proof. Let M ∈ ⊥ C . Let N be a quotient of M . Since there are no non-zeromorphisms from M into any object of C , the same holds for N , so N ∈ ⊥ C .Suppose now that we have M and N in ⊥ C , and an extension:0 → M → E → N → . For any Y ∈ C , we have Hom( M, Y ) = 0 and Hom(
N, Y ) = 0, and it follows fromthe left exactness of the Hom functor that Hom(
E, Y ) = 0 as well. We deduce that E ∈ ⊥ C . ⊥ C satisfies the two defining conditions, and is therefore a torsion class. (cid:3) Torsion classes and torsion free classes
There is a dual notion to that of torsion class, namely that of torsion free class. Atorsion free class in mod A is a subcategory closed under submodules and extensions.We write tf A for the torsion free classes of A , and we think of it as a poset orderedby inclusion.As one should expect, in the setting of finite-dimensional algebras in which wework, the theory of torsion free classes is completely parallel to the theory of torsionclasses. For C a subcategory of mod A , define F ( C ) to be the subcategory of mod A consisting of all modules filtered by submodules of modules from C . Then F ( C ) isthe smallest torsion free class containing C . We can also define C ⊥ = { Y ∈ mod A | Hom(
X, Y ) = 0 for all X ∈ C} . One easily checks that for any subcategory C , the subcategory C ⊥ is a torsion freeclass. Proposition 3.1.
Let T be a torsion class, and let X ∈ mod A . There is amaximum submodule of X contained in T . N INTRODUCTION TO THE LATTICE OF TORSION CLASSES 5
Proof. If M and N are submodules of X , then we have a short exact sequence0 → M → M + N → N/ ( N ∩ M ) → N and M are both in T , it follows that M + N is also. Because X is finite-dimensional by assumption, it therefore has a maximum submodule contained in T . (cid:3) We denote this maximum submodule by t T X . Proposition 3.2.
X/t T X lies in T ⊥ .Proof. Suppose there were a non-zero map f from some M ∈ T to X/t T X . Then im f is a quotient of M , and therefore itself in T . The preimage of im f in X is thenan extension of im f by t T X , and is therefore also in T , contradicting the definitionof t T X . (cid:3) For any X in mod A , we now have a short exact sequence:( ∗ ) 0 → t T X → X → X/t T X → . with the lefthand term in T and the righthand term in T ⊥ . Proposition 3.3.
For X ∈ mod A , any short exact sequence of the form → X ′ → X → X ′′ → with X ′ in T and X ′′ ∈ T ⊥ is isomorphic to ( ∗ ) .Proof. Viewing X ′ as a submodule of X , it must be contained in t T X . If thecontainment were strict, then X ′′ would not lie in T ⊥ . The result follows. (cid:3) We can now prove the following theorem:
Theorem 3.1.
The map
T 7→ T ⊥ is an inclusion-reversing bijection from torsionclasses to torsion free classes. Its inverse is given by the map F 7→ ⊥ F .Proof. Let T be a torsion class. We already pointed out that T ⊥ is torsion free. Itis easy to see that ⊥ ( T ⊥ ) ⊇ T . For the other inclusion, suppose X ∈ ⊥ ( T ⊥ ). Since X/t T X ∈ T ⊥ , there are no non-zero morphisms from X to X/t T X . But this mustmean that X/t T X = 0, so X = t T X , and X ∈ T .Starting with a torsion free class F , we see just as easily that the compositionof the two maps in the other order is also the identity. They are therefore inversebijections. It is easy to see that they are order-reversing. (cid:3) From the previous theorem, together with Proposition 2.2, the following corollaryfollows:
Corollary 3.1.
