DERIVED EQUIVALENCES AND
GORENSTEIN PROJECTIVE DIMENSIONHIROTAKA KOGAAbstract. In this note, we introduce the notion of complexes of finite Gorenstein projective dimension and show that a derived equivalence induces an equivalence between
the full triangulated subcategories consisting of complexes of finite Gorenstein projective
dimension provided that the equivalence satisfies a certain condition.

This work is based on a joint work with M. Hoshino.
Derived equivalences appear in various fields of current research in mathematics. For
instance, in [3] Beilinson showed that there exists an algebra A such that the derived
category of A is triangle equivalent to the derived category of coherent sheaves on Pn , in
[5] Broué conjectured abelian defect group conjecture and in [12] Kontsevich formulated
mirror symmetry in terms of derived equivalences. So it is more and more important
to study derived equivalences. It is natural to ask when two abelian categories are derived equivalent. A way to answer this question is to compare invariants under derived
equivalences. It is well-known that for derived equivalent rings finiteness of selfinjective
dimension is an invariant (see e.g. [11]). Finiteness of selfinjective dimension is closely
related to Gorenstein projective dimension (see [9, 10]). So one can expect that there aresome invariants associated with Gorenstein projective dimension.
In this note, we introduce the notion of complexes of finite Gorenstein projective dimension and show that a derived equivalence induces an equivalence between the full triangulated subcategories consisting of complexes of finite Gorenstein projective dimension
provided that the equivalence satisfies a certain condition. Let A, B be abelian categorieswith enough projectives. Denote by PA the full subcategory of A consisting of projectiveobjects and by GP A the full subcategory of A consisting of Gorenstein projective objects.
A complex X • ∈ Db (A) is said to have finite Gorenstein projective dimension if it isisomorphic to a bounded complex of Gorenstein projective objects in Db (A) (see Definition 11). We denote by Db (A)fGpd the full triangulated subcategory of Db (A) consistingof complexes of finite Gorenstein projective dimension. Let F : Db (A) → Db (B) be atriangle equivalence. Assume that there exists an integer a > 0 such that
HomD(B) (F P, Q[i]) = 0 = HomD(B) (Q, F P [i])for all P ∈ PA and Q ∈ PB unless −a ≤ i ≤ a. Then our main result states that Finduces an equivalence between Db (A)fGpd and Db (B)fGpd (see Theorem 18). Note thatin case A and B are module categories then such an integer a always exists for any derived
The detailed version of this paper will be submitted for publication elsewhere.

–1–equivalence F . As corollaries we have the following: the equivalence F induces a triangle
equivalence between GP A /PA and GP B /PB (see Corollary 19); GP A = PA if and onlyc A = GP A if and only if GPc B = GP B , and GP
c A = PA if and only if
if GP B = PB , GPc B = PB where GPc A stands for the full subcategory of A consisting of objects X ∈ A
GPiwith ExtA (X, PA ) = 0 for i > 0 (see Corollary 20); and letting A, B be rings, A and B
are derived equivalent if and only if Db (Mod-A)fGpd and Db (Mod-B)fGpd are equivalentas triangulated categories (see Corollary 22).
This work is supported by JSPS KAKENHI Grant Number 26887034.
1. Preliminaries
In this note, complexes are cochain complexes and objects are considered as complexes
concentrated in degree zero. Let A be an abelian category with enough projectives. We
denote by PA the full subcategory of A consisting of all projective objects in A. We denoteby D(A) the derived category of complexes over A and by Db (A) the full triangulated
subcategory of D(A) consisting of complexes with bounded cohomology. Also, we denoteby Hom•A (−, −) the associated single complex of the double hom complex.
For an additive category X we denote by K(X ) the homotopy category of cochain
complexes over X and by K+ (X ) and Kb (X ) the full triangulated subcategories of K(X )consisting of bounded below and bounded complexes, respectively.
For a ring A we denote by Mod-A the category of right A-modules.
We refer to [4], [8] and [14] for basic results in the theory of derived categories.
Definition 1. For a complex X • , we denote by Zi (X • ) and Hi (X • ) the ith cycle and theith cohomology of X • , respectively.
Definition 2 ([8]). A complex X • ∈ Db (A) is said to have finite projective dimension
if HomD(A) (X • [−i], −) vanishes on A for i ≫ 0. We denote by Db (A)fpd the épaisse
subcategory of Db (A) consisting of complexes of finite projective dimension.
Note that the canonical functor K(A) → D(A) gives rise to equivalences of triangulated
categories∼Kb (PA ) −→ Db (A)fpd .
Let C be a full subcategory of A.
Definition 3. A complex X • ∈ K(A) is said to be HomA (−, C)-exact if HomA (X • , C) isexact for all C ∈ C.
Definition 4. An exact sequence 0 → M → C 0 → C 1 → · · · → C n → · · · in A is said tobe a C-coresolution of M ∈ A if C i ∈ C for all i and the exact sequence is HomA (−, C)exact.
Definition 5 ([1, 7]). An object M ∈ A is said to be Gorenstein projective if M admitsa P-coresolution. We denote by GP A the full subcategory of A consisting of Gorensteinprojective objects M ∈ A.
We refer to [6] for basic facts on Gorenstein projective dimension.

