aa r X i v : . [ m a t h . R T ] M a r CO-T-STRUCTURES: THE FIRST DECADE
PETER JØRGENSEN
Abstract.
Co-t-structures were introduced about ten years ago as a type of mirror imageof t-structures. Like t-structures, they permit to divide an object in a triangulated category T into a “left part” and a “right part”, but there are crucial differences. For instance,a bounded t-structure gives rise to an abelian subcategory of T , while a bounded co-t-structure gives rise to a so-called silting subcategory.This brief survey will emphasise three philosophical points. First, bounded t-structuresare akin to the canonical example of “soft” truncation of complexes in the derived category.Secondly, bounded co-t-structures are akin to the canonical example of “hard” truncationof complexes in the homotopy category.Thirdly, a triangulated category T may be skewed towards t-structures or co-t-structures,in the sense that one type of structure is more useful than the other for studying T . Inparticular, we think of derived categories as skewed towards t-structures, and of homotopycategories as skewed towards co-t-structures. Introduction
The notion of co-t-structure in a triangulated category T was introduced independently byBondarko and Pauksztello, see Definition 2.1. It is a mirror image of the classic notion oft-structure due to Beilinson, Bernstein, and Deligne, see Definition 1.1.Given an object t ∈ T , both types of structure give a way to divide t into a “left part” anda “right part”. This is exemplified by dividing a complex of modules into a left and a rightpart by “soft” or “hard” truncation, see Figures 1 and 2.Each case gives a triangle u → t → v . Crucially, for a t-structure, u is the left part of t and v the right part; for a co-t-structure, vice versa. This reversal leads to a number of differences,and the theories of t-structures and co-t-structures are far from being simple mirrors of eachother. For instance, while a bounded t-structure induces an abelian subcategory of T , abounded co-t-structure induces a so-called silting subcategory; see Definition 2.3.In the decade since their inception, the theory of co-t-structures has grown considerably.This brief survey is far from encyclopedic, but has the important goal of communicatingthree philosophical points:(i) Bounded t-structures are akin to soft truncation in the bounded derived category ofan abelian category.(ii) Bounded co-t-structures are akin to hard truncation in the bounded homotopy ca-tegory of an additive category. Mathematics Subject Classification.
Key words and phrases.
Abelian subcategory, Auslander–Reiten quiver, co-heart, co-t-structure, com-plex of modules, derived category, heart, homotopy category, silting mutation, silting quiver, silting subcat-egory, simple minded collection, t-structure, triangulated category, truncation. a = · · · (cid:15) (cid:15) / / M − / / (cid:15) (cid:15) M − / / (cid:15) (cid:15) Ker d / / (cid:15) (cid:15) / / (cid:15) (cid:15) / / (cid:15) (cid:15) / / (cid:15) (cid:15) · · · t = · · · (cid:15) (cid:15) / / M − / / (cid:15) (cid:15) M − / / (cid:15) (cid:15) M d / / (cid:15) (cid:15) M / / (cid:15) (cid:15) M / / (cid:15) (cid:15) M / / (cid:15) (cid:15) · · · b = · · · / / / / / / / / Coker d / / M / / M / / · · · Figure 1.
