Silting modules over a class of Morita rings

Dadi Asefa; Qingbing Xu

doi:10.1515/math-2024-0009

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Silting modules over a class of Morita rings

Dadi Asefa and Qingbing Xu

Published/Copyright: May 29, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

Let Δ = A N B A M A B B be a Morita ring, where M ⊗ A N = 0 = N ⊗ B M . Let X be left A -module and Y be left B -module. We prove that ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) is a silting module if and only if X is a silting A -module, Y is a silting B -module, M ⊗ A X is generated by Y , and N ⊗ B Y is generated by X . As a consequence, we obtain that if M A and N B are flat, then ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) is a tilting Δ -module if and only if X is a tilting A -module, Y is a tilting B -module, M ⊗ A X is generated by Y , and N ⊗ B Y is generated by X .

Keywords: (partial)silting modules; tilting modules; Morita rings

MSC 2010: 16D90; 16D80

1 Introduction

Let A and B be rings, N B A and M A B bimodules with M ⊗ A N = 0 = N ⊗ B M such that

Δ = A N B A M A B B = { a n m b ∣ a ∈ A , b ∈ B , m ∈ M , n ∈ N }

is a Morita ring, where the addition is obvious, and the multiplication is given by

a n m b · a ′ n ′ m ′ b ′ = a a ′ a n ′ + n b ′ m a ′ + b m ′ b b ′ .

Morita rings were introduced by Bass [1]. This class of rings contains a very large class of algebras, and many important algebras can be realized as Morita algebras. For example, the 2 × 2 matrix algebra M 2 ( A ) = A A A A over A , the algebra Δ ( 0 , 0 ) = A A A A , the upper triangular matrix algebra A N B A 0 B , and the algebras defined by finite quivers and relations. This provides us a strong motivation to research it.

Tilting modules are essential in the representation theory of algebras. This class of modules receives special attention and is thoroughly studied (see e.g. [2–10]). Angeleri Hügel et al. [3] introduced silting modules, which generalize tilting modules introduced by Auslander and Solberg [11] and support tau-tilting modules introduced by Adachi et al. [12]. Gao and Huang [13] investigated silting modules and then provided necessary and sufficient conditions for a tilting module over triangular matrix rings. This article is motivated by the desire to extend the results in [13] to the case of Morita rings.

The main aim of this article is to study silting and tilting modules over the Morita ring Δ . First, we give sufficient and necessary conditions for a Δ -module ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) to be a partial silting module, which extends [13, Proposition 3.3]. We then prove that ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) is a silting module if and only if X is a silting A -module, Y is a silting B -module, M ⊗ A X is generated by Y , and N ⊗ B Y is generated by X . As a consequence, we obtain that if M A and N B are flat, then ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) is a tilting Δ -module if and only if X is a tilting A -module, Y is a tilting B -module, M ⊗ A X is generated by Y , and N ⊗ B Y is generated by X .

2 Preliminaries

In this section, we recall some basic definitions and facts that will be used throughout the article.

Throughout this article, all rings are associative, and all modules are unitary, and all subcategories are full and closed under isomorphisms. For a ring A , let A - Mod be the category of left A -modules. For a module X ∈ A - Mod , let Add X be the subcategory of A - Mod consisting of direct summands of finite direct sums of X and Gen X the subcategory of A - Mod consisting of quotients of direct sums of copies of X .

Let Δ ≔ A N B A M A B B , where A and B are rings, N B A and M A B are bimodules, where M ⊗ A N = 0 = N ⊗ B M . Then Δ is a Morita ring with zero bimodule homomorphisms, where the addition is as the addition of matrix, and the multiplication is given by

a n m b · a ′ n ′ m ′ b ′ = a a ′ a n ′ + n b ′ m a ′ + b m ′ b b ′ .

This is a special case of the general Morita rings in the sense of Bass [1], see also [14–18].

