Yoneda algebras and their singularity categories
Norihiro Hanihara

TL;DR
This paper studies the properties of Yoneda algebras and their singularity categories for finite dimensional algebras, revealing their Gorenstein nature and establishing equivalences with derived categories of stable categories.
Contribution
It introduces the Yoneda category and proves its coherence and Gorenstein properties, establishing new triangle equivalences for both finite and infinite representation type algebras.
Findings
Yoneda algebra is graded coherent and Gorenstein with self-injective dimension ≤ 1.
Singularity category of Yoneda algebra is triangle equivalent to the derived category of the stable Auslander algebra.
The Yoneda category's singularity category is equivalent to the derived category of the stable module category.
Abstract
For a finite dimensional algebra of finite representation type and an additive generator for , we investigate the properties of the Yoneda algebra . We show that is graded coherent and Gorenstein of self-injective dimension at most , and the graded singularity category of is triangle equivalent to the derived category of the stable Auslander algebra of . These results remain valid for representation-infinite algebras. For this we introduce the Yoneda category of as the additive closure of the shifts of the -modules in the derived category . We show that is coherent and Gorenstein of self-injective dimension at most , and the singularity…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology
Yoneda algebras and their singularity categories
Norihiro Hanihara
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan
Abstract.
For a finite dimensional algebra of finite representation type and an additive generator for , we investigate the properties of the Yoneda algebra . We show that is graded coherent and Gorenstein of self-injective dimension at most , and the graded singularity category of is triangle equivalent to the derived category of the stable Auslander algebra of . These results remain valid for representation-infinite algebras. For this we introduce the Yoneda category of as the additive closure of the shifts of the -modules in the derived category . We show that is coherent and Gorenstein of self-injective dimension at most , and the singularity category of is triangle equivalent to the derived category of the stable category . To give a triangle equivalence, we apply the theory of realization functors. We show that any algebraic triangulated category has an f-category over itself by formulating the filtered derived category of a DG category, which assures the existence of a realization functor.
Key words and phrases:
Yoneda algebra, singularity category, Cohen-Macaulay module, stable category, derived category, tilting theory, realization functor
2010 Mathematics Subject Classification:
16E35, 16G50, 16E65, 16P70, 18E30
This work is supported by JSPS KAKENHI Grant Number JP19J21165
1. Introduction
Yoneda algebras form a class of algebras which has long been studied in ring theory and representation theory. They are defined, for a ring and a -module , by as an abelian group. It has a structure of a graded ring given by the Yoneda product, that is, the concatenation of Yoneda classes of long exact sequences. Usually, Yoneda algebras are studied for the case is a semisimple module, for example, in Koszul duality [BGG, BGSo, Ke2, Ke3, GM1, GM2], and cohomology rings of finite groups in modular representation theory [Ben].
The subject of this paper, on the other hand, is to investigate the properties of Yoneda algebras arising from an additive generator . Although there are some interesting results in this setup for a construction of derived equivalences [HX], only few studies on such have been done. We will give some fundamental ring theoretic (Theorem 1.3) and representation theoretic (Theorem 1.5) properties of these Yoneda algebras. Our fully general results will be summarized in Theorem 1.6.
Let us state our setup precisely. Throughout we fix a field .
Definition 1.1**.**
Let be a finite dimensional -algebra of finite representation type, and an additive generator. We call
[TABLE]
the Yoneda algebra of , which is uniquely determined by up to graded Morita equivalence.
Clearly is finite dimensional if and only if has finite global dimension, but in general is even far from being Noetherian. On the other hand, we have the following notion which generalizes Noetherian property.
Definition 1.2**.**
Let be a graded ring.
- (1)
We say that is left (resp. right) graded coherent if the category (resp. ) of finitely presented graded left (resp. right) -modules is abelian. We say that is graded coherent if it is left and right graded coherent. 2. (2)
A graded coherent ring is called -Gorenstein if it has injective dimension at most in and in .
Clearly, any Noetherian graded ring is graded coherent. The notion of Gorenstein rings is important in ring theory [Ma, EJ] and algebraic geometry [Har], and our definition is an adaptation to coherent rings. Our first main result shows that our Yoneda algebras enjoy nice ring theoretic and homological properties that classical ones do not.
Theorem 1.3**.**
The Yoneda algebra of in Definition 1.1 is graded coherent and -Gorenstein.
The next aim of this paper is to study the representation theory of . Recall that the (graded) singularity category of a graded ring is the Verdier quotient of the bounded derived category by the perfect derived category , which was introduced by Buchweitz in [Bu] and rediscovered by Orlov in the context of mirror symmetry [O]. Over Gorenstein rings, the singularity category has another presentation as the stable category of Cohen-Macaulay modules.
Definition 1.4**.**
Let be a graded coherent Gorenstein ring. A graded module is Cohen-Macaulay if it is finitely presented and satisfies for all .
These Cohen-Macaulay modules over coherent Gorenstein rings are sometimes called Gorenstein-projective (or totally reflexive) modules [AB, EJ] which are defined over arbitrary rings. For a graded Gorenstein ring , we denote by the category of graded Cohen-Macaulay -modules. It is a Frobenius category and hence the stable category has a natural structure of a triangulated category [Hap]. Moreover, there exists a canonical equivalence of triangulated categories [Bu, KV, Ric], by which we will identify these categories. The studies of Cohen-Macaulay modules over Gorenstein rings have attracted enormous attention [CuR, Yo, Si, LW, I3] and recently, various Gorenstein algebras have been discovered and their Cohen-Macaulay representation theory is investigated, for example, in [AIR, BIRS, BIY, DL, GLS, IO, IT, JKS, KST, KMV, KR, Ki1, LZ, MU, MYa, SV, U, Ya].
The second main result of this paper is the following equivalence between the singularity categories of Yoneda algebras and derived categories.
Theorem 1.5**.**
Let be the Yoneda algebra of in Definition 1.1, and the stable Auslander algebra of . Then there exists a triangle equivalence
[TABLE]
which takes to , and to . Here, and .
In fact, the above results can be generalized for arbitrary finite dimensional algebras which are not necessarily representation-finite: we consider the Yoneda category
[TABLE]
which is the full subcategory of the derived category of . This is a categorical analogue of the Yoneda algebra of (see Proposition 2.11). As in the case of algebras, we have the notion of coherence, Gorenstein property, and singularity categories for categories, and Cohen-Macaulay modules over Gorenstein categories, see Definitions 2.4, 2.7, and 2.9. The following is our general main result which contains Theorems 1.3 and 1.5.
Theorem 1.6**.**
Let be an arbitrary finite dimensional algebra and be the Yoneda category of .
- (1)
(Lemma 3.2, Theorem 4.1)* is coherent.* 2. (2)
(Theorem 3.1, 4.3)* is -Gorenstein.* 3. (3)
(Theorem 3.8, 4.23)* There exists a triangle equivalence*
[TABLE]
The equivalence (3) is given in [Ki2] (see also [IO, 4.11]) for the very special case is hereditary. We have two independent strategies to build the triangle equivalence above. In both cases, we depend on the fact that is an algebraic triangulated category, that is, the stable category of a Frobenius category.
The first one, which is valid for the restrictive case is finite, is tilting theory [AHK]. The existence of a tilting object in an algebraic triangulated category implies a triangle equivalence under a mild assumption (see Theorem 2.13). We have the following reformulation of Theorem 1.5 for the case . Here we state the representation-finite case for simplicity, see Theorem 3.8 for the general version.
