1 Introduction
Let R be a commutative artinian ring. An Artin R-algebra is an R-algebra which is finitely generated as an R-module. Let A be an Artin R-algebra. We denote by
$\mathrm {mod} A$
the category of finitely generated right A-modules. Recall that the dominant dimension of an abelian category
$\mathcal {A}$
is the largest
$n\in \mathbb N\cup \{\infty \}$
such that any projective object in
$\mathcal {A}$
has an injective coresolution whose first n terms are projective. The dominant dimension of A is defined as the dominant dimension of
$\mathrm {mod} A$
. An Artin R-algebra A is representation-finite if
$\mathrm {mod} A$
has finitely many isomorphism classes of indecomposable objects. An Artin R-algebra
$\Gamma $
is a (1-)Auslander algebra if we have
$$\begin{align*}\mathrm{gl. dim} \Gamma\leq 2\leq \mathrm{dom. dim} \Gamma. \end{align*}$$
Let us recall Auslander’s classical theorem [Reference Auslander3], which was the starting point and a key tool in developing Auslander–Reiten theory, see for example in [Reference Auslander, Reiten and Smalø2, Chapter VI, Proposition 5.11].
Theorem 1.1 (Auslander correspondence).
There is a bijection between the Morita equivalence classes of representation-finite Artin R-algebras A and the Morita equivalence classes of Auslander R-algebras
$\Gamma $
. The bijection is induced by the map
$M\mapsto \mathrm {End}_{A}(M)$
, where M is an additive generator of
$\mathrm {mod} A$
.
It is natural to ask whether it is possible to replace the number
$2$
, which appears in the definition of Auslander algebras, by any integer
$\geq 2$
. This was done by Iyama [Reference Iyama18, Theorem 0.2]. Let
$d\geq 1$
be any integer. Recall that an A-module M is d-cluster-tilting [Reference Iyama19] if
$$ \begin{align*} \mathrm{add} M&=\{X\in\mathrm{mod} A\mid\mathrm{Ext}^{i}(M,X)=0 \text{ for } 1\leq i\leq d-1 \}\\ &=\{Y\in\mathrm{mod} A\mid \mathrm{Ext}^{i}(Y,M)=0 \text{ for } 1\leq i\leq d-1 \}. \end{align*} $$
Note that the notion of a
$1$
-cluster-tilting module is equivalent to that of an additive generator of
$\mathrm {mod} A$
. In [Reference Iyama19], it was demonstrated that d-cluster-tilting subcategories is a natural domain of a higher-dimensional Auslander–Reiten theory. In [Reference Iyama18, Section 6], amongst other things, Iyama generalized Auslander’s theorem on Mckay correspondence to a higher setting. An Artin R-algebra
$\Gamma $
is d-Auslander if we have
$$\begin{align*}\mathrm{gl. dim} \Gamma \leq d+1\leq \mathrm{dom. dim} \Gamma. \end{align*}$$
Two pairs
$(M,A)$
and
$(N,B)$
, where A and B are Artin R-algebras and M (resp. N) is a d-cluster-tilting A-module (resp. B-module), are equivalent, if there is an equivalence of R-categories
$\mathrm {mod} A\xrightarrow{_\sim } \mathrm {mod} B$
which restricts to an equivalence
$\mathrm {add} M\xrightarrow{_\sim } \mathrm {add} N$
.
Theorem 1.2 ([Reference Iyama18], Auslander–Iyama correspondence).
There is a bijection between the equivalence classes of pairs (A,M), where A is an Artin R-algebra and M is a d-cluster-tilting module, and the Morita equivalence classes of d-Auslander R-algebras
$\Gamma $
. The bijection is induced by the map
$M\mapsto \mathrm {End}_{A}(M)$
, where M is d-cluster-tilting A-module.
In this article, we extend Iyama’s higher Auslander correspondence to the setting of exact dg categories. Let k be a commutative ring. We fix an integer
$d\geq 1$
. We refer to [Reference Chen9], [Reference Chen10] for the necessary definitions and properties of exact dg categories and to [Reference Nakaoka and Palu25] for extriangulated categories.
An extriangulated category with enough projectives is d-Auslander if it has positive global dimension (cf. Definition 3.1) at most
$d+1$
and dominant dimension (cf. Definition 3.2) at least
$d+1$
. It is reduced if moreover the only projective–injectives are
$0$
. An exact dg category
$\mathcal {A}$
is d-Auslander if the extriangulated category
$H^0(\mathcal {A})$
is d-Auslander.
For a triangulated category
$\mathcal T$
and full subcategories
$\mathcal X$
and
$\mathcal Y$
, we write
$\mathcal X\ast \mathcal Y$
for the Verdier product of
$\mathcal X$
with
$\mathcal Y$
, which is the full subcategory of
$\mathcal T$
on the objects
$Z\in \mathcal T$
such that there exists a triangle
$$\begin{align*}X\rightarrow Z\rightarrow Y\rightarrow \Sigma X \end{align*}$$
with
$X\in \mathcal X$
and
$Y\in \mathcal Y$
. Let
$({\mathcal P},\mathcal {I})$
be a pair where
${\mathcal P}$
is a connective additive dg category and
$\mathcal {I}$
is an additive dg subcategory of
${\mathcal P}$
. For
$n\geq 0$
, we denote by
$\mathcal {A}^{(n)}_{{\mathcal P},\mathcal {I}}$
the full dg subcategory of
$\mathrm {pretr} ({\mathcal P})$
consisting of the objects in
$$\begin{align*}{(}H^0({\mathcal P})\ast\Sigma H^0({\mathcal P})\ast\cdots\ast\Sigma^{n-1}H^0({\mathcal P})\ast\Sigma^n H^0({\mathcal P}){)}\cap \mathrm{ker}\mathrm{Ext}^{\geq 1}(-,H^0(\mathcal{I})), \end{align*}$$
where
$H^0({\mathcal P})$
is viewed as a full subcategory of the triangulated category
$\mathrm {tr}({\mathcal P})$
. The pair
$({\mathcal P},\mathcal {I})$
is called d-cluster tilting if
$H^0(\mathcal {I})$
is covariantly finite in
$H^0(\mathcal {A}^{(d)}_{{\mathcal P},\mathcal {I}})$
.
Theorem 1.3 (=Theorem 3.7).
There is a bijective correspondence between the following:
-
(1) equivalence classes of connective exact dg categories
$\mathcal {A}$
which are d-Auslander; -
(2) equivalence classes of d-cluster tilting pairs
$({\mathcal P},\mathcal {I})$
.
The bijection from
$(1)$
to
$(2)$
sends
$\mathcal {A}$
to the pair
$({\mathcal P},\mathcal {I})$
formed by the full dg subcategory
${\mathcal P}$
on the projectives of
$\mathcal {A}$
and its full dg subcategory
$\mathcal {I}$
of projective–injectives. The inverse bijection sends
$({\mathcal P},\mathcal {I})$
to the
$\tau _{\leq 0}$
-truncation of the dg subcategory
$\mathcal {A}^{(d+1)}_{{\mathcal P},\mathcal {I}}$
of
$\mathrm {pretr} ({\mathcal P})$
.
We refer to Definition 3.6 for the definition of the equivalence relations mentioned in Theorem 1.3. Theorem 1.3 also holds when
$d=0$
[Reference Chen10]. It is a key ingredient in solving the problem raised in [Reference Fang, Gorsky, Palu, Plamondon and Pressland13, Problem 3.6] in the algebraic case. In the case when
$d=0$
, the direction from (2) to (1) was known to Fang–Gorsky–Palu–Plamondon–Pressland, (cf. [Reference Fang, Gorsky, Palu, Plamondon and Pressland13, Theorem 4.1]). We show in Example 4.8 that the dg category
$\mathcal {A}$
in Theorem 1.3 (1) is the module category over an Artin algebra
$\Gamma $
(so in particular
$\mathcal {A}$
is concentrated in degree zero) if and only if the corresponding pair
$({\mathcal P},\mathcal {I})$
is such that
${\mathcal P}=\mathrm{add}\ M\subset \mathrm {mod}\ \Lambda $
for a d-cluster tilting module M over some Artin algebra
$\Lambda $
and
$\mathcal {I}$
is the full subcategory of injective
$\Lambda $
-modules.
In [Reference Henrard, Kvamme and van Roosmalen17], the authors introduced the notion of Auslander exact category. Ebrahimi and Nasr-Isfahani [Reference Ebrahimi and Nasr-Isfahani12] generalized it to the notion of d-Auslander exact category. Using the notion of d-cluster tilting subcategory of an exact category introduced by Jasso [Reference Jasso21, Definition 4.13], they proved the following higher Auslander correspondence for exact categories.
Theorem 1.4 [Reference Ebrahimi and Nasr-Isfahani12, Theorem 5.2].
There is a bijection between the following:
-
(1) Equivalence classes of d-cluster tilting subcategories of exact categories with enough injectives.
-
(2) Equivalence classes of exact categories
$\mathcal {E}$
with enough projectives
${\mathcal P}=\mathrm {proj}\,(\mathcal {E})$
satisfying the following conditions:-
(a)
$\mathrm {gl. dim}(\mathcal {E})\leq d+1\leq \mathrm {dom. dim}(\mathcal {E})$
. -
(b) Any morphism
$X\rightarrow E$
with
$E\in {^{\perp }{\mathcal P}}$
is admissible. -
(c)
${\mathcal P}$
is admissibly covariantly finite, (cf. Definition 2.6).
-
The bijection from (1) to (2) sends
$\mathcal {M}$
, which is a d-cluster tilting subcategory of an exact category
$\mathcal {E}$
with enough injectives, to the d-Auslander exact category
$\mathrm {mod}_{\mathrm {adm}}(\mathcal {M})$
, (cf. Definition 2.5).
If we restrict Theorem 1.3 (1) to the class of dg categories concentrated in degree zero, then we obtain an Auslander–Iyama correspondence for Quillen exact categories.
Proposition 1.5 (=Proposition 4.1).
The exact dg categories in Theorem 1.3 (1) which are Quillen exact correspond to the d-cluster tilting pairs
$({\mathcal P},\mathcal {I})$
where
-
1)
${\mathcal P}$
is concentrated in degree
$0$
; -
2) a complex
(1.1)in
$$ \begin{align} 0\rightarrow P_{d+1}^{\wedge}\xrightarrow{f_{d}^{\wedge}} P_{d}^{\wedge}\rightarrow \cdots\xrightarrow{f_{1}^{\wedge}} P_1^{\wedge}\xrightarrow{f_{0}^{\wedge}} P_0^{\wedge} \end{align} $$
$\mathrm {Mod}\,{\mathcal P}$
is exact if the induced complex (1.2)is exact for any
$$ \begin{align} \mathrm{Hom}_{{\mathcal P}}(P_0,I)\rightarrow \mathrm{Hom}_{{\mathcal P}}(P_1,I)\rightarrow \cdots\rightarrow\mathrm{Hom}_{{\mathcal P}}(P_{d},I)\rightarrow \mathrm{Hom}_{{\mathcal P}}(P_{d+1},I)\rightarrow 0 \end{align} $$
$I\in \mathcal {I}$
.
Corollary 1.6 (=Corollary 4.3).
There is a bijection between the following:
-
(1) equivalence classes of exact categories
$\mathcal {A}$
which are d-Auslander as extriangulated categories; -
(2) equivalence classes of d-cluster tilting pairs
$({\mathcal P},\mathcal {I})$
satisfying the conditions in Proposition 1.5.
