
TL;DR
This paper provides examples of finite-dimensional algebras where silting objects are disconnected via silting mutations, highlighting limitations in silting mutation connectivity.
Contribution
It introduces specific algebras with silting objects that cannot be connected through silting mutations, expanding understanding of silting theory limitations.
Findings
Silting objects in certain algebras are not connected by silting mutations.
Invariance under algebra automorphisms affects silting mutation connectivity.
Existence of spherical modules not invariant under twists impacts silting structures.
Abstract
We give examples of finite-dimensional algebras for which the silting objects in are not connected by any sequence of (possibly reducible) silting mutations. The argument is based on the fact that silting mutation preserves invariance under twisting by a fixed algebra automorphism, combined with the existence of spherical modules that are not invariant under such a twist.
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Some algebras that are not silting connected
Alex Dugas
Department of Mathematics, University of the Pacific, 3601 Pacific Ave, Stockton CA 95211, USA
Abstract.
We give examples of finite-dimensional algebras for which the silting objects in are not connected by any sequence of (possibly reducible) silting mutations. The argument is based on the fact that silting mutation preserves invariance under twisting by a fixed algebra automorphism, combined with the existence of spherical modules that are not invariant under such a twist.
1. Introduction
In [2], Aihara and Iyama develop the theory of mutation for silting subcategories of a triangulated category. One of the principal settings in which silting mutation is of interest is the category of perfect complexes over a finite-dimensional algebra . Here, silting complexes provide a nice generalization of tilting complexes that are better behaved under mutation. In particular, it is always possible to mutate a silting complex at any one of its direct summands to obtain a new silting complex; whereas the same is not necessarily true when one considers only tilting complexes. The action of irreducible silting mutation on the set of silting objects in can be visualized by the silting quiver of , which also coincides with the Hasse diagram for a natural partial order on the set of silting objects.
Aihara and Iyama propose the problem of determining which algebras have a connected silting quiver. Such algebras have come to be termed silting connected [1]. While several classes of algebras – including representation finite symmetric algebras, local algebras and piecewise hereditary algebras – are known to be silting connected, less is known about which algebras fail to be silting connected. Two examples of symmetric algebras that are not silting connected, one originally discovered by Aihara, Grant and Iyama, appear in [5, §6.3]. However, in each example the silting objects are all linked by combinations of irreducible silting mutation and powers of the suspension, which is in fact a reducible silting mutation corresponding to the zero summand of the silting object. Aihara and Iyama state that they are aware of no algebras where (not necessarily irreducible) iterated silting mutations do not act transitively on the set of basic silting objects in . This sparsity of known examples is likely due to the difficulty in showing that no possible sequence of mutations can change one silting complex into another; as opposed to such examples being uncommon.
The purpose of this note is to present a family of examples where iterated silting mutation does not act transitively on the set of silting objects in . In fact, we see that the silting quivers in our examples will have infinitely many connected components, even if one includes edges for reducible silting mutations. Our proof that these algebras are not silting connected makes use of a rather elementary observation (Proposition 2.1), which says that if each indecomposable object in a silting subcategory is invariant under an automorphism of the ambient triangulated category , then the same is true for any mutation of . We apply this fact to the algebras under consideration by showing that they admit spherical modules, and hence tilting complexes associated to the corresponding spherical twists, that are not invariant under such an automorphism.
2. Silting Mutation
In this section we review the definition of silting mutation in a triangulated category due to Aihara and Iyama [2], and show that the class of silting subcategories in which every object is invariant under a fixed automorphism of is stable under mutation.
Throughout this section will be a triangulated category, with suspension functor denoted . When we speak of a subcategory of we shall always mean a strict, full subcategory closed under finite direct sums and direct summands. Recall that a subcategory of is silting if and generates as a triangulated category. We let denote the collection of all silting subcategories of . If is a covariantly (resp. contravariantly) finite subcategory of , then the left (resp. right) mutation of at is defined as the subcategory
[TABLE]
where (resp. ) is defined as the cone (resp. co-cone) of a left (resp. right) -approximation of . In other words, and are defined by distinguished triangles of the form
[TABLE]
where and and are left and right -approximations, respectively. An important point here is that these mutated subcategories do not depend on the choices of -approximations in their definition.