Theorem 1.7** (Theorem 3.8).**
Let be a representation-finite algebra of finite global dimension, and let be the Yoneda algebra of in Definition 1.1. Then, we have the following.
- (1)
* is a tilting object.* 2. (2)
* is the stable Auslander algebra of , and has finite global dimension.*
Consequently, there exists a triangle equivalence .
Note that in general there is no tilting object in if is infinite. We therefore need the second strategy which works for arbitrary , namely realization functors [BBD, PV]. Let be a triangulated category endowed with a -structure and let be its heart, which is an abelian category [BBD]. A triangle functor extending the inclusion is called a realization functor. In the appendix we formulate the filtered derived category of a DG category, which assures the existence of a realization functor for algebraic triangulated categories (Corollary A.9). We refer to [KV, Ke1, ChR, M] for different approaches.
To prove Theorem 1.5, we first show that the triangulated category has a -structure, then describe its heart, and finally prove that the realization functor is an equivalence. The results are given as follows, which explain the equivalence in Theorem 1.5. Again, we state them for the representation-finite case for simplicity.
Theorem 1.8**.**
Let be an arbitrary representation-finite algebra and its Yoneda algebra in Definition 1.1.
- (1)
(Theorem 4.11)* Set*
[TABLE]
Then, is a -structure on . 2. (2)
(Proposition 4.17)* The heart of is equivalent to for the stable Auslander algebra of .* 3. (3)
(Theorem 4.23)* The realization functor is a triangle equivalence.*
One can see that under the equivalence in (3), the -structure in (1) becomes the standard one on .
Now we explain the structure of this paper. Section 2 is a preparatory section where we recall some basic concepts on functor categories and tilting theory. Sections 3 and 4 are central ingredients. We treat the case in Section 3, where tilting theory is applied. The general case is treated in Section 4 via realization functors. Note that all the results (except those in Section 3.4) in Section 3 follow from those in Section 4, but we give separate proofs since the arguments in Section 3 are much simpler and also motivate some constructions in Section 4. In Section 5, we discuss the special case . Finally, we give examples in Section 6. In the appendix, we give a concrete description of the filtered derived category of a DG category and give a proof of the existence of a realization functor for algebraic triangulated categories.
Acknowledgement
The author is grateful to his supervisor Osamu Iyama for valuable suggestions and discussions.
2. Preliminaries
2.1. Functor categories
We recall in this subsection the basic concepts on functor categories which will be used throughout this paper.
Let us first fix some notations and conventions. The composition of morphisms is denoted by . A module over a preadditive category means a contravariant additive functor from to the category of abelian groups. In particular, a module over a ring means a left module.
When is essentially small, we denote by the abelian category of modules over , by the -spaces, and by the -spaces in . A -module is finitely presented (resp. finitely generated) if there exists an exact sequence (resp. ) for some . We denote by the full subcategory of consisting of finitely presented modules.
The following notion, introduced in [AR1], provides a fundamental class of categories over which one can develop the representation theory just like over finite dimensional algebras.
Definition 2.1**.**
Let be a field and let be a -linear, -finite category. We say that is a dualizing -variety if induces a duality .
The most basic example of a dualizing variety is the category of finitely generated projective modules over a finite dimensional algebra . If is a dualizing variety, then is an abelian category with enough projectives and injectives. By Yoneda’s lemma, the projective objects are given by the representable functors for , and by the duality, the injective objects are given by for .
For and a full subcategory , we write
[TABLE]
the restricted functors, even if . We recall the following notion from [AS].
Definition 2.2**.**
Let be an additive category and a full subcategory.
- (1)
A morphism in is a right -approximation of if and it induces an exact sequence in . We say is contravariantly finite in if any has a right -approximation. 2. (2)
A morphism in is a left -approximation of if and it induces an exact sequence in . We say is covariantly finite in if any has a left -approximation. 3. (3)
is functorially finite in if it is both contravariantly and covariantly finite.
It is not difficult to see the following, which gives more examples of dualizing varieties.
Proposition 2.3**.**
Any functorially finite subcategory of a dualizing variety is again a dualizing variety.
Let us now recall the following general notion.
Definition 2.4**.**
A category is left (resp. right) coherent if it has weak kernels (resp. weak cokernels). We say is coherent if it is both left and right coherent.
Note that is left (resp. right) coherent if and only if the category (resp. ) is abelian [A]. Therefore dualizing varieties are coherent.
We can speak of the global dimension of a coherent category.
Definition 2.5**.**
The left (resp. right) global dimension of a coherent category is the maximum integer such that on (resp. ). It is if there is no such .
We will often use the following well-known result on abelian categories.
Proposition 2.6** ([A]).**
Any abelian category has global dimension at most (as a coherent category).
We also have the notion of self-injectivity and Gorensteinness of coherent categories.
Definition 2.7**.**
Let be a coherent category. We say is self-injective (resp. -Gorenstein) if any representable -modules and -modules are injective (resp. have injective dimension at most ) in and in .
For example, any triangulated category is self-injective (see [Kr, 4.2]). It is a dualizing variety if and only if it has a Serre functor [IY, 2.11]. We note the following well-known result.
Proposition 2.8** ([Hap]).**
Let be a finite dimensional algebra of finite global dimension. Then, gives a Serre functor on .
Now let us formulate the Cohen-Macaulay modules over Gorenstein categories.
Definition 2.9**.**
Let be a Gorenstein category. We say that is Cohen-Macaulay if for all and .
For a Gorenstein category , we denote by the category of Cohen-Macaulay -modules. Let us collect some basic properties of , which can be shown in a similar way as in the case of Noetherian rings.
Proposition 2.10**.**
Let be a -Gorenstein category.
- (1)
* is Cohen-Macaulay if and only if is a -th syzygy.* 2. (2)
* is a resolving subcategory of , that is, any projective module in belongs to , and is closed under extensions and kernels of epimorphisms.* 3. (3)
* is naturally a Frobenius category.* 4. (4)
The stable category is naturally a triangulated category, and the inverse of its suspension functor is the syzygy functor .
We end this subsection by noting the following fact, which shows that the Yoneda category is indeed a categorical analogue of Yoneda algebras.
Proposition 2.11** (See [Han, 4.1]).**
Let be a category with an automorphism . Suppose there exists such that and set . Then, there exists an equivalence
[TABLE]
such that the action of on commutes with the degree shift on .
2.2. Tilting theory
We recall some basic facts on tilting theory, which have played an essential role in giving a triangle equivalence. For a subcategory of a triangulated category, we denote by the smallest full triangulated subcategory containing and closed under isomorphisms and direct summands.
Definition 2.12**.**
Let be a triangulated category. A full subcategory of is a tilting subcategory if it satisfies the following conditions.
- (1)
for all and . 2. (2)
.
The importance of tilting subcategories are suggested by the following result. We say that an additive category is idempotent-complete if any idempotent morphism has a kernel.
Theorem 2.13** ([Ke2, 4.3]).**
Let be an idempotent-complete algebraic triangulated category with a tilting subcategory . Then, there exists a triangle equivalence
[TABLE]
3. Yoneda algebras for algebras of finite global dimension
We first consider Yoneda categories for algebras of finite global dimension. Note that all the results in this section (except those in Section 3.4) are contained in the corresponding results in Section 4, and the reader can skip this section. The aim of this section is to give a simpler proof of Theorem 1.6 for the special case .
