Silting and Tilting for Weakly Symmetric Algebras
aa r X i v : . [ m a t h . R T ] J a n SILTING AND TILTING FOR WEAKLY SYMMETRIC ALGEBRAS
JENNY AUGUST AND ALEX DUGAS
Abstract.
If A is a finite-dimensional symmetric algebra, then it is well-known that theonly silting complexes in K b (proj A ) are the tilting complexes. In this note we investigateto what extent the same can be said for weakly symmetric algebras. On one hand, weshow that this holds for all tilting-discrete weakly symmetric algebras. In particular, atilting-discrete weakly symmetric algebra is also silting-discrete. On the other hand, wealso construct an example of a weakly symmetric algebra with silting complexes that arenot tilting. Introduction
For a finite-dimensional k -algebra A , the tilting complexes play a central role in thecategory K b (proj A ) of perfect complexes. One of the main tools used in their study is mutation , but to get a well-behaved mutation, one is led to consider the weaker notionof silting complexes instead. While the silting theory of A can be quite complicated ingeneral, the notion of silting-discreteness was introduced by Aihara [A1] as a strong finitenessproperty. This can make it possible to describe all the silting complexes over A and theirbehavior under mutation. For example, under this condition it is well known that A is silting connected [A1] i.e. any two silting complexes of A can be connected by a sequence ofmutations.The silting-discreteness property also has particularly nice implications on the Bridgelandstability manifold associated to D b (mod A ) [AMY, PSZ] – a topological invariant related tothe t -structures in this category. In particular, Pauksztello–Saorin–Zvonareva show that fora silting-discrete algebra, the bounded t -structures in D b (mod A ) are in bijection with basicsilting complexes in K b (proj A ) [PSZ, KY]. Moreover, they use this to show the stabilitymanifold is contractible in this case, something which is often very difficult to determine inmore geometric settings.With this in mind, the results in this article were broadly motivated by the questionof which finite-dimensional algebras are silting-discrete. Because of their connection withderived equivalences, it is often easier to control the tilting complexes of an algebra, ratherthan all the silting complexes. For example, Aihara–Mizuno [AM] use the associated equiv-alences to show that the preprojective algebras of Dynkin type are tilting-discrete, but itremains an open question whether they are all silting-discrete. In particular, the easiestsettings to establish silting-discreteness will be when the notions of silting and tilting (andhence also silting-discrete and tilting-discrete) coincide. This is well-known for symmetricalgebras, and so we asked whether the same is true for weakly symmetric algebras. As weshow below, if a weakly symmetric algebra is tilting-discrete, then it must also be silting-discrete, and in this case all silting complexes are tilting. In particular, this applies to theweakly symmetric preprojective algebras (those of type D n , E and E ), a result which wehave since learned was already known to Aihara [A2], although the proof does not explicitlyappear in [AM]. Furthermore, after writing we became aware of work of Adachi and Kase[AK], which independently proves both of these results as a consequence of a more generaltheory of ν -stable silting.However, we additionally return to the question of whether every silting complex over aweakly symmetric algebra is tilting, and we show that the answer is negative in general. Weachieve this by constructing examples of weakly symmetric algebras with silting complexesthat are not tilting. These examples are modifications of the examples of silting-disconnectedalgebras in [D], and in fact provide further examples of algebras with this property. Preliminaries
We let A be a basic finite-dimensional algebra over an algebraically closed field k with n isomorphism classes of simple (right) modules. We write e , . . . , e n for a complete set ofpairwise orthogonal primitive idempotents for A , and write P i = e i A for the indecomposableprojective right A -modules. We primarily work with right A -modules and use mod A for thecategory of finitely generated right A -modules, D b (mod A ) for the bounded derived categoryand K b (proj A ) for the homotopy category of perfect complexes over A .2.1. Twisted modules.
