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TRIANGULATED CATEGORIES OF PERIODIC COMPLEXES
AND ORBIT CATEGORIES
JIAN LIU
Abstract. We investigate the triangulated hull of the orbit categories of the
perfect derived category and the bounded derived category of a ring concerning
the power of the suspension functor. It turns out that the triangulated hull will
correspond to the full subcategory of compact objects of certain triangulated
categories of periodic complexes. This specializes to Stai and Zhao’s result
when the ring is a finite dimensional algebra with finite global dimension over
a field. As the first application, if A, B are flat algebras over a commutative
ring and they are derived equivalent, then the corresponding derived categories
of n-periodic complexes are triangle equivalent. As the second application, we
get the periodic version of the Koszul duality.
Introduction
Given an additive category Aand an integer n≥1, a complex (X, ∂X) over A
is called n-periodic if Xi=Xi+nand ∂i
X=∂i+n
Xfor all i. A chain map fbetween
n-periodic complexes is a n-periodic morphism if fi=fi+nfor all i. A 1-periodic
complex is just a differential object which first appeared in Cartan and Eilenberg’s
book [12]. It was systematically studied by Avramov, Buchweitz and Iyengar [1].
Two morphisms f, g :X→Yof n-periodic complexes are called homotopic if there
is a homotopy map {σi:Xi→Yi−1}i∈Zfrom fto gsuch that σi=σi+nfor all i.
Then one can form the homotopy category Kn(A) of n-periodic complexes and the
derived category Dn(A) of n-periodic complexes when Ais abelian. They are both
triangulated categories; see [31] or Section 2.
Let Rbe a left noetherian ring. In this article, we will focus on studying the
homotopy category Kn(R-Inj) of n-periodic complexes of injective R-modules and
the derived category Dn(R-Mod) of n-periodic complexes of R-modules. For a
complex Xof R-modules, one can associate a n-periodic complex ∆(X):
· · · −→ a
j≡i−1(mod n)
Xj−→ a
j≡i(mod n)
Xj−→ a
j≡i+1(mod n)
Xj−→ · · · ;
this process is called compression in [1]. As expected, the triangulated category
of periodic complexes and the classical triangulated category are closely linked by
∆. It is known to experts that the homotopy category K(R-Inj) of complexes of
injective R-modules and the derived category D(R-Mod) of complexes of R-modules
are compactly generated; the first one is due to Krause [25]. Inspired by this, we
prove that Kn(R-Inj) and Dn(R-Mod) are compactly generated; see Theorem 2.12.
Date: September 21, 2021.
2020 Mathematics Subject Classification. 18G80 (primary); 16E45, 18E20, 18G35 (secondary).
Key words and phrases. periodic complex, orbit category, triangulated hull, derived category,
derived equivalence, dg category, Koszul duality.
1
arXiv:2109.08831v1 [math.CT] 18 Sep 2021
2 JIAN LIU
Moreover, the full subcategories of compact objects of these two categories are the
triangulated hull of certain orbit categories; see Theorem 1in the introduction. It
is also proved in Theorem 2.12 that the canonical functor Kn(R-Inj)→Dn(R-Mod)
induces a recollement.
Let T:A→Abe an autoequivalence. Following [23], the orbit category A/T is
defined as follows: it has the same objects as Aand the morphism spaces
HomA/T (X, Y ) := a
i∈Z
HomA(X, T iY).
The composition in A/T is defined in a natural way. As the name suggests, the
objects in the same T-orbit are isomorphic.
If Tis a triangulated category with suspension functor Σ, there is a natural
question: does T/Σninherit a triangulated structure from Tsuch that the projec-
tion functor T → T /Σnis exact? Let Rbe a finite dimensional hereditary algebra
over a field. Peng and Xiao [31] observed the orbit category Db(R-mod)/[2] of the
bounded derived category of finitely generated R-modules, introduced by Happel
[17] under the name “root category”, is triangulated. Indeed, they proved that it
is equivalent to the homotopy category of 2-periodic complexes of finitely gener-
ated projective R-modules. This established for the first time a link between the
orbit category and the triangulated category of periodic complexes. By making
use of this triangulated structure, they constructed the so-called Ringel-Hall Lie
algebra determined by Db(R-mod)/[2] and gave a realization of all symmetrizable
Kac-Moody Lie algebras; see [32].
However, Neeman found the answer to the above question is negative; see dis-
cussions in [23]. Inspired by questions from the cluster category in [10], Keller [23]
constructed the triangulated hull of certain orbit categories. As an application, he
proved the cluster category in [10] is triangulated.
Keller’s construction is an abstract embedding of the certain orbit category into
the triangulated hull. It will be nice to know the precise triangulated hull of a given
orbit category. If Ris a finite dimensional algebra with finite global dimension over
a field, it was independently proved by Stai [36] and Zhao [38] that Db(R-mod)/[n]
embeds into its triangulated hull Dn(R-mod), where R-mod is the category of finitely
generated R-modules. We are motivated by the natural question: what is the
triangulated hull of Db(R-mod)/[n]without these assumptions of R?
Our first result Theorem 1answers this question. It extends Stai and Zhao’s
result; see Corollary 3.14.
Recall the perfect derived category per(R) is the full subcategory of D(R-Mod)
formed by complexes that are quasi-isomorphic to bounded complexes of finitely
generated projective R-modules. Recall the embedding i:Db(R-mod)→K(R-Inj)
induced by taking injective resolution; see 3.7. For a triangulated category Twith
corpoducts, an object Xis called compact if HomT(X, −) preserves coproducts.
We let Tcdenote the full subcategory of Tformed by compact objects and note
that Tcis a thick subcategory of T.
Theorem 1. (see 3.11) Let Rbe a left noetherian ring. Induced by the compression
of complexes, the functors
∆◦i:Db(R-mod)/[n]−→ Kn(R-Inj)cand ∆ : per(R)/[n]−→ Dn(R-Mod)c
are the embedding of the orbit categories into their triangulated hull.
TRIANGULATED CATEGORIES OF PERIODIC COMPLEXES AND ORBIT CATEGORIES 3
In order to prove Theorem 1, we realize Kn(R-Inj) and Dn(R-Mod) as derived
categories of orbit categories of certain dg categories; see Theorem 3.8.
Motivated by Theorem 1, we compare the triangle equivalences D(A-Mod)'
D(B-Mod) and Dn(A-Mod)'Dn(B-Mod) for two rings A, B in Section 4. Two rings
that satisfy the first equivalence are called derived equivalent. In general, whether
two rings are derived equivalent is difficult to grasp. Therefore, it is important to
investigate the invariant under the derived equivalence. By introducing the tilting
complex, Rickard [33] established the derived Morita theory of rings. After that,
Keller [21] generalized Rickard’s derived Morita theory through the language of
differential graded categories.
It turns out that the above two equivalences are closely related; see Proposition
4.8. In particular, combine Proposition 4.8 with [21], we get the following result
which extends a result of Zhao [38]; see Corollary 4.11.
Theorem 2. (see 4.10) Let kbe a commutative ring and A, B be flat k-algebras.
If Aand Bare derived equivalent, then Dn(A-Mod) and Dn(B-Mod) are equivalent
as triangulated categories.
In the last section, we give the periodic version of the Koszul duality (Theorem
3). Its proof relies on the classical Koszul duality and the studies in previous
sections.
Let kbe a field and S=k[x1, . . . , xc], where deg(xi) = 1. Denote by Λ the
graded exterior algebra over kon variables ξ1, . . . , ξcof degree −1. Bernstein,
Gel’fand and Gel’fand [7] established the triangle equivalence Db(Λ-gr)'Db(S-gr)
between the bounded derived category of finitely generated graded modules. This
is known as the BGG correspondence.