3.1. Gorenstein property
Let be a finite dimensional -algebra of finite global dimension and . Let
[TABLE]
be the Yoneda category of . The aim of this subsection is to prove the following result.
Theorem 3.1**.**
Let be a finite dimensional algebra of finite global dimension and be the Yoneda category of . Then is a -Gorenstein dualizing variety.
We start the proof of Theorem 3.1 with the following observation which is based on [I2, 5.1].
Lemma 3.2**.**
Let .
- (1)
If , then is a right -approximation of . 2. (2)
If , then is a left -approximation of .
Consequently, is functorially finite, and hence is a dualizing variety.
Proof.
We show (1). Since , any morphism to in can be presented by a morphism of complexes. Therefore, the natural map gives a right -approximation of . The last assertion follows from Proposition 2.3. ∎
A closer look at the above proof shows the following proposition.
Proposition 3.3**.**
For any , the triangle induces a projective resolution
[TABLE]
Therefore, has projective dimension at most .
Proof.
Since , we may assume . By Lemma 3.2(1), we have a right -approximation . This is a monomorphism in and there exists an exact sequence
[TABLE]
in , which induces a triangle
[TABLE]
in . Now, since is stable under , the morphism is also a right -approximation. Therefore, the above triangle yields a short exact sequence
[TABLE]
Since , this proves the assertion. ∎
We can now prove the main theorem of this subsection.
Proof of Theorem 3.1.
We have already seen in Lemma 3.2 that is a dualizing variety. Then the injective modules in are given by for each , which is isomorphic to by the Serre duality. It has projective dimension by Proposition 3.3. Similarly, any projective modules have injective dimension at most . ∎
3.2. Cohen-Macaulay modules
We keep the notations from the previous subsection. Our next aim is to study the category of Cohen-Macaulay -modules.
In the rest, we denote by the suspension functor on a triangulated category . The suspension functor on restricts to an automorphism of , and induces an automorphism of and of , which is also denoted by .
We have the following description of Cohen-Macaulay -modules.
Proposition 3.4**.**
- (1)
Let . Then, if and only if there exists a triangle
[TABLE]
such that and . 2. (2)
We have an isomorphism of functors on .
We need the following observation for the proof.
Lemma 3.5**.**
Let . Then, is projective if and only if .
Proof.
We only have to show the ‘only if’ part. Assume is projective. Then, the projective resolution given in Proposition 3.3 splits. It follows that is a split monomorphism in , hence so is in by Yoneda’s lemma. Therefore, the triangle in splits and . ∎
Proof of Proposition 3.4.
(1) If there exists such a triangle, then is finitely presented and is a submodule of a projective -module , therefore since is -Gorenstein. We next show the ‘only if’ part. Assume and let be a projective presentation of . Complete to a triangle . We want to show that . Consider the following exact sequence in :
[TABLE]
where and . We have . Indeed, is just , and is the second syzygy of . Therefore, . Then, is a Cohen-Macaulay module of finite projective dimension by Proposition 3.3, thus it has to be projective. We conclude that by Lemma 3.5.
(2) This is immediate by (1), indeed, we see by the above exact sequence that is as well as the third syzygy of . ∎
We end this subsection with the following remark.
Proposition 3.6**.**
Let be a finite dimensional algebra of finite global dimension. Then the Yoneda category has finite global dimension if and only if is semisimple.
Proof.
If is semisimple, then clearly is semisimple. If is non-semisimple, there exists a non-split short exact sequence in , hence a non-split triangle in with terms in . By Proposition 3.4, we get a non-projective Cohen-Macaulay -module, so has infinite global dimension. ∎
3.3. A tilting subcategory
In this subsection, we show that the category has a tilting subcategory, hence is equivalent to its perfect derived category.
A -bimodule is a module over . We first define a -bimodule as a categorical analogue of the degree [math] part of Yoneda algebras. This bimodule will play an important role in the sequel.
Definition 3.7**.**
Let be an arbitrary finite dimensional algebra. The -bimodule is given by
[TABLE]
for each indecomposable .
In particular, we obtain and for each .
We fix some more notations. Let be an arbitrary finite dimensional algebra. We set
[TABLE]
Then we have , where the right hand side is the smallest additive category containing the ’s. We put and so on. Note that is equivalent to the ideal quotient , and the bimodule defined above is nothing but the natural bimodule structure induced by the projection .
Now let us state the main result of this subsection.
Theorem 3.8**.**
Let be a finite dimensional algebra of finite global dimension.
- (1)
We have for each . 2. (2)
* is a tilting subcategory of .* 3. (3)
The natural identification induces an equivalence of additive categories.
Consequently, there exists a triangle equivalence .
We can summarize the statements in the following diagram.
[TABLE]
For the proof we fix further notations. The support of a -module is the subcategory
[TABLE]
We say that a -module is concentrated in degree (resp. in degree , and so on) if (resp. , and so on). We note the following fact.
Lemma 3.9**.**
Let be a finite dimensional algebra of finite global dimension.
- (1)
Let . Then is concentrated in degree if and only if . 2. (2)
For any , there exists the unique exact sequence
[TABLE]
*in with *(*resp. ) concentrated in degree *(resp. ).
Proof.
The assertion (1) is clear. In (2), it is clear that the exact sequence (3.2) uniquely exists in . We have to verify that and are finitely presented. Since the functors and on are exact, we may assume that is representable. Let and . Take an injective resolution so that . Consider the triangle
[TABLE]
associated to the standard co--structure on . It induces an exact sequence
[TABLE]
which lies in by Lemma 3.2. Then, since is a complex of injective modules with terms in degree , the -module is concentrated in degree , and therefore so is . Similarly we see that is concentrated in degree . Therefore the exact sequence
[TABLE]
must coincide with (3.2), which shows that . ∎
Note that can be regarded as the ‘degree part’ of . In particular, the functor defined above is nothing but the ‘degree [math] part’ of .
Now let us start the proof with the following important observation.
Lemma 3.10**.**
Let be an arbitrary finite dimensional algebra and let be an exact sequence in .
- (1)
If , then . 2. (2)
If , then .
Proof.
We only prove (1). The exact sequence in induces an exact sequence
[TABLE]
in . Since is concentrated in degree by Lemma 3.9(1), so is . Similarly, since is injective, is concentrated in degree [math], thus so is . Therefore, the exact sequence
[TABLE]
must be the exact sequence in Lemma 3.9(2) for and . We then conclude that . ∎
Let us fix one more notation.
Definition 3.11**.**
Let be an arbitrary finite dimensional algebra.
- (1)
Let be the category of short exact sequences in whose morphisms are triples such that the diagram
[TABLE]
is commutative. 2. (2)
Let be the category of short exact sequences up to homotopy, that is, the ideal quotient of by split short exact sequences.
Note that a morphism in is null-homotopic if and only if factors through .
We now note the following easy observation.
Lemma 3.12**.**
The functor sending to induces a fully faithful functor .
Proof.
By Proposition 3.4, is indeed in . Also, if is split, then is projective, so induces a functor . Therefore it suffices to show that for a morphism in as in (3.3), the corresponding morphism factors through a projective module in if and only if is null-homotopic. We put and .
Suppose that null-homotopic. Then factors through , and as in the diagram below equals the composite .
[TABLE]
Suppose conversely that in factors through a projective -module. Then, it factors through .