Let σ be a k -algebra automorphism of A , acting on the left. Forany right A -module M , we define the twisted module M σ to be M as a k -vector space withthe right action of A given by m · a = mσ ( a ) for all m ∈ M and a ∈ A . Similarly, fora left A -module N , we can define the twisted module σ N as N but with A -action givenby a · m = σ ( a ) m for all a ∈ A and m ∈ N . Observe that we have natural isomorphisms M σ ∼ = M ⊗ A A σ and σ N ∼ = σ A ⊗ A N for all right (resp. left) A -modules M (resp. N ). Thus σ induces an automorphism σ ∗ := − ⊗ A A σ of the category mod A . This action restrictsto an automorphism of proj A and hence also induces automorphisms of K b (proj A ) andD b (mod A ).2.2. Nakayama Automorphism.
Writing D := Hom k ( − , k ) for the standard k -dualitybetween left and right A -modules, the Nakayama functor ν := D Hom A ( − , A ) is a rightexact functor isomorphic to − ⊗ A DA . It induces an equivalence proj A ∼ −→ inj A whosequasi-inverse is ν − := Hom A ( DA, − ).Recall that A is self-injective if and only if there is an isomorphism of right (or left)modules A ∼ −→ DA . In this case, all projective modules are injective and vice versa, andthere always exists an algebra automorphism ν : A → A such that there is a isomorphism of A -bimodules ϕ : A ν → DA.
Note that ν is unique up to inner automorphism, and we call ν the Nakayama auto-morphism of A since ν ∗ := − ⊗ A A ν coincides with the Nakayama functor ν . It is wellknown that A is symmetric if and only if ν is inner which is if and only if ν is isomorphicto the identity functor. Note that ν ( P i ) ∼ = I i for all finite-dimensional algebras, but if A isself-injective, there exists a permutation π such that, for all i , P i ∼ = I π ( i ) , or equivalently , ν P i ∼ = P π − ( i ) . This permutation π is known as the Nakayama permutation of A . Definition 2.1.
An algebra is weakly symmetric if the Nakayama permutation is the identityi.e. P i ∼ = ν P i for all i . Or equivalently, if ν ( e i ) ∼ = e i for all i .Note that the weakly symmetric property is strictly weaker than being symmetric. Theorem 2.2. [BBK, 4.8]
The preprojective algebras of ADE Dynkin type are self-injective.They are weakly symmetric if the Dynkin type is D n , E , or E but these are not symmetricunless char k = 2 . Nakayama and Tilting.
Recall that a complex T ∈ K b (proj A ) is called tilting (resp.silting) if(1) Hom K b (proj A ) ( T, T [ n ]) = 0 for all n = 0 (resp. for all n > b (proj A ) containing T and closedunder forming direct summands is K b (proj A ).We will write tilt A (resp. silt A ) for the set of isomorphism classes of basic tilting (resp.silting) complexes in K b (proj A ). If A is self-injective, the Nakayama functor restricts to anequivalence ν : proj A → proj A and hence there is an induced equivalence ν : K b (proj A ) → K b (proj A ) . It follows that if T is a tilting (resp. silting) complex, then so is ν T . Theorem 2.3. [A1, A.4] If A is self-injective then a basic silting complex T is tilting if andonly if ν T ∼ = T . Recall that when A is symmetric, ν ∼ = id, and hence a direct corollary of this result is thewell-known fact that all silting complexes over a symmetric algebra are tilting complexes.2.4. Silting Mutation.
To create new silting complexes from a given one, Aihara–Iyamaintroduced the notion of mutation [AI].
Definition 2.4.
Suppose that T = X ⊕ Y ∈ K b (proj A ) is a basic silting complex. Thenconsider a triangle X f −→ Y ′ g −→ X ′ → X [1]where f is a left add( Y )-approximation of X . Then µ X ( T ) := X ′ ⊕ Y is a silting complexcalled the left mutation of T with respect to X . There is a dual notion of right mutation.Such mutations are called irreducible if X is indecomposable.For any finite dimensional algebra A , we may view the algebra as a complex centredin degree zero, and this will always be a tilting complex. An algebra A is called siltingconnected (resp. weakly silting connected ) if all basic silting complexes in K b (proj A ) can beobtained from A by a sequence of irreducible (resp. not necessarily irreducible) mutations,left or right at each stage. Note that not all algebras are weakly silting connected [D].If A is self-injective, we say that a complex X ∈ K b (proj A ) is Nakayama stable if ν ( X ) ∼ = X . In other words, Theorem 2.3 says that a silting complex T is tilting if andonly if it is Nakayama stable. We further call T strongly Nakayama stable if each indecom-posable summand of T is Nakayama stable. Proposition 2.5. [D, 2.1] If A is self-injective and T is a strongly Nakayama stable tiltingcomplex, then any (not necessarily irreducible) mutation of T is also strongly Nakayamastable. Tilting Theory for Weakly Symmetric Algebras
In this section, we make some initial observations on the tilting theory of weakly sym-metric algebras, before then placing an extra condition on the algebras, known as tilting-discreteness , and showing that all silting complexes are tilting in this case. As before, we let A be a basic finite-dimensional algebra over a field k with n isomorphism classes of simplemodules.3.1. Initial Observations.