The bijection from
$(1)$
to
$(2)$
sends
$\mathcal {A}$
to the pair
$({\mathcal P}, \mathcal {I})$
where
${\mathcal P}$
is the full subcategory on the projectives of
$\mathcal {A}$
and
$\mathcal {I}$
is the full subcategory of projective–injectives. The inverse bijection sends
$({\mathcal P},\mathcal {I})$
to the
$\tau _{\leq 0}$
-truncation of the dg subcategory
$\mathcal {A}^{(d+1)}_{{\mathcal P},\mathcal {I}}$
of
$\mathrm {pretr} ({\mathcal P})$
.
Note that in Corollary 1.6 we consider exact categories to be equivalent if their weak idempotent completions are equivalent as exact categories, while in Theorem 1.4, the authors consider exact categories to be equivalent if they are equivalent as exact categories. So we restrict ourselves to the class of weakly idempotent complete exact categories. Let
$\mathcal {M}$
be a d-cluster tilting subcategory of a weakly idempotent complete exact category
$\mathcal {E}$
with enough injectives. Let
$\mathcal {I}$
be the full subcategory of injectives in
$\mathcal {E}$
. Then the pair
$(\mathcal {M},\mathcal {I})$
is a d-cluster tilting pair satisfying the conditions in Proposition 1.5, (cf. Proposition 4.4). We have the following diagram
Therefore we obtain a generalization of the higher Auslander correspondence for exact categories in Theorem 1.4 in the case when the exact categories are weakly idempotent complete.
There is another generalization of the class of Auslander algebras, namely the class of Gorenstein algebras
$\Gamma $
which satisfy
$$ \begin{align} \mathrm{dom. dim}(\Gamma)\geq 2\geq \mathrm{id}_{\Gamma}(\Gamma) \end{align} $$
where
$\mathrm {id}$
denotes the injective dimension. The case when both inequalities are equalities was studied in [Reference Auslander and Solberg4] using the notion of
$\tau $
-selfinjective algebras. It was further generalized to the class of d-minimal Auslander–Gorenstein algebras in [Reference Iyama and Solberg20] where the number
$2$
in (1.3) is replaced by
$d+1$
. Iyama–Solberg proved that there is a bijection between the Morita equivalence classes of d-minimal Auslander–Gorenstein algebras and equivalence classes of finite d-precluster tilting subcategories. In [Reference Grevstad14], the bijection is extended to the setting of exact categories. In [Reference Chen11], we will study the Iyama–Solberg correspondence for exact dg categories.
In this article, additive subcategories of additive categories are assumed to be closed under direct summands. Throughout we fix an integer
$d\geq 1$
. For an additive k-category
${\mathcal P}$
, we denote by
$\mathrm {Mod}\, {\mathcal P}$
the category of k-linear functors
${\mathcal P}^{op}\rightarrow k\mbox {-}\mathrm {Mod}\,$
. For each object
$P\in {\mathcal P}$
, put
$P^{\wedge }=\mathrm {Hom}_{{\mathcal P}}(?,P)\in \mathrm {Mod}\,{\mathcal P}$
. If
$\mathcal X$
is a class of objects in
${\mathcal P}$
, we write 
. We denote by
$\mathcal {C}^b_{\mathrm {dg}}({\mathcal P})$
the dg category of bounded complexes over
${\mathcal P}$
. For two complexes V and W over
${\mathcal P}$
, the degree p component of the Hom-complex
$\mathrm {Hom}_{\mathcal {C}^b_{dg}({\mathcal P})}(V,W)$
is given by
$$\begin{align*}\begin{aligned} \mathrm{Hom}_{\mathcal{C}^b_{\mathrm{dg}}({\mathcal P})}(V,W)^p&=\prod_{n\in \mathbb Z}\mathrm{Hom}_{{\mathcal P}}(V^n,W^{n+p})\\ &=\{ f:V\rightarrow W \mid f \text{ is homogeneous of degree } p \}. \end{aligned} \end{align*}$$
The differential is given by
$$\begin{align*}d(f)=d_{W}\circ f-(-1)^{|f|} f\circ d_{V}. \end{align*}$$
2 Preliminaries
Definition 2.1 [Reference Jasso21, Definition 2.2].
Let
$\mathcal {C}$
be an additive k-category and
$d^0:X^0\rightarrow X^1$
a morphism in
$\mathcal {C}$
. An n-cokernel of
$d^0$
is a sequence
such that for all
$Y\in \mathcal {C}$
, the induced sequence
is exact. Equivalently, the sequence
$(d^1,\ldots ,d^n)$
is an n-cokernel of
$d^0$
if for all
$1\leq k\leq n-1$
the morphism
$d^{k}$
is a weak cokernel of
$d^{k-1}$
, and
$d^n$
is moreover a cokernel of
$d^{n-1}$
. The concept of n-kernel of a morphism is defined dually.
Lemma 2.2 (Horseshoe lemma).
Let
$(\mathcal {C},\mathbb E,\mathfrak s)$
be an extriangulated category. Suppose that we have the black part in the following diagram in
$\mathcal {C}$
where both P and Q are projective in
$\mathcal {C}$
. We may complete it as above by the red part, where
$(r,u,s)$
,
$(p,v,q)$
,
$(f',x,f)$
, and
$(g',y,g)$
are morphisms of
$\mathbb E$
-triangles.
Proof. We have the solid part of the following diagram
Since Q is projective in
$\mathcal {C}$
, we have
$q^*(\tau )=0$
. By [Reference Liu and Nakaoka23, Proposition 1.20] and its dual, there exist morphisms
$v'$
and
$v"$
which are part of morphisms of
$\mathbb E$
-triangles and such that both the upper left square and the lower right square in the diagram (2.2) are homotopy cartesian. It follows by [Reference Chen8, Lemma A.8] that they are both
$\mathfrak s$
-deflations. Then 
is also an
$\mathfrak s$
-deflation and
$(p,v,q)$
is a morphism of
$\mathbb E$
-triangles. Now we take the idempotent completion
$(\tilde {\mathcal {C}},\mathbb F,\mathfrak r)$
of
$(\mathcal {C},\mathbb E,\mathfrak s)$
, (cf. [Reference Msapato24]). Note that an extriangulated category
$(\mathcal {C}, \mathbb E,\mathfrak s)$
satisfies the condition
$(\mathrm {WIC})$
[Reference Nakaoka and Palu25, Condition 5.8] if and only if the underlying additive category
$\mathcal {C}$
is weakly idempotent complete, (cf. [Reference Klapproth22, Proposition 2.7] and also [Reference Msapato24, Proposition 3.33]). Then we apply [Reference Nakaoka and Palu25, Lemma 5.9] to the following diagram
so that we get the diagram (2.1) in
$\tilde {\mathcal {C}}$
with the required properties. It follows from the commutativity of diagram (2.2) that the morphism
$Q\rightarrow Z$
completing the above diagram is q. Since q is an
$\mathfrak s$
-deflation, its cocone
$Z'$
lies in
$\mathcal {C}$
, and this completes the proof.
Lemma 2.3. Let
$(\mathcal {C},\mathbb E,\mathfrak s)$
be an extriangulated category. Let
$f\in \mathcal {C}(A,B)$
and
$g\in \mathcal {C}(B,C)$
be any composable pair of morphisms. If
$g\circ f$
is an
$\mathfrak s$
-deflation and f is an
$\mathfrak s$
-inflation, then g is an
$\mathfrak s$
-deflation.
Proof. We take the idempotent completion
$(\tilde {\mathcal {C}},\mathbb F,\mathfrak r)$
of
$(\mathcal {C},\mathbb E,\mathfrak s)$
, (cf. [Reference Msapato24]). Then g is an
$\mathfrak r$
-deflation, and by [Reference Nakaoka and Palu25, Proposition 3.17] we have
The object F is an extension of D and E and hence lies in
$\mathcal {C}$
. Therefore g is an
$\mathfrak s$
-deflation.
Definition 2.4 [Reference Jasso21, Definition 4.13].
Let
$\mathcal {E}$
be a small exact category and
$\mathcal {M}$
a subcategory of
$\mathcal {E}$
. We say that
$\mathcal {M}$
is a d-cluster tilting subcategory of
$\mathcal {E}$
if the following conditions are satisfied:
-
(i) Every object
$E\in \mathcal {E}$
admits a left
$\mathcal {M}$
-approximation by an inflation
$E\rightarrowtail M$
. -
(ii) Every object
$E\in \mathcal {E}$
admits a right
$\mathcal {M}$
-approximation by a deflation
$M'\twoheadrightarrow E$
. -
(iii) We have
$$ \begin{align*} \mathcal{M}&=\{X\in\mathcal{E} \mid \mathrm{Ext}^{i}(\mathcal{M},X)=0 \text{ for } 1\leq i\leq d-1 \}\\ &=\{Y\in\mathcal{E} \mid \mathrm{Ext}^{i}(Y,\mathcal{M})=0 \text{ for } 1\leq i\leq d-1 \}. \end{align*} $$
A morphism
$f:X\rightarrow Y$
in an exact category
$\mathcal {E}$
is admissible if f admits a factorization
$X\twoheadrightarrow I\rightarrowtail Y$
. A cochain complex over
$\mathcal {E}$
$$\begin{align*}\ldots\rightarrow X^{n-1}\xrightarrow{d^{n-1}} X^n\xrightarrow{d^n} X^{n+1}\rightarrow \ldots \end{align*}$$
is called acyclic or exact if each
$d^i$
is admissible and
$\mathrm {ker}(d^{i+1})=\mathrm {im} (d^i)$
.
Definition 2.5 [Reference Ebrahimi and Nasr-Isfahani12, Definition 3.1(2)].
Let
$\mathcal {M}$
be a full subcategory of an exact category
$\mathcal {E}$
. Let
$\mathrm {mod}_{\mathrm {adm}}(\mathcal {M})$
be the full subcategory of
$\mathrm {Mod}\,\mathcal {M}$
consisting of those functors F that admit a projective presentation
$$\begin{align*}X^{\wedge}\xrightarrow{f^{\wedge}} Y^{\wedge}\rightarrow F\rightarrow 0 \end{align*}$$
for some morphism
$f:X\rightarrow Y$
in
$\mathcal {M}$
which is admissible in
$\mathcal {E}$
.
Definition 2.6 [Reference Henrard, Kvamme and van Roosmalen17, Definition 4.16].
Let
$\mathcal {M}$
be a full subcategory of an exact category
$\mathcal {E}$
. A morphism
$f:X\rightarrow M$
in
$\mathcal {E}$
with
$M\in \mathcal {M}$
is called an admissible left
$\mathcal {M}$
-approximation if f is an admissible morphism and any morphism
$X\rightarrow M'$
with
$M'\in \mathcal {M}$
factors through f. The subcategory
$\mathcal {M}$
is called admissibly covariantly finite if for all objects X in
$\mathcal {E}$
, there exists an admissible left
$\mathcal {M}$
-approximation
$X\rightarrow M$
.
3 Auslander–Iyama correspondence
Definition 3.1 [Reference Gorsky, Nakaoka and Palu15, Definition 3.28].
A small extriangulated category
$(\mathcal {C},\mathbb E,\mathfrak s)$
has positive global dimension m for some integer
$m\geq 0$
, if
$\mathbb E^m\neq 0$
and
$\mathbb E^{m+1}=0$
. If such an m does not exist, we say that
$(\mathcal C,\mathbb E,\mathfrak s)$
has infinite positive global dimension.
Definition 3.2 [Reference Gorsky, Nakaoka and Palu16, Definition 3.5].