[TABLE]
Using the fact that , we see that the triangle above the dashed line is commutative. Now, since is an inflation in and is a projective -module, factors through , and therefore, factors through . ∎
Now we are ready to prove the theorem.
Proof of Theorem 3.8.
(1) Consider the exact sequence
[TABLE]
in with . Then we have by Lemma 3.10(1). Therefore it is in by Proposition 3.4.
(2) We have to show the following two statements:
- (a)
for all and . 2. (b)
.
The claim (a) follows as in [Ya, 3.4]: since the syzygies of are concentrated in degree , the supports of and are disjoint, thus there is no nonzero homomorphism of -modules between them. This shows the vanishing of the extensions.
Now we turn to (b). Note that since by Proposition 3.4, lies in the thick subcategory of generated by for all and . Consider as the singularity category . We show that generates by showing that the subcategory of generates . By , any -module is concentrated in bounded degrees, so by Lemma 3.9(2), it suffices to show that any -module concentrated in a certain degree lies in the thick subcategory of generated by . But such modules can be view as -modules, so we have the assertion since by Proposition 2.6.
(3) Consider the left square of the diagram (3.1). It suffices to show that for each in , the corresponding morphism factors through a projective module in if and only if factors through an injective module in . Consider as in (1) the exact sequences and in with being injective. By Lemma 3.10(1), we have and . Note that for each in , there is the unique morphism in and that factors through an injective -module if and only if it factors through if and only if is null-homotopic. Therefore, our assertion is a consequence of Lemma 3.12.
The rest of the statement follows from Theorem 2.13 and the fact that if is finite, then so is [AR1, 10.2]. ∎
3.4. Veronese subrings of Yoneda algebras
We note the following interesting property of Veronese subalgebras of Yoneda algebras. It turns out, contrary to Theorem 3.1, that they have finite global dimension.
Theorem 3.13**.**
Let be a finite dimensional algebra with . For , consider the subcategory
[TABLE]
of . Then, the category has global dimension at most .
Note that this is not true for ; in this case in fact, has infinite global dimension by Proposition 3.6.
If is of finite representation type, we have the following reformulation in terms of algebras.
Corollary 3.14**.**
Let be a finite dimensional algebra of global dimension , and of finite representation type with an additive generator for . For , set
[TABLE]
Then, .
We start with the following lemma on homological dimensions of -modules.
Lemma 3.15**.**
Let and . Then, has projective dimension at most .
Proof.
We show by induction on . If is injective, then and the assertion is clear. Assume and consider the exact sequence
[TABLE]
with . Then we have . For , we have , hence the assertion by the induction hypothesis. It remains to consider . By the short exact sequence above, we have a long exact sequence
[TABLE]
since by . Now, since by the induction hypothesis, we see that . ∎
We fix some notations as in Definition 3.7. Define the ‘degree [math] part’ of for each by
[TABLE]
for each indecomposable . These are the projective modules when considered as -modules.
Lemma 3.16**.**
For each non-injective , we have .
Proof.
Consider as above the exact sequence
[TABLE]
with . Note that . This induces an exact sequence
[TABLE]
Then, as in the proof of Lemma 3.10(1), we have . Since and by Lemma 3.15, we obtain . ∎
Remark 3.17*.*
If is injective, then is projective.
Now we are ready to prove Theorem 3.13.
Proof of Theorem 3.13.
We have to show that any -module has projective dimension at most . It suffices to consider the case is concentrated in a certain degree, which we may assume to be [math]. Note that such modules can be viewed as a -module. Therefore, for concentrated in degree [math], there exists an exact sequence with since by Proposition 2.6. By Lemma 3.16 and Remark 3.17, we have . We therefore deduce that . ∎
4. Yoneda algebras for arbitrary algebras
In this section, contrary to Section 3, we discuss Yoneda categories for algebras of possibly infinite global dimension. Let be an arbitrary finite dimensional algebra. As before, consider the subcategory
[TABLE]
of .
4.1. Coherence
We first prove in this subsection that the category is always coherent, generalizing Lemma 3.2. We remark that contrary to Lemma 3.2, is not a dualizing variety if .
Theorem 4.1**.**
Let be a finite dimensional algebra, and let be the Yoneda category. Then, we have the following.
- (1)
* is functorially finite in .* 2. (2)
* *(resp. ) is finitely generated for all . 3. (3)
* is coherent and thus is abelian.*
The proof is based on the following observation.
Lemma 4.2**.**
Let with for . Then, gives a right -approximation.
Proof.
We may assume . Setting , we have a triangle
[TABLE]
in . Indeed, consider the triangle associated to the standard co--structure on and note that in .
Consider the morphism in for each and . If , we see by the triangle (4.1) that factors through . If , we see as in Lemma 3.2 that factors through . ∎
Now our result is a consequence of this lemma.
Proof of Theorem 4.1.
We only show the ‘left’ version, the ‘right’ version is dual. We have (2) by Lemma 4.2. The other statements are direct consequences, indeed, (1) is just a reformulation of (2). Also, (1) (3) is clear since if is contravariantly finite, then it has weak kernels. ∎
4.2. Gorenstein property
We give an analogue of Theorem 3.1 in this subsection. The following result states that the category has self-injective dimension at most .
Theorem 4.3**.**
Let be a finite dimensional algebra and be the Yoneda category of . Then we have the following.
- (1)
* in for all .* 2. (2)
* in for all .*
We need several lemmas analogous to the previous section. We first have the following observation generalizing Proposition 3.3.
Proposition 4.4**.**
Let . There exists a triangle in with which induces a projective resolution
[TABLE]
Therefore, has projective dimension at most .
Proof.
Replacing by its injective resolution, we may assume and . Then by Lemma 4.2, gives a right -approximation of . This injective map in induces an exact sequence as in the diagram below.
[TABLE]
Note that in , therefore, the above exact sequence induces the triangle
[TABLE]
in . Then, this is a desired triangle. Indeed, this triangle yields the projective resolution
[TABLE]
of since is also a right -approximation. ∎
We also note the following analogue of Lemma 3.5, which will be used later.
Lemma 4.5**.**
Let . Then, is projective if and only if .
Proof.
This is same as Lemma 3.5: if is projective, the triangle as well as the induced resolution given in Lemma 4.4 splits, hence . ∎
Now we consider the functor
[TABLE]
sending to . Note that for each . This is a left exact functor, whose -th right derived functor is
[TABLE]
sending to . In what follows, we consider these functors for .
For each , the functor maps to , so we have a morphism for each , which gives a morphism
[TABLE]
of -modules. The following observation is crucial.
Lemma 4.6**.**
The morphism is surjective for all .
Proof.
By Lemma 4.4, we have a triangle in which induces a projective resolution
[TABLE]
Now, applying to this resolution and comparing with the exact sequence obtained from the triangle, we have a commutative diagram with exact rows
[TABLE]
which shows that is surjective. ∎
Lemma 4.7**.**
Let be a left coherent category. If is surjective for all , then for all .
Proof.
We denote by the transpose duality [AB]. By the Auslander-Bridger sequence
[TABLE]
we have , and hence the assertion. ∎
Now we are ready to prove Theorem 4.3.
Proof of Theorem 4.3.