The following pairs of subcategories are the same: • { ( T , T ⊥ ) | T ∈ tors A } , • { ( ⊥ F , F ) | F ∈ tf A } , • { ( X , Y ) | X = ⊥ Y , Y = X ⊥ ) } . HUGH THOMAS Posets and lattices
A possible reference for basis material on lattices is [Gr].A poset is a partially ordered set. In a poset, we say that x covers y if x isgreater than y and there is no element z such that x > z > y . In this case we write x ⋗ y .A lattice is a poset in which any two elements x and y have a unique leastupper bound (their “join”) denoted x ∨ y , and a unique greatest lower bound (their“meet”) denoted x ∧ y .A complete lattice is a lattice such that any subset S of L has a unique leastupper bound, which we denote either W x ∈ S x or W S , and a unique greatest lowerbound, which we denote V x ∈ S x or V S .A finite lattice is necessarily complete. The perspective taken in this note is thatthe desirable infinite generalization of finite lattices are the complete lattices.A complete lattice necessarily has a minimum element ˆ0 (the meet of all theelements of L ) and similarly a maximum element ˆ1.5. Torsion classes form a complete lattice
The poset tors A clearly has a meet operation given by intersection, since theintersection of two torsion classes again satisfies the defining properties of a torsionclass. The same is true for meets of arbitrary collections of torsion classes, for thesame reason.To see the other lattice operation, there are three approaches which all work.Since the left perpendicular/right perpendicular operations are order-reversing bi-jections between torsion-classes and torsion-free classes, we have that _ T ∈ S T = ⊥ ^ T ∈ S T ⊥ ! . Since the V on the righthand side exists (being given by intersection), so does the W on the lefthand side.We can also define the join operation in tors A implicitly. Any poset with amaximum element and a V also has a W , which can be defined as follows: _ T ∈ S T = ^ {Y∈ tors A | ∀T ∈ S, Y⊇T } Y Finally, we can also describe the join explicitly using Proposition 2.1: _ T ∈ S T = T [ T ∈ S T ! We therefore have the following result:
Proposition 5.1. tors A is a complete lattice. join irreducible elements in lattices An element x of a lattice L is called join irreducible if it cannot be written as thejoin of two elements both strictly smaller than it, and it is also not the minimumelement of the lattice. Especially for finite lattices, the join irreducible elementscan be viewed as “building blocks” of the lattice. N INTRODUCTION TO THE LATTICE OF TORSION CLASSES 7
Proposition 6.1.
In a finite lattice L , any element is the join of the join irreducibleelements below it.Proof. Suppose x ∈ L were a minimal counter-example to the statement of theproposition. If x were join irreducible, it is obviously not a counter-example, sosuppose that it is not join irreducible. We can therefore write x = y ∨ z with y, z < x .By the assumption that x is a minimal counter-example, y and z can each be writtenas a join of join irreducible elements. Joining together these two expressions, weget an expression for x as a join of join irreducible elements, contradicting ourassumption that x was a counter-example. (cid:3) The situation for infinite lattices is more complicated. It can still be interestingto consider join irreducible elements defined as above. However, for our purposes,the following definition is more important. We say that x ∈ L is completely joinirreducible if W y Recall that a module B is called a brick if every non-zero endomorphism of B is invertible. A brick is necessarily indecomposable, since projection onto a properindecomposable summand is a non-invertible endomorphism. Write br A for the A -modules which are bricks.In the case of the Kronecker quiver, the bricks are the indecomposable modulesfrom the preprojective and preinjective components, together with the quasi-simplemodule at the bottom of each tube. HUGH THOMAS In this section, we shall show an important result by Barnard–Carroll–Zhu [BCZ,Theorem 1.5], that there is a bijection between br A and the completely join irre-ducible elements of tors A .The same result holds for tf A , and, by the order-reversing bijection between tf A and tors A , the same result also holds for the meet irreducible elements of tors A and tf A . For simplicity, we will focus our attention on tors A and its completelyjoin irreducible elements; everything we prove has analogues in the other settings.The following lemma says that a torsion class is characterized by the