It is proved in [16] that the category Δ ( 0 , 0 ) is equivalent to the category Ω whose objects are tuples ( X , Y , f , g ) , where X ∈ A - Mod , Y ∈ B - Mod , and f ∈ Hom B ( M ⊗ A X , Y ) , g ∈ Hom A ( N ⊗ B Y , X ) and whose morphisms from ( X , Y , f , g ) to ( X ′ , Y ′ , f ′ , g ′ ) are pairs ( a , b ) such that a ∈ Hom A ( X , X ′ ) , b ∈ Hom B ( Y , Y ′ ) and the following diagrams are commutative:

A sequence of Ω -homomorphisms

0 ⟶ ( X 1 , Y 1 , f 1 , g 1 ) ⟶ ( X 2 , Y 2 , f 2 , g 2 ) ⟶ ( X 3 , Y 3 , f 3 , g 3 ) ⟶ 0

is exact if and only if the sequence of A -homomorphisms 0 ⟶ X 1 ⟶ X 2 ⟶ X 3 ⟶ 0 is exact in A - Mod and the sequence of B -homomorphisms 0 ⟶ Y 1 ⟶ Y 2 ⟶ Y 3 ⟶ 0 is exact in B - Mod .

In what follows, we identify Δ - Mod by Ω .

Proposition 2.1

[19, Prop. 3.1] Let Δ = A N B A M A B B be a Morita ring. The indecomposable projective Δ -modules are exactly

( P , M ⊗ A P , 1 , 0 ) or ( N ⊗ B Q , Q , 0 , 1 ) ,

where P runs over indecomposable projective A-modules, and Q runs over indecomposable projective B-modules.

Let A be a ring and

σ : P 1 ⟶ P 0

a homomorphism in A - Mod , where P 1 and P 0 are projective A -modules. We write

D σ ≔ { K ∈ A - Mod ∣ Hom A ( σ , K ) is epic } .

Recall that if a subcategory T of A - Mod is closed under images, direct sums, and extensions, then it is referred to as a torsion class [20].

Definition 2.2

[2, Definition 3.7] Let T be an A -module.

T is called partial silting if there exists a projective presentation σ of T such that D σ is torsion class and T ∈ D σ .
T is called silting if there exists a projective presentation σ of T such that Gen T = D σ .

D σ is closed under images and extensions (see [2, Lemma 3.6(1)]). Thus, D σ is a torsion free class if and only it is closed under direct sums. Furthermore, T ∈ D σ implies Gen T ∈ D σ .

Given a subcategory X of A - Mod , recall that a right approximation of M in X is a morphism f : X → M with X ∈ X , such that the induced morphism Hom A ( X ′ , X ) → Hom A ( X ′ , M ) is surjective for each X ′ ∈ X .

The following result gives a relation between partial silting and silting modules.

Proposition 2.3

[2, Proposition 3.11] Let T ∈ A - Mod with a projective presentation σ . Then T is a silting module with respect to σ if and only if T is a partial silting module with respect to σ and there exists an exact sequence

A ⟶ ϕ T 0 ⟶ T 1 ⟶ 0

with T 0 , T 1 ∈ Add X such that ϕ is a right D σ -approximation.

3 Silting modules over Morita rings

Let Δ = A N B A M A B B be the Morita ring with M ⊗ A N = 0 = N ⊗ B M . Assume X ∈ A -Mod and Y ∈ B -Mod. Let

(3.1) P 1 ⟶ σ X P 0 ⟶ X ⟶ 0

and

(3.2) Q 1 ⟶ σ Y Q 0 ⟶ Y ⟶ 0

be projective presentations of X and Y , respectively, where P 1 and P 0 are projective A -modules, and Q 1 and Q 0 are projective B -modules. If we apply the functor M ⊗ A − to (3.1), we obtain the following exact sequence of abelian groups:

(3.3) M ⊗ A P 1 ⟶ M ⊗ σ X M ⊗ A P 0 ⟶ M ⊗ A X ⟶ 0 .

Similarly, if we apply the functor N ⊗ B − to (3.2), we obtain the following exact sequence of abelian groups:

(3.4) N ⊗ B Q 1 ⟶ N ⊗ σ Y N ⊗ B Q 0 ⟶ N ⊗ B Y ⟶ 0 .

Thus, the sequence

( P 1 , M ⊗ A P 1 , 1 , 0 ) ⊕ ( N ⊗ B Q 1 , Q 1 , 0 , 1 ) ⟶ σ ( P 0 , M ⊗ A P 0 , 1 , 0 ) ⊕ ( N ⊗ B Q 0 , Q 0 , 0 , 1 ) ⟶ ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) ⟶ 0

is a projective presentation of ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) with σ = ( a 0 0 b ) such that a = ( σ X , M ⊗ σ X ) and b = ( N ⊗ σ Y , σ Y ) .