Let
$(\mathcal {C},\mathbb E,\mathfrak s)$
be an extriangulated category with enough projectives. We define its dominant dimension
$\mathrm {dom.dim}(\mathcal {C},\mathbb E,\mathfrak s)$
to be the largest integer n such that for any projective object P, there exist n
$\mathfrak s$
-triangles
with
$I^k$
being projective–injective for
$0\leq k\leq n-1$
. If such an n does not exist, we set
$\mathrm {dom.dim}(\mathcal {C}, \mathbb E,\mathfrak s)=\infty $
. Dually, one can define its codominant dimension.
Definition 3.3 [Reference Gorsky, Nakaoka and Palu16].
Let
$d\geq 1$
be an integer. An extriangulated category with enough projectives is d-Auslander if it has positive global dimension at most
$d+1$
and has dominant dimension at least
$d+1$
. It is reduced if moreover the only projective–injectives are
$0$
.
Proposition 3.4. The notion of d-Auslander extriangulated category is self-dual.Footnote
1
In other words, if an extriangulated category
$(\mathcal {C},\mathbb E,\mathfrak s)$
is d-Auslander, then it has enough injectives and has codominant dimension at least
$d+1$
.
Proof. Let
$(\mathcal {C},\mathbb E,\mathfrak s)$
be a d-Auslander extriangulated category. We say that
$Y\in \mathcal {C}$
admits an injective coresolution of length i provided that there exists i conflations
where
$J^{k}$
is injective in
$\mathcal {C}$
for
$0\leq k\leq i-1$
. If furthermore in this sequence of conflations
$J^{k}$
is injective–projective for
$k\leq l-1$
, then we say that Y admits a projective–injective coresolution of length
$(i,l)$
. We say that the dominant dimension of Y is greater or equal to l if it admits a projective–injective coresolution of length
$(l,l)$
. Dually, we define its codominant dimension.
We first show that
$\mathcal {C}$
has enough injectives. Let X be an arbitrary object in
$\mathcal {C}$
. We have conflations in
$\mathcal {C}$
where
$P_i$
is projective in
$\mathcal {C}$
for
$0\leq i\leq d+1$
. We show by descending induction that for each
$i\geq 1$
we have a conflation
where
$I_i$
is projective–injective and
$L_i$
admits a projective–injective coresolution of length
$(i,i-1)$
. This clearly holds for
$i=d+1$
, where
$X_{d+1}=P_{d+1}$
is projective, since
$\mathcal {C}$
is d-Auslander. Suppose it holds for some
$i\geq 2$
. We will show that the claim also holds for
$i-1$
. By the dual of [Reference Chen10, Lemma 2.2], we have the following diagram
By Lemma 2.2, the object
$M_i$
admits a projective–injective coresolution of length
$(i,i-1)$
thanks to the second column of diagram (3.2). Since
$I_i$
is injective in
$\mathcal {C}$
, the second row of diagram (3.2) splits. So we have the following diagram by [Reference Nakaoka and Palu25, Axiom (ET4)]
where
$U_i$
admits a projective–injective coresolution of length
$(i-1,i-2)$
. By Lemma 2.2, the object
$V_i$
also admits a projective–injective coresolution of length
$(i-1,i-2)$
thanks to the bottom row of diagram (3.3). This finishes the proof of the induction step. When
$i=0$
, we see that there exists an inflation
$X=X_0\rightarrowtail J$
where J is injective. Therefore, the extriangulated category
$\mathcal {C}$
has enough injectives.
Now we show that
$\mathcal {C}$
has codominant dimension at least
$d+1$
. Let us assume that X is injective. By the above, we have an inflation
$X\rightarrowtail J$
, where J is injective and is a direct sum of objects
$J'$
with conflations
where
$I_i$
is projective–injective for
$0\leq i\leq d$
and
$P_{d+1}$
is projective. In other words, the object
$J'$
has codominant dimension at least
$d+1$
. Such class of objects is closed under direct sums and hence we may assume that J admits such sequences of conflations. We have a conflation
which splits since X is injective. By [Reference Chen10, Lemma 2.2], we have the following diagram
where the object
$N_1$
has codominant dimension at least d. The second row of diagram (3.5) shows that each injective object in
$\mathcal {C}$
has codominant dimension at least
$1$
. Now the object Y is injective as a direct summand of J, and therefore it has codominant dimension at least
$1$
. By Lemma 2.2, we infer that the object
$N_1'$
has codominant dimension at least
$1$
thanks to the leftmost column in diagram (3.5). By the second row of diagram (3.5), we see that X has codominant dimension at least
$2$
. We repeat the above procedure and we see that X has codominant dimension at least
$d+1$
. Therefore the codominant dimension of
$(\mathcal {C},\mathbb E,\mathfrak s)$
is at least
$d+1$
.
Definition 3.5. An exact dg category
$\mathcal {A}$
is d-Auslander if the extriangulated category
$H^0(\mathcal {A})$
is d-Auslander.
Recall [Reference Chen10, Theorem 3.1] that each exact dg category
$\mathcal {A}$
admits a universal exact morphism
$\mathcal {A}\rightarrow \mathcal {D}^b_{\mathrm {dg}}(\mathcal {A})$
to pretriangulated dg categories in
$\mathrm {Hqe}$
, where
$\mathrm {Hqe}$
is the localization of the model category
$\mathrm {dgcat}$
of small dg categories at the quasi-equivalences, (cf. [Reference Tabuada26]). Put
$\mathcal {D}^b(\mathcal {A})=H^0(\mathcal {D}^b_{\mathrm {dg}}(\mathcal {A}))$
. For a connective exact dg category
$\mathcal {A}$
, we denote by
$\overline {\mathcal {A}}$
the
$\tau _{\leq 0}$
-truncation of the full dg subcategory of
$\mathcal {D}^{b}_{\mathrm {dg}}(\mathcal {A})$
consisting of the objects in the closure under kernels of retractions of
$H^0(\mathcal {A})$
in
$\mathcal {D}^{b}(\mathcal {A})$
. It is not hard to see that
$\overline {\mathcal {A}}$
is stable under extensions in
$\mathcal {D}^b_{\mathrm {dg}}(\mathcal {A})$
and hence inherits a canonical exact structure. It is clear that the inclusion
$\mathcal {A}\rightarrow \overline {\mathcal {A}}$
is an exact morphism which induces a quasi-equivalence
$\mathcal {D}^b_{\mathrm {dg}}(\mathcal {A})\xrightarrow{_\sim } \mathcal {D}^b_{\mathrm {dg}}(\overline {\mathcal {A}})$
.
Let
$({\mathcal P},\mathcal {I})$
be a pair where
${\mathcal P}$
is a connective additive dg category and
$\mathcal {I}$
is an additive dg subcategory of
${\mathcal P}$
. For
$n\geq 0$
, we denote by
$\mathcal {A}^{(n)}_{{\mathcal P},\mathcal {I}}$
the full dg subcategory of
$\mathrm {pretr} ({\mathcal P})$
consisting of the objects in
$$\begin{align*}{(}H^0({\mathcal P})\ast\Sigma H^0({\mathcal P})\ast\cdots\ast\Sigma^{n-1}H^0({\mathcal P})\ast\Sigma^nH^0({\mathcal P}){)}\cap \mathrm{ker}\mathrm{Ext}^{\geq 1}(-,H^0(\mathcal{I})). \end{align*}$$
We denote by
$\mathcal {Q}_{{\mathcal P},\mathcal {I}}$
(resp.
$\mathcal {J}_{{\mathcal P},\mathcal {I}}$
) the full dg subcategory of
$\mathrm {pretr} ({\mathcal P})$
whose objects are the direct summands in
$H^0(\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)})$
of objects in
$H^0({\mathcal P})\subset \mathrm {tr}({\mathcal P})$
(resp.
$H^0(\mathcal {I})\subset \mathrm {tr}({\mathcal P})$
). Clearly the dg subcategory
$\mathcal {Q}_{{\mathcal P},\mathcal {I}}$
is stable under kernels of retractions in
$$\begin{align*}\mathrm{pretr} (\mathcal{Q}_{{\mathcal P},\mathcal{I}})\xrightarrow{_\sim} \mathrm{pretr} ({\mathcal P}). \end{align*}$$
Definition 3.6.
-
1) Two connective exact dg categories
$\mathcal {A}$
and
$\mathcal {B}$
are equivalent if there is an exact quasi-equivalence
$\overline {\mathcal {A}}\xrightarrow{_\sim } \overline {\mathcal {B}}$
. -
2) Let
$({\mathcal P},\mathcal {I})$
be a pair consisting of a connective additive dg category
${\mathcal P}$
and an additive dg subcategory
$\mathcal {I}\subset {\mathcal P}$
. It is called d-cluster tilting if
$H^0(\mathcal {I})$
is covariantly finite in
$H^0(\mathcal {A}^{(d)}_{{\mathcal P},\mathcal {I}})$
. -
3) Two d-cluster tilting pairs
$({\mathcal P},\mathcal {I})$
and
$({\mathcal P}',\mathcal {I}')$
are equivalent if there is a quasi-equivalence
$\mathcal {Q}_{{\mathcal P},\mathcal {I}}\xrightarrow{_\sim } \mathcal {Q}_{{\mathcal P}',\mathcal {I}'}$
which restricts to a quasi-equivalence
$ \mathcal {J}_{{\mathcal P},\mathcal {I}}\xrightarrow{_\sim } \mathcal {J}_{{\mathcal P}',\mathcal {I}'}$
.
The following theorem is the main result in this paper. The remainder of this section is devoted to proving it.
Theorem 3.7 (Auslander–Iyama correspondence for exact dg categories).
There is a bijective correspondence between the following:
-
(1) equivalence classes of connective exact dg categories
$\mathcal {A}$
which are d-Auslander; -
(2) equivalence classes of d-cluster tilting pairs.
The bijection from
$(1)$
to
$(2)$
sends
$\mathcal {A}$
to the pair
$({\mathcal P},\mathcal {I})$
formed by the full dg subcategory
${\mathcal P}$
of projectives in
$\mathcal {A}$
and its full dg subcategory
$\mathcal {I}$
of projective–injectives. The inverse bijection sends
$({\mathcal P},\mathcal {I})$
to the
$\tau _{\leq 0}$
-truncation of the dg subcategory
$\mathcal {A}^{(d+1)}_{{\mathcal P},\mathcal {I}}$
of
$\mathrm {pretr} ({\mathcal P})$
.
Clearly, under the correspondence of Theorem 3.7,
-
a) the exact dg categories in
$(1)$
whose corresponding extriangulated category is moreover reduced correspond to the pairs in
$(2)$
where
$\mathcal {I}=0$
; -
b) the exact dg categories in
$(1)$
with the trivial exact structure (so that each object is both projective and injective) correspond to the pairs in
$(2)$
where
$\mathcal {I}={\mathcal P}$
.
Remark 3.8. Recall that an extriangulated category is algebraic if it is equivalent as an extriangulated category to
$H^0(\mathcal {A})$
for an exact dg category
$\mathcal {A}$
, (cf. [Reference Chen10, Proposition–Definition 3.14]). Since
$H^0(\tau _{\leq 0}\mathcal {A})$
and
$H^0(\mathcal {A})$
are isomorphic as extriangulated categories, we may assume in the above definition that
$\mathcal {A}$
is connective. Therefore by Theorem 3.7, each algebraic d-Auslander extriangulated category is of the form
$H^0(\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)})$
for some d-cluster tilting pair
$({\mathcal P},\mathcal {I})$
.