The BGG correspondence can be lifted to the compact completions. That is,
there is a triangle equivalence K(Λ-GrInj)'D(S-Gr) (see [25] or 5.5), where A-Gr
is the category of graded modules over the graded algebra Aand Λ-GrInj is the full
subcategory of Λ-Gr formed by injective objects. The corresponding equivalence
will be called the Koszul duality. The Koszul duality phenomenon has played
an important role in representation theory. For instance, the DG version of the
Koszul duality was used by Benson, Iyengar and Krause [6] to stratify the modular
representation theory of finite groups.
Theorem 3. (see 5.1 and 5.2) There is a triangle equivalence
Kn(Λ-GrInj)∼
−→ Dn(S-Gr).
Moreover, this equivalence induces an embedding
Db(Λ-gr)/[n]−→ Dn(S-gr)
of the orbit category into its triangulated hull.
Acknowledgements. Part of the work was done during the author’s visit to the
University of Utah. The author would like to thank Srikanth Iyengar for his kind
hospitality and the China Scholarship Council for their financial support. Special
thanks to Benjamin Briggs for providing a similar project related to Section 5which
makes the article possible. The author thanks Xiao-Wu Chen, Srikanth Iyengar,
Janina Letz, and Josh Pollitz for their discussions on this work.
4 JIAN LIU
1. Notations and Preliminaries
Throughout the article, Ris a left noetherian ring. R-Mod (resp. R-mod) will be
the category of left (resp. finitely generated left) R-modules. The full subcategory
of R-Mod consisting all projective (resp. injective) R-modules is denoted by R-Proj
(resp. R-Inj). For an additive category A,C(A) will be the category of complexes
over Awith suspension functor [l] (X[l]i:= Xi+l, ∂i
X[l]:= (−1)l∂i+l
X). Denote by
K(A) the homotopy category of complexes over A. When Ais abelian, let D(A)
denote the derived category of complexes over A.
A complex of R-modules is perfect provided that it is quasi-isomorphic to a
bounded complex of finitely generated projective R-modules. per(R) will be the
full subcategory of D(R-Mod) consisting of all perfect complexes.
1.1. Thick subcategories and localizing subcategories. Let Tbe triangu-
lated category and Cbe a triangulated subcategory of T. We say Cis thick (resp.
localizing) if it is closed under direct summands (resp. coproducts). For a set S
of object in T, we let thickT(S) denote the smallest thick subcategories of Tcon-
taining S. This can be realized as the intersection of all thick subcategories of T
containing S; it has an inductive construction (see [2, 2.2.4]).
If Thas coproducts, then a technique of Eilenberg’s swindle implies that any
localizing subcategory is thick.
It is well-known that per(R) = thickD(R-Mod)(R) is thick; see [11, Lemma 1.2.1].
1.2. Let F:T → T 0be an exact functor between triangulated categories. Then
the kernel of Fdefined by
Ker F:= {X∈ T | F(X)∼
=0}
is a thick subcategory of T. When the functor Fis full, the essential image of F
defined by
Im F:= {Y∈ T 0|Y∼
=F(X) for some X∈ T }
is a triangulated subcategory of T0.
1.3. Recollement. Following Beilinson, Bernstein and Deligne [4], we call a dia-
gram
T0T T 00
i∗
i∗
i!
j∗
j!
j∗
of triangulated categories and exact functors a recollement if the following condi-
tions are satisfied.
(1) (i∗, i∗),(i∗, i!),(j!, j∗) and (j∗, j∗) are adjoint pairs.
(2) i∗, j!and j∗are fully faithful.
(3) Im i∗= Kerj∗, that is, j∗(X) = 0 if and only if X∼
=i∗(Y) for some Y∈ T 0.
Next, we record a useful result; its proof can refer [25, Section 3].
1.4. Let Sbe a thick subcategory of T. Then the inclusion functor S → T has
a right (resp. left) adjoint if and only if the quotient functor Q:T → T /Shas a
right (resp. left) adjoint. In this case, the right (resp. left) adjoint of the quotient
functor is fully faithful. The sequence Sinc
−−→ T Q
−→ T /Sin this case is called a
localization sequence (resp. colocalization sequence).
TRIANGULATED CATEGORIES OF PERIODIC COMPLEXES AND ORBIT CATEGORIES 5
Assume the sequence Sinc
−−→ T Q
−→ T /Sis a localization sequence. Denote by
π(resp. ι) the right adjoint of the functor inc (resp. Q). Then the sequence
T/Sι
−→ T π
−→ S is a colocalization sequence. In particular, πinduces a triangle
equivalence
T/Im ι∼
−→ S.
Note that a sequence T0→ T → T 00 induces a recollement as 1.3 if and only if
the sequence is both a localization sequence and a colocalization sequence.
1.5. Compactly generated triangulated categories. Let Tbe a triangulated
category with coproducts. An object X∈ T is called compact provided that the
Hom functor HomT(X, −) commutes with coproducts. That is, for any class of
objects Yi(i∈I) in T, the canonical map
can:a
i∈I
HomT(X, Yi)−→ HomT(X, a
i∈I
Yi)
is isomorphic. We let Tcdenote the full subcategory of Tformed by compact
objects in T. It is not hard to show that Tcis a thick subcategory of T.
Tis said to be compactly generated if there exists a set Sof compact objects
such that any object Ysatisfying HomT(X, Y [i]) = 0 for all X∈Sand i∈Zis
a zero object; the condition is equivalent to Tis equal to the smallest localizing
subcategory containing S(see [28, Lemma 3.2]). In this case, Tc=thickT(S); see
[27, Lemma 2.2]. For instance, D(R-Mod) is compactly generated by the compact
object R.
A set Sof objects in Tis called a compact generating set provided that S⊆ T c
and Tis compactly generated by S. The following result is well-known. For its
proof, we refer the reader to [6, Lemma 4.5]; compare [3, Lemma 1] and [21, Lemma
4.2].
Lemma 1.6. Let F:T → T 0be an exact functor between compactly generated
triangulated categories. Assume Fpreserves coproducts and S⊆ T cis a compact
generating set. Then Fis fully faithful if and only if the induced maps
HomT(X, Σi(Y)) −→ HomT0(F X, F Σi(Y))
are isomorphic for all X, Y ∈Sand i∈Z. In this case, Fis dense if and only if
Im Fcontains a compact generating set of T0.
1.7. DG categories and dg functors. An additive category Ais called a dg
category provided that for each X, Y ∈ A, the morphism space HomA(X , Y ) is a
complex and the compositions
HomA(Y, Z )⊗ZHomA(X, Y )−→ HomA(X, Z)
are chain maps. An additive functor F:A→Bis called a dg functor provided that
Fcommutes with the differential.
Let Abe an additive category. Denote by Cdg(A) the dg category of complexes
over Awhose mophism spaces are Hom complex defined by
HomA(X, Y )i=Y
p∈Z
HomA(Xp, Y p+i)
with differential ∂(f) = ∂Y◦f+ (−1)|f|f◦∂X.
6 JIAN LIU
The homotopy category H0(A) of Ais defined to be the category with same ob-
jects of Aand the morphism spaces are the zeroth cohomology of the corresponding
Hom complexes in A. Observe that
H0(Cdg(A)) = K(A).
1.8. Derived categories of dg categories. We briefly discuss the derived cate-
gory of a dg category. See [21] for more details.
Let Abe a small dg category. A dg module over Ais a dg functor
M:Aop −→ Cdg(Z-Mod).