Aihara and Iyama show that for any silting subcategory and any covariantly (resp. contravariantly) finite subcategory , the mutation (resp. ) is also silting and satisfies
[TABLE]
In addition, if is a Krull-Schmidt triangulated category, then Aihara and Iyama define irreducible silting mutations of a silting subcategory as those mutations with respect to a subcategory for which contains a unique indecomposable object (up to isomorphism). As usual, we write for the set of isomorphism classes of indecomposable objects in a category . Under the assumption
[TABLE]
irreducible left and right silting mutations are defined for all silting subcategories of . This assumption holds, in particular, if is a Hom-finite -category (over a field ) and has a silting object. In this case, we abbreviate by for indecomposable objects in .
Finally, recall the definition of the silting quiver of , under the assumption (F). Its vertices are the (equivalence classes of) basic silting objects in , and each has arrows to for each . Thus, two silting objects are in the same connected component of this quiver if and only if they are linked by iterated irreducible (left or right) silting mutation. We point out that the silting quiver of only conveys information about irreducible silting mutation in , and the silting connectedness of usually refers to the connectedness of this quiver, i.e., to the connectedness of under irreducible mutation. In this paper, however, we are really interested in the connectedness of with respect to all mutations. Thus we will consider the extended silting quiver of , obtained from by adding arrows for each silting object and each subcategory for a basic direct summand of .
Now consider an automorphism of . We say that is -invariant if . We say that a subcategory of is -invariant if each object is -invariant.
Proposition 2.1**.**
Let be a Krull-Schmidt category and let be a silting subcategory of that is -invariant. Then for any covariantly (resp. contravariantly) finite subcategory of the silting subcategory (resp. ) is also -invariant.
Proof.
We give the proof for left silting mutation only, as the other half of the argument is dual. Consider a triangle
[TABLE]
with , and a minimal left -approximation. Since is -invariant, applying yields another minimal left -approximation
[TABLE]
Since is -invariant there is an isomorphism , and uniqueness of minimal approximations implies the existence of an isomorphism making the diagram commute
[TABLE]
Thus there is an induced isomorphism from to as required. Since any arises in this way (by choosing a right -approximation of to get a left -approximation of with cone ), we see that each indecomposable in is -invariant. ∎
As a consequence, we obtain the following criterion for to fail to be silting connected.
Corollary 2.2**.**
Assume that has an -invariant silting subcategory and another silting subcategory that is not -invariant. Then cannot be obtained from by iterated silting mutation. In particular, the action of iterated silting mutation on is not transitive.
3. Spherical modules
In this section we develop some simple properties of spherical modules in the derived category of a finite-dimensional algebra of finite global dimension. First recall that if is a Hom-finite triangulated -category with Serre functor , then an object in is -spherical if and
[TABLE]
We let be a finite-dimensional -algebra of finite global dimension, and write for the category of finitely-generated right -modules. We will focus on the case where is the bounded derived category , which we may also abbreviate as . Recall that has a Serre functor where denotes the duality .
Associated to any -spherical object in , Seidel and Thomas have defined an exact auto-equivalence of , known as a spherical twist [13]. For any in we can compute as the cone of the natural evaluation map in the distinguished triangle below
[TABLE]
One easily checks that . Additionally, because is an auto-equivalence will be a tilting complex with endomorphism ring isomorphic to . In order to say more about , we now assume that , i.e., that is a spherical module. We also let be a complete set of pairwise orthogonal primitive idempotents for .
Lemma 3.1**.**
Assume the -module is -spherical in . Then for all and each with we have isomorphisms
[TABLE]
Proof.
We prove the claim by induction on . For , we have by Serre duality
[TABLE]
Now one uses that .
Now assume the claim holds for some . We have
[TABLE]
which vanishes unless , or equivalently , in which case it is isomorphic to as a -vector space. The proof for is similar. ∎
Next we describe the homology of the iterated spherical twists of each indecomposable projective .