Since a basic silting complex T ∈ K b (proj A ) has n indecom-posable summands by [AI, 2.28], we can write T = ⊕ ni =1 T i where each T i is indecom-posable. If A is self-injective and T is tilting, then Theorem 2.3 shows ν must permutethese summands. In this case, the associated standard derived equivalence D b (mod A ) → D b (mod End( T )) maps the T i to the distinct indecomposable projective modules over End( T ),and commutes with the Nakayama functors of the two algebras [Ri2, 5.2]. Hence it followsthat the permutation of the T i induced by ν will correspond with the Nakayama permutationof End( T ). Proposition 3.1.
Let A be weakly symmetric. Then any tilting complex T ∈ K b (proj A ) isstrongly Nakayama stable. Consequently, any algebra derived equivalent to A is also weaklysymmetric.Proof. The Grothendieck group of the triangulated category K b (proj A ) is a free abeliangroup with basis elements [ P i ] for each indecomposable projective A -module P i . Since atilting complex T = ⊕ ni =1 T i ∈ K b (proj A ) with B = End( T ) induces an equivalence oftriangulated categories K b (proj B ) → K b (proj A ) taking B to T , it induces an isomorphismof Grothendieck groups taking the natural basis over B to { [ T i ] } ni =1 . Thus the latter is abasis for the Grothendieck group of K b (proj A ) (this is in fact true if T is any silting complexby [AI, 2.27]). However, if A is weakly symmetric, we have ν P i ∼ = P i for all i , and thus ν acts as the identity on the Grothendieck group. Since ν permutes the T i , if ν T i ∼ = T j , then[ T i ] = [ ν T i ] = [ T j ] in the Grothendieck group, which means that T i = T j ∼ = ν T i , as required. The second statement of the proposition, now follows from the fact mentioned above thatthe action of ν on the T i induces the Nakayama permutation of End( T ). (cid:3) Our next observation is the following direct corollary of Proposition 2.5.
Proposition 3.2. If A is a weakly symmetric algebra, then all silting complexes reachablefrom A via iterated mutation are strongly Nakayama stable tilting complexes. Moreover,their endomorphism algebras will all be weakly symmetric algebras.Proof. Since A is weakly symmetric, by definition we have ν P i ∼ = P i for all indecomposableprojective modules and thus A is a strongly Nakayama stable tilting complex. By Proposition2.5, any mutation of A is again a strongly Nakayama stable tilting complex, and henceiterating this result shows any silting complex reachable from A is a strongly Nakayamastable tilting complex. The second statement again uses the fact mentioned above that theaction of ν on a tilting complex T induces the Nakayama permutation of End( T ). (cid:3) Corollary 3.3. If A is weakly symmetric and weakly silting connected, then every siltingcomplex for A is a strongly Nakayama stable tilting complex.Proof. By Proposition 3.2, since A is weakly symmetric, all silting complexes reachable from A by mutation are strongly Nakayama stable tilting complexes. Since A is weakly siltingconnected, these are all the silting complexes of A and hence the result follows. (cid:3) Tilting-discreteness.
Silting- and tilting-discreteness are notions which were devel-oped by Aihara–Mizuno [AM] using the partial order on silting complexes introduced byAihara-Iyama [AI].
Definition 3.4. If T, S ∈ K b (proj A ) are two silting complexes, then we say T ≥ S ifHom K b (proj A ) ( T, S [ n ]) = 0 for all n > τ -tilting theory and cluster-tilting theory. Definition 3.5.