Remark 3.9. Theorem 3.7 also holds when
$d=0$
[Reference Chen10, Theorem 5.1]. It is a key ingredient in solving the problem raised in [Reference Fang, Gorsky, Palu, Plamondon and Pressland13, Problem 3.6] in the algebraic case, (cf. [Reference Chen10, Theorem 5.7]). In the case when
$d=0$
, the direction from (2) to (1) was known to Fang–Gorsky–Palu–Plamondon–Pressland, (cf. [Reference Fang, Gorsky, Palu, Plamondon and Pressland13, Theorem 4.1]).
Proposition 3.10. Let
$\mathcal {A}$
be a connective d-Auslander exact dg category. Then
$\overline {\mathcal {A}}$
is also d-Auslander.
Proof. Let
${\mathcal P}$
(resp.
$\mathcal {I}$
) be the full dg subcategory of projectives (resp. projective–injectives) of
$\mathcal {A}$
. Let
$X\in \overline {\mathcal {A}}$
. Then we have Y and Z in
$\mathcal {A}$
such that Y is isomorphic to
$X\oplus Z$
. We have a conflation in
$\mathcal {A}$
$$\begin{align*}M\rightarrow P\rightarrow Y, \end{align*}$$
where P is projective in
$H^0(\mathcal {A})$
. Recall that we denote by
$\mathcal {D}^b(\mathcal {A})$
the
$H^0$
-category of the bounded dg derived category
$\mathcal {D}^b_{\mathrm {dg}}(\mathcal {A})$
of
$\mathcal {A}$
. Then by the octahedral axiom, we have the following diagram in
$\mathcal {D}^b(\mathcal {A})$
Then it is clear that N lies in
$\mathcal {A}$
and hence
$\overline {\mathcal {A}}$
has enough projectives.
Since
$\mathcal {D}^b_{\mathrm {dg}}(\mathcal {A})$
is quasi-equivalent to
$\mathcal {D}^b_{\mathrm {dg}}(\overline {\mathcal {A}})$
and the higher extensions of
$H^0(\mathcal {A})$
(resp.
$H^0(\overline {\mathcal {A}})$
) can be computed as the suspended groups in their bounded dg derived categories, (cf. [Reference Chen10, Proposition 3.17], it is immediate that the positive global dimension of
$\overline {\mathcal {A}}$
is at most
$d+1$
.
Suppose
$X\in \overline {\mathcal {A}}$
is projective. Since
$\mathcal {A}$
has enough projectives, we have Y and Z in
${\mathcal P}$
such that Y is isomorphic to
$X\oplus Z$
: in diagram (3.6), the middle column splits and we have that
$N\in {\mathcal P}$
and P is isomorphic to
$X\oplus N$
. We have a conflation in
$\mathcal {A}$
$$\begin{align*}Y\rightarrow I\rightarrow U, \end{align*}$$
where
$I\in \mathcal {I}$
and U admits a projective–injective coresolution of length d. So we have the following diagram in
$\mathcal {D}^b(\mathcal {A})$
Then by Lemma 2.2, the object V admits a projective–injective coresolution of length d, and the second column shows that X admits a projective–injective coresolution of length
$d+1$
.
The following lemma is well-known, (cf. [Reference Aihara and Iyama1, Lemma 2.15]). We provide a proof for the convenience of the reader.
Lemma 3.11. Let
${\mathcal P}$
be a connective additive dg category. Then for any
$i\geq 0$
, the full subcategory
${\mathcal P}\ast \Sigma {\mathcal P}\ast \cdots \ast \Sigma ^{i}{\mathcal P}$
is stable under extensions in
$\mathrm {tr}({\mathcal P})$
.
Proof. By [Reference Beilinson, Bernstein and Deligne5, lemme 1.3.10], the Verdier product
$\ast $
is associative. For any integers
$n\geq m$
, any triangle in
$\mathrm {tr}({\mathcal P})$
where P and Q both are objects in
$H^0({\mathcal P})$
, satisfies
$f=0$
since the dg category
${\mathcal P}$
is connective. Therefore we have
$\Sigma ^{n}{\mathcal P}\ast \Sigma ^{m}{\mathcal P}\subseteq \Sigma ^{m}{\mathcal P}\ast \Sigma ^{n}{\mathcal P}$
and
$\Sigma ^{n}{\mathcal P}\ast \Sigma ^{n}{\mathcal P}=\Sigma ^{n}{\mathcal P}$
. It follows that for any
$0\leq j\leq i$
we have
$$ \begin{align*} & \quad({\mathcal P}\ast \Sigma {\mathcal P}\ast\cdots\ast \Sigma^{i-1}{\mathcal P} \ast \Sigma^{i}{\mathcal P})\ast \Sigma^{j}{\mathcal P}\\ & = {\mathcal P}\ast \Sigma {\mathcal P}\ast\cdots\ast \Sigma^{i-1}{\mathcal P}\ast (\Sigma^{i}{\mathcal P} \ast \Sigma^{j}{\mathcal P}) \\ & \subseteq {\mathcal P}\ast \Sigma {\mathcal P}\ast\cdots \ast\Sigma^{i-1}{\mathcal P}\ast (\Sigma^{j}{\mathcal P} \ast \Sigma^{i}{\mathcal P})\\ & \subseteq \cdots\\ & \subseteq {\mathcal P}\ast \Sigma {\mathcal P}\ast \cdots\ast \Sigma^{j-1}{\mathcal P}\ast\Sigma^{j}{\mathcal P}\ast \Sigma^{j}{\mathcal P}\ast \Sigma^{j+1}{\mathcal P}\ast \cdots\ast \Sigma^{i}{\mathcal P}\\ & = {\mathcal P}\ast \Sigma {\mathcal P}\ast \cdots\ast \Sigma^{j-1}{\mathcal P}\ast \Sigma^{j}{\mathcal P}\ast \Sigma^{j+1}{\mathcal P}\ast\cdots\ast \Sigma^{i}{\mathcal P}. \end{align*} $$
Therefore we have
$$ \begin{align*} &({\mathcal P}\ast \Sigma {\mathcal P}\ast\cdots\ast \Sigma^{i-1}{\mathcal P} \ast \Sigma^{i}{\mathcal P})\ast ({\mathcal P}\ast \Sigma {\mathcal P}\ast\cdots\ast \Sigma^{i-1}{\mathcal P} \ast \Sigma^{i}{\mathcal P})\\ \subseteq&{\mathcal P}\ast \Sigma {\mathcal P}\ast\cdots\ast \Sigma^{i-1}{\mathcal P} \ast \Sigma^{i}{\mathcal P} \end{align*} $$
and hence
${\mathcal P}\ast \Sigma {\mathcal P}\ast \cdots \ast \Sigma ^{i-1}{\mathcal P} \ast \Sigma ^{i}{\mathcal P}$
is stable under extensions in
$\mathrm {tr}({\mathcal P})$
.
Proposition 3.12. For a d-cluster tilting pair
$({\mathcal P},\mathcal {I})$
, the category
$H^0(\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)})$
is stable under kernels of retractions in
$\mathrm {tr}({\mathcal P})$
.
Proof. Let
$\mathcal {U}$
be the full subcategory of
$\mathrm {tr}({\mathcal P})$
on the objects in
$$\begin{align*}{\mathcal P}\ast\Sigma{\mathcal P}\ast\cdots\ast\Sigma^{d}{\mathcal P}. \end{align*}$$
By Lemma 3.11, it is stable under extensions in
$\mathrm {tr}({\mathcal P})$
. Let Z be an object in
$\mathrm {tr}({\mathcal P})$
such that there exists objects X and Y in
$\mathcal {U}$
satisfying
$Y\xrightarrow{_\sim } X\oplus Z$
. Clearly, we have
$\mathrm {Hom}_{\mathrm {tr}({\mathcal P})}(\Sigma ^{-1}{\mathcal P}, Y)=0$
and hence
$\mathrm {Hom}_{\mathrm {tr}({\mathcal P})}(\Sigma ^{-1}{\mathcal P}, Z)=0$
. We have a triangle in
$\mathrm {tr}({\mathcal P})$
Therefore we have
$Z\in \Sigma ^{-1}\mathcal {U}\ast \mathcal {U}=\Sigma ^{-1}{\mathcal P}\ast \mathcal {U}$
. We thus have a triangle in
$\mathrm {tr}({\mathcal P})$
where P lies in
${\mathcal P}$
and U lies in
$\mathcal {U}$
. Since the Hom space
$\mathrm {Hom}_{\mathrm {tr}({\mathcal P})}(\Sigma ^{-1}{\mathcal P}, Z)$
vanishes, the triangle (3.8) splits and we have that Z is an extension of U and
$\Sigma P$
. Therefore the object Z lies in
$\mathcal {U}$
and the subcategory
$\mathcal {U}$
is stable under kernels of retractions in
$\mathrm {tr}({\mathcal P})$
. Since
$\mathcal {A}^{(d+1)}_{{\mathcal P},\mathcal {I}}=\mathcal {U}^{(d+1)}\cap \mathrm {ker}\mathrm {Ext}^{\geq 1}(-,\mathcal {I})$
and
$\mathrm {ker} \mathrm {Ext}^{\geq 1}(-,\mathcal {I})$
is stable under the passage to direct summands, we have that
$H^0(\mathcal {A}^{(d+1)}_{{\mathcal P},\mathcal {I}})$
is stable under kernels of retractions in
$\mathrm {tr}({\mathcal P})$
.
Let
$({\mathcal P},\mathcal {I})$
be a pair where
${\mathcal P}$
is a connective additive dg category and
$\mathcal {I}$
is an additive dg subcategory of
${\mathcal P}$
. Let
$\mathcal {B}^{(n)}_{{\mathcal P},\mathcal {I}}$
be the full dg subcategory of
$\mathrm {pretr} ({\mathcal P})$
consisting of objects in
$$\begin{align*}{(}H^0(\mathcal{I})\ast\Sigma H^0(\mathcal{I})\ast\cdots\ast\Sigma^{n-1} H^0(\mathcal{I})\ast\Sigma^n H^0({\mathcal P}){)}\cap \mathrm{ker}\mathrm{Ext}^{\geq 1}(-,H^0(\mathcal{I})). \end{align*}$$
Let
$\mathcal {C}^{(n)}_{{\mathcal P},\mathcal {I}}$
be the full dg subcategory of
$\mathrm {pretr} ({\mathcal P})$
consisting of the objects in the union of the
$\mathcal {B}^{(i)}_{{\mathcal P},\mathcal {I}}$
for
$0\leq i\leq n$
.
Lemma 3.13. The pair
$({\mathcal P},\mathcal {I})$
is d-cluster tilting if and only if
$H^0(\mathcal {I})$
is covariantly finite in
$H^0(\mathcal {C}^{(d)}_{{\mathcal P},\mathcal {I}})$
.
Proof. Suppose that
$H^0(\mathcal {I})$
is covariantly finite in
$H^0(\mathcal {C}^{(d)}_{{\mathcal P},\mathcal {I}})$
. Let P be an object in
$H^0({\mathcal P})$
. By assumption, it admits triangles
where
$f^{k}$
is a left
$H^0(\mathcal {I})$
-approximation for
$Y^k$
for
$0\leq k\leq d$
. Let
$X_1$
be any object in
$H^0(\mathcal {A}^{(d)}_{{\mathcal P},\mathcal {I}})$
. Then it admits a sequence of triangles (3.1) where
$P_i\in H^0({\mathcal P})$
for
$1\leq i\leq d+1$
. Similarly to the proof of Proposition 3.4, one observes that
$X_1$
admits a triangle
$$\begin{align*}X_1\xrightarrow{f_1} I_1\rightarrow L_1\rightarrow \Sigma X_1, \end{align*}$$
where
$I_1\in H^0(\mathcal {I})$
and
$L_1\in H^0(\mathcal {A}^{(d+1)}_{{\mathcal P},\mathcal {I}})$
. It follows that
$f_1$
is a left
$H^0(\mathcal {I})$
-approximation for
$X_1$
. Therefore the pair
$({\mathcal P},\mathcal {I})$
is d-cluster tilting. The converse is easy to see.