Then the category of dg A-modules, denoted Moddg (A), is still a dg category; see
[21, Section 1.2]. Its homotopy category H0(Moddg(A)) is a triangulated category;
see [21, Lemma 2.2]. A dg A-module is called acyclic if M(X) is acyclic for each
object X∈ A. The derived category of Ais defined to be the Verdier quotient of
H0(Moddg (A)) by its full subcategory of acyclic dg A-modules.
We have the Yoneda embedding
Y : H0(A)−→ D(A), X 7→ HomA(−, X).
It is well-known that D(A) is compactly generated by the image of Y; see [21, 4.2].
1.9. Pretriangulated category. Keep the notation as above. The dg category
Ais called pretriangulated if Im Yis a triangulated category. In this case, H0(A)
inherits a natural triangulated structure and there is (up to direct summands) a
triangle equivalence
H0(A)∼
−→ D(A)c.
1.10. DG enhancement. Let Tbe a triangulated category and Abe a dg cate-
gory. Ais said to be a dg enhancement of Tprovided that Ais pretriangulated and
Tis triangle equivalent to H0(A) endowed with the natural triangulated structure
(see 1.9). In this case, any triangulated subcategory Sof Thas a dg enhance-
ment. Indeed, denote by A0the full dg subcategory of Aconsisting of objects in
the essential image of S. Then A0is a dg enhancement of S; see [20, Section 2.2].
Let Abe an additive category. Then Cdg(A) is pretriangulated and is a dg
enhancement of K(A).
Example 1.11. By above, Cdg(R-Mod) (resp. Cdg(R-Inj)) is a dg enhancement
of K(R-Mod) (resp. K(R-Inj)). Denote by perdg (R) the full dg subcategory of
Cdg(R-Mod) consisting of all perfect complexes. Then perdg(R) is a dg enhancement
of per(R).
Next we give an example that is used in Section 3. We write C+,f
dg (R-Inj) to
be the full subcategory of Cdg (R-Inj) formed by bounded below complexes whose
total cohomology are finitely generated R-modules. Induced by taking injective
resolution, there exists a triangle equivalence
Db(R-mod)∼
−→ H0(C+,f
dg (R-Inj)).
2. Triangulated categories of periodic complexes
Throughout the article, n≥1 is an integer. In this section, we investigate
periodic complexes. Remarkably, there exists an adjoint pair between the classi-
cal triangulated category and the corresponding triangulated category of periodic
TRIANGULATED CATEGORIES OF PERIODIC COMPLEXES AND ORBIT CATEGORIES 7
complexes. It is proved that many properties of the latter can be determined by
the former. The main result in this section is Theorem 2.12.
Let Abe an additive category, denote by Cn(A) the category of n-periodic
complexes over Awhose morphism spaces are n-periodic morphisms; see the intro-
duction. For each l∈Z, there is a canonical suspension functor [l] on Cn(A) which
maps Xto X[l] (X[l]i:= Xi+l,∂i
X[l]:= (−1)l∂i+l
X) and acts trivially on morphisms.
2.1. Homotopy category of n-periodic complexes. Let Abe an additive
category and X, Y ∈Cn(A). Two morphisms f, g :X→Yare called homotopic
if there exists a sequence {σi:Xi→Yi−1}i∈Zof morphisms over Asuch that
fi−gi=σi+1 ◦∂i
X+∂i−1
Y◦σiand σi=σi+nfor all i∈Z.
The homotopy category of n-periodic complexes over A, denoted Kn(A), is de-
fined by identifying homotopy in Cn(A). It is a triangulated category with suspen-
sion functor [1]; see [31, Section 7].
Let f:X→Ybe a morphism in Cn(A). The mapping cone C(f) of fis
C(f)i:= Xi+1 aYi, ∂i
C(f):= −∂i+1
X0
fi+1 ∂i
Y.
In Kn(A), fcan be embedded in a canonical exact triangle
Xf//Y0
1//C(f)(1 0) //X[1]
As Peng and Xiao [31, 7.1] mentioned, Cn(A) is a subcategory of C(A) (usually
not full) and Kn(A) is usually not a subcategory of K(A).
2.2. Derived category of n-periodic complexes. Let Abe an abelian cat-
egory. A n-periodic complex Xis called acylcic if it is acyclic as complex, i.e.
Hi(X) := Ker(∂i
X)/Im(∂i−1
X) = 0 for all i∈Z. The derived category of n-periodic
complexes over A, denoted Dn(A), is the Verdier quotient category of Kn(A) by its
full subcategory of acyclic n-periodic complexes.
Following the definition of the compression for the case n= 1 in [1, 1.3], we
define the compression for arbitrary n≥1; see also [36].
2.3. Compression. Let Abe an additive category with coproducts. For a complex
X∈C(A)
· · · //Xi−1∂i−1
X//Xi∂i
X//Xi+1 //· · · ,
The compression ∆(X) of Xis defined by
· · · −→ a
j≡i−1(mod n)
Xj−→ a
j≡i(mod n)
Xj−→ a
j≡i+1(mod n)
Xj−→ · · ·
with the natural differential induced by the differential of X, where the i-th com-
ponent of ∆(X) is `j≡i(mod n)Xj.
This gives an additive functor ∆ : C(A)→Cn(A).
Clearly, there is a natural exact functor ∇:Cn(A)→C(A) which maps a periodic
complex to itself. We observe that (∆,∇) is an adjoint pair. For each Xin C(A),
it is not hard to see there is an isomorphism ∇∆(X)∼
=`i∈ZX[ni]. Moreover, the
unit ηX:X→ ∇∆(X) corresponding to the adjoint pair is the composition
Xcan
−→ a
i∈Z
X[ni]∼
=∇∆(X).
8 JIAN LIU
2.4. Keep the notation as 2.3. One can check directly that ∆ and ∇preserves
homotopy, suspensions and mapping cones. Hence they induce an adjoint pair of
exact functors between the homotopy categories
K(A)Kn(A).
∆
∇
If Ais also abelian, we observe ∇induces an exact functor ∇:Dn(A)→D(A).
If further Ais an AB4 category (i.e. an abelian category with coproducts and
the coproduct is an exact functor), then ∆ preserves acyclic objects. Thus ∆
naturally induces an exact functor ∆: D(A)→Dn(A). Moreover, (∆,∇) is an
adjoint between the derived categories; see [30, Lemma 1].
2.5. If Ais an additive category with coproducts, then it can be checked directly
that both K(A) and Kn(A) have coproducts. If Ais an AB4 category, then both
D(A) and Dn(A) have coproducts; see [26, Proposition 3.5.1].
In addition, in these cases, the degree-wise coproduct of objects in K(A) (resp.
Kn(A), D(A), Dn(A)) is the categorical coproduct.
Similar results of 2.5 hold when we replace the coproduct by the product and re-
place an AB4 category by an AB4* category (i.e. an abelian category with products
and the product is an exact functor).
Lemma 2.6. (1) Let Abe an additive category with coproducts and Xbe an object
in K(A). Then Xis compact in K(A)if and only if ∆(X)is compact in Kn(A).
(2) Let Abe an AB4 category and Xbe an object in D(A). Then Xis compact
in D(A)if and only if ∆(X)is compact in Dn(A).
Proof. We prove (1). The proof of (2) is similar. First, assume Xis compact
in K(A). Since (∆,∇) is an adjoint pair and ∇preserves coproducts (c.f. 2.5), ∆
preserves compact objects; see [28, Theorem 5.1]. Thus ∆(X) is compact in Kn(A).