Lemma 3.1

Let X 1 be an A-module and Y 1 a B-module.

( X 1 , Y 1 , f , g ) ∈ D σ if and only if X 1 ∈ D σ X and Y 1 ∈ D σ Y .
If X 1 ∈ D σ X , then ( X 1 , 0 , 0 , 0 ) ∈ D σ .
If Y 1 ∈ D σ Y , then ( 0 , Y 1 , 0 , 0 ) ∈ D σ .
If X 1 ∈ D σ X and M ⊗ A X 1 ∈ D σ Y , then ( X 1 , M ⊗ A X 1 , 1 , 0 ) ∈ D σ .
If Y 1 ∈ D σ Y and N ⊗ B Y 1 ∈ D σ X , then ( N ⊗ A Y 1 , Y 1 , 0 , 1 ) ∈ D σ .

Proof

( 1 ) Suppose ( X 1 , Y 1 , f , g ) ∈ D σ . Let α ∈ Hom A ( P 1 , X 1 ) . Then ( ( α , f ∘ ( M ⊗ α ) ) , ( 0.0 ) ) ∈ Hom Δ ( ( P 1 , M ⊗ A P 1 , 1 , 0 ) ⊕ ( N ⊗ B Q 1 , Q 1 , 0 , 1 ) , ( X 1 , Y 1 , f , g ) ) . Since ( X 1 , Y 1 , f , g ) ∈ D σ there exists ( ( f 1 , g 1 ) , ( h 1 , k 1 ) ) ∈ Hom Δ ( ( P 0 , M ⊗ A P 0 , 1 , 0 ) ⊕ ( N ⊗ B Q 0 , Q 0 , 0 , 1 ) , ( X 1 , Y 1 , f , g ) ) such that the diagram

commutes. Thus, it follows that ( ( α , f ∘ ( M ⊗ α ) ) , ( 0.0 ) ) = ( ( f 1 , g 1 ) , ( h 1 . k 1 ) ) ∘ σ . Hence, ( α , f ∘ ( M ⊗ α ) ) = ( f 1 , g 1 ) ∘ a = ( f 1 , g 1 ) ∘ ( σ X , M ⊗ σ X ) , i.e., α = f 1 ∘ σ X . Thus, X 1 ∈ D σ X .

Let β ∈ Hom B ( Q 1 , Y 1 ) . Then ( ( 0 , 0 ) , ( g ∘ ( N ⊗ β ) , β ) ) ∈ Hom Δ ( ( P 1 , M ⊗ A P 1 , 1 , 0 ) ⊕ ( N ⊗ B Q 1 , Q 1 , 0 , 1 ) , ( X 1 , Y 1 , f , g ) ) . Since ( X 1 , Y 1 , f , g ) ∈ D σ there exists ( ( f 2 , g 2 ) , ( h 2 . k 2 ) ) ∈ Hom Δ ( ( P 0 , M ⊗ A P 0 , 1 , 0 ) ⊕ ( N ⊗ B Q 0 , Q 0 , 0 , 1 ) , ( X 1 , Y 1 , f , g ) ) such that the diagram

commutes. Thus, it follows that ( ( 0 , 0 ) , ( g ∘ ( N ⊗ β ) , β ) ) = ( ( f 2 , g 2 ) , ( h 2 . k 2 ) ) ∘ σ . Hence, ( g ∘ ( N ⊗ β ) , β ) = ( h 2 , k 2 ) ∘ b = ( h 2 , k 2 ) ∘ ( N ⊗ σ Y , σ Y ) , i.e., β = k 2 ∘ σ Y . Thus Y 1 ∈ D σ Y .