A complex t = · · · → M − → M d → M → M → · · · hassoft truncations a and b . There is a triangle a → t → b in the derived cate-gory. Each vertical module homomorphism is either the identity, a canonicalinclusion or surjection, or zero.(iii) A triangulated category T may be skewed in the direction of t-structures or co-t-structures, in the sense that one type of structure is more useful than the other forstudying T .We explain these points in the next three subsections. Throughout, T is a triangulatedcategory with Hom-spaces T ( − , − ) and suspension functor Σ. If M is an abelian categorythen D b ( M ) is the derived category of bounded complexes over M , and if P is an additivecategory then K b ( P ) is the homotopy category of bounded complexes over P . (i) Bounded t-structures are akin to soft truncation in the bounded derivedcategory. Let R be a ring, M = Mod R the category of left modules over R . Each complex t ∈ D b ( M ) has soft truncations a and b as shown in Figure 1, and there is a triangle a → t → b in D b ( M ). The full subcategories A and B consisting of complexes isomorphicto such truncations satisfy Definition 1.1 and hence form a bounded t-structure, sometimesknown as the standard t-structure, see Example 1.2.Up to isomorphism, the heart H = A ∩ Σ B consists of complexes concentrated in degree 0.The heart is an abelian subcategory of D b ( M ) which is equivalent to M . Each t ∈ D b ( M )permits a “tower” as shown in Proposition 3.1 due to Beilinson, Bernstein, and Deligne.This expresses how to build t from objects of the form Σ i h with h ∈ H . The objects can betaken to be Σ i H − i ( t ) and then the tower shows how t is built from its cohomology modules,see Example 3.3.In general, if ( A , B ) is a bounded t-structure in a triangulated category T , then each t ∈ T still permits a triangle a → t → b with a ∈ A , b ∈ B . The heart H is still an abeliansubcategory of T by Theorem 1.3 due to Beilinson, Bernstein, and Deligne, and each t ∈ T still has the tower in Proposition 3.1.Working in such a setup is akin to working with soft truncation in D b ( M ). (ii) Bounded co-t-structures are akin to hard truncation in the bounded homo-topy category. Let R be a ring, P = Prj R the category of projective left modules over R . Each complex t ∈ K b ( P ) has hard truncations x and y as shown in Figure 2, andthere is a triangle x → t → y in K b ( P ). The full subcategories X and Y consisting of O-T-STRUCTURES DECADE 3 x = · · · (cid:15) (cid:15) / / / / (cid:15) (cid:15) / / (cid:15) (cid:15) / / (cid:15) (cid:15) P / / (cid:15) (cid:15) P / / (cid:15) (cid:15) P / / (cid:15) (cid:15) · · · t = · · · (cid:15) (cid:15) / / P − / / (cid:15) (cid:15) P − / / (cid:15) (cid:15) P − / / (cid:15) (cid:15) P / / (cid:15) (cid:15) P / / (cid:15) (cid:15) P / / (cid:15) (cid:15) · · · y = · · · / / P − / / P − / / P − / / / / / / / / · · · Figure 2.
A complex t = · · · → P − → P − → P → P → · · · has hardtruncations x and y . There is a triangle x → t → y in the homotopy category.Each vertical module homomorphism is the identity or zero.complexes isomorphic to such truncations satisfy Definition 2.1 and hence form a boundedco-t-structure, sometimes known as the standard co-t-structure, see Example 2.2.Up to isomorphism, the co-heart C = X ∩ Σ − Y consists of complexes concentrated indegree 0. The co-heart is an additive subcategory of K b ( P ) which is equivalent to P .The co-heart is not in general abelian, but it does have the strong property of being a so-called silting subcategory of K b ( P ), see Definition 2.3 and Theorem 2.4 due to MendozaHern´andez et.al. Such subcategories are structurally important.Each t = · · · → P − → P − → P − → P → P → P → · · · in K b ( P ) permits a “tower”as shown in Proposition 3.2 due to Bondarko. This expresses how to build t from objects ofthe form Σ i c with c ∈ C . The objects can be taken to be Σ i P − i and then the tower showshow t is built from its constituent modules, see Example 3.4.In general, if ( X , Y ) is a bounded co-t-structure in a triangulated category T , then each t ∈ T still permits a triangle x → t → y with x ∈ X , y ∈ Y . The co-heart C is still asilting subcategory of T by Theorem 2.4, and each t ∈ T still has the tower in Proposition3.2.Working in such a setup is akin to working with hard truncation in K b ( P ). (iii) Triangulated categories skewed in the direction of t-structures or co-t-structures. When studying a given triangulated category, bounded co-t-structures maybe more useful than bounded t-structures, simply because there are none of the latter. SeeSection 4 for a toy example.Section 5 shows a subtler skewing phenomenon. Let Λ be a finite dimensional C -algebra, M = mod Λ the category of finite dimensional left modules over Λ, and P = prj Λ thecategory of finite dimensional projective left modules over Λ. Theorem 5.2, due to K¨onig andYang, shows a bijection between all bounded co-t-structures in K b ( P ) and the bounded t-structures in D b ( M ) whose hearts are length categories (these are, in a sense, the “algebraic”t-structures; for instance, they permit a nice mutation theory).In itself, this does not imply that either category has more t-structures than co-t-structuresor vice versa, but we think of it as indicating that when going from derived to homotopycategories, the role played by t-structures is taken over by co-t-structures. Summing up,
Sections 1 and 2 show the definitions of t-structures and co-t-structures,emphasising the similarities with soft and hard truncation of complexes of modules. Section
PETER JØRGENSEN t-structures
The following definition was made in [4, def. 1.3.1].