bricks itcontains. Lemma 7.1. Let T ∈ tors A . Then T = _ B ∈T ∩ br A T ( B ) Proof. Let us write U = _ B ∈T ∩ br A T ( B )Clearly, U ⊆ T . Now suppose that we have some X which is in T but not in U ,and among such X , choose one of minimal dimension. X is clearly not a brick,since otherwise it would be contained in U . Thus it has a non-zero non-invertibleendomorphism f . We get a short exact sequence:0 → f ( X ) → X → X/f ( X ) → X ∈ T , we have X/f ( X ) ∈ T , and since the dimension of X/f ( X ) is lessthan that of X , it follows that X/f ( X ) ∈ U .Similarly, though, since f ( X ) is also a quotient of X , we know f ( X ) ∈ T andsince f ( X ) is in fact a proper quotient of X , the minimality assumption on X thenimplies that f ( X ) ∈ U . We now see that X is the extension of two objects from U ,so it is itself in U , contrary to our assumption. (cid:3) We also need the following lemma due to Sota Asai. Lemma 7.2 ([As, Lemma 1.7(1)]) . If X ∈ T ( B ) , then either X admits a surjectiononto B or Hom( X, B ) = 0 .Proof. Suppose that f ∈ Hom( X, B ) is non-zero. Since X is filtered by quotientsof B , we can write 0 = X ⊆ X ⊆ · · · ⊆ X r = X , with X i /X i − isomorphic to aquotient of B . Consider the smallest i such that f | X i is non-zero. Since f | X i − = 0, f induces a map from X i /X i − to B , and thus from B to B . Since B is a brick,this map must be surjective, so f | X i is surjective, and thus f is surjective. (cid:3) We can now prove the main result of the section: Theorem 7.1 ([BCZ, Theorem 1.5]) . The map B T ( B ) is a bijection from br A to Ji c tors A .Proof. First of all, we want to show that, for B a brick, T ( B ) is a completely joinirreducible torsion class. This requires showing that there is a unique maximumelement among all those torsion classes strictly below T ( B ). We claim that thistorsion class can be described as T ( B ) ∩ ⊥ F ( B ).Since B ⊥ F ( B ), it is clear that T ( B ) ∩ ⊥ F ( B ) is a torsion class strictlycontained in T ( B ). On the other hand, any torsion class strictly contained in T ( B )cannot contain B , and thus cannot contain any module X admitting a surjective N INTRODUCTION TO THE LATTICE OF TORSION CLASSES 9 map onto B . Thus, by Lemma 7.2, any such torsion class must be contained in ⊥ B . Clearly ⊥ B ⊇ ⊥ F ( B ), and the reverse inclusion follows because any elementof F ( B ) is filtered by subobjects of B , so if X has no non-zero morphisms into B ,it has no non-zero morphisms into any element of F ( B ). Therefore, any torsionclass properly contained in T ( B ) is contained in T ( B ) ∩ ⊥ F ( B ). This proves theclaim, thus establishing that T ( B ) is a completely join irreducible torsion class.On the other hand, by Lemma 7.1, any torsion class can be written as the join of T ( B ) as B runs through all bricks in the torsion class. This shows that any torsionclass can be written as a join of the completely join irreducible torsion classes wehave already identified (those of the form T ( B ) for B a brick) so there cannot beany completely join irreducible torsion classes not of this form.Finally, we want to check that the map from bricks to torsion classes is injective.Suppose that T ( B ) = T ( B ′ ), for B and B ′ two bricks. B ′ cannot be contained in T ( B ) ∗ . Thus there is a surjection from B ′ to B by Lemma 7.2. Reversing the rˆolesof B ′ and B , there is also a surjection from B to B ′ . Therefore B and B ′ must beisomorphic. (cid:3) This theorem is one of the key justifications for the impression that when con-sidering lattices of torsion classes, it is most appropriate to think in terms of thecomplete versions of lattice-theoretic phenomena. As we saw in the example of theKronecker quiver, there is a join irreducible torsion class which is not completelyjoin irreducible, namely, the additive hull of the preinjective component. In accor-dance with Theorem 7.1, it does not correspond to any brick in mod A . This raisesthe following interesting question: Question 7.1. Is there any way to extend Theorem 7.1 to characterize the joinirreducible but not completely join irreducible elements of tors A ? The proof of the following theorem is dual to the proof of Theorem 7.1. Theorem 7.2. The map B F ( B ) is a bijection from br A to Ji c tf A. Then, applying Theorem 3.1, we deduce: Corollary 7.1. The map B ⊥ F ( B ) is a bijection from br A to Mi c tors A . From Theorem 7.1, Corollary 7.1, and their proofs, we can say that associatedto a brick B , there are four torsion classes, arranged as in the following diagram,where the join of the two torsion classes on the middle layer equals the top torsionclass, and their meet equals the bottom torsion class.