Conversely, suppose that X 1 ∈ D σ X and Y 1 ∈ D σ Y . Let ( h , k ) ∈ Hom Δ ( ( P 1 , M ⊗ A P 1 , 1 , 0 ) ⊕ ( N ⊗ B Q 1 , Q 1 , 0 , 1 ) , ( X 1 , Y 1 , f , g ) ) . Write h = ( f 3 , g 3 ) and k = ( f 4 , g 4 ) with f 3 ∈ Hom A ( P 1 , X 1 ) , g 3 ∈ Hom B ( M ⊗ A P 1 , Y 1 ) , f 4 ∈ Hom A ( N ⊗ B Q 1 , X 1 ) , and g 4 ∈ Hom B ( Q 1 , Y 1 ) . Then by definition of morphisms in Δ -Mod, we have the following commutative diagrams:

Thus, g 3 = f ∘ ( M ⊗ f 3 ) and f 4 = g ∘ ( N ⊗ g 4 ) . Since X 1 ∈ D σ X and Y 1 ∈ D σ Y , there exist f 3 ′ ∈ Hom A ( P 0 , X 1 ) and g 4 ′ ∈ Hom B ( Q 0 , Y 1 ) such that f 3 = f 3 ′ ∘ σ X and g 4 = g 4 ′ ∘ σ Y . Then

( ( f 3 ′ , f ∘ ( M ⊗ f 3 ′ ) ) , ( g ∘ ( N ⊗ g 4 ′ ) , g 4 ′ ) ) ∘ σ = ( ( f 3 ′ , f ∘ ( M ⊗ f 3 ′ ) ) , ( g ∘ ( N ⊗ g 4 ′ ) , g 4 ′ ) ) ∘ ( a 0 0 b ) = ( ( f 3 ′ , f ∘ ( M ⊗ f 3 ′ ) ) ∘ a , ( g ∘ ( N ⊗ g 4 ′ ) , g 4 ′ ) ∘ b ) = ( ( f 3 ′ , f ∘ ( M ⊗ f 3 ′ ) ) ∘ ( σ X , M ⊗ σ X ) , ( g ∘ ( N ⊗ g 4 ′ ) , g 4 ′ ) ∘ ( N ⊗ σ Y , σ Y ) ) = ( ( f 3 ′ ∘ σ X , f ∘ ( M ⊗ ( f 3 ′ ∘ σ X ) ) ) , ( g ∘ ( N ⊗ ( g 4 ′ ∘ σ Y ) ) , g 4 ′ ∘ σ Y ) ) = ( ( f 3 , f ∘ ( M ⊗ f 3 ) ) , ( g ∘ ( N ⊗ g 4 ) , g 4 ) ) = ( ( f 3 , g 3 ) , ( f 4 , g 4 ) ) .

Thus, ( X 1 , Y 1 , f , g ) ∈ D σ .

The assertions (2), (3), (4), and (5) follow from (1).□

Let I be a set and { ( X i , Y i , f i , g i ) } a Δ -module with all X i ∈ A -Mod and Y i ∈ B -Mod. Since tensor functor commutes with direct sums, we have that

⊕ i ∈ I { ( X i , Y i , f i , g i ) } ≅ { ( ⊕ i ∈ I X i , ⊕ i ∈ I Y i , ⊕ i ∈ I f i , ⊕ i ∈ I g i ) } .

Lemma 3.2

D σ is torsion class if and only if D σ X and D σ Y are torsion classes.

Proof

Suppose that D σ is a torsion class. Let { X i } i ∈ I be a family of modules in D σ X . Then, by Lemma 3.1(2) ( X i , 0 , 0 , 0 ) ∈ D σ X for any i ∈ I . Thus, { ( ⊕ i ∈ I X i , 0 , 0 , 0 ) } ≅ ⊕ i ∈ I { ( X i , 0 , 0 , 0 ) } ∈ D σ . So by Lemma 3.1(1) we have ⊕ i ∈ I X i ∈ D σ X . Thus, D σ X is a torsion class by [2, Lemma 3.6(1)]. Similarly, D σ Y is also a torsion class.