Definition 1.1 (Beilinson, Bernstein, and Deligne) . A t-structure in the triangulated cate-gory T is a pair ( A , B ) of full subcategories, closed under isomorphisms, direct sums, anddirect summands, which satisfy the following conditions.(i) Σ A ⊆ A and Σ − B ⊆ B .(ii) T ( A , B ) = 0.(iii) For each object t ∈ T there is a triangle a → t → b with a ∈ A , b ∈ B .The heart is H = A ∩ Σ B .The t-structure is called bounded if [ i ∈ Z Σ i A = [ i ∈ Z Σ i B = T . The objects a and b in Definition 1.1(iii) depend functorially on t . The resulting functor t a is a right-adjoint to the inclusion A ֒ → T . Similarly, t b is a left adjoint to theinclusion B ֒ → T . See [4, prop. 1.3.3].The following is the canonical example of a t-structure. Example 1.2 (The standard t-structure) . Let R be a ring, M = Mod R the category ofleft modules over R . Let A and B be the isomorphism closures in the bounded derivedcategory D b ( M ) of the subsets { · · · / / M − / / M − / / M / / / / / / / / · · · | M i ∈ M , M i = 0 for i ≪ } , { · · · / / / / / / / / M / / M / / M / / · · · | M i ∈ M , M i = 0 for i ≫ } . We will show that if A and B are viewed as full subcategories, then ( A , B ) is a boundedt-structure with heart H equivalent to M .It is easy to show A = { M ∈ D b ( M ) | H i ( M ) = 0 for i > } , B = { M ∈ D b ( M ) | H i ( M ) = 0 for i } . (1.1)This description clearly implies that A and B are closed under isomorphisms, direct sums,and direct summands. O-T-STRUCTURES DECADE 5
We next check the conditions in Definition 1.1. Condition (i) is immediate. Condition (ii)requires that if objects a = · · · / / M − / / M − / / M / / / / / / / / · · · ,b = · · · / / / / / / / / M / / M / / M / / · · · in A and B are given, then Hom D b ( M ) ( a, b ) = 0. This can be shown by noting that a has aprojective resolution p = · · · → P − → P − → P → → → → · · · , and that Hom D b ( M ) ( a, b ) = Hom K ( M ) ( p, b ) = 0 . Here K ( M ) is the homotopy category of complexes over M , and the second = holds becausein each degree, either the complex p or the complex b is zero. The triangle in Definition 1.1(iii)can be obtained by soft truncation of the object t = · · · → M − → M d → M → M → · · · in D b ( M ), see Figure 1 in the introduction.It is immediate that the t-structure ( A , B ) is bounded.Finally, it follows from Equation (1.1) that the heart H = A ∩ Σ B is H = { M ∈ D b ( M ) | H i ( M ) = 0 for i = 0 } , and this subcategory of D b ( M ) is equivalent to M . The following pivotal result is one of the motivations for the definition of t-structures. Itwas proved in [4, thm. 1.3.6].
Theorem 1.3 (Beilinson, Bernstein, and Deligne) . Let ( A , B ) be a t-structure in T . Thenthe heart H = A ∩ Σ B is an abelian subcategory of T . Example 1.4.