For the converse, assume ∆(X) is compact in Kn(A).For a class of objects
Yi(i∈I) in K(A), consider the commutative diagram
a
i∈I
HomK(A)(X, Yi)
`i∈I(ηYi)∗
can //HomK(A)(X, a
i∈I
Yi)
(η`i∈IYi)∗
a
i∈I
HomK(A)(X, ∇∆(Yi)) HomK(A)(X, ∇∆(a
i∈I
Yi))
a
i∈I
HomKn(A)(∆(X),∆(Yi))
∼
=
OO
∼
=//HomKn(A)(∆(X),∆(a
i∈I
Yi))
∼
=
OO
where the vertical isomorphisms are induced by adjoint pair (∆,∇) and the hori-
zontal one is based on the assumption. Since the unit ηM:M→ ∇∆(M) is split
injection for each M∈K(A) (see 2.3), we conclude that can is isomorphism.
TRIANGULATED CATEGORIES OF PERIODIC COMPLEXES AND ORBIT CATEGORIES 9
Example 2.7. Let n= 1. Following [1], a differential R-module (P, δP) admits a
finite projective flag if P=P0`P1`· · · `Pland δPis of the form
0∂1,0∂2,0· · · ∂l−1,0∂l,0
0 0 ∂2,1· · · ∂l−1,1∂l,1
0 0 0 · · · ∂l−1,2∂l,2
.
.
..
.
..
.
.....
.
..
.
.
0 0 0 · · · 0∂l,l−1
0 0 0 · · · 0 0
where each Piis finitely generated projective R-module. Set Fi= (P0`· · · `Pi, δP)
(0 ≤i≤l). These are differential submodules of (P, δP). It follows that (P, δP)
has a filtration
F0⊆F1⊆ · · · ⊆ Fl= (P, δP)
such that Fi/F i−1∼
=(Pi,0) for each i. Since ∆(R)=(R, 0) ∈D1(R-Mod)c(see
Lemma 2.6), the differential modules that admit finite projective flags are compact
objects in D1(R-Mod).
If Ais an abelian category, then an object Xin Dn(A) is zero if and only if
∇(X) is zero in D(A). Similar result holds in the homotopy category; see next
lemma. For the case of n= 1 and Ais the category of modules over a ring, it was
obtained by Avramov, Buchweitz and Iyengar [1, Proposition 1.8].
Lemma 2.8. Let Abe an additive category and Xbe an object in Kn(A). Then
Xis zero in Kn(A)if and only if ∇(X)is zero in K(A).
Proof. The forward direction is trivial. For the converse, assume ∇(X) is zero in
K(A). Then there exists si∈HomA(Xi, Xi−1) for all i∈Zsuch that
(1) idXi=si+1 ◦∂i
X+∂i−1
X◦si.
We define σi:Xi→Xi−1as follows
σi=(sn◦∂−1
X◦s0,if i≡0(mod n)
sjif i≡j(mod n) and 1 ≤j≤n−1.
Our aim is to show that this gives the homotopy map from idXto 0 in Kn(A).
Due to the choice of σi, it remains to check that idX0=σ1◦∂0
X+∂−1
X◦σ0and
idXn−1=σ0◦∂n−1
X+∂n−2
X◦σn−1. Indeed, these are direct consequences of (1).
Thus Xis zero in Kn(A).
Lemma 2.9. (1) Let Abe an additive category with coproducts. If K(A)is com-
pactly generated, then so is Kn(A)and it’s compactly generated by the image of
K(A)cunder the compression functor.
(2) Let Abe an AB4 category. If D(A)is compactly generated, then so is Dn(A)
and it’s compactly generated by the image of D(A)cunder the compression functor.
Proof. We prove (1). The proof of (2) is similar. Suppose K(A) is compactly
generated. Lemma 2.6 yields ∆(K(A)c)⊆Kn(A)c. Let X∈Kn(A) and
HomKn(A)(∆(K(A)c), X) = 0.
In order to show Kn(A) is compactly generated by ∆(K(A)c), we need to prove
X= 0 in Kn(A). By adjoint we have
HomK(A)(K(A)c,∇(X)) = 0.
10 JIAN LIU
Then the assumption implies ∇(X) = 0. It follows immediately from Lemma 2.8
that X= 0 in Kn(A). As required.
The following result is due to Neeman; see [28, Theorem 4.1] and [29, Theorem
8.6.1].
2.10. Let Sbe a compactly generated triangulated category and F:S → T be an
exact functor between triangulated categories. Then
(1) Fhas a right adjoint if and only if Fpreserves coproducts.
(2) Fhas a left adjoint if and only if Fpreserves products.
As we assume Ris a left noetherian ring, the direct sum of injective R-modules
is still injective; see [16, Theorem 3.1.17]. Hence R-Inj is an additive category with
coproducts.
2.11. Krause proved that K(R-Inj) and the full subcategory of K(R-Inj) formed
by acyclic complexes (denoted Kac(R-Inj)) are compactly generated triangulated
categories; see [25, Proposition 2.3, Corollary 5.4]. Moreover, Krause [25, Corollary
4.3] observed that the canonical sequence
Kac(R-Inj)inc
−→ K(R-Inj)Q
−→ D(R-Mod)
induces a recollement; the definition of recollement is recalled in 1.3.
Let Kac
n(R-Inj) denote the full subcategory of Kn(R-Inj) formed by acyclic com-
plexes. This is a localizing subcategory of K(R-Inj). Next, we give the periodic
version of Krause’s result in 2.11.
Theorem 2.12. Let Rbe a left noetherian ring. Then
(1) Kac
n(R-Inj),Kn(R-Inj)and Dn(R-Mod)are compactly generated triangulated
categories.
(2) The sequence
Kac
n(R-Inj)inc
−→ Kn(R-Inj)Q
−→ Dn(R-Mod)
induces a recollement
Kac
n(R-Inj)Kn(R-Inj)Dn(R-Mod).
inc Q
Proof. (1) Combine with 2.11, it follows immediately from Lemma 2.9 that Kn(R-Inj)
and Dn(R-Mod) are compactly generated. Also, with the same proof of Lemma 2.9,
Kac
n(R-Inj) is compactly generated.
(2) Next, we borrow Krause’s idea in the proof of [25, Corollary 4.3]. Since Q
preserves coproducts and products (c.f. 2.5), Q has both a left adjoint and a right
adjoint by (1) and 2.10. Combine with 1.4, it remains to show Qinduces a triangle
equivalence Kn(R-Inj)/Kac
n(R-Inj)∼
=Dn(R-Mod). Again 1.4 yields this is equivalent
to show the right adjoint of Qis fully faithful.
Denote by Qρthe right adjoint of Q. It is clear that the inclusion functor
J:Kn(R-Inj)→Kn(R-Mod) preserves products. Then Lemma 2.9 and 2.10 imply
that Jhas a left adjoint Jλ. Hence there are adjoint pairs
Kn(R-Mod)Jλ//Kn(R-Inj)
J
oo
Q//Dn(R-Mod).
Qρ
oo
TRIANGULATED CATEGORIES OF PERIODIC COMPLEXES AND ORBIT CATEGORIES 11
Since Jis a fully faithful right adjoint of Jλ, HomKn(R-Mod)(KerJλ,Kn(R-Inj)) = 0.
This implies KerJλ⊆Kac
n(R-Mod). Thus for each M∈Kn(R-Mod), the unit
ηM:M→Jλ(M) is a quasi-isomorphism. It follows that Q(M)∼
=(Q◦Jλ)(M).
That is, Q◦Jλis isomorphic to the localization functor Kn(R-Mod)→Dn(R-Mod).
By 1.4,J◦Qρis fully faithful. As Jis also fully faithful, we infer that Qρis too.