We start the proof of Theorem 3.7 by showing that for a d-cluster tilting pair
$({\mathcal P},\mathcal {I})$
, the dg subcategory
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
of
$\mathrm {pretr} ({\mathcal P})$
inherits a canonical exact dg structure whose corresponding extriangulated category is d-Auslander.
Proposition 3.14. Let
$({\mathcal P},\mathcal {I})$
be a d-cluster tilting pair, and let
$\mathcal {A}=\tau _{\leq 0}\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
.
-
(1) The dg subcategory
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
of
$\mathrm {pretr} ({\mathcal P})$
inherits a canonical exact dg structure whose corresponding extriangulated category is d-Auslander. -
(2) We have that
$\mathcal {Q}_{{\mathcal P},\mathcal {I}}$
is the full dg subcategory of projectives in
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
and
$\mathcal {J}_{{\mathcal P},\mathcal {I}}$
is the full dg subcategory of projective–injectives in
$\mathcal {A}_{{\mathcal P},\mathcal {I}}$
. -
(3) The bounded dg derived category of
$\mathcal {A}$
is
$\mathrm {pretr} ({\mathcal P})$
.
Proof. By Lemma 3.11, the full dg subcategory
${\mathcal P}\ast \Sigma {\mathcal P}\ast \cdots \ast \Sigma ^{d+1}{\mathcal P}$
is stable under extensions in
$\mathrm {pretr} ({\mathcal P})$
and hence
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
is also stable under extensions in
$\mathrm {pretr} ({\mathcal P})$
. Therefore, by [Reference Chen9, Example–Definition 4.8], it inherits a canonical exact dg structure.
We first show that
$\mathcal {Q}_{{\mathcal P},\mathcal {I}}$
is the full dg subcategory of projective objects. Suppose that X is a projective object in
$\mathcal {A}$
. We have triangles in
$\mathrm {tr}({\mathcal P})$
$$\begin{align*}\ldots \end{align*}$$
Since
$\mathrm {Ext}^2(X,\mathcal {I})=0$
, from the triangle (3.9) we have
$\mathrm {Ext}^1(X_1,\mathcal {I})=0$
and therefore
$X_1$
also lies in
$\mathcal {A}$
. Similarly, we have that all
$X_i$
lie in
$\mathcal {A}$
for
$1\leq i\leq d+1$
. Then since X is projective in
$\mathcal {A}$
, triangle (3.9) splits and X is a direct summand of
$P_0$
. So X lies in
$\mathcal {Q}_{{\mathcal P},\mathcal {I}}$
and hence
$\mathcal {Q}_{{\mathcal P},\mathcal {I}}$
is the full dg subcategory of
$\mathcal {A}_{{\mathcal P},\mathcal {I}}$
consisting of the projective objects.
It is clear that the objects in
$\mathcal {I}$
are projective–injective in
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
. By definition,
$H^0(\mathcal {J}_{{\mathcal P},\mathcal {I}})$
is the closure of
$H^0(\mathcal {I})$
under kernels of retractions in
$\mathrm {tr}({\mathcal P})$
. Therefore
$\mathcal {J}_{{\mathcal P},\mathcal {I}}$
is a dg subcategory of
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
and consists of projective–injective objects.
Let us show that
$H^0(\mathcal {A})$
has dominant dimension at least
$d+1$
. Suppose that X is projective in
$\mathcal {A}$
with the triangles (3.9–3.11), then the triangle (3.9) splits. By the above, we may assume
$X_1\in {\mathcal P}$
. We have a triangle
$$ \begin{align} P_0\xrightarrow{u} I\xrightarrow{v} Y\xrightarrow{w} \Sigma X, \end{align} $$
where
$I\in \mathcal {I}$
and u is a left
$\mathcal {I}$
-approximation of
$P_2$
. Then it is clear that Y lies in
$\mathcal {A}$
. From the octahedral axiom, we have the following diagram
Since Y admits an
$\mathcal {I}$
-coresolution of length d and
$X_1\in {\mathcal P}$
has an
$\mathcal {I}$
-coresolution of length
$d+1$
, the object U admits an
$\mathcal {I}$
-coresolution of length d by Lemma 2.2. Hence the middle column of the diagram (3.13) shows that X admits an
$\mathcal {I}$
-coresolution of length
$d+1$
. Be the definition of
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
, it has enough projectives. Hence, the exact dg category
$\mathcal {A}$
is d-Auslander.
If X is moreover injective, then the triangle in the middle column of (3.13) splits and hence X lies in
$\mathcal {J}_{{\mathcal P},\mathcal {I}}$
. So
$\mathcal {J}_{{\mathcal P},\mathcal {I}}$
is the full dg subcategory of projective–injectives in
$\mathcal {A}_{{\mathcal P},\mathcal {I}}$
.
Let us show that the bounded dg derived category of
$\mathcal {A}$
is
$\mathrm {pretr} ({\mathcal P})$
. By [Reference Chen10, Proposition 3.17], it is enough to show that the higher extensions of the extriangulated category
$H^0(\mathcal {A})$
are naturally isomorphic to the corresponding suspended Hom groups of
$\mathrm {tr}({\mathcal P})$
. This is clear from the fact that
${\mathcal P}$
is projective in
$H^0(\mathcal {A})$
, that is,
$\mathrm {Hom}_{\mathrm {tr}({\mathcal P})}(P,\Sigma ^{\geq 1} X)=0$
for
$P\in {\mathcal P}$
and
$X\in H^0(\mathcal {A})$
, and
$\mathrm {Hom}_{\mathrm {tr}({\mathcal P})}(X,\Sigma ^{i} Y)=0$
for X,
$Y\in {\mathcal P}$
and
$i\geq 1$
.
Proposition 3.15. Let
$\mathcal {A}'$
be a connective exact dg category such that
$H^0(\mathcal {A}')$
is d-Auslander. Suppose that
${\mathcal P}\subset \mathcal {A}'$
is the full dg subcategory of projective objects and
$\mathcal {I}\subset {\mathcal P}$
is the full dg subcategory of projective–injective objects. Then the following statements hold.
-
(a) The pair
$({\mathcal P},\mathcal {I})$
is d-cluster tilting. -
(b)
$\mathcal {A}'$
is equivalent (in the sense of Definition 3.6) to
$\mathcal {A}=\tau _{\leq 0}(\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)})$
. When
$\mathcal {I}=0$
, we have that
$\mathcal {A}'$
is quasi-equivalent to
$\mathcal {A}$
.
Proof. We have the following diagram
The induced morphism
${\mathcal P}\rightarrow \mathcal {D}^b_{\mathrm {dg}}(\mathcal {A}')$
is quasi-fully faithful and
$\mathcal {D}^b(\mathcal {A}')$
is generated as a triangulated category by the objects in
${\mathcal P}$
. Therefore the morphism
$\mu :\mathrm {pretr} ({\mathcal P})\rightarrow \mathcal {D}^b_{\mathrm {dg}}(\mathcal {A}')$
is a quasi-equivalence. Recall that
$\mathrm {pretr} ({\mathcal P})$
is the bounded dg derived category of
$\mathcal {A}$
by Proposition 3.14 (3).
We first show statement (a). By Lemma 3.13, it is enough to show that each object in
$H^0(\mathcal {C}_{{\mathcal P},\mathcal {I}}^{(d)})$
admits a left
$\mathcal {I}$
-approximation. Let
$X_1$
be an arbitrary object in
$H^0(\mathcal {C}_{{\mathcal P},\mathcal {I}}^{(d)})$
. We have conflations in
$H^0(\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d)})$
where
$P_{d+1}\in {\mathcal P}$
and
$I_i\in \mathcal {I}$
for
$1\leq i\leq d$
. Since
$H^0(\mathcal {A}')$
is d-Auslander, there exists conflations in
$H^0(\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d)})$
where
$I_i'\in \mathcal {I}$
for
$0\leq i\leq d$
. So we have the following diagram
Since
$I_d'\in \mathcal {I}$
is injective in
$H^0(\mathcal {A}')$
, we have
$I_d'\oplus X_d\xrightarrow{_\sim } I_d\oplus X_d'$
. We replace
$X_d$
(resp.
$X_d'$
) by
$X_d\oplus I_d'$
(resp.
$X_d'\oplus I_d$
) and we may assume that
$X_d\xrightarrow{_\sim } X_d'$
. Then we repeat the above argument. In the end, we see that
$X_1\oplus I\xrightarrow{_\sim } X_1'\oplus I'$
for some objects I and
$I'$
in
$\mathcal {I}$
. Since
$X_1'$
admits an inflation into the object
$I_1'$
, it follows that
$X_1$
also admits an inflation into an object in
$\mathcal {I}$
. This shows that
$H^0(\mathcal {I})$
is covariantly finite in
$H^0(\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d)})$
and hence the pair
$({\mathcal P},\mathcal {I})$
is d-cluster tilting.
Let us show statement (b). It is enough to show that the quasi-essential image of
$\mathcal {A}\subset \mathrm {pretr} ({\mathcal P})$
under the morphism
$\mu $
is in the closure of
$\mathcal {A}'$
under kernels of retractions in
$\mathcal {D}^b_{\mathrm {dg}}(\mathcal {A}')$
. Let
$X\in \mathcal {D}^b(\mathcal {A}')$
be an object with triangles (3.9), (3.10), and (3.11). Assume that
$$\begin{align*}\mathrm{Hom}_{\mathcal{D}^b(\mathcal{A}')}(X,\Sigma^{\geq 1} \mathcal{I})=0. \end{align*}$$
We claim that X lies in the closure of
$H^0(\mathcal {A}')$
under kernels of retractions in
$\mathcal {D}^b(\mathcal {A}')$
. Indeed, since
$H^0(\mathcal {A}')$
is d-Auslander and the object
$X_{d+1}=P_{d+1}$
in the triangle (3.11) is projective in
$H^0(\mathcal {A}')$
, it admits a triangle as follows
$$\begin{align*}P_{d+1}\rightarrow I_{d+1}\rightarrow U_{d+1}\rightarrow \Sigma P_{d+1}, \end{align*}$$
where
$I_{d+1}\in \mathcal {I}$
and
$U_{d+1}$
is in
$H^0(\mathcal {A}')$
. From the octahedral axiom, we have the following diagram in
$\mathcal {D}^b(\mathcal {A}')$
By assumption, we have
$\mathrm {Hom}(X_d,\Sigma I_{d+1})=0$
and hence
$X_{d}\oplus I_{d+1}$
is isomorphic to
$V_{d}\in \mathcal {A}'$
. The object
$P_d$
admits an
$\mathcal {I}$
-coresolution of length
$d+1$
and
$U_{d+1}$
admits an
$\mathcal {I}$
-coresolution of length d. From the middle column, we see that
$V_d$
admits an
$\mathcal {I}$
-coresolution of length d. So we have the following diagram
where
$V_{d-1}$
lies in
$\mathcal {A}'$
as an extension of
$P_{d-1}\oplus I_{d+1}$
and
$U_{d}$
, and the second row splits since
$\mathrm {Hom}(X,\Sigma I_{d})=0$
. We repeat the above procedure and we see that X lies in
$\overline {\mathcal {A}'}$
.
On the other hand, it is clear that each object in
$\mathcal {A}'$
is from an object in
$\mathcal {A}$
.