Let Λ = C A be the path algebra of the quiver A = 1 → M = mod Λ be the category of finite dimensional left modules over Λ.There is a bounded t-structure ( A , B ) in D b ( M ) defined by Equation (1.1); this is shownby the same method as in Example 1.2.There are three isomorphism classes of indecomposable objects in M given by the followingrepresentations of the quiver A .0 → C , C id → C , C → x , x , x of indecomposable objects in D b ( M ), and wedefine further isomorphism classes recursively by Σ x i = x i +3 for i ∈ Z .The Auslander–Reiten quiver of D b ( M ) looks as follows, where red and green vertices show A and B . x − (cid:31) (cid:31) ❄❄❄❄❄❄❄ x − (cid:31) (cid:31) ❄❄❄❄❄❄❄ x (cid:31) (cid:31) ❄❄❄❄❄❄❄ x (cid:31) (cid:31) ❄❄❄❄❄❄❄ x (cid:31) (cid:31) ❄❄❄❄❄❄❄ · · · ? ? ⑧⑧⑧⑧⑧⑧⑧ x − ? ? ⑧⑧⑧⑧⑧⑧⑧ x ? ? ⑧⑧⑧⑧⑧⑧⑧ x ? ? ⑧⑧⑧⑧⑧⑧⑧ x ? ? ⑧⑧⑧⑧⑧⑧⑧ · · · (1.3)Note that if t ∈ D b ( M ) is indecomposable, then t is in A or in B , so the triangle inDefinition 1.1(iii) is trivial in the sense that it reads t → t → → t → t . PETER JØRGENSEN
The heart H = A ∩ Σ B is determined by H = add( x ⊕ x ⊕ x ) . Co-t-structures
The following definition was made in [5, def. 1.1.1] and [13, def. 2.4].
Definition 2.1 (Bondarko and Pauksztello) . A co-t-structure in T is a pair ( X , Y ) of fullsubcategories, closed under isomorphisms, direct sums, and direct summands, which satisfythe following conditions.(i) Σ − X ⊆ X and Σ Y ⊆ Y .(ii) T ( X , Y ) = 0.(iii) For each object t ∈ T there is a triangle x → t → y with x ∈ X , y ∈ Y .The co-heart is C = X ∩ Σ − Y .The co-t-structure is called bounded if [ i ∈ Z Σ i X = [ i ∈ Z Σ i Y = T . In contrast to t-structures, the objects x and y in Definition 2.1(iii) do not in general dependfunctorially on t , see [5, rmk. 1.2.2].The following is the canonical example of a co-t-structure. Example 2.2 (The standard co-t-structure) . Let R be a ring, P = Prj R the category ofprojective left-modules over R . Let X and Y be the isomorphism closures in the boundedhomotopy category K b ( P ) of the subsets { · · · / / / / / / / / P / / P / / P / / · · · | P i ∈ P , P i = 0 for i ≫ } , { · · · / / P − / / P − / / P − / / / / / / / / · · · | P i ∈ P , P i = 0 for i ≪ } . (2.1)We will show that if X and Y are viewed as full subcategories, then ( X , Y ) is a boundedco-t-structure with co-heart C equivalent to P .Recall that X and Y are required to be closed under isomorphisms, direct sums, and directsummands. The two former properties are immediate, and the latter follows from the resultsin [14, secs. 3 and 4].We next check the conditions in Definition 2.1. Conditions (i) and (ii) are clear. The trianglein Definition 2.1(iii) can be obtained by hard truncation of the object t = · · · → P − → P − → P → P → · · · in K b ( P ), see Figure 2 in the introduction.It is immediate that the co-t-structure ( X , Y ) is bounded.Finally, it follows from [14, cor. 4.11] that the coheart C = X ∩ Σ − Y is equivalent to P . The term silting set was coined in [9]. The following definition was made in [1, def. 2.1].
Definition 2.3. A silting subcategory C of T is a full subcategory, closed under isomor-phisms, direct sums, and direct summands, which satisfies O-T-STRUCTURES DECADE 7 (i) T ( C , Σ > C ) = 0.(ii) Each object in T can be obtained from C by taking finitely many (de)suspensions,triangles, and direct summands.A silting object s of T is an object such that add( s ) is a silting subcategory. The following was proved in [12, cor. 5.9].
Theorem 2.4 (Mendoza Hern´andez et.al.) . The map ( X , Y ) C = X ∩ Σ − Y is a bijection between bounded co-t-structures and silting subcategories of T . Remark 2.5.
The inverse map sends a silting subcategory C to a pair ( X , Y ) where X isthe smallest full subcategory, closed under isomorphisms, direct sums, and direct summands,which is closed under Σ − and contains C . Similarly, Y is the smallest full subcategory,closed under isomorphisms, direct sums, and direct summands, which is closed under Σ andcontains Σ C . Example 2.6.