Conversely, suppose that both D σ X and D σ Y are torsion classes. Let { ( X i , Y i , f i , g i ) } i ∈ I be a family of modules in D with X i ∈ A -Mod and Y i ∈ B -Mod. Then by Lemma 3.1(1), we have X i ∈ D σ X and Y i ∈ D σ Y for each i ∈ I . Thus, it follows that ⊕ i ∈ I X i ∈ D σ X and ⊕ i ∈ I Y i ∈ D σ Y . Let ( h , k ) ∈ Hom Δ ( ( P 0 , M ⊗ A P 0 , 1 , 0 ) ⊕ ( N ⊗ B Q 0 , Q 0 , 0 , 1 ) , ( ⊕ i ∈ I X i , ⊕ i ∈ I Y i , ⊕ i ∈ I f i , ⊕ i ∈ I g i ) ) . Write h = ( h 1 , k 1 ) and k = ( h 2 , k 2 ) with h 1 ∈ Hom A ( P 1 , ⊕ i ∈ I X i ) , k 1 ∈ Hom B ( M ⊗ A P 1 , ⊕ i ∈ I Y i ) , h 2 ∈ Hom A ( N ⊗ B Q 1 , ⊕ i ∈ I X i ) , and k 2 ∈ Hom B ( Q 1 , ⊕ i ∈ I Y i ) . Then by definition of morphisms in Δ -Mod we have the following commutative diagrams.

Thus, k 1 = ( ⊕ i ∈ I f i ) ∘ ( M ⊗ h 1 ) and h 2 = ( ⊕ i ∈ I g i ) ∘ ( N ⊗ k 2 ) . Since ⊕ i ∈ I X i ∈ D σ X and ⊕ i ∈ I Y i ∈ D σ Y , there exist h 1 ′ ∈ Hom A ( P 1 , ⊕ i ∈ I X i ) and k 2 ′ ∈ Hom B ( Q 1 , ⊕ i ∈ I Y i ) such that h 1 = h 1 ′ ∘ σ X and k 2 = k 2 ′ ∘ σ Y . Then

( ( h 1 ′ , ⊕ i ∈ I f i ∘ ( M ⊗ h 1 ′ ) ) , ( ⊕ i ∈ I g i ∘ ( N ⊗ k 2 ′ ) , k 2 ′ ) ) ∘ σ = ( ( h 1 ′ , ⊕ i ∈ I f i ∘ ( M ⊗ h 1 ′ ) ) , ( ⊕ i ∈ I g i ∘ ( N ⊗ k 2 ′ ) , k 2 ′ ) ) ∘ ( a 0 0 b ) = ( ( h 1 ′ , ⊕ i ∈ I f i ∘ ( M ⊗ h 1 ′ ) ) ∘ a , ( ⊕ i ∈ I g i ∘ ( N ⊗ k 2 ′ ) , k 2 ′ ) ∘ b ) = ( ( h 1 ′ , ⊕ i ∈ I f i ∘ ( M ⊗ h 1 ′ ) ) ∘ ( σ X , M ⊗ σ X ) , ( ⊕ i ∈ I g i ∘ ( N ⊗ k 2 ′ ) , k 2 ′ ) ∘ ( N ⊗ σ Y , σ Y ) ) = ( ( h 1 ′ ∘ σ X , ⊕ i ∈ I f i ∘ ( M ⊗ ( h 1 ′ ∘ σ X ) ) ) , ( ⊕ i ∈ I g i ∘ ( N ⊗ ( k 2 ′ ∘ σ Y ) ) , k 2 ′ ∘ σ Y ) ) = ( ( h 1 , ⊕ i ∈ I f i ∘ ( M ⊗ h 1 ) ) , ( ⊕ i ∈ I g i ∘ ( N ⊗ k 2 ) , k 2 ) ) = ( ( h 1 , k 1 ) , ( h 2 , k 2 ) ) = ( h , k ) .

Thus, ⊕ i ∈ I ( X i , Y i , f i , g i ) ≅ ( ⊕ i ∈ I X i , ⊕ i ∈ I Y i , ⊕ i ∈ I f i , ⊕ i ∈ I g i ) ∈ D σ . So D σ is a torsion class by [2, Lemma 3.6(1)] again.□

By Lemmas 3.1 and 3.2, we obtain the following sufficient and necessary conditions of partial silting modules over a Morita ring Δ = A N B A M A B B .

Theorem 3.3

Let X be an A-module with a projective presentation σ X and Y a B-module with a projective presentation σ Y . Then ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) is a partial silting Δ -module with respect to σ if and only if the following conditions are satisfied.

X is a partial silting A-module with respect to σ X .
Y is a partial silting B-module with respect to σ Y .
M ⊗ A X ∈ D σ Y .
N ⊗ B Y ∈ D σ X .