( ⊥ F ( B )) ∗ T ( B ) ⊥ F ( B ) T ( B ) ∗ In the diagram, the edges drawn as undashed lines are cover relations in the lat-tice of torsion classes. The edges drawn using dashed lines are weak poset relations.In particular, the torsion classes at the endpoints of a dotted line may be equal.Also, the pair of torsion classes not connected by a line are not comparable in thelattice of torsion classes. We follow these conventions in subsequent diagrams. Parenthesis: τ -tilting We include the following section because it makes the link to another topic ofcurrent research related to torsion classes, which was also presented during thespring school. A possible reference is the survey by Iyama and Reiten [IR].A torsion class T is called functorially finite if there is some X ∈ mod A suchthat T = Gen( X ), where Gen( X ) is by definition the collection of quotients ofdirect sums of copies of X .In the Kronecker case, which ones are functorially finite? Exactly those not in theBoolean lattice. There is no single module which generates the whole preinjectivecomponent and nothing more, and there is no single module which generates anytube without in fact being preprojective (and thus generating all the tubes andmore).As shown by Adachi, Iyama, and Reiten, in the paper [AIR] which introducedthe topic of τ -tilting theory, functorially finite torsion classes correspond bijectivelyto a certain class of modules called basic support τ -tilting modules; the bijectionfrom basic support τ -tilting modules to torsion classes is Gen.Functorially finite torsion classes need not form a lattice. There is nothing thatguarantees that the intersection of two functorially finite torsion classes will befunctorially finite, so in order for them to form a lattice anyway, there would haveto be a biggest functorially finite torsion class contained in the intersection, andthis does not always hold. Generally, for hereditary algebras not of finite type, thefunctorially finite torsion classes do not form a lattice [IR+, Ri]. Thus, for lattice-theoretic study, it seems preferable not to restrict to functorially finite torsionclasses. 9. Semidistributivity In this section we introduce the notion of semidistributivity of a lattice. See[AN, RST] for more on the subject.A lattice L is called join semidistributive if x ∨ y = x ∨ y ′ implies that x ∨ ( y ∧ y ′ )is also equal to both of them. It is called completely join semidistributive if given x ∈ L and a set S ⊆ L , such that x ∨ y = z for all y ∈ S , then x ∨ V S = z .Join semidistributivity and complete join semidistributivity are equivalent forfinite lattices. As usual for us, in the infinite setting, the version which we preferis the complete one.Complete join semidistributivity is equivalent to saying that, given x, z ∈ L ,if we consider { y | x ∨ y = z } , then this set, if it is non-empty, has a minimumelement. When we say “minimum element,” we do not mean only “minimal” (i.e.,an element such that there is no element strictly below it), we mean an elementwhich is weakly below all the elements in the set.Similarly, a lattice is called meet semidistributive if x ∧ y = x ∧ y ′ implies that x ∧ ( y ∨ y ′ ) is also equal to both of them. It is called completely meet semidistributiveif given x ∈ L and a set S ⊆ L , such that x ∧ y = z for all y ∈ S , then x ∧ W S = z .Equivalently, given x, z ∈ L , if we consider { y | x ∧ y = z } , then this set, ifnon-empty, has a maximum element.A lattice is called semidistributive if it is join semidistributive and meet semidis-tributive. It is called completely semidistributive if it is completely join semidis-tributive and completely meet semidistributive. N INTRODUCTION TO THE LATTICE OF TORSION CLASSES 11 Complete semidistributivity is the property which we are going to focus on. Weare now going to develop some properties of completely semidistributive lattices. Proposition 9.1. In any completely join semidistributive lattice L , every cover y ⋗ x has a unique completely join irreducible element j such that x ∨ j = y and x ∨ j ∗ = x .Proof. Let S = { z | x ∨ z = y } . This set is non-empty, since y ∈ S . Thus, bycomplete join semidistributivity, it has a minimum element. Call it j .Any z < j satisfies that x ∨ z < y , and thus that x ∨ z = x . It follows that any z < j satisfies that z ≤ x . Therefore, any