We continue Example 1.4, so Λ = C A is the path algebra of the quiver A from Equation (1.2) and P = prj Λ is the category of finite dimensional projective leftmodules over Λ.There is a bounded co-t-structure ( X , Y ) in K b ( P ) where X and Y are the isomorphismclosures in K b ( P ) of the subsets in Equation (2.1); this is shown by the same method asin Example 2.2.Recall that Λ also has a bounded derived category D b ( M ), see Example 1.4. Since Λhas global dimension 1, the triangulated categories K b ( P ) and D b ( M ) are equivalent, so K b ( P ) has the Auslander–Reiten quiver shown in Equation (1.3). We redraw the quiver,this time with red and green vertices showing X and Y . x − (cid:31) (cid:31) ❄❄❄❄❄❄❄ x − (cid:31) (cid:31) ❄❄❄❄❄❄❄ x (cid:31) (cid:31) ❄❄❄❄❄❄❄ x (cid:31) (cid:31) ❄❄❄❄❄❄❄ x (cid:31) (cid:31) ❄❄❄❄❄❄❄ · · · ? ? ⑧⑧⑧⑧⑧⑧⑧ x − ? ? ⑧⑧⑧⑧⑧⑧⑧ x ? ? ⑧⑧⑧⑧⑧⑧⑧ x ? ? ⑧⑧⑧⑧⑧⑧⑧ x ? ? ⑧⑧⑧⑧⑧⑧⑧ · · · (2.2)Note that x is neither in X nor Y . Indeed, if we abuse notation to confuse isomorphismclasses with individual objects, then we can set t = x and the triangle in Definition 2.1(iii)becomes x → x → x .The coheart C = X ∩ Σ − Y is determined by C = add( x ⊕ x ) . Theorem 2.4 implies that C is a silting subcategory of K b ( P ). The corresponding isomor-phism class of silting objects is x ⊕ x . Towers which build an arbitrary object from objects of the (co-)heart
The following two results were proved in [4, p. 34] and [5, prop. 1.5.6]. A wavy arrow s / / /o/o/o t denotes a morphism s → Σ t . PETER JØRGENSEN
Proposition 3.1 (Beilinson, Bernstein, and Deligne) . Let ( A , B ) be a bounded t-structurein T with heart H . For each object t ∈ T , there is an integer n > and a diagramconsisting of triangles, t / / t / / (cid:127) (cid:127) ⑧⑧⑧⑧⑧ t / / (cid:127) (cid:127) ⑧⑧⑧⑧⑧ · · · / / t n − / / t n = t, (cid:127) (cid:127) ⑧⑧⑧⑧⑧ Σ i h _ _ _(cid:31) _(cid:31) _(cid:31) Σ i h _ _ _(cid:31) _(cid:31) _(cid:31) Σ i n h n _ _ _(cid:31) _(cid:31) with h m ∈ H for each m and i > i > · · · > i n . Proposition 3.2 (Bondarko) . Let ( X , Y ) be a bounded co-t-structure in T with co-heart C . For each object t ∈ T , there is an integer n > and a diagram consisting of triangles, t / / t / / (cid:127) (cid:127) ⑧⑧⑧⑧⑧ t / / (cid:127) (cid:127) ⑧⑧⑧⑧⑧ · · · / / t n − / / t n = t, (cid:127) (cid:127) ⑧⑧⑧⑧⑧ Σ i c _ _ _(cid:31) _(cid:31) _(cid:31) Σ i c _ _ _(cid:31) _(cid:31) _(cid:31) Σ i n c n _ _ _(cid:31) _(cid:31) with c m ∈ C for each m and i < i < · · · < i n . Example 3.3.
Consider the t-structure in Example 1.2. If t ∈ D b ( M ) is given, then thereis a diagram as in Proposition 3.1 where the objects Σ i m h m are of the form Σ i H − i ( t ). Thediagram expresses that t can be built from its cohomology modules H − i ( t ). Example 3.4.
Consider the co-t-structure in Example 2.2. If t = · · · → P − → P − → P → P → · · · in K b ( P ) is given, then there is a diagram as in Proposition 3.2 wherethe objects Σ i m c m are of the form Σ i P − i . The diagram expresses that t can be built fromits constituent modules P − i . Categories skewed towards t- or co-t-structures
In this section, d is a fixed integer. Definition 4.1. If T is C -linear, then an object s ∈ T is called d -spherical if there is anisomorphism T ( s, Σ ∗ s ) ∼ = C [ X ] / ( X )of graded algebras where deg X = d . Remark 4.2. If d = 0, then s is d -spherical if and only if there are isomorphisms of C -vectorspaces T ( s, Σ i s ) ∼ = (cid:26) C for i ∈ { , d } , The following was proved in [10, thm. 2.1].