We prove the ‘left’ version. Let and be a projective presentation of . Complete to a triangle in so that we have exact sequences
[TABLE]
in and
[TABLE]
in , indeed, the image of the middle morphism is since it is the kernel of . Applying to the left half of the second exact sequence and comparing it with the right half of the first one, we have a commutative diagram with exact rows
[TABLE]
Now the middle vertical map is surjective by Lemma 4.6, thus so is the left vertical map. We therefore obtain the result by Lemma 4.7. ∎
4.3. Cohen-Macaulay modules
Our next aim is to study the category of Cohen-Macaulay modules. We have the following analogue of Proposition 3.4.
Proposition 4.8**.**
Let be a finite dimensional algebra, and let be its Yoneda category.
- (1)
* is Cohen-Macaulay if and only if there exists a triangle*
[TABLE]
such that and . 2. (2)
We have an isomorphism of functors on .
Proof.
Using Theorem 4.3 and Lemma 4.5 instead of Theorem 3.1 and Lemma 3.5, the same proof as in Proposition 3.4 applies. ∎
We also have a generalization of Proposition 3.6 with the same proof.
Proposition 4.9**.**
Let be an arbitrary finite dimensional algebra. Then the Yoneda category has finite global dimension if and only if is semisimple.
We later use the following computation of the duality functor .
Lemma 4.10**.**
Let and take a triangle (4.2) such that . Then, we have in .
[TABLE]
Proof.
We have an exact sequence
[TABLE]
in . Applying to this sequence yields the assertion. ∎
4.4. A -structure
Let be a finite dimensional algebra and be its Yoneda category. We show that the category has a natural -structure and describe its heart. This will lead to the construction of a realization functor.
Recall the -bimodule from Definition 3.7. By abuse of notation we often view as a subcategory and , or their images in the stable categories. Our main result of this subsection is the following.
Theorem 4.11**.**
Let be an arbitrary finite dimensional algebra. Set
[TABLE]
Then, forms a -structure on . Moreover, its heart is equivalent to .
Remark 4.12*.*
The above description of the -structure suggests that is ‘injective-like’ in the following sense: Let be a finite dimensional algebra, and let be its derived category. Consider the standard -structure on , which is given by
[TABLE]
The description of the subcategories in Theorem 4.11 are analogous, with in the place of the injective module . It turns out that the ‘projective-like’ subcategory is . These are justified in Lemma 4.18.
The plan of our proof is as follows: using Wakamatsu-type lemma (Proposition 4.20), we show in Lemma 4.21 that is an aisle in , that is, is a -structure, and in Lemma 4.19 that . We therefore obtain the desired result.
Put as usual , for each . We start the proof with the following immediate consequence of the definition.
Lemma 4.13**.**
We have and .
Let us introduce a piece of notation.
Notation 4.14**.**
For , we write with .
We note the following observation.
Lemma 4.15**.**
We have for each and as -modules.
Proof.
Indeed, using Yoneda’s lemma, we see that the -module maps to . ∎
Recall from Proposition 4.8 that comes from a triangle (4.2). If has no projective direct summand, we may assume that all belong to the radical of . We keep the notations from Section 3.3, for example, and so on.
An important step toward proving Theorem 4.11 is the following description of each subcategories.
Proposition 4.16**.**
Let be an object coming from a triangle
[TABLE]
in with each term in , each map is a radical map, and . Then, the following hold.
- (1)
* if and only if .* 2. (2)
* if and only if .*
Proof.
We first prove (2) since this is simpler. We have a projective resolution
[TABLE]
of . Applying to this sequence yields a complex
[TABLE]
which is isomorphic by Lemma 4.15 to
[TABLE]
Now, is equivalent to the acyclicity of this complex (4.3). It is clear that if , then the above complex is acyclic. We show the converse. Suppose the complex (4.3) is acyclic. Then we see that is projective (in ) for every since by Proposition 2.6, hence is a split epimorphism to its image for each . On the other hand, since the maps are the summands of , , and , they are radical maps. Therefore, must be zero for all , and we conclude that , , and .
Now we turn to (1). We similarly have the following exact sequence:
[TABLE]
Note that the complex
[TABLE]
obtained by applying to (4.4), which is isomorphic by Lemma 4.15 to
[TABLE]
computes the spaces , that is, the cohomologies of (4.5) are . Therefore, the vanishing of the extensions are equivalent to the acyclicity of the complex (4.5).
We first prove the ‘only if’ part. Suppose the complex (4.5) is acyclic. Then, as in the proof of (2), it is a minimal projective resolution of by the assumption that and are radical maps. We conclude that , , and by (Proposition 2.6).
We next prove the ‘if’ part. Suppose that . We have to show that the sequence
[TABLE]
is exact. Taking the 0-th cohomology of the triangle and using , we see that is exact in . This yields the desired exactness. ∎
We give a description of the ‘heart’ . Recall from Definition 3.11 that is the category of short exact sequences up to homotopy. We also need the well-known equivalence
[TABLE]
Proposition 4.17**.**
- (1)
The functor sending to is an equivalence. 2. (2)
We have equivalences of categories .
Proof.
(1) is fully faithful by Lemma 3.12. It is dense by Proposition 4.16.
(2) This follows from (1) and (4.6). ∎
We need some more information of . Recall that there exist dualities
[TABLE]
which satisfy for each . This duality plays a key role in the sequel.
Lemma 4.18**.**
Let . Consider the equivalence in Proposition 4.17.
- (1)
The image of the projective -module under is . In particular, we have . 2. (2)
The image of the injective -module under is , where is the AR-translation in .
Consequently, has enough projectives and enough injectives .
Proof.
We consider the images of certain objects in under and .
(1) Consider the exact sequence with . Its image in under is the projective module .
On the other hand, by Lemma 3.10(2), we have an exact sequence
[TABLE]
in . Applying , we have
[TABLE]
Therefore the image of under is .
(2) Note that by the AR-duality, we have . Consider the exact sequence with . We see as above that it is mapped to by , and to by . ∎
We next show the orthogonality of the subcategories. For a subcategory of an additive category , we denote by
[TABLE]
the orthogonal categories. Also, we identify the opposite category of with the Yoneda category of by the usual duality as in the diagram below.
[TABLE]
Lemma 4.19**.**
We have .
Proof.
We first show the inclusion ‘’. For this we show . Let and . By Proposition 4.16, we can take a triangle with such that , and a triangle with such that . Then, we have , and hence the assertion.
We next show ‘’. Suppose . By Lemma 4.18(1) we have , so . Applying yields . Now, let be a triangle in with terms in such that are radical maps and . Then by Lemma 4.10, is a triangle in with radical maps and . By Proposition 4.16, we see that . Therefore, we deduce that and hence again by Proposition 4.16. ∎
Now we recall Wakamatsu’s lemma for triangulated categories. We say that a subcategory of a triangulated category is extension-closed if two end terms of a triangle is in , then so is the middle term.
Proposition 4.20**.**
[IY, 2.3]** Let be a Krull-Schmidt triangulated category, and let be a subcategory which is closed under . Assume is contravariantly finite and extension-closed in . Then is a -structure on .
We are now at the final step of the proof.
Lemma 4.21**.**
* is an aisle in , that is, is a -structure.*
Proof.
The subcategory is clearly extension-closed, and is closed under by Lemma 4.13. Therefore by Proposition 4.20, it suffices to show that is contravariantly finite in .
Let and be a triangle in with terms in such that . Let and be the triangles associated to the standard -structure on . Note that these triangles split and have terms in . By [BBD, 1.1.11], there exists a diagram of triangles
[TABLE]
namely, the rows and columns are triangles, each square but the bottom right corner is commutative, and the bottom right square is anti-commutative.