z < j satisfies z ≤ x ∧ j . Since j < x , wehave x ∧ j < j . Thus every element strictly below j is weakly below x ∧ j < j . Itfollows that j is completely join irreducible, and j ∗ = j ∧ x .Now suppose that we had some other completely join irreducible element j ′ suchthat x ∨ j ′ = y and x ∨ j ′∗ = x . Since j is the minimum element of S , we must have j ′ > j . But then x ≥ j ′∗ ≥ j , which contradicts x ∨ j > x . Thus j is unique. (cid:3) Write γ ( y ⋗ x ) for the completely join irreducible element defined in the previousproposition.Similarly, in a completely meet semidistributive lattice L , every cover y ⋗ x has aunique completely meet irreducible element m such that m ∧ y = x and m ∗ ∧ y = y .Write µ ( y ⋗ x ) for this completely meet irreducible element. Proposition 9.2. In a completely semidistributive lattice L , there are inverse bi-jections κ and κ d : Ji c ( L ) Mi c ( L ) κκ d such that κ ( j ) = µ ( j ⋗ j ∗ ) and κ d ( m ) = γ ( m ∗ ⋗ m ) . It is standard to call these two maps κ and κ d but different sources disagree asto which is which. Proof. Let j be a completely join irreducible element of L , and let m = κ ( j ) = µ ( j ⋗ j ∗ ). We therefore have the following diagram: m ∗ jm j ∗ But now it is clear that κ d ( m ∗ ⋗ m ) = j , so κ d ◦ κ is the identity. The dualargument shows that κ ◦ κ d is the identity, and we have shown that κ and κ d areinverse bijections. (cid:3) We now have the following theorem, which shows that the two labellings of thecovers of L differ only by a bijection. Theorem 9.1. Let L be a completely semidistributive lattice. Then µ ( y ⋗ x ) = κ ( γ ( y ⋗ x )) Proof. For any y ⋗ x , let j = γ ( y ⋗ x ) and m = µ ( y ⋗ x ). We therefore have thefollowing diagram, from which the result follows. m ∗ ym x jj ∗ (cid:3) Complete semidistributivity of tors A The fact that lattices of torsion classes are semidistributive was first proved byGarver and McConville [GM]. For not necessarily finite lattices of torsion classes,it turns out to be natural to consider complete semidistributivity. Theorem 10.1 ([DI+, Theorem 3.1(a)]) . tors A is completely semidistributive.Proof. We will prove complete meet semidistributivity. Complete join semidistribu-tivity follows from the complete meet semidistributivity of tf A , which is establishedby a dual argument.Let X ∈ tors A , and let S ⊆ tors A such that for all Y ∈ S , we have X ∧Y is equal.Let Z be their common value. Since the meet of torsion classes is intersection, wehave that Z = X ∩ Y for any Y ∈ S .We want to show that X ∩ W S = Z also.Clearly X ∩ W S ≥ Z . To prove the opposite inclusion, let M ∈ X ∩ W S be aminimal-dimensional counter-example.Since M ∈ W S , there is a filtration of M M ⊂ M · · · ⊂ M r = M with M i /M i − ∈ Y i , with Y i ∈ S .Consider the short exact sequence:0 → M → M → M/M → M/M ∈ X ∩ W S since M is. Since M is non-zero, the dimension of M/M is less than that of M , and thus by our choice of M , we know that M/M is not a counter-example. Therefore, M/M ∈ Z , so in particular M/M ∈ Y . Onthe other hand, we also know that M ∈ Y . Because Y is a torsion class, andtherefore closed under extensions, M ∈ Y . We also know M ∈ X . Therefore M ∈X ∩ Y = Z . This contradicts our choice of M , so it must be that X ∩ W S = Z . (cid:3) Consequences of the complete semidistributivity of tors A As we showed in Section 9, a completely semidistributive lattice has a labellingof every cover relation y ⋗ x by a completely join irreducible element γ ( y ⋗ x ),and a labelling of every cover relation by a completely meet irreducible element µ ( y ⋗ x ), and these two labellings are related by the maps κ and κ d . We would liketo understand what this means in the case of the lattice of torsion classes. N INTRODUCTION TO THE LATTICE OF TORSION CLASSES 13 Since we know that the completely join irreducible torsion classes correspondto bricks by Theorem 7.1, for Y ⋗ X in tors A , define ˆ γ ( Y ⋗ X ) = B , such that γ ( Y ⋗ X ) = T ( B ). The following proposition defines ˆ γ ( Y ⋗ X ) directly. Proposition 11.1. ˆ γ ( Y ⋗ X ) is the unique brick B which is contained in Y butnot in X .Proof. By the complete semidistributivity of tors A , we know that there is a uniquecompletely join irreducible torsion class, γ ( Y ⋗ X ), such that Y ≥ γ ( Y ⋗ X ) but X 6≥ γ ( Y ⋗ X ). By Theorem 7.1, the completely join irreducible elements are ofthe form T ( B ), for B a brick. We have that Y ⊇ T ( B ) and X 6⊇ T ( B ) iff B ∈ Y and B 6∈ X . So there is a unique brick