This completes the proof.
2.13. Let Abe an abelian category. A n-periodic complex X∈Kn(A) is called
homotopy injective (resp. homotopy projective) if
HomKn(A)(Y, X ) = 0 (resp. HomKn(A)(X, Y ) = 0)
for each acyclic complex Y∈Kn(A). Denote by Ki
n(A) (resp. Kp
n(A)) the full sub-
category of Kn(A) consisting of all homotopy injective (resp. homotopy projective)
complexes. They naturally inherit the structure of triangulated categories.
Let Qρdenote the right adjoint of Q:Kn(R-Inj)→Dn(R-Mod). Using the
adjointness, it is easy to check Qρ(X) is homotopy injective for each n-periodic
complex Xand the unit X→Qρ(X) is a quasi-isomorphism. Thus we get:
Corollary 2.14. Qρinduces a triangle equivalence
Dn(R-Mod)∼
−→ Ki
n(R-Mod).
Remark 2.15. (1) Tang and Huang [37, Theorem 5.11] proved an analog of the
above result for higher differential objects. Two results coincide when n= 1.
(2) Stai [36, Section 3] obtained the dual version of the above result. That is,
the localization functor Q:Kn(R-Mod)→Dn(R-Mod) has a left adjoint and the
left adjoint induces a triangle equivalence
Dn(R-Mod)∼
−→ Kp
n(R-Mod).
3. The triangulated hull of the orbit categories
In this section, we prove Theorem 1from the introduction.
3.1. Let Abe an additive category and T:A→Abe an autoequivalence. As
mentioned in the introduction, the objects in the same T-orbit are isomorphic in
the orbit category A/T . We remind the reader that in general Fis not isomorphic
to the identity functor in the orbit category A/T ; see [24] and [36, Proposition 5.6].
However, there is a natural isomorphism π∼
=π◦T, where π:A→A/T is the
projection functor. Moreover, this gives rise to the universal property of the orbit
category:
If the functor F:A→Bsatisfies F◦T∼
=F, then there exists a natural functor
F:A/T → B such that F◦π=F.
3.2. Let Abe an additive category. Recall the degree shift functor (n) on C(A):
for a complex X,X(n)i:= Xi+n,∂i
X(n):= ∂i+n
X; (n) acts trivially on morphisms.
There is a natural isomorphism X[n]∼
=
−→ X(n) which maps x∈Xito (−1)nix.
If further Ais an additive category with coproducts (resp. AB4 category), then
∆◦[n]∼
=∆◦(n) = ∆. By the universal property of the orbit category, ∆ induces
∆: K(A)c/[n]−→ Kn(A)c(resp. ∆ : D(A)c/[n]−→ Dn(A)c).
We first strength Lemma 2.9 to the following result; compare [36, Lemma 3.13].
12 JIAN LIU
Proposition 3.3. (1) Let Abe an additive category with coproducts. If K(A)is
compactly generated, then there is a fully faithful embedding
∆: K(A)c/[n]−→ Kn(A)c
and Kn(A)is compactly generated by its image.
(2) Let Abe an AB4 category. If D(A)is compactly generated, then there is a
fully faithful embedding
∆: D(A)c/[n]−→ Dn(A)c
and Dn(A)is compactly generated by its image.
Proof. We prove (1). The proof of (2) is similar. By Lemma 2.9, it remains to show
∆ is fully faithful. For X, Y ∈K(A)c, we have
HomKn(A)(∆(X),∆(Y)) ∼
=HomK(A)(X, ∇∆(Y))
∼
=a
i∈Z
HomK(A)(X, Y [ni]),
where the second isomorphism is because Xis compact and ∇∆(Y)∼
=`i∈ZY[ni]
(see 2.3). It follows immediately from the isomorphism above that the induced
functor ∆: K(A)c/[n]→Kn(A)cis fully faithful.
3.4. A homogeneous morphism fin a dg category is called closed if fis of degree
0 and ∂(f) = 0. We call a natural transformation ηbetween dg functors closed if
ηXis closed for all X∈ A.
The following is the universal property of the orbit category of the dg category.
Lemma 3.5. Let T:A → A be a dg autoequivalence of a dg category Aand
F:A → B be a dg functor between dg categories such that there exists a closed
natural isomorphism η:F◦T→F. Then Finduces a dg functor F:A/T → B
such that F◦π=F, where π:A→A/T is the projection functor.
Proof. By assumption, ηinduces closed isomorphisms F◦Ti∼
→F(denoted ηi).
We define F:A/T → B as follows: F(M) = F(M) for all M∈ A; for each
homogeneous morphism α:M→Tj(N) in HomA/T (M, N ), F(α) is defined by the
following composition
F(M)F(α)//F(Tj(N)) ηj
N
∼
=//F(N).
By assumption Fis dg functor and ηis closed, we have a commutative diagram
HomA(X, T j(Y)) F//
∂
HomB(F(X), F (Tj(Y))) (ηj
Y)∗//
∂
HomB(F(X), F (Y))
∂
HomA(X, T j(Y)) F//HomB(F(X), F (Tj(Y))) (ηj
Y)∗//HomB(F(X), F (Y)).
This means Fis a dg functor. Clearly F◦π=F.
Example 3.6. Let Cbe an additive category. Set A=Cdg(C) and B=Cdg (Z).
The suspension functor [n] : Cdg (C)→Cdg(C) is a dg autoequivalence. If Xis a
n-periodic complex in A, then
HomA(Y[n], X)∼
=HomA(Y(n), X)∼
=HomA(Y, X ),
TRIANGULATED CATEGORIES OF PERIODIC COMPLEXES AND ORBIT CATEGORIES 13
where the first isomorphism is induced by Y[n]∼
=Y(n) and the second one maps
α:Y(n)i→Xjto α:Yn+i→Xn+j. Set F= HomA(−, X ) and T= [n]. We
conclude that there is a closed natural isomorphism F◦T∼
=F. This is an example
that satisfies the assumption of Lemma 3.5.
3.7. Let Rbe a left noetherian ring. Krause [25, Proposition 2.3] proved that
K(R-Inj) is a compactly generated. Moreover, he observed that the localization
functor K(R-Mod)→D(R-Mod) induces a triangle equivalence
K(R-Inj)c∼
−→ Db(R-mod).
The inverse is induced by taking injective resolution. In particular, K(R-Inj)cis the
full subcategory of K(R-Inj) consisting of complexes with finitely generated total
cohomology.
Recall that perdg (R) is the dg category of perfect complexes over Rand C+,f
dg (R-Inj)
is the dg category of bounded below complexes of injective R-modules with finitely
generated total cohomology. They are dg enhancements of per(R) and Db(R-mod)
respectively; see 3.7 and Example 1.11.
Next, we realize examples of triangulated categories in Theorem 2.12 as derived
categories of dg categories; compare [13, Theorem 2.2] and [25, Appendix A].
Theorem 3.8. Let Rbe a left noetherian ring. There are triangle equivalences
Kn(R-Inj)∼
−→ D(C+,f
dg (R-Inj)/[n]) and Dn(R-Mod)∼
−→ D(perdg(R)/[n]).
Proof. We prove the first equivalence. The proof of the second one is similar.
For each complex Xof R-modules, set X∧= HomR(−, X ). By Lemma 3.5 and
Example 3.6, the map I7→ I∧|C+,f
dg (R-Inj)/[n]induces an exact functor
Φ: Kn(R-Inj)−→ D(C+,f
dg (R-Inj)/[n]).