The following result gives sufficient and necessary conditions for a silting module over a Morita ring Δ = A N B A M A B B .

Theorem 3.4

Let X be an A-module and Y a B-module. Then ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) is a silting Δ -module if and only if the following conditions are satisfied.

X is a silting A-module.
Y is a silting B-module.
M ⊗ A X ∈ Gen Y .
N ⊗ B Y ∈ Gen X .

Proof

Suppose that ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) is a silting Δ -module with respect to σ . Thus, we have

(3.5) D σ = Gen ( ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) ) = Gen ( X , M ⊗ A X , 1 , 0 ) ⊕ Gen ( N ⊗ B Y , Y , 0 , 1 ) .

Since X is partial silting by Theorem 3.3, we have Gen X ⊆ D σ X . Let X 1 be in D σ X . Then by Lemma 3.1(2), ( X 1 , 0 , 0 , 0 ) is in D σ . If ( X 1 , 0 , 0 , 0 ) has a direct summand ( X 1 ′ , 0 , 0 , 0 ) in Gen ( N ⊗ B Y , Y , 0 , 1 ) , then we have the following commutative diagram with exact columns:

So X 1 ′ = 0 . Thus, ( X 1 , 0 , 0 , 0 ) ∈ Gen ( X , 0 , 0 , 0 ) and X 1 ∈ Gen X . This implies Gen X = D σ X and X is a silting Δ -module. By Theorem 3.3, we have N ⊗ A Y ∈ D σ X = Gen X . Similarly, we have that Y is a silting B -module and M ⊗ A X ∈ Gen Y .

Conversely, suppose that X is a silting A -module with respect to σ X such that M ⊗ A X ∈ Gen Y and Y is a silting B -module with respect to σ Y such that N ⊗ B Y ∈ Gen X . Then by Theorem 3.3 we have ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) is a partial silting Δ -module with respect to σ . By Proposition 2.3, there exist the following exact sequences:

A ⟶ ϕ T 1 ⟶ T 2 ⟶ 0 and B ⟶ ψ E 1 ⟶ E 2 ⟶ 0

with T 1 , T 2 ∈ Add X and E 1 , E 2 ∈ Add Y such that ϕ is a right D σ X -approximation and ψ is a right D σ Y -approximation. Set

a ′ ≔ ( ϕ , M ⊗ ϕ ) , b ′ ≔ ( N ⊗ ψ , ψ ) , and α = ( a ′ 0 0 b ′ ) .

Then, we obtain the following exact sequence:

( A , M , 1 , 0 ) ⊕ ( N , B , 0 , 1 ) ⟶ α ( T 1 , M ⊗ A T 1 , 1 , 0 ) ⊕ ( N ⊗ B E 1 , E 1 , 0 , 1 ) ⟶ ( T 2 , M ⊗ A T 2 , 1 , 0 ) ⊕ ( N ⊗ B E 2 , E 2 , 0 , 1 ) ⟶ 0 .

Clearly both ( T 1 , M ⊗ A T 1 , 1 , 0 ) ⊕ ( N ⊗ B E 1 , E 1 , 0 , 1 ) and ( T 2 , M ⊗ A T 2 , 1 , 0 ) ⊕ ( N ⊗ B E 2 , E 2 , 0 , 1 ) belong to Add ( ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) ) . Let ( X 1 , Y 1 , f 1 , g 1 ) ∈ D σ and ( h , k ) ∈ Hom Δ ( ( A , M , 1 , 0 ) ⊕ ( N , B , 0 , 1 ) , ( X 1 , Y 1 , f , g ) ) . Then by Lemma 3.1(1), we have X 1 ∈ D σ X , and Y 1 ∈ D σ Y . Write h = ( f 2 , f 3 ) and k = ( g 2 , g 3 ) with f 2 ∈ Hom A ( A , X 1 ) , f 3 ∈ Hom B ( M , Y 1 ) , g 2 ∈ Hom A ( N , X 1 ) and g 3 ∈ Hom B ( B , Y 1 ) . Then by definition of morphisms in Δ -Mod we have the following commutative diagrams.