Proof of Theorem 3.7
Let
$\mathcal {A}$
be a connective d-Auslander exact dg category. By Proposition 3.10, the connective exact dg category
$\overline {\mathcal {A}}$
is also d-Auslander. Notice that the inclusion
$\mathcal {A}\rightarrow \overline {\mathcal {A}}$
induces a quasi-equivalence
$\mathcal {D}^b_{\mathrm {dg}}(\mathcal {A})\xrightarrow{_\sim } \mathcal {D}^b_{\mathrm {dg}}(\overline {\mathcal {A}})$
and
$\mathcal {D}^b_{\mathrm {dg}}(\mathcal {A})$
is quasi-equivalent to the pretriangulated hull of the full dg subcategory
${\mathcal P}$
of
$\mathcal {A}$
consisting of the projectives. By the proof of Proposition 3.10, we have that the full dg subcategory
${\mathcal P}_{\overline {\mathcal {A}}}$
of
$\overline {\mathcal {A}}$
of the projectives is quasi-equivalent to
$\mathcal {Q}_{{\mathcal P},\mathcal {I}}$
. A similar argument shows that the full dg subcategory
$\mathcal {I}_{\overline {\mathcal {A}}}$
of the projective–injectives is quasi-equivalent to
$\mathcal {J}_{{\mathcal P},\mathcal {I}}$
. Therefore the map sending the equivalence class of
$\mathcal {A}$
to the equivalence class of
$({\mathcal P},\mathcal {I})$
is well-defined.
For a d-cluster tilting pair
$({\mathcal P},\mathcal {I})$
, the inclusion dg functor
${\mathcal P}\hookrightarrow \mathcal {Q}_{{\mathcal P},\mathcal {I}}$
induces a quasi-equivalence
$\mathrm {pretr} ({\mathcal P})\xrightarrow{_\sim } \mathrm {pretr} (\mathcal {Q}_{{\mathcal P},\mathcal {I}})$
. Let us show that this induces a quasi-equivalence
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}\xrightarrow{_\sim }\mathcal {A}_{\mathcal {Q}_{{\mathcal P},\mathcal {I}},\mathcal {J}_{{\mathcal P},\mathcal {I}}}^{(d+1)}$
. We observe
${\mathcal P}\ast \Sigma {\mathcal P}=\mathcal {Q}_{{\mathcal P},\mathcal {I}}\ast \Sigma \mathcal {Q}_{{\mathcal P},\mathcal {I}}$
and then by the associativity of the Verdier product
$\ast $
, we have the following equality by induction on n:
$$ \begin{align*} (\mathcal{Q}_{{\mathcal P},\mathcal{I}}\ast\cdots\ast \Sigma^{n-1}\mathcal{Q}_{{\mathcal P},\mathcal{I}})\ast\Sigma^n\mathcal{Q}_{{\mathcal P},\mathcal{I}} &=({\mathcal P}\ast\cdots\ast\Sigma^{n-1}{\mathcal P})\ast\Sigma^n\mathcal{Q}_{{\mathcal P},\mathcal{I}}\\ &={\mathcal P}\ast\cdots\ast(\Sigma^{n-1}{\mathcal P}\ast\Sigma^{n}\mathcal{Q}_{{\mathcal P},\mathcal{I}})\\ &={\mathcal P}\ast\cdots\ast(\Sigma^{n-1}{\mathcal P}\ast\Sigma^{n}{\mathcal P}). \end{align*} $$
It is not hard to see that
$\mathrm {ker}\mathrm {Ext}^{\geq 1}(-,\mathcal {J}_{{\mathcal P},\mathcal {I}})=\mathrm {ker}\mathrm {Ext}^{\geq 1}(-,\mathcal {I})$
by the definition of
$\mathcal {J}_{{\mathcal P},\mathcal {I}}$
. Hence, the induced morphism
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}\xrightarrow{_\sim }\mathcal {A}_{\mathcal {Q}_{{\mathcal P},\mathcal {I}},\mathcal {J}_{{\mathcal P},\mathcal {I}}}^{(d+1)}$
is a quasi-equivalence. Therefore the map sending the equivalence class of a pair
$({\mathcal P},\mathcal {I})$
to the equivalence class of
$\tau _{\leq 0}\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
is well-defined.
Combining Propositions 3.14 and 3.15, we see that the above maps are inverse to each other.
4 Connection to Quillen exact categories.
In this section we give necessary and sufficient conditions on the d-cluster tilting pair
$({\mathcal P},\mathcal {I})$
for when the connective d-Aulander exact dg category
$\tau _{\leq 0}\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
is concentrated in degree zero (resp. is an abelian category). We therefore obtain a generalization of the higher Auslander correspondence for exact categories (resp. abelian categories) in [Reference Ebrahimi and Nasr-Isfahani12, Theorem 5.2] in the case of exact categories with split retractions (resp. in [Reference Beligiannis6, Theorem 8.23]).
4.1 Exact categories
Proposition 4.1. The exact dg categories in Theorem 3.7 (1) which are Quillen exact correspond to the d-cluster tilting pairs
$({\mathcal P},\mathcal {I})$
where
-
1)
${\mathcal P}$
is concentrated in degree
$0$
; -
2) a complex
(4.1)in
$$ \begin{align} 0\rightarrow P_{d+1}^{\wedge}\xrightarrow{f_{d}^{\wedge}} P_{d}^{\wedge}\rightarrow \cdots\xrightarrow{f_{1}^{\wedge}} P_1^{\wedge}\xrightarrow{f_{0}^{\wedge}} P_0^{\wedge} \end{align} $$
$\mathrm {Mod}\,{\mathcal P}$
is exact if the induced complex (4.2)is exact for any I in
$$ \begin{align} \mathrm{Hom}_{{\mathcal P}}(P_0,I)\rightarrow \mathrm{Hom}_{{\mathcal P}}(P_1,I)\rightarrow \cdots\rightarrow\mathrm{Hom}_{{\mathcal P}}(P_{d},I)\rightarrow \mathrm{Hom}_{{\mathcal P}}(P_{d+1},I)\rightarrow 0 \end{align} $$
$\mathcal {I}$
.
For such a pair
$({\mathcal P},\mathcal {I})$
, if we identify
$\mathrm {pretr} ({\mathcal P})$
with
$\mathcal {C}^b_{\mathrm {dg}}({\mathcal P})$
thanks to item 1), then
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
is identified with the full dg subcategory of complexes
where
$P_i$
is in degree
$-i$
and such that the induced complex (4.2) is exact for any
$I\in \mathcal {I}$
.
Proof. Let us assume that
$\tau _{\leq 0}\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
is concentrated in degree 0. Since
${\mathcal P}$
is contained in
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
, it is concentrated in degree 0. This shows item 1).
If a complex (4.1) in
$\mathrm {Mod}\,{\mathcal P}$
is such that the complex (4.2) is exact, then it defines an object
$X=X_0$
in
$\mathcal {A}^{(d+1)}_{{\mathcal P},\mathcal {I}}$
as follows. First we complete the morphism
$f_{d}$
to a triangle:
$$\begin{align*}P_{d+1}\xrightarrow{f_{d}} P_{d}\rightarrow X_{d}\rightarrow \Sigma P_{d+1}. \end{align*}$$
Then we have
$X_{d}\in \mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
. Since
${\mathcal P}$
is concentrated in degree
$0$
, for each
$I\in \mathcal {I}$
we have a short exact sequence
$$ \begin{align} 0\rightarrow \mathrm{Hom}_{\mathrm{tr}({\mathcal P})}(X_{d},I)\rightarrow \mathrm{Hom}_{\mathrm{tr}({\mathcal P})}(P_{d},I)\rightarrow \mathrm{Hom}_{\mathrm{tr}({\mathcal P})}(P_{d+1},I)\rightarrow 0. \end{align} $$
Since
${\mathcal P}$
is concentrated in degree
$0$
, the morphism
$f_{d-1}$
induces a unique map
$X_{d}\rightarrow P_{d-1}$
and we complete it into a triangle
$$\begin{align*}X_{d}\rightarrow P_{d-1}\rightarrow X_{d-1}\rightarrow \Sigma X_{d}. \end{align*}$$
By the sequences (4.2) and (4.4), we have that the morphism
$$\begin{align*}\mathrm{Hom}(P_{d-1},\mathcal{I})\rightarrow \mathrm{Hom}(X_d,\mathcal{I}) \end{align*}$$
is a surjection and hence
$X_{d-1}\in \mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
. We repeat the above procedure and we get a sequence of triangles
$$ \begin{align} X_{i+1}\xrightarrow{f_i} P_{i}\rightarrow X_{i}\rightarrow \Sigma X_{i+1} \end{align} $$
for
$0\leq i\leq d$
where
$X_{d+1}=P_{d+1}$
and
$X_{i}\in \mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
. Since
$\tau _{\leq 0}\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
is concentrated in degree
$0$
, we have
$\mathrm {Hom}_{\mathrm {tr}({\mathcal P})}(\Sigma ^{j}Q,X_i)=0$
for
$Q\in {\mathcal P}$
and
$j\geq 1$
and
$0\leq i\leq d+1$
. For each
$Q\in {\mathcal P}$
we apply the cohomological functor
$\mathrm {Hom}(Q,-)$
to the sequences (4.5) for
$0\leq i\leq d+1$
and we see that the complex (4.1) is exact. This shows item 2).
Conversely, suppose the pair
$({\mathcal P},\mathcal {I})$
in Theorem 3.7 (2) moreover satisfies conditions 1) and 2). Then each object
$X=X_0$
in
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
can be obtained by a sequence of triangles (4.5) for
$0\leq i\leq d+1$
where
$X_{d+1}=P_{d+1}$
and
$P_i\in {\mathcal P}$
and
$X_i\in \mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
for each i. So the complex (4.1) satisfies that the induced complex (4.2) is exact for any I in
$\mathcal {I}$
and hence by 2), the complex (4.1) is exact. We identify
$\mathrm {tr}({\mathcal P})$
with the homotopy category of bounded complexes over
${\mathcal P}$
and X is identified with the complex
$$ \begin{align} \ldots\rightarrow 0\rightarrow P_{d+1}\xrightarrow{f_{d}} P_d\xrightarrow{f_{d-1}}\cdots\xrightarrow{f_1} P_1\xrightarrow{f_0} P_0\rightarrow 0\rightarrow \ldots \end{align} $$
where
$P_i$
is in degree i. By the exactness of the complex (4.1), it is straightforward to verify that
$\tau _{\leq 0}\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
is concentrated in degree
$0$
.
Remark 4.2. If a pair
$({\mathcal P},\mathcal {I})$
satisfies the conditions in Proposition 4.1 for an integer d, then it does so for all integers
$\leq d$
.
Corollary 4.3. There is a bijective correspondence between the following:
-
(1) equivalence classes of exact categories
$\mathcal {A}$
which are d-Auslander as extriangulated categories; -
(2) equivalence classes of d-cluster tilting pairs
$({\mathcal P},\mathcal {I})$
satisfying the conditions in Proposition 1.5.
The bijection from
$(1)$
to
$(2)$
sends
$\mathcal {A}$
to the pair
$({\mathcal P}, \mathcal {I})$
where
${\mathcal P}$
is the full subcategory on the projectives of
$\mathcal {A}$
and
$\mathcal {I}$
is the full subcategory of projective–injectives. The inverse bijection sends
$({\mathcal P},\mathcal {I})$
to the
$\tau _{\leq 0}$
-truncation of the dg subcategory
$\mathcal {A}^{(d+1)}_{{\mathcal P},\mathcal {I}}$
of
$\mathrm {pretr} ({\mathcal P})$
.