Lemma 3.2**.**
Assume that is -spherical in for some . Then for all with and each with we have
[TABLE]
Proof.
We again argue by induction on . For , the claim is trivial. Now assume that it holds for some , and consider the triangle used to define
[TABLE]
The corresponding long exact sequence in homology (obtained by applying ) shows that
[TABLE]
for all . For the remaining , we know that vanishes by the induction hypothesis (this is where we need ). Thus
[TABLE]
which is either or [math] depending on whether or not. ∎
4. Examples
The goal of this section is to describe concrete examples of finite-dimensional algebras over a field for which iterated silting mutation does not act transitively on the set of (equivalence classes of) silting objects in . To this end, we fix an integer and let be the path algebra of the following quiver
[TABLE]
modulo the relations . We write for the primitive idempotent of corresponding to vertex (for ), and and for the corresponding simple, indecomposable projective and indecomposable injective right -modules, respectively. It is not hard to see that .
We let be the order two automorphism induced by the automorphism of that fixes each vertex and swaps each pair of and arrows. We view as acting on on the right, so that it induces an automorphism of (acting on the left). By definition, sends a right -module to the twisted module which equals as an abelian group and has -action given by for all and all . Since as sets for any module , we can define (as functions) for any morphism . This action restricts to an automorphism of and hence also induces automorphisms of and , which we continue to write as . Since for all , it is clear that each indecomposable projective is -invariant.
We set , which is a uniserial module of length . Note that , so is not -invariant. When we consider we may identify with its minimal projective resolution
[TABLE]
which has in degree [math].
Proposition 4.1**.**
- (1)
If is even, then and are Hom-orthogonal -spherical objects in . 2. (2)
If is odd, then is exceptional in and .
Proof.
To compute we tensor the projective resolution of with to obtain the complex
[TABLE]
with where denotes the map given by right multiplication by . A simple calculation shows that maps the left branch of to [math] and maps the right branch of onto the left branch of as in the figure:
[TABLE]
It is clear that this complex is exact, except in degree , where its homology is . This is a length uniserial module with a length two submodule annihilated by . If is even, this uniserial module is isomorphic to , while if is odd it is isomorphic to . Hence
[TABLE]
To compute we apply to the projective resolution of . Note that is one-dimensional for each . Moreover, the induced map
[TABLE]
can be identified with the map given by right multiplication by from to . Thus this map is an isomorphism if is even or else the zero map when is odd. It follows that the complex has nonzero homology (isomorphic to ) only in degree [math] if is odd, or else only in degrees [math] and if is even. Similarly if we apply to , we get a complex of one-dimensional vector spaces with maps corresponding to right multiplication by from to . This time these maps are isomorphisms whenever is odd and zero otherwise. Thus we see that for all in case is even. For completeness, we also note that if is odd, we have for all , while . ∎
Remark 4.2**.**
When is even, is the Beilinson algebra (see [4]) of the dihedral algebra
[TABLE]
with the usual grading that places and in degree . This connection provides another way to see that is a spherical object and to study the action of the spherical twist using results of [6, 7]. We will elaborate in the Appendix.* *
Remark 4.3**.**
When is odd, part (b) of the above Proposition shows that is an exceptional 2-cycle, following the terminology of [3], with . It is shown in [3] that the twist with respect to the direct sum of objects in an exceptional cycle is an auto-equivalence of . However, since is -invariant, this twist will only produce other -invariant tilting complexes.**
Now assume that is even. As we saw in the previous section, there is an auto-equivalence of . Naturally, it will induce an automorphism of the silting quiver of , which takes to the tilting complex . Likewise applying powers of (or its quasi-inverse) to yields tilting complexes for each .
Proposition 4.4**.**
Let and be as above, with even. Then for any distinct integers the tilting complexes and are not connected by iterated silting mutation. In particular, the extended silting quiver of has infinitely many connected components.
Proof.