Thus, f 3 = f ∘ ( M ⊗ f 2 ) and g 2 = g ∘ ( N ⊗ g 3 ) . Since ϕ is a right D σ X -approximation and ψ is a right D σ Y -approximation, there exist f 2 ′ ∈ Hom A ( T 1 , X 1 ) and g 3 ′ ∈ Hom B ( E 1 , Y 1 ) such that f 2 = f 2 ′ ∘ σ X and g 3 = g 3 ′ ∘ σ Y . Then we have the following equality.

( ( f 2 ′ , f ∘ ( M ⊗ f 2 ′ ) ) , ( g ∘ ( N ⊗ g 3 ′ ) , g 3 ′ ) ) ∘ σ = ( ( f 2 ′ , f ∘ ( M ⊗ f 2 ′ ) ) , ( g ∘ ( N ⊗ g 3 ′ ) , g 3 ′ ) ) ∘ ( a ′ 0 0 b ′ ) = ( ( f 2 ′ , f ∘ ( M ⊗ f 2 ′ ) ) ∘ a ′ , ( g ∘ ( N ⊗ g 3 ′ ) , g 3 ′ ) ∘ b ′ ) = ( ( f 2 ′ , f ∘ ( M ⊗ f 2 ′ ) ) ∘ ( ϕ , M ⊗ ϕ ) , ( g ∘ ( N ⊗ g 3 ′ ) , g 3 ′ ) ∘ ( N ⊗ ψ , ψ ) ) = ( ( f 2 ′ ∘ ϕ , f ∘ ( M ⊗ ( f 2 ′ ∘ ϕ ) ) ) , ( g ∘ ( N ⊗ ( g 3 ′ ∘ ψ ) ) , g 3 ′ ∘ ψ ) ) = ( ( f 2 , f ∘ ( M ⊗ f 2 ) ) , ( g ∘ ( N ⊗ g 3 ) , g 3 ) ) = ( ( f 2 , f 3 ) , ( g 2 , g 3 ) ) = ( h , k ) .

Thus, the diagram

commutes. Hence, α is a right D σ -approximation. So by Proposition 2.3, ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) is a silting Δ -module.□

Recall from [6] that an A -module X is called a tilting if Gen X = { N ∈ A - Mod A ∣ Ext A 1 ( X , N ) = 0 } or equivalently, if the following conditions are satisfied.

The projective dimension of X is at most one.
Ext A 1 ( X , X ( I ) ) = 0 for any set I .
There exists an exact sequence 0 ⟶ A ⟶ T 0 ⟶ T 1 ⟶ 0 in A -Mod with T 0 , T 1 ∈ Add X .

An A -module X is tilting if and only if it is silting with respect to a monomorphic projective presentation [2, Proposition 3.13(1)]. The following corollary follows from this fact and Theorem 3.4.

Corollary 3.5

Let X be a silting A-module with respect to a monomorphic projective presentation σ X and Y a silting B-module with respect to monomorphic projective presentation σ Y . If M ⊗ σ X and N ⊗ σ Y are monic, M ⊗ A X ∈ Gen X and N ⊗ B Y ∈ Gen Y , then ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) is a tilting Δ -module.

We then have the following result:

Theorem 3.6

Let X be an A-module and Y a B-module. If M A and N B are flat, then the following statements are equivalent.

( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) is a tilting Δ -module
X is a tilting A -module, Y is a tilting B-module, M ⊗ A X ∈ Gen Y , and N ⊗ B Y ∈ Gen X .

Proof

Assume M A and N B are flat. Then σ is monic if and only if σ X and σ Y are monic. Thus, the assertion follows from Theorem 3.4.□

Remark 3.7

It remains open whether the main results of this article hold true for a Morita ring with non-zero bimodule homomorphisms.

Acknowledgements

We sincerely thank the referees for their valuable suggestions and comments.