The functor Φ preserves coproducts. Indeed, for each object J∈C+,f
dg (R-Inj) and
a family Ii∈Kn(R-Inj) (i∈S), we have isomorphisms
Hl(Φ(a
i∈S
Ii)(J)) ∼
=HomK(R-Inj)(J[−l],a
i∈S
Ii)
∼
=a
i∈S
HomK(R-Inj)(J[−l], Ii)
∼
=a
i∈S
Hl(Φ(Ii)(J))
∼
=Hl(a
i∈S
Φ(Ii)(J))
for each l∈Z, where the second isomorphism is because Jis a compact object in
K(R-Inj); see 3.7. Hence in D(C+,f
dg (R-Inj)/[n]),
Φ(a
i∈S
Ii)∼
=a
i∈S
Φ(Ii).
14 JIAN LIU
We observe that there exists a commutative diagram
(2) H0(C+,f
dg (R-Inj/[n]) ∆//
Y
Kn(R-Inj)c
inc
D(C+,f
dg (R-Inj)/[n]) Kn(R-Inj)
Φ
oo
,
where Y is the Yoneda embedding. From Proposition 3.3,
∆: H0(C+,f
dg (R-Inj)/[n]) −→ Kn(R-Inj)c
is fully faithful and Kn(R-Inj) is compactly generated by the image of ∆. As
D(C+,f
dg (R-Inj)/[n]) is compactly generated by the image of Y, we conclude that
Φ is an equivalence by Lemma 1.6.
Remark 3.9. (1) Let kbe a field. When Ris a finite dimensional k-algebra with
finite global dimension, the triangle equivalence Dn(R-Mod)∼
→D(perdg (R)/[n])
was proved by Stai with a different method; see [36, Section 4].
(2) Let B(resp. A) denote perdg (R)/[n] (resp. C+,f
dg (R-Inj)/[n]). We can regard B
as a full dg subcategory of A; see 3.7. Then we can form a dg quotient category A/B;
see Keller’s construction in [22, Section 4]. The restriction functor D(A/B)→D(A)
is fully faithful and it’s essential image is equal to the kernel of the restriction functor
D(A)→D(B); see [22, Section 4] and [14, Proposition 4.6]. Combine this with
Theorem 2.12 and Theorem 3.8, we conclude that there is a triangle equivalence
Kac
n(R-Inj)'D(A/B).
3.10. Let Abe a dg enhancement of a triangulated category T. Assume the
functor F:T → T is an autoequivalence and it lifts to a dg equivalence A → A
(still denoted F). Then we can form an orbit category A/F which naturally inherits
a structure of the dg category and gives the desired enhancement of T/F . Hence
T/F ∼
−→ H0(A/F)Y
−→ D(A/F),
where Y is the Yoneda embedding. The triangulated hull of T/F is chosen to be
the triangulated subcategory of D(A/F ) generated by the image of Y. It is up to
direct summands equivalent to D(A/F )c. Thus we use D(A/F )cto represent the
triangulated hull of T/F in the article; see Keller’s definition in [23, Section 5] for
a broader definition of the triangulated hull.
The inverse of the equivalence K(R-Inj)c∼
→Db(R-mod) (see 3.7) is induced by
taking injective resolution. We denote it by i.
Corollary 3.11. Let Rbe a left noetherian ring. Induced by the compression of
complexes, the functors
∆◦i:Db(R-mod)/[n]−→ Kn(R-Inj)cand ∆: per(R)/[n]−→ Dn(R-Mod)c
are the embedding of the orbit categories into their triangulated hull.
Proof. For the first one: C+,f
dg (R-Inj) is the dg enhancement of Db(R-mod). Then
C+,f
dg (R-Inj)/[n] is the desired dg enhancement of Db(R-mod)/[n]. Given this and
3.10, the desired result follows from Theorem 3.8.
For the second one: perdg(R) is the dg enhancement of per(R). Then the re-
maining proof is parallel to the first one.
TRIANGULATED CATEGORIES OF PERIODIC COMPLEXES AND ORBIT CATEGORIES 15
Remark 3.12. Fix a locally noetherian Grothendieck category A. That is, Ais an
AB4 category with exact direct colimit, and Ahas a set A0of noetherian objects
such that every object in Ais a quotient of a coproduct of objects in A0. Denote
by A-noeth the full subcategory of Aformed by noetherian objects, and by A-Inj
the full subcategory of Aformed by injective objects. With the same argument
of Theorem 3.8, we have Kn(A-Inj)∼
→D(C+,f
dg (A-Inj)/[n]),where C+,f
dg (A-Inj) is the
dg category of bounded below complexes of injective objects with noetherian total
cohomology. Then the same proof of Corollary 3.11 yields that the compression of
complexes
∆◦i:Db(A-noeth)/[n]−→ Kn(A-Inj)c
induces an embedding of the orbit category into its triangulated hull.
3.13. It was proved by Stai [36, Lemma 3.5] when Rhas finite global dimension,
then every object in C1(R-mod) is quasi-isomorphic to one admitting a finite pro-
jective flag (see definition in Example 2.7). Thus D1(R-mod) is equal to the thick
subcategory generated by ∆(R). As he also mentioned, this extends to any n≥1.
Combine with Proposition 3.3, there is a natural triangle equivalence
Dn(R-mod)∼
−→ Dn(R-Mod)c.
With the same method of Stai, one can show: if Ais an abelian category with
enough projective objects and every object in Ahas finite projective dimension,
then
Dn(A) = thickDn(A)({∆(P)|Pis projective in A}).
The following result was proved independently by Stai [36, Theorem 4.3] and
Zhao [38, Theorem 2.10] when Ris a finite dimensional algebra with finite global
dimension over a field.
Corollary 3.14. Let Rbe a left noetherian ring with finite global dimension, then
the compression of complexes
∆: Db(R-mod)/[n]−→ Dn(R-mod)
is an embedding of the orbit category into its triangulated hull.
Proof. When Rhas finite global dimension, Db(R-mod) = per(R). Combine with
3.13, the desired result follows from Corollary 3.11.
3.15. When Ris hereditary, Db(R-mod)/[n] is triangulated and hence it is (up to
direct summands) equivalent to its triangulated hull; see [23, Theorem 1] and [36,
Proposition 5.3]. When n= 1 and Ris a path algebra of finite connected acyclic
quiver, Ringel and Zhang [35, Theorem 1] proved that Db(R-mod)/[1] is equivalent
to a stable category of certain Frobenius category.
4. Derived equivalence as derived tensor product
For two rings Aand B, the purpose of this section is to compare the triangle
equivalences D(A-Mod)'D(B-Mod) and Dn(A-Mod)'Dn(B-Mod). It turns out
that these two equivalences are closely related; see Proposition 4.8.
4.1. Tensor products. Let Xbe a complex of B-Abimodules. For a n-periodic
complex Yin Cn(A-Mod), the tensor product X⊗AYis a n-periodic complex in
Cn(B-Mod). Thus X⊗A−gives a functor
XA−:Cn(A-Mod)−→ Cn(B-Mod).
16 JIAN LIU
The notation XA−is to distinguish it from X⊗A−:C(A-Mod)→C(B-Mod).
Moreover, the following diagram
(3) C(A-Mod)
∆
X⊗A−//C(B-Mod)
∆
Cn(A-Mod)XA−//Cn(B-Mod).
is commutative; see [1, (1.9.4)] for the case n= 1.