contained in Y but not in X , and it isˆ γ ( Y ⋗ X ). (cid:3) Dually, µ ( Y ⋗ X ) = ⊥ F (ˆ γ ( Y ⋗ X )). Example 11.1 (Type A ) . The brick labelling of the covers of tors kQ for Q =1 ← is as follows: h [10] , [11] , [01] ih [11] , [01] ih [01] i h [10] i [10][11][01] [10][01] Example 11.2 (Kronecker quiver) . We revisit the Kronecker quiver from Example1.3. The brick labels of some of the covers were already shown in Figure 2. Insidethe interval that is isomorphic to a Boolean lattice on the set of tubes, one torsionclass covers another if they differ exactly in that there is one tube present in one butnot the other. In this case the brick labelling the cover relation is the quasi-simpleat the bottom of that tube. Algebra quotients and lattice quotients A surjective map of lattices L ։ L ′ is called a (complete) lattice quotient if itrespects the (complete) lattice operations.For I an ideal of an algebra A , consider the algebra quotient φ : A ։ A/I . Wecan view mod A/I as the subcategory of mod A consisting of modules annihilatedby I . We will be interested in the map sending T in mod A to T ∩ mod A/I . Proposition 12.1. T ∩ mod A/I is a torsion class for A/I .Proof. It is easy to check that it satisfies the two defining conditions. (cid:3) Proposition 12.2 ([DI+, Proposition 5.7(a)]) . If ( T , F ) is a torsion pair of mod A ,then ( T ∩ mod A/I, F ∩ mod A/I ) is a torsion pair of mod A/I .Proof. In this proof, when we write C ⊥ or ⊥ C , we always intend it in the ambientcategory mod A .Consider ( T ∩ mod A/I ) ⊥ . Clearly this contains F . Now suppose we have somemodule M ∈ mod A/I , M 6∈ F . There is therefore some N ∈ T and some non-zero f ∈ Hom( N, M ) = 0. Since IM = 0, we must have f ( IN ) = 0, so f descends toa map in Hom( N/IN, M ). But N/IN ∈ ( T ∩ mod A/I ) . This shows that in fact M ( T ∩ mod A/I ) ⊥ . We conclude that the torsion free class in mod A/I whichcorresponds to T ∩ mod A/I is F ∩ mod A/I . (cid:3) For T a torsion class in mod A , write φ ( T ) for T ∩ mod A/I . Proposition 12.3 ([DI+, Proposition 5.7(d)]) . If φ is the quotient A ։ A/I , then φ is a lattice quotient from tors A to tors A/I .Proof. From the definition, it is clear that φ respects the meet operation on tors A .To see that φ respects join, we recall that _ T ∈ S T = ⊥ \ T ∈ S T ⊥ ! and the result now follows from Proposition 12.2. (cid:3) We are interested in understanding this lattice quotient better. In particular,we will address the question of when two torsion classes in mod A have the sameimage under this quotient. For this purpose, we need the following lemma. Lemma 12.1. For U ≤ V in tors A , the following are equivalent: (1) U < V , (2) U ⊥ ∩ V 6 = { } , (3) U ⊥ ∩ V contains a brick.Proof. The implications (3) implies (2) and (2) implies (1) are obvious.To see that (1) implies (2), let X ∈ V \ U . We have a short exact sequence0 → t U X → X → X/t U X → . Since X 6∈ U , we know that t U X = X , so X/t U X is a non-zero module in U ⊥ . Onthe other hand, X ∈ V , so X/t U X is also. Thus X/t U X witnesses (2).We now show that (2) implies (3). Suppose that X ∈ U ⊥ ∩ V , and suppose thatthe dimension of X is minimal among non-zero modules in U ⊥ ∩ V . If X is a brick,we are done, so suppose that X is not a brick. It therefore has a non-invertiblenon-zero endomorphism f . Let Y = f ( X ). Now Y is at the same time a quotientand a submodule of X . Since Y is a quotient of X , we know that Y ∈ V . On theother hand, since Y is a submodule of X , we know that Y ∈ U ⊥ . Therefore Y isan element of U ⊥ ∩ V of dimension smaller than X , contradicting our choice of X .Thus X must have been a brick. (cid:3) From Lemma 12.1, the following proposition is immediate: Proposition 12.4 ([DI+, Theorem 5.15(b)]) . For U ≤ V in tors A , φ ( U ) = φ ( V ) if and only if U ⊥ ∩ V contains no modules annihilated by I , or equivalently containsno bricks annihilated by I . N INTRODUCTION TO THE LATTICE OF TORSION CLASSES 15 Another way to formulate the proposition is that if U ≤ V , then φ ( U ) = φ ( V )precisely if there is some module in U ⊥ ∩ V which is annihilated by I .Also, we have the following proposition. We write ˆ γ A and ˆ γ A/I for the labellingsassociated to covers in tors A and tors A/I , respectively. Proposition 12.5 ([DI+, Theorem 5.15(a)]) . If Y ⋗ X in tors A and φ ( Y ) ⋗ φ ( X ) in tors A/I , then ˆ γ A ( Y ⋗ X ) = ˆ γ A/I ( φ ( Y ) ⋗ φ ( X )) .Proof. If Y ⋗ X in tors A , then there is a unique brick from mod A in X ⊥ ∩Y , namelyˆ γ A ( Y ⋗ X ). Given that φ ( Y ) = φ ( X ), this brick must in fact lie in mod A/I . It istherefore the unique brick in φ ( X ) ⊥ ∩ φ ( Y ), and thus equals ˆ γ A/I ( Y ⋗ X ). (cid:3) Example 12.1 (Type A ) . Let A = kQ , where Q = 1 ← . Let I be the ideal of A generated by the arrow. A/I is the path algebra of two vertices and no arrows; tors mod A/I is as follows: h [01] , [10] ih [01] i h [10] i We see that it is obtained from the lattice tors A by identifying the two torsionclasses h [01] , [11] i and h [01] i , which differ only in modules which are not in mod A/I .We further see that the labels of the cover relations which remain cover relations in tors A/I receive the same labels as cover relations in tors A and as cover relationsin tors A/I , consistent with Proposition 12.5. In the next section, we will see how to combine Proposition 12.4 with the labellingˆ γ . In order to do that, we need another important structural result about tors A .13. tors A is weakly atomic A lattice is called weakly atomic if in any interval [ u, v ] with u < v , there is somepair of elements x, y with x ⋖ y . (This property is referred to as arrow-separatednessin the current version of [DI+] and as cover-separatedness in the current version of[RST], but they will be updated to reflect the standard terminology.) The interval[0 , 1] in R , with the usual order, is an example of a lattice which is not weaklyatomic (since it has no cover relations at all).In this section, we will prove the following two theorems. Theorem 13.1 ([DI+]) . tors A is weakly atomic. Theorem 13.2 ([DI+]) . Let φ : A ։ A/I be an algebra quotient. For U ⊆ V , wehave that φ ( U ) = φ ( V ) iff all covers in [ U , V ] are labelled by bricks which are notannihilated by I . On the way to proving these theorems, we first prove the following proposition,which can be viewed as a relative version of Theorem 7.1. Proposition 13.1 ([DI+, Theorem 3.4]) . Let U ≤ V be two torsion classes. Themap B T ( B ) ∨ U is a bijection from br ( U ⊥ ∩ V ) to Ji c [ U , V ] .Proof. Let B ∈ br ( U ⊥ ∩ V ). Let Y = U ∨ T ( B ). Also consider the torsion class X = Y ∩ ⊥ F ( B ). Since B ∈ U ⊥ , we have that ⊥ F ( B ) ⊇ U , so X also lies in [ U , V ].Now X is strictly contained in Y since it does not contain B . But any torsion classcontaining U which is strictly contained in Y cannot include any module admitting asurjective map onto B . By Lemma 7.2, any such torsion class is therefore containedin ⊥ B = ⊥ F ( B ). This shows that Y is completely join irreducible in [ U , V ] andthat Y covers X .We now show that all the completely join irreducible elements of [ U , V ] corre-spond to some brick as above. Any torsion class can be written as the join of thetorsion classes corresponding to the bricks it contains, so any torsion class in [ U , V ]can be written as the join of U and a set of torsion classes of the form T ( B ) for B lying in some subset of U ⊥ ∩ V . It follows that the only completely join irreducibleelements of [ U , V ] are those of the form U ∨ T ( B ).Finally, the map from bricks to torsion classes is invertible. If Y is a completelyjoin irreducible torsion class in [ U , V ], with X the unique torsion class in [ U , V ]which it covers, then the brick corresponding to Y is ˆ γ ( Y ⋗ X ). (cid:3) Based on this, we can now easily establish the following proposition: Proposition 13.2 ([DI+]) . Let U < V be two torsion classes in mod A . Thenthere are covers in [ U , V ] labelled by each brick in U ⊥ ∩ V , and no others. Note that U ⊥ ∩ V is non-empty by Lemma 12.1. Proof. It is clear that no other brick can appear as a label since if V ≥ Y ⋗ X ≥ U ,and ˆ γ ( Y ⋗ X ) = B then B ∈ Y ⊆ V and B ∈ X ⊥ ⊆ U ⊥ .For the converse direction, if B is a brick in U ⊥ ∩ V , then by Proposition 13.1,there is a completely join irreducible in [ U , V ] corresponding to B , and the coverrelation down from it in [ U , V ] is labelled by B . (cid:3) Theorem 13.1 follows directly from Proposition 13.2, since if U < V , then byLemma 12.1, U ⊥ ∩ V contains a brick.Theorem 13.2 follows as well, by combining Proposition 12.4 with Proposition13.2.14. A combinatorial application: finite semi-distributive lattices Consider the following lattice:This lattice is semidistributive. Suppose that it were isomorphic to tors A for some A . We see that this lattice has four (completely) join irreducible elements and four(completely) meet irreducible elements, so mod A would necessarily have four bricks N INTRODUCTION TO THE LATTICE OF