The following proposition shows that Corollary 4.3 is a generalization of the higher Auslander correspondence for exact categories in Theorem 1.4 in the case of exact categories with split retractions.
Proposition 4.4. Let
$\mathcal {E}$
be a weakly idempotent complete exact category with enough injectives. Let
$\mathcal {I} $
be the class of injectives of
$\mathcal {E}$
. Let
$\mathcal {M}$
be a d-cluster tilting subcategory of
$\mathcal {E}$
, (cf. Definition 2.4). Then the following statements hold.
-
(a) The pair
$(\mathcal {M},\mathcal {I})$
is d-cluster tilting and satisfies the conditions in Proposition 4.1. -
(b) We have an equivalence of exact categories
$H^0(\mathcal {A}_{\mathcal {M},\mathcal {I}}^{(d+1)})\xrightarrow{_\sim } \mathrm {mod}_{\mathrm {adm}}(\mathcal {M})$
.
Proof. We view the additive category
$\mathcal {M}$
as a dg category concentrated in degree zero and so condition 1) in Proposition 4.1 holds. Consider a complex over
$\mathcal {M}$
$$ \begin{align} 0\rightarrow M_{d+1}\xrightarrow{f_d} M_d\xrightarrow{f_{d-1}} \cdots\xrightarrow{f_2} M_2\xrightarrow{f_1} M_1\xrightarrow{f_0} M_0 \end{align} $$
such that the induced complex
$$ \begin{align} {\mathcal{M}}(M_0,I)\rightarrow {\mathcal{M}}(M_1,I)\rightarrow {\mathcal{M}}(M_2,I)\rightarrow\cdots\rightarrow {\mathcal{M}}(M_{d},I)\rightarrow {\mathcal{M}}(M_{d+1},I)\rightarrow 0 \end{align} $$
is exact for any I in
$\mathcal {I}$
. Since
$\mathcal {E}$
has enough injectives, there is an inflation
$i_{d+1}:M_{d+1}\rightarrowtail I_{d+1}$
where
$I_{d+1}\in \mathcal {I}$
. By the exactness of the complex (4.8), there is a morphism
$h:M_{d}\rightarrow I_{d+1}$
such that following diagram commutes
Since
$\mathcal {E}$
is weakly idempotent complete, it follows that
$f_d$
is an inflation, (cf. [Reference Bühler7, Proposition 7.6]). Let
be a conflation in
$\mathcal {E}$
. Then the complex (4.7) can be expanded to
Similarly we have that
$v_{d-1}:L_d\rightarrow M_{d-1}$
is an inflation. Repeating the above argument, we see that the complex (4.7) is exact. Since
$\mathrm {Ext}^{i}(\mathcal {M},\mathcal {M})=0$
for
$1\leq i\leq d-1$
, we see that
$$ \begin{align} 0\rightarrow \mathcal{M}(M,M_{d+1})\rightarrow \mathcal{M}(M,M_d)\rightarrow \cdots\rightarrow \mathcal{M}(M,M_1)\rightarrow \mathcal{M}(M,M_0) \end{align} $$
is exact for each
$M\in \mathcal {M}$
. This shows condition 2).
We identify
$\mathrm {pretr} (\mathcal {M})$
with
$\mathcal {C}^b_{\mathrm {dg}}(\mathcal {M})$
thanks to condition 1). Let X be an object in
$$\begin{align*}\mathcal{A}_{\mathcal{M},\mathcal{I}}^{(d)}={(}\mathcal{M}\ast\Sigma\mathcal{M}\ast\cdots\ast\Sigma^{n-1}\mathcal{M}\ast\Sigma^d\mathcal{M}{)}\cap \mathrm{ker}\mathrm{Ext}^{\geq 1}(-,\mathcal{I}). \end{align*}$$
Then X can be identified with a complex
$$ \begin{align} 0\rightarrow M_{d+1}\xrightarrow{f_d} M_d\xrightarrow{f_{d-1}} \cdots\xrightarrow{f_2} M_2\xrightarrow{f_1} M_1 \end{align} $$
such that the induced complex
$$ \begin{align} {\mathcal{M}}(M_1,I)\rightarrow {\mathcal{M}}(M_2,I)\rightarrow {\mathcal{M}}(M_3,I)\rightarrow\cdots\rightarrow {\mathcal{M}}(M_{d},I)\rightarrow {\mathcal{M}}(M_{d+1},I)\rightarrow 0 \end{align} $$
is exact for any
$I\in \mathcal {I}$
, where
$M_i$
is in degree
$-i+1$
. By the above, we see that the complex (4.11) is exact and in particular the morphism
$f_1$
is admissible. Let
$v_0:M_1\rightarrow L_1$
be its cokernel. Since
$\mathcal {E}$
has enough injectives, there is an inflation
$L_1\rightarrowtail I_0$
. Then the following diagram
gives rise to a morphism
$X\rightarrow I_0$
in
$\mathcal H^b(\mathcal {M})$
which is a left
$\mathcal {I}$
-approximation. Here we denote by
$\mathcal H^b(\mathcal {M})$
the bounded homotopy category of the additive category
$\mathcal {M}$
. This shows that the pair
$(\mathcal {M},\mathcal {I})$
is d-cluster tilting.
The Yoneda embedding
$\mathcal {M}\rightarrow \mathrm {Mod}\,\mathcal {M}$
induces a fully faithful functor
$G:\mathcal H^b(\mathcal {M})\rightarrow \mathcal H^b(\mathrm {Mod}\,\mathcal {M})$
. Taking
$0$
-cohomology gives rise to a functor
$H: \mathcal H^b(\mathrm {Mod}\,\mathcal {M})\rightarrow \mathrm {Mod}\,\mathcal {M}$
. Let F be the restriction of the functor
$HG$
to the full subcategory
$H^0(\mathcal {A}_{\mathcal {M},\mathcal {I}}^{(d+1)})$
of
$\mathcal H^b(\mathcal {M})$
. Then we obtain a functor
$F:H^0(\mathcal {A}_{\mathcal {M},\mathcal {I}}^{(d+1)})\rightarrow \mathrm {mod}_{\mathrm {adm}}(\mathcal {M})$
. Indeed, let X be an object in
$\mathcal {A}_{\mathcal {M},\mathcal {I}}^{(d+1)}$
which corresponds to the complex (4.7). Then
$F(X)$
is the cokernel of the morphism
$M_1^{\wedge }\xrightarrow{f_0^{\wedge }} M_0^{\wedge }$
in
$\mathrm {Mod}\,\mathcal {M}$
. Since
$f_0$
is an admissible morphism, it follows that
$F(X)\in \mathrm {mod}_{\mathrm {adm}}(\mathcal {M})$
. Since the complex (4.10) is exact for each
$M\in \mathcal {M}$
, we see that the functor F is fully faithful. For any admissible morphism
$f_0:M_1\rightarrow M_0$
, we may complete it to an exact sequence in
$\mathcal {E}$
where
$M_i\in \mathcal {M}$
for
$0\leq i\leq d$
and which remains exact after applying the functor
$\mathrm {Hom}_{\mathcal {E}}(M,-)$
for each
$M\in \mathcal {M}$
, by item (ii) in Definition 2.4. It follows that
$\mathrm {Ext}^{i}(M,L_{d+1})=0$
for each
$1\leq i\leq d-1$
and
$M\in \mathcal {M}$
. Therefore we have that
$L_{d+1}\in \mathcal {M}$
and hence the functor F is dense.
Let us show that F is an exact functor. Let X be an object in
$H^0(\mathcal {A}_{\mathcal {M},\mathcal {I}}^{(d+1)})$
which is identified with the complex (4.7) under the equivalence
$\mathrm {tr}(\mathcal {M})\xrightarrow{_\sim } \mathcal {H}^b(\mathcal {M})$
. We have seen that the induced sequence (4.10) is exact for each
$M\in \mathcal {M}$
. Therefore the complex
$G(X)\in \mathcal H^b(\mathrm {Mod}\,\mathcal {M})$
is exact except in degree zero. Let
be a triangle in
$\mathcal H^b(\mathcal {M})$
with X, Y and Z all lie in
$H^0(\mathcal {A}_{\mathcal {M},\mathcal {I}}^{(d+1)})$
. By applying the functor G to the triangle (4.13), we obtain a triangle in
$\mathcal H^b(\mathrm {Mod}\,\mathcal {M})$
. By the above, we have
$H^{-1}(G(Z))=0$
and
$H^1(G(X))=0$
. Therefore, we obtain a short exact sequence in
$\mathrm {Mod}\,\mathcal {M}$
Hence, the functor F is exact. Finally, let us show that the functor F is an equivalence of exact categories. Let
$$\begin{align*}0\rightarrow F(X)\rightarrow F(Y)\rightarrow F(Z)\rightarrow 0 \end{align*}$$
be a conflation in
$\mathrm {mod}_{\mathrm {adm}}(\mathcal {M})$
where X, Y and Z are in
$H^0(\mathcal {A}_{\mathcal {M},\mathcal {I}}^{(d+1)})$
. By Lemma 2.2, it can be lifted to a conflation in
$\mathcal H^b(\mathrm {Mod}\,\mathcal {M})$
. Since G is a fully faithful triangle functor, it then can be lifted to a conflation in
$H^0(\mathcal {A}_{\mathcal {M},\mathcal {I}}^{(d+1)})$
. Therefore, the functor F is an equivalence of exact categories.
4.2 Abelian categories
Proposition 4.5. The exact dg categories in Theorem 3.7
$(1)$
which are abelian categories correspond to the d-cluster tilting pairs
$({\mathcal P},\mathcal {I})$
(assumed to be weakly idempotent complete) in
$(2)$
where
-
1)
${\mathcal P}$
is concentrated in degree
$0$
and has d-kernels in the sense of Definition 2.1 (in particular the category
$\mathrm {mod}{\mathcal P}$
of finitely presented right
${\mathcal P}$
-modules is abelian, cf. [Reference Auslander3]); -
2) a complex
(4.14)in
$$ \begin{align} 0\rightarrow P_{d+1}^{\wedge}\xrightarrow{f_{d}^{\wedge}} P_{d}^{\wedge}\rightarrow \cdots\xrightarrow{f_{1}^{\wedge}} P_1^{\wedge}\xrightarrow{f_{0}^{\wedge}} P_0^{\wedge} \end{align} $$
$\mathrm {mod}{\mathcal P}$
is exact if and only if the induced complex (4.15)is exact for any
$$ \begin{align} \mathrm{Hom}_{{\mathcal P}}(P_0,I)\rightarrow \mathrm{Hom}_{{\mathcal P}}(P_1,I)\rightarrow \cdots\rightarrow\mathrm{Hom}_{{\mathcal P}}(P_{d},I)\rightarrow \mathrm{Hom}_{{\mathcal P}}(P_{d+1},I)\rightarrow 0 \end{align} $$
$I\in \mathcal {I}$
.
In this case, the exact dg category
$\tau _{\leq 0}\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
is equivalent to the abelian category
$\mathrm {mod}{\mathcal P}$
with
$I^{\wedge }$
,
$I\in \mathcal {I}$
, the projective–injective objects.