Since is one-dimensional for each , by Lemma 3.2 we have , which is not -invariant. Thus, by Corollary 2.2, is not connected to by iterated silting mutation. Similarly, no can be connected to for . Consequently, for any , the tilting complexes and are not connected by iterated silting mutation. For if they were, applying would yield a way of connecting and by iterated silting mutation, which we know is not possible. ∎
By Proposition 4.1, and are Hom-orthogonal spherical objects. Hence, by [13], the equivalences and commute. Moreover, it is easy to see that . We further note that is again -invariant. However, we do not know if it is connected to by iterated (irreducible) silting mutation. One can check further that in , where denotes the Auslander-Reiten translation. We verify this for by a direct calculation below, and justify it more generally in the Appendix.
Example 4.5**.**
To give a more concrete illustration of some of these tilting complexes which are not connected by silting mutation, we now describe the tilting complexes and when . For each , can be described as a mapping cone of a map . In the complexes below, we indicate the degree-[math] term by underlining it.**
[TABLE]
Each complex has homology isomorphic to in degree [math] and to in degree . In particular they are not -invariant.**
Next we compute of each of the above complexes. For each , is realized as the mapping cone of the unique (up to a scalar multiple) map .**
[TABLE]
Each of the above indecomposable complexes has homology isomorphic to in degree [math] and to in degree . Furthermore, one can check that each is -invariant. We do not know if the corresponding tilting complex is connected to via silting mutation. However, as mentioned earlier, we can observe that gives the injective coresolution of . Hence**
[TABLE]
Question 1**.**
For the algebra above, is connected to by (irreducible) silting mutation?
The same idea we have used here to show that the algebra is not silting connected can also be applied to show that its trivial extension is not tilting connected (note that every silting complex in is tilting since is a symmetric algebra). The quiver of is obtained from the quiver of by adding two arrows, also labeled and , from vertex back to vertex . In addition to the relations and of , which now extend to include the new arrows as well, also has the relations at each vertex. In particular, there is an order-two automorphism of that swaps and at each vertex. Abusing notation, will continue to write for this automorphism and for the induced automorphisms of and .
Rickard [12] has shown that if is a tilting complex in for any finite-dimensional -algebra , then is a tilting complex in . Moreover, the endomorphism ring of can be identified with the trivial extension of . Thus, for each , the induced tilting complex is not -invariant, and hence not connected to through any sequence of tilting mutations, irreducible or otherwise. Unfortunately, here it is not obvious whether each of these tilting complexes lies in a different connected component of the tilting quiver of . For, while Rickard’s results show that each induces an auto-equivalence of , we do not know whether for each , and consequently we do not know if . One may check that it is no longer the case that these auto-equivalences are spherical twists. One might wonder if the , being derived auto-equivalences of symmetric algebras, could be instances of periodic twists studied by Grant [8]. However, this also appears to not be the case, since Grant shows that a periodic twist factors as a sequence of tilts by 2-term Okuyama-Rickard complexes [8, Theorem B]. Consequently, any tilting complex associated to a periodic twist must be connected to via tilting mutation. We are thus not aware of whether the auto-equivalences of have any characterization intrinsic to this derived category that does not rely on tilting complexes over .
Finally, we recall that for any finite-dimensional -algebra , Koenig and Yang [11] have established mutation preserving bijections between (equivalence classes of) silting complexes in , simple-minded collections in , bounded t-structures of with length heart, and bounded co-t-structures of . Thus, the algebras as in Proposition 4.4 and their trivial extensions yield examples of algebras for which mutation does not act transitively on simple-minded collections or on bounded t-structures with length heart in , or on bounded co-t-structures in .
5. Appendix: Connection to dihedral algebras
The examples of spherical objects presented in this article were not discovered randomly, but rather correspond naturally to certain spherical stable twists introduced in [6]. While this correspondence is not necessary in the above exposition, it does present an alternative means of computing the actions of the spherical twist functors and also illustrates how further examples may be found. For these reasons, we believe it is worthwhile to provide more details about this connection.