We first claim that . We know that and since each column is a triangle. Since and , the triangle is a direct sum of triangles and , where . Now, since it is a direct summand of , and also since it is the mapping cocone of . Therefore , and hence .
Set . By the claim above, we have and by Proposition 4.16, we see that . Note that is a submodule of and setting , we have a commutative diagram with exact rows
[TABLE]
We claim that gives a right -approximation of in . Let whose image in lies in , and be a morphism in . It suffices to show . Since in , there is a surjection in with by Proposition 4.16. Therefore, we have , and hence the assertion. ∎
We have completed the proof of the main result of this subsection. Let us summarize the proof below.
Proof of Theorem 4.11.
By Lemma 4.21 and 4.19, is certainly a -structure . The statement on the heart is Proposition 4.17. ∎
We note the boundedness of the -structure constructed above.
Proposition 4.22**.**
The -structure given in Theorem 4.11 is bounded.
Proof.
We have to show that for any , there exists such that and . Note that if is concentrated in degree , then is concentrated in degree since by Proposition 2.6. It follows that is concentrated in degree for sufficiently large , so we have the second statement since is concentrated in degree [math]. Also, we have the first statement as in [Han, 7.2]. ∎
4.5. The triangle equivalence
We are now ready to prove a main result of this paper. Let be an arbitrary finite dimensional algebra. Once we know that there is a -structure on with heart , there exists by Corollary A.9, a realization functor .
Theorem 4.23**.**
Let be a finite dimensional algebra. The realization functor
[TABLE]
is a triangle equivalence.
Proof.
We first verify the condition in Theorem A.3(3c) that the realization functor is fully faithful. Let , and be a morphism in . By Lemma 4.18, we can take an injection in with . Then, the composition is zero since . Now, we have by Proposition 4.22. We therefore conclude that is an equivalence. ∎
5. Yoneda algebras for hereditary algebras
The aim of this section is to discuss in further detail the Yoneda algebras for hereditary algebras. In the first subsection, we note some consequences of the previous sections, and apply these results in the following two subsections.
5.1. Basic properties
We first record the self-injectivity of Yoneda algebras in the hereditary case, which also characterizes hereditary algebras. The proof is based on the following well-known fact. Recall that we denote by the derived category of .
Proposition 5.1** ([Hap, I.5.2]).**
We have if and only if is hereditary.
We immediately obtain the following result.
Proposition 5.2**.**
Let be an arbitrary finite dimensional algebra and be the Yoneda category of . Then, is self-injective if and only if is hereditary.
Proof.
If is hereditary, then by Proposition 5.1, thus is self-injective. If is not hereditary, then there exists by Proposition 5.1. Then, has projective dimension precisely by Lemma 4.4 and 4.5, so cannot be self-injective. ∎
From now on, we restrict ourselves to representation-finite case: Let be a finite dimensional hereditary algebra of finite representation type with an additive generator for , and be its Yoneda algebra. Then, we have . If is the path algebra for a Dynkin quiver , then the derived category is presented by the infinite translation quiver (see [ASS, Hap]) with mesh relations [Hap, Rie]. Thus we have the following explicit description of the Yoneda algebra of .
Proposition 5.3**.**
Let be a Dynkin quiver. Then, the Yoneda algebra of the path algebra is presented by the quiver and the mesh relations.
Set and . Moreover, let be the idempotent of corresponding to the maximal injective summand of . Then, is the stable Auslander algebra, and we have natural surjections
[TABLE]
Note that , so is a projective -module and holds in .
We record the following special case of Proposition 3.4 and Theorem 3.8, upon which the results of the following subsections are build. Since is self-injective by Proposition 5.2, we have in this case.
Theorem 5.4**.**
Let be a representation-finite hereditary algebra, its Yoneda algebra, and the stable Auslander algebra of .
- (1)
We have an isomorphism of functors on , where is the degree shift. 2. (2)
There exists a triangle equivalence
[TABLE]
taking to .
Remark 5.5*.*
The equivalence in (2) can also be deduced using the results of [Ya].
In the following subsections, we give two independent applications of this equivalence.
5.2. Fractional Calabi-Yau property of stable Auslander algebras
As a first application of the triangle equivalence in Theorem 5.4, we deduce that the stable Auslander algebra of a representation-finite hereditary algebra has the fractional Calabi-Yau property. For integers and , we say that a triangulated category is fractionally Calabi-Yau of dimension (-CY for short) if it has a Serre functor such that . A finite dimensional algebra (of finite global dimension) is fractionally CY if its bounded derived category is fractionally CY.
We first note the well-known result on fractional CY property of representation-finite hereditary algebras.
Proposition 5.6** ([MYe]).**
Let be a representation-finite hereditary algebra of Dynkin type , and let be its Coxeter number, which is given as in the following table:
[TABLE]
Then, is -CY.
We have the following result as an application.
Theorem 5.7**.**
Let be a representation-finite hereditary algebra of Dynkin type , and let be its Coxeter number. Then, the stable Auslander algebra of is -CY.
We need the following observation which is a variant of [AR2][Ke4, 8.5].
Lemma 5.8**.**
Let be an -CY triangulated category. Then, is -CY.
Now we are ready to prove our result.
Proof of Theorem 5.7.
Note that by , we have . Together with Theorem 5.4, we deduce . Now the result is a consequence of Proposition 5.6 and Lemma 5.8. ∎
5.3. Rigid modules over Yoneda algebras
So far we have considered the category of graded modules over the graded algebra . In this subsection, we discuss the category of ungraded modules over . Let us first recall the related notions.
Definition 5.9**.**
Let be a full subcategory of a triangulated category and .
- (1)
is -rigid if for all . 2. (2)
is maximal -rigid if it is -rigid and any -rigid subcategory containing equals . 3. (3)
[I1] is -cluster tilting if it is functorially finite in and satisfies the following.
[TABLE]
Our main result of this section states the is endowed with a maximal -rigid object.
Theorem 5.10**.**
- (1)
We have
[TABLE] 2. (2)
* is maximal -rigid.* 3. (3)
There is no -cluster tilting subcategory in containing unless is semisimple.
Although the theorem can be stated only in terms of , our proof is based on the equivalence in Theorem 5.4. We start with the following proposition by which we can regard as the full subcategory of .
Proposition 5.11**.**
- (1)
The composition
[TABLE]
is fully faithful. 2. (2)
The above functor coincides as a map on objects with the composition
[TABLE]
induced by the canonical surjections .
Proof.
(1) Let and be the corresponding objects under the equivalence . It suffices to show for . Indeed, since on , we have , which is zero for by .
(2) This can be seen by observing that both compositions take to , and take short exact sequences to triangles. ∎
In what follows, we identify with the full subcategory of via Proposition 5.11. We note the following consequence of the above proposition.
Lemma 5.12**.**
We have . In particular, for , if and only if is annihilated by .
Proof.
Since , it is enough to show that contains . Let be an injective envelope of the additive generator . Then, we have a surjection , thus . ∎
The following description of inside is crucial for the proof of Theorem 5.10.
Proposition 5.13**.**
Under the identification in Proposition 5.11, we have
[TABLE]
We need several lemmas for the proof. The first one shows the one of the inclusions stated in the proposition.
Lemma 5.14**.**
For any , we have for .
Proof.
For , we have , which is zero by . ∎
Lemma 5.15**.**
- (1)
* and .* 2. (2)
* and .*
Proof.