A basic silting complex T ∈ K b (proj A ) is called two-term if A ≥ T ≥ A [1]or equivalently, T only has nonzero terms in degrees 0 and − Proposition 3.6.
For a weakly symmetric algebra A , all two-term silting complexes aretilting.Proof. Suppose that A = L ni =1 P i and that T is a two-term silting complex for A . Then,since the [ P i ] give a basis of the Grothendieck group of K b (proj A ), we may write[ T ] = n M i =1 a i [ P i ]and, using the language of [DIJ], we say that the g -vector of T is ( a , . . . , a n ) ∈ Z n . Now ν T is another two-term silting complex for A , and since A is weakly symmetric ( ν P i ∼ = P i for all i ), ν T must have the same g -vector. However, by [DIJ, 6.5], g -vectors completelydetermine two-term silting complexes and thus T ∼ = ν T and T is tilting by Theorem 2.3. (cid:3) If an algebra A has finitely many basic two-term silting complexes, the algebra is called τ -tilting finite . Aihara [A1] generalised this notion, with Aihara–Mizuno then developing itfurther. Definition 3.7. [AM, 2.4, 2.11] A self-injective finite-dimensional algebra A is called tilting-discrete (resp. silting-discrete) if the set { T ∈ tilt A | P ≥ T ≥ P [1] } (resp. { T ∈ silt A | P ≥ T ≥ P [1] } )is finite for any tilting (resp. silting) complex P obtained from A by iterated irreducible leftmutation. It is clear that silting-discrete implies tilting-discrete and if the algebra A is symmetric, thetwo notions are equivalent. It is also known that silting-discrete implies silting connected[AM, 3.9] and tilting-discrete implies tilting-connected [CKL, 5.14]. However, if we onlyknow an algebra is tilting-discrete, it is generally unknown whether the algebra is alsosilting-discrete/silting connected. Proposition 3.8. (Cf. [AK, Cor. 2.25]) If A is a tilting-discrete weakly symmetric algebra,then A is in fact silting-discrete and all silting complexes for A are tilting.Proof. Suppose that P is a silting complex obtained from A by iterated irreducible leftmutation. Then, since A is weakly symmetric, P is a strongly Nakayama stable tiltingcomplex, and B := End A ( P ) is a weakly symmetric algebra using Proposition 3.2. Thus,there is a standard derived equivalence F : D b (mod A ) → D b (mod B ) P B and this preserves silting (resp. tilting) complexes and the silting order (see e.g. [Au, 2.8]).In particular, F induces a bijection { T ∈ silt A | P ≥ T ≥ P [1] } ↔ { S ∈ silt B | B ≥ S ≥ B [1] } (3.A)which further restricts to a bijection { T ∈ tilt A | P ≥ T ≥ P [1] } ↔ { S ∈ tilt B | B ≥ S ≥ B [1] } . (3.B)By the tilting-discreteness of A , the left hand side of (3.B) is finite and hence so is the righthand side. However, as B is weakly symmetric, Proposition 3.6 shows that { S ∈ tilt B | B ≥ S ≥ B [1] } = { S ∈ silt B | B ≥ S ≥ B [1] } and thus, both sides in (3.A) are also finite, proving that A is silting-discrete. Then by[A1, 3.9], this implies A is silting connected and thus all silting complexes can all be obtainedfrom A by iterated mutation. Using Proposition 3.3 this shows that all silting complexesare strongly Nakayama stable tilting complexes. (cid:3) Corollary 3.9. (Cf. [A2, Ex. 22], [AK, Ex. 2.26])
The preprojective algebras of Dynkin type D n , E and E are silting-discrete algebras, where every silting complex is a tilting complex.Proof. By Theorem 2.2, these algebras are weakly symmetric and [AM, 5.1] shows that theyare tilting-discrete. The result then follows directly from Proposition 3.8. (cid:3)