Proof. Suppose that
$\tau _{\leq 0}\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
is abelian. Then for each morphism
$Q_1\rightarrow Q_2$
in
${\mathcal P}$
, there is a kernel
$X\in \tau _{\leq 0}\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
. Let us assume that X is given by the complex (4.6). Then we have an exact sequence of Hom spaces in
$\mathrm {tr}({\mathcal P})$
$$\begin{align*}\mathrm{Hom}(P,P_1)\rightarrow \mathrm{Hom}(P,P_0)\rightarrow \mathrm{Hom}(P,X)\rightarrow 0, \end{align*}$$
and hence an exact sequence of Hom spaces in
${\mathcal P}$
$$\begin{align*}\mathrm{Hom}(P,P_0)\rightarrow \mathrm{Hom}(P,Q_1)\rightarrow \mathrm{Hom}(P,Q_2). \end{align*}$$
This shows that the category
${\mathcal P}$
has weak kernels.
If a morphism
$Q_1\rightarrow Q_2$
in
${\mathcal P}$
is such that the induced morphism in
$\mathrm {mod}{\mathcal P}$
is a monomorphism, then it is a monomorphism in
$\tau _{\leq 0}(\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)})$
and hence an inflation in the exact dg structure. Since the exact dg structure is inherited from
$\mathrm {pretr} ({\mathcal P})$
, it follows that the mapping cone of the morphism
$Q_1\rightarrow Q_2$
is in
$\mathcal {A}^{(d+1)}_{{\mathcal P},\mathcal {I}}$
. Then we have that the map
$\mathrm {Hom}_{{\mathcal P}}(Q_2,I)\rightarrow \mathrm {Hom}_{{\mathcal P}}(Q_1,I)$
is a surjection for each
$I\in \mathcal {I}$
. This shows that
$I^{\wedge }$
is injective in
$\mathrm {mod}{\mathcal P}$
for
$I\in \mathcal {I}$
.
For the “only if” part of item 2), we observe that if the complex (4.14) is exact, then the object given by the complex
$$ \begin{align} \ldots\rightarrow 0\rightarrow P_{d+1}\rightarrow P_d\rightarrow\cdots\rightarrow P_{i}\rightarrow 0\rightarrow \ldots, \end{align} $$
where
$P_j$
lies in degree
$j-i$
for each
$j\geq i\geq 1$
, is an object in
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
, which one shows by induction on i. The morphism from the above object to
$P_{i-1}$
is a monomorphism in
$\tau _{\leq 0}(\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)})$
. Hence we argue as above that its mapping cone also lies in
$\tau _{\leq 0}(\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)})$
, that is, the complex (4.16) is an object in
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
where i is replaced by
$i-1$
. Then by the definition of
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
, we see that the complex (4.15) is exact. This shows item 2).
We claim that
$\tau _{\leq 0}(\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)})$
is equivalent to
$\mathrm {mod}{\mathcal P}$
as an exact category.
The map sending a complex (4.6) to the cokernel of the map
$P_1^{\wedge }\rightarrow P_0^{\wedge }$
in
$\mathrm {mod}{\mathcal P}$
induces a fully faithful exact functor
$\tau _{\leq 0}(\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)})\rightarrow \mathrm {mod}{\mathcal P}$
. Since both categories are abelian categories, it is dense and hence an equivalence of categories. So
$\mathrm {mod}{\mathcal P}$
is a d-Auslander abelian category. It follows that the category
${\mathcal P}$
has d-kernels.
Conversely, assume that items 1) and 2) hold. By item 1), we identify
$\mathrm {tr}({\mathcal P})$
with the homotopy category of bounded complexes over
${\mathcal P}$
. Then by defintion the objects in
$\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
are exactly given by the complexes (4.6) such that the induced complex (4.15) is exact and hence by item 2) if and only if the complex (4.14) is exact. The above functor
$\tau _{\leq 0}(\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)})\rightarrow \mathrm {mod}{\mathcal P}$
is thus a fully faithful functor. By item 1), objects in
$\mathrm {mod}{\mathcal P}$
have projective dimension at most
$d+1$
and hence the above functor is dense. Therefore
$\tau _{\leq 0}(\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)})\rightarrow \mathrm {mod}{\mathcal P}$
is an equivalence of categories.
The following is a direct consequence of Propositions 4.4 and 4.5.
Corollary 4.6. Let
$\mathcal {E}$
be an abelian category with enough injectives and let
$\mathcal {I}$
be the class of injectives of
$\mathcal {E}$
. Let
$\mathcal {M}$
be a d-cluster tilting subcategory of
$\mathcal {E}$
, (cf. Definition 2.4). Then the pair
$(\mathcal {M},\mathcal {I})$
is d-cluster tilting and satisfies the conditions in Proposition 4.5.
It follows from Corollary 4.6 that the following is a generalization of the higher Auslander correspondence for abelian categories [Reference Beligiannis6, Theorem 8.23].
Corollary 4.7. There is a bijective correspondence between the following:
-
(1) equivalence classes of abelian categories
$\mathcal {A}$
which are d-Auslander as extriangulated categories; -
(2) equivalence classes of d-cluster tilting pairs
$({\mathcal P},\mathcal {I})$
which satisfy the conditions in Proposition 4.5.
The bijection from
$(1)$
to
$(2)$
sends
$\mathcal {A}$
to the pair
$({\mathcal P},\mathcal {I})$
formed by the full subcategory
${\mathcal P}$
on the projectives of
$\mathcal {A}$
and its full subcategory
$\mathcal {I}$
of projective–injectives. The inverse bijection sends
$({\mathcal P},\mathcal {I})$
to the
$\tau _{\leq 0}$
-truncation of the dg subcategory
$\mathcal {A}^{(d+1)}_{{\mathcal P},\mathcal {I}}$
of
$\mathrm {pretr} ({\mathcal P})$
.
Example 4.8 (Auslander–Iyama correspondence).
Let R be a commutative Artin ring. Let us fix an Artin R-algebra
$\Lambda $
. Let
$\mathcal {I}$
be the full subcategory of injective
$\Lambda $
-modules. Let
${\mathcal P}=\mathrm {add} M$
for some
$\Lambda $
-module M. We will prove that the d-Auslander category
$\tau _{\leq 0}\mathcal {A}_{{\mathcal P},\mathcal {I}}^{(d+1)}$
is equivalent to
$\mathrm {mod} {\mathcal P}\xrightarrow{_\sim }\mathrm {mod}\Gamma $
, where
$\Gamma $
is the endomorphism algebra of M, if and only if M is a d-cluster tilting module. In this case, the abelian category
$\mathrm {mod} {\mathcal P}$
is d-Auslander.
The “if” part follows from Corollary 4.6. Let us show the “only if” part. First, we observe that
$\mathrm {Ext}^{i}(M,M)=0$
for
$1\leq i\leq d-1$
. We take an injective coresolution of M
$$\begin{align*}0\rightarrow M\rightarrow I_{d}\rightarrow \cdots\rightarrow I_{0}. \end{align*}$$
By Proposition 4.5 2), it remains exact after applying the functor
$\mathrm {Hom}_{\Lambda }(M,-)$
. Therefore
$\mathrm {Ext}^{i}(M,M)=0$
for
$1\leq i\leq d-1$
.
We observe that each
$\Lambda $
-module K admits a resolution
$$ \begin{align} 0\rightarrow M_{d+1}\rightarrow M_{d}\rightarrow \cdots\rightarrow M_2\rightarrow K\rightarrow 0, \end{align} $$
which remains exact when we apply to it the functor
$\mathrm {Hom}(M,-)$
. To see this, let us take an injective copresentation of K
$$\begin{align*}0\rightarrow K\xrightarrow{h} I_1\xrightarrow{f_0} I_0. \end{align*}$$
Then we complete it to a projective resolution of
$\mathrm {cok} (f_{0}^{\wedge })\in \mathrm {mod}{\mathcal P}$
$$\begin{align*}0\rightarrow M_{d+1}^{\wedge}\rightarrow\cdots\rightarrow M_2^{\wedge}\xrightarrow{f_1^{\wedge}} I_1^{\wedge}\xrightarrow{f_0^{\wedge}} I_0^{\wedge}. \end{align*}$$
Notice that the functor
$\mathrm {Hom}(-,I)$
preserves and reflects exactness for I an injective cogenerator in
$\mathrm {mod}\Lambda $
. Therefore by Proposition 4.5 2), the image of the map
$f_1:M_2\rightarrow I_1$
is K and we get the required resolution (4.17).
If we have
$\mathrm {Ext}^{i}_{\Lambda }(K,M)=0$
for
$1\leq i\leq d-1$
, then by applying a dimension shift argument to the sequence (4.17) we see that
$K\in \mathrm {add} M$
.
Suppose we have
$\mathrm {Ext}^{i}_{\Lambda }(M,K)=0$
for
$1\leq i\leq d-1$
. Since the subcategory
$\mathcal {I}$
of injectives is contained in
${\mathcal P}=\mathrm {add} M$
, we have an exact sequence
$$ \begin{align} 0\rightarrow K\xrightarrow{h} M_{d}\rightarrow M_{d-1}\rightarrow \cdots\rightarrow M_{2}\rightarrow L\rightarrow 0, \end{align} $$
where
$M_i\in {\mathcal P}$
for
$2\leq i\leq d$
and which remains exact after applying the functor
$\mathrm {Hom}(-,M)$
. Then we have
$\mathrm {Ext}^j(L,M)=0$
for
$1\leq j\leq d-1$
and hence
$L\in \mathrm {add} M$
. Then we apply the functor
$\mathrm {Hom}(-,K)$
to the sequence (4.18) and it remains exact. Therefore the morphism
$h:K\rightarrow M_d$
splits and K is in
$\mathrm {add} M$
. Therefore M is a d-cluster tilting module.
Suppose
$\Gamma $
is a d-Auslander R-algebra. Let I be an additive generator for the subcategory
$\mathcal {I}$
of projective–injectives. Put
$\Lambda =\mathrm {End}_{\Gamma }(I)$
. Then the functor
$D\mathrm {Hom}_{\Gamma }(-,I):\mathrm {add} \Gamma \rightarrow \mathrm {mod}\Lambda $
is fully faithful where
$D:\mathrm {mod}\Lambda ^{op}\rightarrow \mathrm {mod}\Lambda $
is the duality functor. Indeed, since
$\Gamma $
is d-Auslander, we have an injective copresentation of
$\Gamma $
$$\begin{align*}0\rightarrow \Gamma\rightarrow I_0\rightarrow I_1 \end{align*}$$
which, by the injectivity of I, induces an exact sequence of left
$\Lambda $
-modules
$$\begin{align*}\mathrm{Hom}_{\Gamma}(I_1,I)\rightarrow \mathrm{Hom}_{\Gamma}(I_0,I)\rightarrow \mathrm{Hom}_{\Gamma}(\Gamma,I)\rightarrow 0. \end{align*}$$
We apply the functor
$\mathrm {Hom}_{\Lambda ^{op}}(-,\mathrm {Hom}_{\Gamma }(\Gamma ,I))$
to the above sequence and we get the following diagram
where the Hom spaces in the first row are in
$\mathrm {mod} \Lambda ^{op}$
. Therefore the leftmost column is an isomorphism and we see that the functor
$D\mathrm {Hom}_{\Gamma }(-,I):\mathrm {add} \Gamma \rightarrow \mathrm {mod}\Lambda $
is fully faithful. It induces an equivalence between
$\mathcal {I}$
and the full subcategory of injective
$\Lambda $
-modules. By the above, the
$\Lambda $
-module
$D\mathrm {Hom}_{\Gamma }(\Gamma ,I)$
is d-cluster tilting.
Acknowledgements
The author thanks the referee for numerous useful suggestions. The author thanks his Ph.D supervisor Bernhard Keller and his Ph.D co-supervisor Xiao-Wu Chen for their constant support and their encouragements. The author thanks Yann Palu and Mikhail Gorsky for helpful discussions.
Funding
This work is supported by Natural Science Foundation of China (Grant No. 11971449).