Note that by and , these statements are equivalent, so we prove the first one. By Proposition 5.14, it is enough to show . But this is clear since corresponds to under the equivalence . ∎
Now we are in the position to prove Proposition 5.13.
Proof of Proposition 5.13 .
The inclusion ‘’ is proved in Lemma 5.14. We will show the converse inclusion. Let be an object with for . Consider the exact sequence of -modules. Applying , we have an exact sequence
[TABLE]
in , where is the submodule of annihilated by , and . Note that and . Indeed, holds since is annihilated by , and also holds by Lemma 5.12 since .
To prove that , we may assume has no -projective (=injective) summands, since such summands are also summands of .
Applying the functor to the sequence (5.1) yields an exact sequence . We have by Lemma 5.14 and by assumption, and hence .
Now, let be a projective resolution of the -module and consider it as an exact sequence in (concentrated in degree 0). For let be the object corresponding to under . Note that . Then, the complex in corresponding to is with at degree 0. Indeed, letting , it is the mapping cone of in , so the corresponding object in is the mapping cone of , which is the complex with at degree [math]. Similarly, is the mapping cone of in , so the corresponding object in is the mapping cone of , which is precisely the complex .
We have
[TABLE]
and similarly, and . Now, shows that is acyclic at degree . Then, since , we see that . Note that since is a split epimorphism to its image. Therefore, in with and . Note that also in , since we assumed has no projective summands.
Consider the following diagram in obtained by the push-out of (5.1) along .
[TABLE]
We claim that is projective. Since , this claim will yield in , hence the theorem. To show that is projective, we show that . For this, it is sufficient to show that for , since is generated by .
Applying to the above diagram yields
[TABLE]
where stands for . Now, the map is zero since , and hence is an isomorphism by . Note also that is an isomorphism since and . Therefore, is also an isomorphism. This shows that and , since (by ; Lemma 5.15(2)) and (by and Lemma 5.14). follows from (by Lemma 5.15(2)) and ( and Lemma 5.14). This completes the proof. ∎
Now the main result of this section is a consequence of Proposition 5.13.
Proof of Theorem 5.10.
(1) The inclusion ‘’ is done in Lemma 5.15(1). We show the inclusion ‘’. Let be in the right-hand-side. We see that by Proposition 5.13. Then, we have . (Here, denotes the identified objects through the functors in Proposition 5.11.) The vanishing of this for shows that for . Therefore we conclude that by .
(3) Assume there exists a 3-cluster tilting subcategory in containing . Then, its preimage under the functor is a 3-cluster tilting subcategory of containing and stable under . On the other hand, the -orbit
[TABLE]
of is a 3-cluster tilting subcategory of [I2, 1.23], where is the Serre functor for and . Then, since is -stable, we have . Moreover, since and are cluster tilting subcategories, the equality must hold, and in particular, is -stable. Then by [IO, 3.1], is 3-representation-finite. Since , has to be semisimple, or equivalently, is semisimple. ∎
6. Examples
We give some examples to illustrate our results. We use the following notation for graded modules: the number of the symbols (resp. ) over the composition factor indicates that the factor lies in the positive (resp. negative) degree of that value.
The first one is a hereditary algebra, and we explain the results in Section 5.
Example 6.1**.**
Let be the path algebra of linearly oriented type :
[TABLE]
It is a representation-finite hereditary algebra and its derived category is presented by the quiver with mesh relations:
[TABLE]
where are the objects from . Then by Proposition 5.3, the Yoneda algebra of is presented by the quiver :
[TABLE]
with mesh relations, and the arrows from and have degree , while the others have degree [math]. The composition series of the projectives look
[TABLE]
We see is indeed self-injective, as stated in Proposition 5.2. The AR-quiver of is computed to be
[TABLE]
We deduce a triangle equivalence , which certifies Theorem 3.8.
We next consider the ungraded module category , whose AR-quiver is the following:
[TABLE]
where two ends are identified along the dotted lines. Then, has indecomposable summands, which are in the boxes.
On the one hand, consists of -modules which are annihilated by the idempotents of corresponding to injective -modules by Lemma 5.12. Therefore, they are precisely -modules having only 1, 2, or 3 as composition factors, hence are the shaded modules. On the other hand, a computation of the subcategory shows that it actually coincides with . This demonstrates Proposition 5.13.
If there is a 3-cluster tilting subcategory containing , since is -stable, we see that it has to be the whole , which is absurd. This shows Theorem 5.10(3) in this case.
The next two examples are algebras of finite global dimension and we demonstrate the results in Section 3.
Example 6.2**.**
Let and be as in the previous example. Let be an algebra presented by the following quiver with relations:
[TABLE]
It has global dimension and is of finite representation type. The category is located in in from Example 6.1. Therefore, the Yoneda algebra of is isomorphic to for the idempotent corresponding to these five summands. Renumbering the vertices, is presented by the following quiver
[TABLE]
with relations induced from the mesh relations on , and the arrows and have degree , and others have degree [math]. The projective and injective -modules look
[TABLE]
It is easily checked that is indeed a Gorenstein algebra of dimension , as in Theorem 3.1. The AR-quiver of is computed to be
[TABLE]
The Cohen-Macaulay -modules are the circled modules and the projective modules. Consequently, we have , as was proved in Theorem 3.8.
As in Section 3.4, we now consider the second Veronese subalgebra of . We see as above that it is presented by the following quiver with relations:
[TABLE]
It is easily verified that this algebra has global dimension , which illustrates Theorem 3.13.
Example 6.3**.**
Let be an algebra presented by the following quiver with relations:
[TABLE]
It has global dimension 2 and is of finite representation type. Its Yoneda algebra is presented by the following graded quiver with relations:
[TABLE]
[TABLE]
[TABLE]
The composition series of the projective and the injective modules look
[TABLE]
Using this we can check that is indeed 1-Gorenstein (Theorem 3.1). Moreover, the AR-quiver of is as follows:
[TABLE]
We can verify the triangle equivalence (Theorem 3.8).
Now let us consider the second Veronese subalgebra of . It is presented by
[TABLE]
We can easily check that has global dimension as in Theorem 3.13.
The next examples are self-injective algebras of infinite global dimension. We illustrate the results in Section 4.
Example 6.4**.**
Let be a self-injective Nakayama algebra with vertices and Loewy length , so it is presented with the following cyclic quiver with relations:
[TABLE]
Its Yoneda algebra is -Gorenstein by Theorem 4.3. By Theorem 4.23, we have a triangle equivalence , with the stable Auslander algebra of presented by the following quiver with mesh relations:
[TABLE]
Example 6.5**.**
We specialize the above example to ; Let . Then the Yoneda algebra of is presented by the following graded quiver with relations:
[TABLE]
[TABLE]
[TABLE]
By Theorem 4.3, this is a Gorenstein algebra of dimension . Also, the stable Auslander algebra of is the preprojective algebra of type . Therefore by Theorem 4.23, we have a triangle equivalence .
We take a closer look for the case ; Let . This is a self-injective algebra with two indecomposable modules, namely the simple module and the free module . Its Yoneda algebra is -Gorenstein by Theorem 4.3. It is presented by the following graded quiver with relations:
[TABLE]
The composition series of of the projective -modules look
[TABLE]
Now, by the triangle in , we get three non-projective Cohen-Macaulay -modules;
[TABLE]
By the triangle equivalence of Theorem 4.23, we see that these are all the graded Cohen-Macaulay -modules up to degree shift.