We start by reviewing some general results of Happel and Chen, which we shall need. For a -graded algebra , we write for the category of finitely generated -graded right -modules and degree preserving morphisms, and for the associated stable category obtained by factoring out the ideal of morphisms that factor through a projective module. For graded modules and , we will write (resp. ) and for the spaces of degree-[math] morphisms (resp. stable morphisms) and degree-[math] extensions. If is concentrated in degrees [math] through (with ), Chen defines the Beilinson algebra of to be the matrix algebra
[TABLE]
Furthermore, is well-graded if and are nonzero for each primitive idempotent . We note that has finite global dimension if and only if does. Finally, we note that one way of producing well-graded self-injective (in fact, symmetric) algebras is as trivial extensions. If is any algebra, we will regard its trivial extension as a graded algebra with in degree [math] and in degree .
Theorem 5.1**.**
[4, Theorem 1.1]** For a well-graded self-injective algebra , there is an equivalence of categories . (However, this equivalence typically does not commute with the grading shift.)
Now, combining with this Happel’s theorem which states that for any algebra of finite global dimension there is an equivalence of triangulated categories , we obtain the following.
Corollary 5.2**.**
[4, Corollary 1.2]** Let be a well-graded self-injective algebra such that has finite global dimension. Then we have equivalences of triangulated categories
[TABLE]
We continue to assume that is a well-graded self-injective algebra such that has finite global dimension, and we set . Each of the above categories has a Serre functor, and by uniqueness of Serre functors, these equivalences commute with these Serre functors up to natural isomorphism. For , the Serre functor is given by . On , the Serre functor is given by where is the Nakayama functor. Observe that with in degree [math] and in degree . Hence as graded -bimodules. Consequently, is isomorphic to the grading shift functor on . Hence the Serre functor on is isomorphic to . Passing to , the Nakayama functor will correspond to the grading shift , which we note is not the Nakayama functor of , while the Serre functor on must be isomorphic to .
In [9], Guo uses covering theory to describe the relationship between and more concretely. Nameley, he defines an algebra as the orbit algebra of the category with respect to the Nakayama automorphism. If is symmetric, its Nakayama automorphism coincides with the grading shift , and hence in this case the algebra is a Galois covering of with group . Thus may also be realized as a smash product with respect to the natural grading on viewed as a -grading. By [9, Theorems 5.8, 5.1], is isomorphic to a twisted trivial extension of . If is also symmetric, it must be isomorphic to the (untwisted) trivial extension .
Thus, in case and are both symmetric, writing for the equivalence , we have
[TABLE]
for all .
We now return to the notation used earlier in the paper. In particular is an even integer, is the algebra introduced in Section 4, and
[TABLE]
is a local dihedral algebra. Note that can be graded by placing and in degree . With this grading, is a well-graded symmetric algebra concentrated in degrees [math] through . By Theorem 5.1 of [9], it is easy to see that is the Beilinson algebra of . Moreover, here and are both symmetric, so we see that is a Galois covering of with Galois group , and the categories of -graded modules over and are equivalent, with the grading shift over corresponding to the power of the grading shift over .
An object is -Calabi-Yau if and only if , or equivalently, . Equivalently, if we write for the corresponding object in , we see that is -Calabi-Yau if and only if . The -spherical object in corresponds to the module in . It is easy to see directly that as graded -modules, and hence that is -Calabi-Yau. Furthermore, it is straightforward to verify that where is the unique (up to a scalar multiple) degree endomorphism of . Consequently, in we have
[TABLE]
We denote the corresponding spherical twist in by . In [6], we defined a spherical stable twist which is an auto-equivalence of the (ungraded) stable category . As both and are defined using cones of right -approximations, it is clear that their actions agree on graded -modules. Likewise, the spherical object corresponds to , and the spherical twist on can be viewed as a graded version of the spherical stable twist on . Now, as coincides with on objects [6, Example 7.1], it follows that also coincides with on objects in . Carrying this information back to , we see that also coincides with on objects. While we do not know if this gives a functorial factorization of , the fact that similar factorizations have appeared elsewhere (see [3, Cor. 5.5] for instance) suggests that something deeper may underlie this phenomenon.
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