Funding information: The publication of this article was supported by the Natural Science Foundation of Universities in Anhui Province (Key University Science Research Project) (Grant No.: 2023AH053092, KJ2020ZD74, and KJ2021A1096).
Author contributions: Conceptualization: Dadi Asefa; writing–original draft preparation: Dadi Asefa; writing–review and editing: Dadi Asefa and Qingbing Xu; funding acquisition: Qingbing Xu.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] H. Bass, The Morita Theorems, Mimeographed Notes, University of Oregon, Eugene, Oregon, 1962. Search in Google Scholar

[2] L. Angeleri Hügel and M. Hrberk, Silting modules over commutative ring, Int. Math. Res. Not. IMRN 2017 (2017), no. 13, 4131–4151. Search in Google Scholar

[3] L. Angeleri Hügel, F. Marks, and J. Victória, Silting modules, Int. Math. Res. Not. IMRN 2016 (2016), no. 4, 1251–1284. 10.1093/imrn/rnv191Search in Google Scholar

[4] I. Assem and N. Marmaridis, Tilting modules over split-by-nilpotent extensions, Comm. Algebra 26 (1998), no. 5, 1547–1555. 10.1080/00927879808826219Search in Google Scholar

[5] Q. Chen, M. Gong, and W. Rump, Tilting and trivial extensions, Arch. Math. 93 (2009), 531–540. 10.1007/s00013-009-0064-xSearch in Google Scholar

[6] R. Colpi and J. Trlifaj, Tilting modules and tilting torsion theories, J. Algebra 178 (1995), no. 2, 614–634. 10.1006/jabr.1995.1368Search in Google Scholar

[7] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Soc. 274 (1982), 399–443. 10.1090/S0002-9947-1982-0675063-2Search in Google Scholar

[8] D. Happel and L. Unger, On a partial order of tilting modules, Algebr. Represent. Theory 8 (2005), no. 2, 147–156. 10.1007/s10468-005-3595-2Search in Google Scholar

[9] Y. Miyashita, Tilting modules of finite projective dimension, Math. Z. 193 (1986), no. 1, 113–146. 10.1007/BF01163359Search in Google Scholar

[10] S. O. Smalø, Torsion theory and tilting modules, Bull. London Math. Soc. 16 (1984), 518–522. 10.1112/blms/16.5.518Search in Google Scholar

[11] M. Auslander and Ø. Solberg, Relative homology and representation theory I. Relative homology and homological subcategories, Comm. Algebra 21 (1993), 2995–3031. 10.1080/00927879308824717Search in Google Scholar

[12] T. Adachi, O. Iyama, and I. Reiten, τ-tilting theory, Compos. Math. 150 (2014), 415–467. 10.1112/S0010437X13007422Search in Google Scholar

[13] H. Gao and Z. Huang, Silting modules over triangular matrix rings, Taiwanese J. Math. 24 (2020), no. 6, 1417–1437. 10.11650/tjm/200204Search in Google Scholar

[14] N. Gao and C. Psaroudakis, Gorenstein homological aspects of monomorphism categories via morita rings, Algebr. Represent. Theory 20 (2017), no. 2, 487–529. 10.1007/s10468-016-9652-1Search in Google Scholar

[15] N. Gao, J. Ma, and X. Y. Liu, RSS equivalences over a class of Morita rings, J. Algebra 573 (2021), 336–363. 10.1016/j.jalgebra.2020.12.037Search in Google Scholar

[16] E. Green, On the representation theory of rings in matrix form, Pacific J. Math. 100 (1982), no. 1, 123–138. 10.2140/pjm.1982.100.123Search in Google Scholar

[17] D. Asefa, Ding projective modules over Morita context rings, Comm. Algebra 52 (2024), no. 1, 79–87. 10.1080/00927872.2023.2233032Search in Google Scholar

[18] D. Asefa, Construction of Gorenstein-projective modules over Morita rings, J. Algebra Appl. 22 (2023), no. 11, 2350247. 10.1142/S021949882350247XSearch in Google Scholar

[19] E. Green and C. Psaroudakis, On Artin algebras arising from Morita contexts, Algebr. Represent. Theory 17 (2014), no. 5, 1485–1525. 10.1007/s10468-013-9457-4Search in Google Scholar

[20] I. Assem, D. Simson, and A. Skowroński, Elements of the representation theory of associative algebras, Vol 1: Techniques of Representation Theory, London Mathematical Society Student Texts, Vol. 65, Cambridge University Press, Cambridge, 2006. 10.1017/CBO9780511614309Search in Google Scholar

Received: 2023-10-19

Revised: 2024-03-14

Accepted: 2024-03-17

Published Online: 2024-05-29

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2024-0009

Keywords for this article

(partial)silting modules; tilting modules; Morita rings

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