We end this paper by the following example of which is not even Gorenstein.
Example 6.6**.**
Let be an algebra presented by the following quiver with relations:
[TABLE]
This is of finite representation type and its Yoneda algebra is presented by the graded quiver
[TABLE]
[TABLE]
with relations given as follows:
- •
the mesh relations along the dotted lines.
- •
a commutativity relation .
- •
any path of degree which strictly contains the AR sequences , , or is zero.
- •
vanishing of extensions:
The stable Auslander algebra of is the path algebra of non-linearly oriented type . Therefore, by Theorem 4.23, we deduce a triangle equivalence .
Appendix A Existence of realization functors
In this appendix we give a proof of the existence of a realization functor for algebraic triangulated categories. This is well-known to experts, but we include a proof for the convenience of the reader. Similar constructions are done in [BGSc, Section 3], [ChR, Section 3]. In the first subsection we recall the notion of f-categories and in the second subsection we give a formulation of a filtered derived category of a DG category.
A.1. F-categories and realization functors
Let be a triangulated category with a -structure, and let be its heart. It is a classical problem in the theory of triangulated categories to compare with the derived category . However, it is not clear whether we can extend the inclusion to a triangle functor because of the lack of the functoriality of mapping cones in a triangulated category. It was shown in [BBD] that a realization functor exists under the assumption that has an ‘f-category’ over itself.
We first recall the notion of f-categories (or filtered enhancements) of a given triangulated category.
Definition A.1** ([Bei, Appendix][PV, 3.1]).**
Let be a triangulated category. An f-category over is a triangulated category endowed with full triangulated subcategories , such that , an autoequivalence of , and a natural transformation , satisfying the following conditions, where we put and .
- (1)
. 2. (2)
For any , there exists a triangle
[TABLE]
with and . 3. (3)
. 4. (4)
and . 5. (5)
in . 6. (6)
and are isomorphisms for all and . 7. (7)
For any morphism in , the commutative diagram of triangles
[TABLE]
can be completed to a diagram of triangles. Here, and are the truncations of with respect to the stable -structure
Remark A.2*.*
We require the additional axiom (7) from [Sc][PV, 3.10], which ensures that the realization functor is triangulated, see [PV, Appendix].
The following result shows that the existence of an f-category over a given triangulated category assures the existence of a realization functor.
Theorem A.3** ([BBD, Section 3.1][Bei, Appendix][PV, 3.11]).**
Let be a triangulated category with a -structure with heart . Assume there is an f-category over . Then, the following hold.
- (1)
There exists a triangle functor extending the inclusion . 2. (2)
* induces isomorphisms for all and .* 3. (3)
The following are equivalent:
- (a)
* is fully faithful.* 2. (b)
For any , , and a morphism in , there exists an epimorphism in such that the composition is zero. 3. (c)
For any , , and a morphism in , there exists a monomorphism in such that the composition is zero.
A.2. Filtered derived category of a DG category
In this subsection, we formulate the filtered derived category of a DG category, and thereby proving that any algebraic triangulated category admits an f-category over itself. We refer to [Ke2, Ke5] for general backgrounds on DG categories.
We follow the construction in [Ke2] of the derived a DG category, together with that of filtered derived category of an abelian category to obtain the filtered derived category of a DG category.
Construction A.4**.**
(1) First we construct , the category of filtered DG -modules. The objects are finitely filtered DG -modules, that is, a DG -module together with a finite descending filtration
[TABLE]
of DG submodules of . The morphisms of filtered DG modules are those of DG modules which preserve filtrations.
We have an automorphism called the shift functor on which assigns to each the DG module with the filtration .
We also have an automorphism of called the shift of filtration, defined for each by as a DG module with the filtration . It naturally associates a morphism of functors induced by the inclusions in filtrations.
We moreover have the -functor
[TABLE]
to the category of DG -modules for each . A filtered DG module is filtered acyclic if each is an acyclic DG module, or equivalently, each is an acyclic DG module. We denote by the full subcategory of consisting of filtered acyclic DG -modules.
(2) Next we construct , the filtered homotopy category of . We denote by the category of filtered graded -modules. We have the forgetful functor . Endow with the exact structure whose conflations are short exact sequences in which are split in .
Proposition A.5** (cf. [Ke2, 2.2]).**
* with the above exact structure is a Frobenius category.*
Let be a graded -module. Following [Ke2, 2.2], we define the DG module as follows;
- •
as a graded abelian group for each .
- •
Make into a graded -module by setting for each morphism of degree .
- •
Make into a DG -module by setting as the differential.
For , we give the filtration on by , which yields the functor . The following lemma shows that is the right adjoint of the forgetful functor .
Lemma A.6**.**
For each and , we have an isomorphism
[TABLE]
Proof.
For , the construction of gives functorial conflations
[TABLE]
where and . Then the map gives a desired isomorphism. ∎
Proof of Proposition A.5.
We deduce from the above lemma that is injective in . By the conflations (A.1), we see that has enough injectives. We can dually construct the projective objects and the conflations . The constructions show that the projective and the injective objects coincide. ∎
Let be the stable category of , hence is naturally a triangulated category with suspension .
The shift of filtration on clearly takes projectives to projectives in , and induces an automorphism, again denoted by on together with a natural transformation .
The same holds for -functors, indeed, . Therefore, there is a family of triangle functors to the homotopy category of . The subcategory of corresponding to is denoted by , whose objects are again called the filtered acyclic DG modules. Note that is a thick subcategory of .
Let be a morphism in . The mapping cone of in is, the mapping cone of as a DG module, which we denote by , together with the filtration .
Let again be a morphism in . We call a filtered quasi-isomorphism if it satisfies the following equivalent conditions:
- •
induces quasi-isomorphisms in for all .
- •
The mapping cone of is filtered acyclic.
- •
is a quasi-isomorphism in for each .
(3) Finally we obtain , the filtered derived category of , as the Verdier quotient of by , that is, the localization of with respect to the filtered quasi-isomorphisms.
Since the functors and take filtered quasi-isomorphisms to (filtered) quasi-isomorphisms, they induce an automorphism together with a natural transformation , and triangle functors to the derived category of .
Set
[TABLE]
These are thick subcategories of . Also define the functor by giving the trivial filtration, that is, assign to each DG module a filtration by for and for . This functor yields an equivalence .
Now we are ready to state the following main result, whose proof is completely analogous to the case of filtered derived categories of abelian categories.
Theorem A.7**.**
Let be a DG category. Then, the filtered derived category is an f-category over .
Proof.
Let be the shift of filtration and be the natural transformation constructed above. Then, these data together with the subcategories (A.2) and the equivalence defines an f-category structure. Indeed, the axioms (3), (4), (5) are clear. We refer to [Sc, 6.3] for (1), (2), (6), and to [Sc, 7.4] for (7). ∎
Corollary A.8**.**
Any idempotent-complete algebraic triangulated category has an f-category over itself.
Proof.
By [Ke2, 4.3], any idempotent-complete algebraic triangulated category is the perfect derived category a DG category . By [PV, 3.8], the f-category over restricts to that over . ∎
We summarize the important consequence in the following corollary.
Corollary A.9**.**
Let be an idempotent-complete algebraic triangulated category. Assume has a -structure with heart . Then, there exists a realization functor .
Proof.
This is a direct consequence of Theorem A.3 and Corollary A.8. ∎
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