Theorem 4.3 (Keller, Yang, and Zhou) . There is a triangulated category S d which isalgebraic, C -linear with finite dimensional Hom -spaces, and contains a d -spherical object s such that each object in S d can be obtained from s by taking finitely many (de)suspensions,triangles, and direct summands.Up to triangulated equivalence, S d is unique. The following was proved in [7, thm. A]. It clearly implies that if d
0, then boundedco-t-structures are more useful than bounded t-structures for the study of S d . O-T-STRUCTURES DECADE 9
Theorem 4.4 (Holm, J, and Yang) . If d then S d has no bounded t-structures. It hasone familiy of bounded co-t-structures, all of which are (de)suspensions of a canonical one.If d > then S d has no bounded co-t-structures. It has one familiy of bounded t-structures,all of which are (de)suspensions of a canonical one. The bijections of K¨onig and Yang
Definition 5.1. A simple minded collection in T is a set { t , . . . , t n } of objects of T withthe following properties.(i) T ( t i , Σ < t j ) = 0 for all i and j .(ii) T ( t i , t i ) is a division ring for each i and T ( t i , t j ) = 0 when i = j .(iii) Each object in T can be obtained from t , . . . , t n by taking finitely many (de)sus-pensions, triangles, and direct summands. Let Λ be a finite dimensional C -algebra, M = mod Λ the category of finite dimensional leftmodules over Λ, and P = prj Λ the category of finite dimensional projective left modulesover Λ. The following was proved in [11, thm. (6.1)]. We interpret it as indicating thatthe role of t-structures in derived categories is taken over by co-t-structures in homotopycategories. Theorem 5.2.
There are bijections between the following sets.(i) Bounded t-structures in D b ( M ) whose hearts are length categories (“length category”means that each object has finite length).(ii) Bounded co-t-structures in K b ( P ) .(iii) Isomorphism classes of simple minded collections in D b ( M ) .(iv) Isomorphism classes of basic silting objects in K b ( P ) (basic means no repeatedindecomposable summands). Remark 5.3.
There is an extensive body of work on the bijections of Theorem 5.2 whichpredates [11], see [1], [2], [3], [5], [6], [8], [9], and [12]. The contributions of these papers toTheorem 5.2 are explained in the introduction to [11]. Remark 5.4.
The proof of Theorem 5.2 occupies a significant part of [11], and we onlyshow how some of the bijections are defined.(i) to (iii): Let ( A , B ) be a bounded t-structure in D b ( M ) whose heart H = A ∩ Σ B is alength category. Take a simple object from each isomorphism class of simple objects in H .This gives a simple minded system in D b ( M ), see [11, sec. 5.3].(ii) to (iv): Let ( X , Y ) be a bounded co-t-structure in K b ( P ). The co-heart C = X ∩ Σ − Y is a silting subcategory by Theorem 2.4, and there is a silting object s such that C = add( s ), see [11, sec. 5.2].(i) to (ii): Let ( X , Y ) be a bounded co-t-structure in K b ( P ). Set A = { a ∈ D b ( M ) | Hom D b ( M ) ( x, a ) = 0 for each x ∈ X } , B = { b ∈ D b ( M ) | Hom D b ( M ) ( y, b ) = 0 for each y ∈ Y } and view these two sets as full subcategories of D b ( M ). Then ( A , B ) is a bounded t-structure in D b ( M ) with length heart, see [11, sec. 5.7]. The silting mutation of Aihara and Iyama
Silting mutation is an operation which changes one silting subcategory into another. Byvirtue of Theorem 2.4, it can be viewed as changing one bounded co-t-structure into another.This leads to the definition of the so-called silting quiver of the triangulated category T which shows how silting subcategories, and hence co-t-structures, fit together inside T .In this section, T is C -linear with finite dimensional Hom-spaces and split idempotents,and m = m ⊕ m is a basic silting object of T with m indecomposable. The followingdefinition and theorem are special cases of [1, sec. 2.4]. Definition 6.1 (Aihara and Iyama) . Let r ρ → m λ → ℓ be a minimal right add( m )-approximation and a minimal left add( m )-approximation of m . Complete these morphismsto triangles m ∼ → r ρ → m , m λ → ℓ → m † in T and set µ − ( m, m ) = m ∼ ⊕ m , µ + ( m, m ) = m † ⊕ m . These are called right and left silting mutations of m . Theorem 6.2 (Aihara and Iyama) . (i) The silting mutations µ − ( m, m ) and µ + ( m, m ) are basic silting objects of T .(ii) Right and left silting mutations are inverse in the sense that µ − (cid:0) µ + ( m, m ) , m (cid:1) ∼ = m , µ + (cid:0) µ − ( m, m ) , m (cid:1) ∼ = m. The following is a special case of [1, def. 2.41].