–2–2. Gorenstein projective dimension
In this section, we study Gorenstein projective objects and introduce the notion ofcomplexes of finite Gorenstein projective dimension.
c A the full subcategory of A consisting of objects X ∈ A
Definition 6. We denote by GPiwith ExtA (X, PA ) = 0 for i > 0.
c 0 := GPc A . For n ≥ 1 we denote by GP
c n the full subcategory ofDefinition 7. Let GP
cGP A consisting of objects X admitting right resolutions in A 0 → X → P 1 → · · · →c A and P i ∈ P for 1 ≤ i ≤ n.
P n → Y → 0 with Y ∈ GP
∩c n.
Proposition 8. We have GP A = n≥0 GP
Theorem 9. Let X • ∈ Kb (GP A ) with X i = 0 unless 0 ≤ i ≤ l. Then there exists
a quasi-isomorphism X • → P • with P • ∈ K+ (PA ) such that Zl+1 (P • ) ∈ GP A and
H−i (Hom•A (P • , PA )) = 0 for i > l.
Proposition 10. Let X • ∈ Db (A). The followings are equivalent:(1) X • ∼= Y • in Db (A) for some Y • ∈ Kb (GP A ).
(2) There exists a distinguished triangle X • → Y • → Z[l] → in Db (A) with Y • ∈Db (A)fpd , Z ∈ GP A and l ∈ Z.
(3) X • ∼= Z[l] in Db (A)/Db (A)fpd with Z ∈ GP A and l ∈ Z.
Definition 11. A complex X • ∈ Db (A) is said to have finite Gorenstein projective dimension if X • satisfies the equivalent condition in Proposition 10. We denotes by Db (A)fGpd
the full subcategory of Db (A) consisting of all complexes in Db (A) having finite Gorenstein
projective dimension.
Theorem 12 (cf. [2] and [9, Proposition 3.5]). The followings hold:c A → Db (A) induces a fully faithful functor(1) The embedding GP
c A /PA → Db (A)/Db (A)fpd
GP(2) The embedding GP A → Db (A)fGpd induces an equivalence
GP A /PA → Db (A)fGpd /Db (A)fpd
At the end of this section, using the quotient category, we characterize Gorenstein
projective objects.
c A . Then X ∈ GP A if and only if for each i > 0 there exists
Theorem 13. Let X ∈ GPc A such that X ∼Yi ∈ GP= Yi [−i] in Db (A)/Db (A)fpd .
3. Derived equivalences
In this section, we deal with derived equivalences of abelian categories with enoughprojectives. Let B be an abelian category with enough projectives. Throughout thissection we assume that there exists an equivalence of triangulated categories F : Db (A) →Db (B) with an integer a > 0 such that(∗)

HomD(B) (F P, Q[i]) = 0 = HomD(B) (Q, F P [i])–3–for all P ∈ PA and Q ∈ PB unless −a ≤ i ≤ a and G stands for a quasi-inverse of F .
Lemma 14. Hi (F P ) = 0 for all P ∈ PA unless −a ≤ i ≤ a.
Remark 15. Hi (GQ) = 0 for all Q ∈ PB unless −a ≤ i ≤ a.
Proposition 16. The equivalence F induces an equivalence of triangulated categories
between Db (A)fpd and Db (B)fpd .
□Proof. See [13, Proposition 8.2].

c B such that F X ∼c A there exists X ′ ∈ GP
Lemma 17. For each X ∈ GP= X ′ [a] in
Db (B)/Db (B)fpd .
Theorem 18. Let F : Db (A) → Db (B) be an equivalence of triangulated categories. If
there exists a > 0 such that
HomD(B) (F P, Q[i]) = 0 = HomD(B) (Q, F P [i])for all P ∈ PA and Q ∈ PB unless −a ≤ i ≤ a then F induces an equivalence oftriangulated categories between Db (A)fGpd and Db (B)fGpd .
Corollary 19. The equivalence F induces an equivalence between GP A /PA and GP B /PB .
Corollary 20. The following hold.
(1) GP A = PA if and only if GP B = PB .
c A = GP A if and only if GP
c B = GP B .
(2) GPc A = PA if and only if GP
c B = PB .
(3) GP
Proposition 21. Let F ′ : Db (A)fGpd → Db (B)fGpd be an equivalence of triangulated categories. Then F ′ induces an equivalence of triangulated categories Db (A)fpd → Db (B)fpd
if both A and B satisfy the condition Ab4.
Corollary 22. Let A, B be rings. Then A and B are derived equivalent, i.e., Db (Mod-A)
and Db (Mod-B) are equivalent as triangulated categories if and only if Db (Mod-A)fGpdand Db (Mod-B)fGpd are equivalent as triangulated categories.
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School of Information Environment
Tokyo Denki UniversityChiba, 270-1382, Japan
E-mail address: koga@mail.dendai.ac.jp–5–