One application of silting-discreteness is in the study of Bridgeland stability. Given atriangulated category, in this case the bounded derived category of our finite dimensionalalgebra, Bridgeland stability constructs a complex manifold associated to this category. If A is a finite dimensional silting-discrete algebra, then [PSZ] show that this manifold will becontractible, and combining this with Proposition 3.8 immediately gives the following. Corollary 3.10. If A is a finite dimensional weakly symmetric tilting-discrete algebra, thenthe Bridgeland stability manifold of D b (mod A ) is contractible.Proof. This follows directly from Proposition 3.8 and [PSZ]. (cid:3) Examples
We now give examples of weakly symmetric algebras with silting complexes that are nottilting. The examples are based on those in [D], so we begin by reviewing the necessarydetails from that work.We fix an even integer n ≥ A be the path algebra of the quiver1 x / / y / / x / / y / / · · · x / / y / / n modulo the relations x = y = 0. We write e i for the primitive idempotent of A correspond-ing to vertex i (for 1 ≤ i ≤ n ). As A has finite global dimension, we can identify K b (proj A )with D b (mod A ), and we write S := − ⊗ L A DA for the Serre functor on this category. We let σ ∈ Aut k ( A ) be the order two automorphism induced by the automorphism of Q that fixes each vertex and swaps each pair of x and y arrows. We write σ ∗ for theinduced automorphisms on the categories mod A , K b (proj A ), or D b (mod A ) depending oncontext. We set E = e A/e yA , which is a uniserial module of length n , and note that σ ∗ E ∼ = e A/e xA ≇ E . Proposition 4.1. [D, 4.1] E and σ ∗ E are Hom-orthogonal ( n − -spherical objects in D b (mod A ) . Now E defines a spherical twist functor, which we can apply to A to obtain a tiltingcomplex T that fits into an exact triangle E [1 − n ] n → A → T → E [2 − n ] n (4.A)in D b (mod A ). By applying σ ∗ , and using the fact that σ ∗ A ∼ = A we obtain another triangle σ ∗ E [1 − n ] n → A → σ ∗ T → σ ∗ E [2 − n ] n . (4.B)To get a weakly symmetric algebra, we can form the twisted trivial extension of A usingthe automorphism σ . Thus we define Λ := T σ A = A ⋉ σ DA , where the latter denotes theusual bimodule extension of A by the bimodule σ DA . The idempotents e i of A induce acomplete set of primitive orthogonal idempotents ( e i ,
0) of Λ, which we will continue to writeas e i . In general, by [G1, Prop. 2.2] the Nakayama automorphism ν of T σ A is given by ν ( a, f ) = ( σ ( a ) , f σ − ) . (4.C)In particular, since σ fixes the idempotents e i of A , we see also that ν ( e i ) = e i for all i , andthus Λ is weakly symmetric. The quiver and relations of a twisted trivial extension can becomputed as described in [G2, § x / / y / / x / / y / / · · · x / / y / / n u f f v x x with relations x = y = 0 and xv = ux = yu = vy = 0, together with additional relationsexpressing equality of the two (remaining) nonzero paths of length n at each vertex: for0 ≤ r < q := n/ xy ) r v ( xy ) q − r − x = ( yx ) r u ( yx ) q − r − y and ( xy ) r xu ( yx ) q − r − = ( yx ) r yv ( xy ) q − r − . Furthermore, we can see from (4.C) that ν swaps each pair x and y (of parallel arrows),while also swapping u and v . For n = 4, the tilting complex T ∈ K b (proj A ) from (4.A) isdescribed in [D]. The corresponding complex T ⊗ A Λ ∈ K b (proj Λ) will look the same, butwith each A replaced by Λ. Its indecomposable summands are as follows, where we indicatethe degree-0 term by underlining it:0 / / / / e Λ y / / e Λ y / / e Λ / / / / e Λ (cid:18) xy (cid:19) / / ( e Λ) ( y ) / / e Λ y / / e Λ / / / / e Λ (cid:18) yxy (cid:19) / / e Λ ⊕ e Λ ( y ) / / e Λ y / / e Λ / / / / e Λ (cid:18) xyxy (cid:19) / / e Λ ⊕ e Λ ( y ) / / e Λ y / / e Λ / / ν , it is not a tiltingcomplex. However, it is silting. Proposition 4.2.
Let
A, σ and T be as above, and let Λ = T σ A . Then T ⊗ A Λ is a siltingcomplex in K b (proj Λ) that is not tilting. Proof.