Definition 6.3 (Aihara and Iyama) . The silting quiver of T has a vertex for each isomor-phism class of basic silting objects of T , and an arrow [ m ] → [ m ∗ ] if m ∗ is a left siltingmutation of m , where square brackets denote isomorphism class. Remark 6.4.
The silting quiver gives a picture of how silting mutation moves from onesilting object to another, hence from one silting subcategory to another. By virtue of The-orem 2.4, it gives a picture of how silting mutation moves from one bounded co-t-structureto another. Example 6.5.
We continue Example 2.6, so Λ = C A is the path algebra of the quiver A from Equation (1.2). The bounded homotopy category K b ( P ) has the Auslander–Reitenquiver shown in Equation (2.2).Recall from Example 2.6 that K b ( P ) has the isomorphism class of silting objects x ⊕ x .Indeed, x ⊕ x is a vertex in the silting quiver of K b ( P ). The full quiver was determinedin [1, exa. 2.45], see Figure 3. As the quiver shows, there is a left silting mutation of x ⊕ x which gives x ⊕ x . The isomorphism classes of silting objects x ⊕ x and x ⊕ x give riseto silting subcategories add( x ⊕ x ) and add( x ⊕ x ) which, under the bijection of Theorem2.4, correspond to two bounded co-t-structures. The first of these is shown on the AR quiver O-T-STRUCTURES DECADE 11 · · · (cid:15) (cid:15) (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ · · · (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎ · · · ☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎ x − ⊕ x (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎ x − ⊕ x (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎ x ⊕ x (cid:15) (cid:15) (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ x − ⊕ x (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎ · · · ☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎ x − ⊕ x (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎ x − ⊕ x (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎ x ⊕ x (cid:15) (cid:15) (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ x − ⊕ x (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎ · · · ☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎ x ⊕ x (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎ x − ⊕ x (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎ x ⊕ x (cid:15) (cid:15) (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ x − ⊕ x (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎ · · · ☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎ x ⊕ x (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ x − ⊕ x (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿☎☎☎☎☎☎☎ (cid:2) (cid:2) ☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ x ⊕ x (cid:15) (cid:15) x ⊕ x · · · x ⊕ x x − ⊕ x · · · · · · · · · Figure 3.
The silting quiver of the bounded homotopy category K b ( P )where P is the category of finite dimensional projective modules over C A .of K b ( P ) in Equation (2.2). The second co-t-structure ( X ′ , Y ′ ) can be shown as follows,where the red and green vertices show X ′ and Y ′ . · · · (cid:31) (cid:31) ❄❄❄❄❄❄❄ x − (cid:31) (cid:31) ❄❄❄❄❄❄ x (cid:31) (cid:31) ❄❄❄❄❄❄ x (cid:31) (cid:31) ❄❄❄❄❄❄ x (cid:31) (cid:31) ❄❄❄❄❄❄ x (cid:31) (cid:31) ❄❄❄❄❄❄ x − ? ? ⑧⑧⑧⑧⑧⑧ x ? ? ⑧⑧⑧⑧⑧⑧ x ? ? ⑧⑧⑧⑧⑧⑧ x ? ? ⑧⑧⑧⑧⑧⑧ x ? ? ⑧⑧⑧⑧⑧⑧ · · · Acknowledgement.
I thank Kiyoshi Igusa, Alex Martsinkovsky, and Gordana Todorov forinviting me to speak at the Maurice Auslander International Conference and to write thissurvey.
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