Tilted Algebras and Crossed Products

Yi, Lin; Zhou, Zhenqiang

doi:10.1017/s0017089515000336

Cited by 2 publications

(2 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recall that a k-algebra A is called tilted if there is a hereditary algebra B and a tilting B-module T such that A ≃ End B (T ) op . The following corollary is the field extension version compared to the skew group algebra extension version proved in [9]. Corollary 3.8.…”

Section: Directing Objects and Galois Extensionsmentioning

confidence: 99%

See 1 more Smart Citation

Piecewise hereditary algebras under field extensions

2020

Preprint

View full text Add to dashboard Cite

Let A be a finite-dimensional k-algebra and K/k be a finite separable field extension. We prove that A is derived equivalent to a hereditary algebra if and only if so is A ⊗ k K. introductionLet A be a finite-dimensional algebra over a field k and A-mod be the category of finitely generated left A-modules. Recall that A is called piecewise hereditary if there is a hereditary, abelian category H such that the bounded derived category D b (A-mod) is equivalent to D b (H) as triangulated categories.Piecewise hereditary algebras are important and well-studied in representation theory. A homological characteristic via strong global dimensions of piecewise hereditary algebras was given by Happel and Zacharia in [4]. Using this characteristic, Li proved in [8] that the piecewise hereditary property is compatible under certain skew group algebra extensions. Similarly, we prove that it is also compatible under finite separable field extensions (see Corollary 3.3), which is a special case of [7, Proposition 5.1].According to [4], a connected piecewise hereditary k-algebra is derived equivalent to either a hereditary k-algebra or a canonical k-algebra. Notice that the homological characteristic and hence the compatibilities mentioned above do not distinguish these two situations. In this paper, we look for a refinement. We prove that these two kinds of piecewise hereditary algebras are closed under certain base field change. More precisely, Main Theorem. Let K/k be a finite separable filed extension and A a k-algebra. Then A is derived equivalent to a hereditary algebra if and only if so is A ⊗ k K.As a corollary, A is derived equivalent to a canonical algebra if and only if so is A ⊗ k K. We also prove that A is a tilted algebra if and only if so is A ⊗ k K.By [6], if an algebra is derived equivalent to a hereditary algebra (or a canonical algebra), then so is its skew group algebra extension under certain condition. However, the converse of this statement has not been proved. Our theorem is the field extension version of this statement with a confirmation of the converse.Our proof of the main theorem is based on the description of hereditary triangulated categories using directing objects in [2]. We are inspired by the proof of a theorem in [9] saying that tilted algebras are compatible under certain skew group algebra extensions.

show abstract

Section: Directing Objects and Galois Extensionsmentioning

confidence: 99%

“…Our theorem is the field extension version of this statement with a confirmation of the converse.Our proof of the main theorem is based on the description of hereditary triangulated categories using directing objects in [2]. We are inspired by the proof of a theorem in [9] saying that tilted algebras are compatible under certain skew group algebra extensions.…”

mentioning

confidence: 99%