4.2. Keep the same assumption as 4.1. Since XA−preserves homotopy, sus-
pensions and mapping cones, it induces an exact functor XA−:Kn(A-Mod)→
Kn(B-Mod). We define the derived tensor product XL
A−by the following com-
position
Dn(A-Mod)p//Kn(A-Mod)XA−//Kn(B-Mod)Q//Dn(B-Mod)
where pis the left adjoint of the canonical functor Kn(A-Mod)→Dn(A-Mod);
see Remark 2.15 for its existence. The compression functor ∆: K(A-Mod)→
Kn(A-Mod) preserves homotopy projective objects because its right adjoint pre-
serves acyclic complexes. Combine this with (3), we observe that there exists a
commutative diagram
D(A-Mod)
∆
X⊗L
A−//D(B-Mod)
∆
Dn(A-Mod)XL
A−//Dn(B-Mod).
For a triangulated category T, we write ΣTto be the suspension functor of T.
Lemma 4.3. Let F:T → T 0be an exact functor between triangulated categories.
Then Fis fully faithful if and only if the induced functor F:T/Σn
T→ T 0/Σn
T0is
fully faithful. Moreover, Fis an equivalence if and only if Fis an equivalence.
Proof. Since F(X) = F(X) for each object X∈ T , the second statement follows
from the first one. Fix objects X, Y ∈ T , we observe that the map
F:a
i∈Z
HomT(X, Σni
TY)−→ a
i∈Z
HomT0(F(X),Σni
T0F(Y))
is the direct sum of the following composition maps
HomT(X, Σni
TY)F
−→ HomT0(F(X), F (Σni
TY)) ∼
=HomT0(F(X),Σni
T0F(Y)),
where the isomorphisms are induced by the canonical isomorphism FΣT∼
=ΣT0F.
The desired result follows.
Lemma 4.4. Let F, G, Φ1,Φ2are exact functors between compactly generated tri-
angulated categories such that the following diagram
S1
F//
Φ1
S2
Φ2
T1
G//T2
TRIANGULATED CATEGORIES OF PERIODIC COMPLEXES AND ORBIT CATEGORIES 17
commutes. Assume F , G preserves coproducts and Φipreserves compact objects for
i= 1,2. Moreover, we assume Φiinduces a fully faithful functor
Φi:Sc
i/Σn
Si−→ T c
i
such that Tiis compactly generated by its image. Then we have implications:
(1) Fis an equivalence ⇒Gis an equivalence.
(2) Fpreserves compact objects and Gis an equivalence ⇒Fis fully faithful.
Proof. Combine with the assumption, the condition of (1) or (2) implies the diagram
Sc
1/Σn
S1
F//
Φ1
Sc
2/Σn
S2
Φ2
Tc
1
G//Tc
2.
commutes. Indeed, this is trivial for (2). For (1), it remains to show that Gpreserves
compact objects. The assumption and the condition of (1) yield G(Im Φ1)⊆ T c
2.
Since T1is compactly generated by Im Φ1, we have thickT1(Im Φ1) = Tc
1; see 1.5.
On the other hand, the full subcategory {X∈ T1|G(X)∈ T c
2}of T1is thick. Thus
Gpreserves compact objects.
(1) Assume Fis equivalence. Then Fis an equivalence. For i= 1,2, Im Φiis a
compact generating set of Ti. Clearly Im Φiis closed under suspensions. Then we
apply Lemma 1.6 to conclude that G:T1→ T2is an equivalence.
(2) By assumption, Ginduces an equivalence G:Tc
1
∼
−→ T c
2.Combine with Φi
is fully faithful for i= 1,2, we get that Fis fully faithful. Then Lemma 4.3 yields
the functor F:Sc
1→ Sc
2is fully faithful. According to Lemma 1.6,Fis fully
faithful.
Example 4.5. Let Rbe a commutative noetherian ring with a dualizing complex
ω. Iyengar and Krause [18, Theorem I] proved that
ω⊗R−:K(R-Proj)−→ K(R-Inj)
is a triangle equivalence. Combine this result with Proposition 3.3 and Lemma 4.4,
we immediately get that there is a triangle equivalence
ωR−:Kn(R-Proj)∼
−→ Kn(R-Inj).
Let Thick Tbe the lattice of thick subcategories of a triangulated category T.
4.6. Suppose Ais an additive category with coproducts (resp. AB4 category).
We write Tto be K(A) (resp. D(A)) and T0to be Kn(A) (resp. Dn(A)). For a
thick subcategory Sof Tc, we let F(S) be the smallest thick subcategory of T0c
containing all objects ∆(X) such that X∈ S. For a thick subcategory S0of T0c,
we let G(S0) be the smallest thick subcategory of Tccontaining all objects Xin
Tcsuch that ∆(X)∈ S0.Thus we have maps of lattices
Thick TcThick T0c
F
G
.
Next result is inspired by a recent result of Iyengar, Letz, Pollitz and the author
[19, Corollary 5.9]. It is important in the proof of Proposition 4.8.
Lemma 4.7. Keep the assumptions as 4.6. Then G◦F= id. In particular, the
map of lattices F:Thick Tc→Thick T0cis injective.
18 JIAN LIU
Proof. Fix a thick subcategory Sof Tc. In order to show GF (S) = S, it suffices to
show for X, Y ∈ T c,
X∈thickT(Y)⇐⇒ ∆(X)∈thickT0(∆(Y)).
The forward direction is trivial; see [2, Lemma 2.4]. For the converse, assume
∆(X) is an object in thickT0(∆(Y)). Then we have ∇∆(X)∈thickT(∇∆(Y)).
Since ∇∆(M)∼
=`i∈ZM[ni] for each M∈ T (see 2.3), Xis in the localizing
subcategory of Tgenerated by Y. As X, Y are compact objects in T, we conclude
by 1.5 that Xis in thickT(Y). As required.
Proposition 4.8. Let A, B be two rings and Xbe a complex of B-A-bimodules.
Then the functor X⊗L
A−:D(A-Mod)→D(B-Mod)is a triangle equivalence if and
only if the functor XL
A−:Dn(A-Mod)→Dn(B-Mod)is a triangle equivalence.
Proof. First, assume X⊗L
A−is a triangle equivalence. It follows immediately from
Proposition 3.3,4.2 and Lemma 4.4 that XL
R−is a triangle equivalence.
Now, assume XL
A−is a triangle equivalence. It restricts to an equivalence
between the full categories of compact objects. Combine with the commutative
diagram in 4.2, we conclude by Lemma 2.6 that X⊗L
A−:D(A-Mod)→D(B-Mod)
preserves compact objects. It follows from Proposition 3.3 and Lemma 4.4 that
X⊗L
A−is fully faithful.
To show X⊗L
A−is equivalence, by Lemma 1.6 it remains to show the essential
image of X⊗L
A−:D(A-Mod)c→D(B-Mod)c, denoted S, is a compact generating
set of D(B-Mod). Consider the commutative diagram
D(A-Mod)c
∆
X⊗L
A−//D(B-Mod)c
∆
Dn(A-Mod)c∼
XL
A−//Dn(B-Mod)c
we apply Proposition 3.3 to get thickDn(B-Mod)(∆(S)) = Dn(B-Mod)c. Then Lemma
4.7 yields the smallest thick subcategory of D(B-Mod)ccontaining Sis the whole of
D(B-Mod)c. Hence Sis a compact generating set of D(B-Mod). As required.
Two rings are derived equivalent provided that D(A-Mod) and D(B-Mod) are
equivalent as triangulated categories.
4.9. It is an open question that whether any triangle equivalence
D(A-Mod)∼
−→ D(B-Mod)
is isomorphic to a derived tensor functor X⊗L
A−, where Xis a complex of B-A
bimodules. Such derived equivalence is called a standard equivalence.
However, if A, B are two algebras over a commutative ring ksuch that they
are flat as k-modules, then any triangle equivalence D(A-Mod)∼
→D(B-Mod) is
standard; see [21, Corollary 9.2] and [34, Section 3].
