Derived equivalences and stable equivalences of Morita type, II

Hu, Wei; Xi, Changchang

doi:10.4171/rmi/981

Cited by 9 publications

(9 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This was first observed by Miyachi in [, Theorem 4.11] for symmetric algebras (that is,

_{A} A_{A} ≃_{A} D {(A)}_{A}

as bimodules). We state the following generalization of Miyachi's result (see ). Proposition Let

A

be a finite‐dimensional algebra over a field, and let

e

be a

ν

‐stable idempotent element in

A

.…”

Section: Derived Equivalences and Stable Equivalences Of Morita Typementioning

confidence: 83%

“…This was first observed by Miyachi in [90,Theorem 4.11] for symmetric algebras (that is, A A A A D(A) A as bimodules). We state the following generalization of Miyachi's result (see [63]).…”

Section: Extending Tilting Complexes Over Corner Algebras To Algebrasmentioning

confidence: 87%

“…It is a Frobenius algebra, of course, a self‐injective algebra. For a proof, one may see [, Lemma 2.5; ]. Note that a Frobenius part of

A

is uniquely determined by

A

(up to isomorphism).…”

Section: Derived Equivalences and Stable Equivalences Of Morita Typementioning

confidence: 99%

“…Case (ii): Both

σ

and

σ^{'}

are non‐empty. By [, Lemma 3.4], the functor

normalΦ

restricts to a stable equivalence

Φ_{1}

of Morita type between

e_{σ} A e_{σ}

and

e_{σ^{'}} B e_{σ^{'}}

. Moreover, the idempotent elements

e_{σ}

and

e_{σ^{'}}

are projectively

ν

‐stable and the algebras

e_{σ} A e_{σ}

and

e_{σ^{'}} B e_{σ^{'}}

are self‐injective with fewer simple modules.…”

Section: Applicationsmentioning

confidence: 99%

“…Theorem 5.2 [63]. Let A and B be finite-dimensional k-algebras over an algebraically closed field k and without non-zero semisimple direct summands.…”

Section: Lifting Stable To Derived Equivalencesmentioning

confidence: 99%

See 4 more Smart Citations

Derived equivalences of algebras

2018

Bull. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

Derived categories and equivalences between them are the pièce de résistance of modern homological algebra. They are widely used in many branches of mathematics, especially in algebraic geometry and representation theory. In this note, we shall survey some recently developed construction methods of derived equivalences for algebras and rings, with applications to homological conjectures, such as Broué's abelian defect group conjecture and the finitistic dimension conjecture, and to computation of higher algebraic K‐groups of algebras and rings.

show abstract

“…This was first observed by Miyachi in [, Theorem 4.11] for symmetric algebras (that is,

_{A} A_{A} ≃_{A} D {(A)}_{A}

as bimodules). We state the following generalization of Miyachi's result (see ). Proposition Let

A

be a finite‐dimensional algebra over a field, and let

e

be a

ν

‐stable idempotent element in

A

.…”

Section: Derived Equivalences and Stable Equivalences Of Morita Typementioning

confidence: 83%

Section: Extending Tilting Complexes Over Corner Algebras To Algebrasmentioning

confidence: 87%

“…It is a Frobenius algebra, of course, a self‐injective algebra. For a proof, one may see [, Lemma 2.5; ]. Note that a Frobenius part of

A

is uniquely determined by

A

(up to isomorphism).…”

Section: Derived Equivalences and Stable Equivalences Of Morita Typementioning

confidence: 99%

“…Case (ii): Both

σ

and

σ^{'}

are non‐empty. By [, Lemma 3.4], the functor

normalΦ

restricts to a stable equivalence

Φ_{1}

of Morita type between

e_{σ} A e_{σ}

and

e_{σ^{'}} B e_{σ^{'}}

. Moreover, the idempotent elements

e_{σ}

and

e_{σ^{'}}

are projectively

ν

‐stable and the algebras

e_{σ} A e_{σ}

and

e_{σ^{'}} B e_{σ^{'}}

are self‐injective with fewer simple modules.…”

Section: Applicationsmentioning

confidence: 99%

“…Theorem 5.2 [63]. Let A and B be finite-dimensional k-algebras over an algebraically closed field k and without non-zero semisimple direct summands.…”

Section: Lifting Stable To Derived Equivalencesmentioning

confidence: 99%

See 3 more Smart Citations