TORSION CLASSES 17 by Theorem 7.1. We see that two of the bricks would have to be simple, call them S and S , and there would be maps as follows, with X and Y being the other twobricks: S XY S There is no such module category. Results of [AP], extending [Ja], can also be usedto construct many examples of finite semidistributive lattices which are not latticesof torsion classes.In light of this, it would seem unlikely that representation theory could help usto understand general finite semidistributive lattices. Nonetheless, it turns out thatit can. Indeed, in [RST], inspired by properties of lattices of torsion classes, we gavea construction which yields exactly the finite semidistributive lattices. I will closeby describing this construction.Given any finite set, which we will call X , and a reflexive relation → on X ,for a subset C ⊂ X , we can define C ⊥ = { Y ∈ X | ∀ X ∈ C , X Y } , and ⊥ C = { X | ∀ Y ∈ C , X Y } . Torsion pairs in X are then defined to be pairs ofsubsets ( T , F ) such that T ⊥ = F and T = ⊥ F . We can then define tors ( X , → ) tobe the set of torsion pairs ( T , F ), ordered by inclusion on T . If we allow ourselvesto start with any set X and reflexive relation → , this construction is so general asto be able to construct any finite lattice, as was discovered by Markowsky [Ma].Therefore, if we want to get only semidistributive lattices, we need to put somefurther conditions on → . It turns out that the way to do this is the insist thatthe relation → on the set X be more like the relation “there exists a non-zeromorphism” on the set of bricks of a module category.We make this precise as follows. Starting from a reflexive relation → , define twoother relations, ։ and ֒ → . We define X ։ Y iff whenever Y → Z then X → Z .Similarly, we define X ֒ → Y iff whenever Y → Z then X → Z . Again, the intuitionfrom representation theory is clear: if M and N are A -modules and there is asurjection from M to N then whenever there is a non-zero map from N to some L , then there is also a non-zero map from M to L and dually for injections. (Note,though, that if we take X = br mod A and take → to be “there exists a non-zeromorphism”, the relations ։ and ֒ → defined as above are not exactly “there exists asurjection” and “there exists an injection”. See [RST, Section 8] for more details.)We say that a reflexive relation → on X is factorizable if it satisfies the followingtwo conditions: • For any X, Z ∈ X with X → Z , there exists Y ∈ X such that X ։ Y ֒ → Z. • Any of X ։ Y ։ X or X ֒ → Y ։ X , or X ֒ → Y ֒ → X imply X = Y .As is probably clear, the motivating intuition for the first condition is that a non-zero morphism can be factored as a surjection followed by an injection.We can now state the main result of [RST]: Theorem 14.1 ([RST, Theorem 1.2]) . Let X be a finite set, and → a reflexivefactorizable relation on X . Then tors ( X , → ) is a semidistributive lattice, and every semidistributive lattice arises in this way for a choice of X and → which is uniqueup to isomorphism. I close with the following question: Question 14.1. Is there a way to interpret any finite semidistributive lattice asthe lattice of torsion classes of a “real” category? The question is deliberately worded somewhat imprecisely. Another way toask the question would be to ask for a representation-theoretic meaning to theconstruction of finite semidistributive lattices of [RST]. Acknowledgements I would like to thank my coauthors on [IR+, DI+, RST], from whom I havelearned a great deal. It is my pleasure to acknowledge NSERC and the CanadaResearch Chairs program for their financial support. Thanks to Nathan Reading,Alexander Garver, and two anonymous referees for helpful comments on this man-uscript. I am extremely grateful to have been given the opportunity to presentthis material at Zhejiang University in 2018 and at the Isfahan School on Repre-sentations of Algebras in 2019. Thanks to Fang Li for the invitation to Zhejiangand to the organizers of the Isfahan School and Conference on Representations ofAlgebras, and in particular Javad Asadollahi, for the invitation to speak in Isfahan,and for the invitation to prepare this contribution to the special issue. References [AIR] Takahide Adachi, Osamu Iyama, and Idun Reiten, τ -tilting theory, Compos. Math. (2014), no. 3, 415–452.[AN] Kira Adaricheva and J.B. Nation, Classes of semidistributive lattices. Lattice theory: specialtopics and applications. 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