The proof is similar to Rickard’s that T ⊗ A T A is a tilting complex over the trivialextension algebra
T A for any tilting complex T over A [Ri1]. We begin by noting that T ⊗ A Λ generates K b (proj Λ). This can be seen using that T generates K b (proj A ) and − ⊗ A Λ : K b (proj A ) → K b (proj Λ) is an exact functor of triangulated categories taking A to Λ. It remains to show thatHom K b (Λ) ( T ⊗ A Λ , T ⊗ A Λ[ i ]) = 0 for all i > . To this end, observe that for all i = 0Hom K b (Λ) ( T ⊗ A Λ , T ⊗ A Λ[ i ]) ∼ = Hom K b ( A ) ( T, T ⊗ A Λ[ i ])= Hom K b ( A ) ( T, T [ i ] ⊕ T ⊗ A σ DA [ i ]) ∼ = Hom K b ( A ) ( T, T [ i ]) ⊕ Hom K b ( A ) ( T, ( σ ∗ ) − S T [ i ]) ∼ = 0 ⊕ Hom K b ( A ) ( σ ∗ T, S T [ i ]) ∼ = D Hom K b ( A ) ( T, σ ∗ T [ − i ]) , where the penultimate isomorphism is from the fact that T A is a tilting complex and thelast is by Serre duality. Thus it suffices to show that Hom K b ( A ) ( T, σ ∗ T [ j ]) = 0 for all j < b (proj A ), and so we willomit the corresponding subscripts in our Hom-spaces. Applying Hom( − , σ ∗ E [ j ]) to (4.A)and using the fact that E and σ ∗ E are Hom-orthogonal, we get isomorphismsHom( T, σ ∗ E [ j ]) ∼ = Hom( A, σ ∗ E [ j ]) (4.D)for all j , and the latter vanishes for all j = 0 since the homology of σ ∗ E is concentrated indegree 0. Now we apply Hom( T, − ) to (4.B), which yields isomorphismsHom( T, A [ j ]) ∼ = Hom( T, σ ∗ T [ j ])for all j = n − , n −
1. In particular, for all j <
0, we have Hom(
T, σ ∗ T [ j ]) ∼ = Hom( T, A [ j ]).We now show that Hom( T, A [ j ]) = 0 for j <
0. Apply Hom( − , A [ j ]) to (4.A) to get an exactsequence Hom( E [2 − n ] n , A [ j ]) → Hom(
T, A [ j ]) → Hom(
A, A [ j ]) . (4.E)By Serre duality, the first term is isomorphic to D Hom( A [ j ] , S ( E )[2 − n ] n ) ∼ = D Hom( A [ j ] , E [1] n ) ∼ = D Hom(
A, E [1 − j ] n ) = 0for j = 1. As the last term of (4.E) vanishes for j = 0, we see that Hom( T, A [ j ]) = 0 for all j <
0, as required (in fact, for all j = 0 , T, σ ∗ T [ j ]) = 0 for all j < T ⊗ A Λ is a siltingcomplex. While, we can see that T ⊗ A Λ is not a tilting complex since it is not invariantunder the Nakayama functor of Λ, which switches x and y , we also provide a direct proofby showing that it has nonzero self-extensions in degree 2 − n .Applying Hom( T, − ) to (4.B), and using Hom( T, A [ j ]) = 0 for j = 0 ,
1, and then (4.D),gives Hom(
T, σ ∗ T [ n − ∼ = Hom( T, σ ∗ E n ) ∼ = Hom( A, σ ∗ E n ) ∼ = σ ∗ E n . ThusHom K b (Λ) ( T ⊗ A Λ , T ⊗ A Λ[2 − n ]) ∼ = D Hom(
T, σ ∗ T [ n − ∼ = D ( σ ∗ E ) n = 0 . (cid:3) As a consequence of Proposition 3.8, the algebra Λ is not tilting-discrete. In fact, com-bining with Corollary 3.3, we see that it is not even weakly silting connected. We concludeby pointing out another interesting property of the silting complex T , which follows fromPropositions 3.1 and 2.5, and to our knowledge has not been observed in other examples. Corollary 4.3.
For Λ and T as defined above, T is a silting complex which is not connectedto any tilting complex by iterated silting mutations. References [AK] T. Adachi and R. Kase,
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