Combine with Proposition 4.8, the statement in 4.9 implies the following result.
Corollary 4.10. Let kbe a commutative ring and A, B be flat k-algebras. If A
and Bare derived equivalent, then Dn(A-Mod)and Dn(B-Mod)are equivalent as
triangulated categories.
TRIANGULATED CATEGORIES OF PERIODIC COMPLEXES AND ORBIT CATEGORIES 19
If Ais a left noetherian ring with finite global dimension, then Dn(A-Mod)c=
Dn(A-mod); see 3.13. As a consequence of Corollary 4.10, we have:
Corollary 4.11. Let kbe a commutative ring and A, B be flat k-algebras. If A, B
are noetherian with finite global dimensions and A, B are derived equivalent, then
Dn(A-mod)and Dn(B-mod)are equivalent as triangulated categories.
Remark 4.12. The above corollary extends a result of Zhao [38, Theorem]. In her
paper, she proved the above result holds for finite dimensional algebras with finite
global dimensions over a field.
5. Koszul duality for periodic complexes
Throughout this section, kis a field and Sis the graded polynomial algebra
k[x1, . . . , xc] with deg(xi) = 1. We let Λ denote the Koszul dual of S. More
precisely, Λ is the graded exterior algebra over kon variables ξ1, . . . , ξcof degree
−1.
For a graded algebra A, denote by A-Gr (resp. A-gr) the category of left (resp.
finitely generated left) graded A-modules. A graded A-module is called graded-
injective provided that it is an injective object in A-Gr. It is well-known that A-Gr
has enough projective objects and enough injective objects; see [9, Section 1.5 and
Theorem 3.6.2].
Let A-GrInj denote the category of graded-injective A-module. As Λ is noe-
therian, one can show that the direct sum of graded-injective Λ-module is graded-
injective; the proof is parallel to the non-graded version [16, Theorem 3.1.17].
The main purpose of this section is to give the following periodic version of the
Koszul duality.
Theorem 5.1. There exists a triangle equivalence
Kn(Λ-GrInj)∼
−→ Dn(S-Gr).
We give the proof of the above result at the end of this section. As a consequence,
we have:
Corollary 5.2. There is an embedding
Db(Λ-gr)/[n]−→ Dn(S-gr)
of the orbit category into its triangulated hull.
Before giving the proof of the corollary, we recall a result.
5.3. Due to Krause [25, Proposition 2.3], K(Λ-GrInj) is compactly generated. More-
over, the localization functor K(Λ-Gr)→D(Λ-Gr) induces a triangle equivalence
K(Λ-GrInj)c∼
−→ Db(Λ-gr).
Its inverse is induced by taking grade-injective resolution, denoted i.
Proof of Corollary 5.2.Keep the notation as 5.3, Remark 3.12 implies that the
compression
∆◦i:Db(Λ-gr)/[n]−→ Kn(Λ-GrInj)c
induces an embedding of Db(Λ-gr)/[n] into its triangulated hull. It follows from
Theorem 5.1 that Kn(Λ-GrInj)cis triangle equivalent to Dn(S-Gr)c. Choose A=
S-gr in 3.13, we conclude that Dn(S-gr) is the smallest thick subcategory containing
20 JIAN LIU
∆(S(i)) for all i∈Z. It is precisely Dn(S-Gr)c; see 1.5 and Proposition 3.3. This
completes the proof.
5.4. Recall the functor Φ: C(S-Gr)→C(Λ-Gr); see [5] or [15] for more details. Set
(−)∗:= Homk(−, k). For a graded S-module M=`i∈ZMi, Φ(M) is defined by
the complex
· · · ∂
−→ Λ∗⊗kMi−1
∂
−→ Λ∗⊗kMi
∂
−→ Λ∗⊗kMi+1
∂
−→ · · · ,
where ∂(f⊗m) := (−1)l+iPc
j=1 ξjf⊗xjmfor f∈(Λ∗)land m∈Mi; the sign
makes sure that ∂is Λ-linear. For a complex M:· · · d
−→ Mj−1d
−→ Mjd
−→ Mj+1 d
−→
· · · in C(S-Gr), Φ(M) is defined by the total complex of the double complex
(4) .
.
.
1⊗d
.
.
.
1⊗d
· · · ∂//Λ∗⊗kMj
i
∂//
1⊗d
Λ∗⊗kMj
i+1
1⊗d
∂//· · ·
· · · ∂//Λ∗⊗kMj+1
i
∂//
1⊗d
Λ∗⊗kMj+1
i+1
∂//
1⊗d
· · ·
.
.
..
.
.
where the l-th component of Φ(M) is `i+j=lΛ∗⊗kMj
i.
5.5. Keep the notation as above. Since Φ preserves homotopy, suspensions and
mapping cones, it induces an exact functor Φ: K(S-Gr)→K(Λ-Gr). The image of
this functor lies in K(Λ-GrInj) because Λ∗is graded-injective. Bernstein, Gel’fand
and Gel’fand [7, Theorem 3] proved that Φ naturally induces a triangle equivalence
Φ: Db(S-gr)∼
−→ Db(Λ-gr);
see also [5, Theorem 2.12.1]. This is known as the BGG correspondence. Moreover,
it fits into the following commutative diagram
(5) Db(S-gr)∼//
inc
Db(Λ-gr)
i
D(S-Gr)//K(Λ-GrInj),
where the bottom map is the following composition
D(S-Gr)p
−→ K(S-Gr)Φ
−→ K(Λ-GrInj),
here pis the left adjoint of the localization functor K(S-Gr)→D(S-Gr) (see [8,
Proposition 2.12] for its exisence).
The essential images of the vertical functors in (5) are precisely the full subcat-
egories of compact objects in the bottom categories. This is clear for the left one
as the global dimension of Sis finite. See 5.3 for the right one. Combine with
that Φ ◦ppreserves coproducts, Lemma 1.6 yields Φ ◦pis an equivalence; see [25,
Example 5.7].
TRIANGULATED CATEGORIES OF PERIODIC COMPLEXES AND ORBIT CATEGORIES 21
Now we define the exact functor Dn(S-Gr)→Kn(Λ-GrInj).
5.6. For a n-periodic complex M∈Cn(S-Gr), the total complex Φ(M) (see (4)) is
an-periodic complex in Cn(Λ-Gr). Therefore this gives a functor
Φ0:Cn(S-Gr)−→ Cn(Λ-Gr)
which maps Mto Φ(M). Also, Φ0induces an exact functor Φ0:Kn(S-Gr)→
Kn(Λ-Gr) between the homotopy categories and its image lies in Kn(Λ-GrInj). Con-
sider the following composition
Dn(S-Gr)p0
−→ Kn(S-Gr)Φ0
−→ Kn(Λ-GrInj),
where p0is the left adjoint of the localization functor Kn(S-Gr)→Dn(S-Gr); its
existence can refer the non-graded version of Remark 2.15.
Proof of Theorem 5.1.It follows from Proposition 3.3 and 5.3 that Dn(S-Gr) and
Kn(Λ-GrInj) are compactly generated triangulated categories. Combine with 5.5,
we observe that there exists a commutative diagram
D(S-Gr)Φ◦p
∼//
∆
K(Λ-GrInj)
∆
Dn(S-Gr)Φ0◦p0
//Kn(Λ-GrInj).
Since Φ0◦p0preserves coproducts, Proposition 3.3 and Lemma 4.4 imply Φ0◦p0is
a triangle equivalence.
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School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240,
P.R. China.
Email address:liuj231@sjtu.edu.cn
