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Tilting pairs and Wakamatsu tilting subcategories over triangular matrix algebras

  • Yafeng Zhang , Yajun Ma and Tiwei Zhao EMAIL logo
Published/Copyright: March 29, 2023
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Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 30 Issue 3

Abstract

Let A and B be Artin algebras and let M be an ( A , B ) -bimodule with M A and M B finitely generated. In this paper, we construct tilting pairs of subcategories and Wakamatsu tilting subcategories over an upper triangular matrix Artin algebra Ξ› = ( A M 0 B ) using tilting pairs and Wakamatsu tiling subcategories over A and B. Let π’ž be a subcategory of A ⁒ -mod and let π’Ÿ be a subcategory of B ⁒ -mod . Consider the subcategory of left Ξ›-modules 𝔅 π’Ÿ π’ž = { ( X Y ) f : f Β is a monomorphism,Β  Y ∈ π’Ÿ Β andΒ  Coker ⁑ f ∈ π’ž } . We prove the following results: (1) Assume that M βŠ— B 𝒯 β€² βŠ† 𝒯 , M βŠ— B π’ž β€² βŠ† π’ž and Tor i B ⁒ ( M , 𝒯 β€² ) = 0 for all i β‰₯ 1 . Then ( π’ž , 𝒯 ) and ( π’ž β€² , 𝒯 β€² ) are n-tilting pairs respectively in A ⁒ - ⁒ mod and B ⁒ - ⁒ mod if and only if ( 𝔅 π’ž β€² π’ž , 𝔅 𝒯 β€² 𝒯 ) is an n-tilting pair in Ξ› ⁒ - ⁒ mod . (2) Assume that M βŠ— B 𝒱 βŠ† 𝒲 and Tor i B ⁒ ( M , 𝒱 βŠ₯ ) = 0 for all i β‰₯ 1 . If 𝒲 and 𝒱 are Wakamatsu tilting subcategories respectively in A ⁒ - ⁒ mod and B ⁒ - ⁒ mod , then 𝔅 𝒱 𝒲 is a Wakamatsu tilting subcategory in Ξ› ⁒ - ⁒ mod .

Keywords: Triangular matrix algebra; tilting pair; Wakamatsu tiling subcategory
MSC 2010: 16S70; 16E05; 16G10

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11901341

Award Identifier / Grant number: 11971225

Funding statement: This work was partially supported by NSFC (No. 11901341, No. 11971225), and the project of Youth Innovation Team of Universities of Shandong Province (No. 2022KJ314).

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Received: 2022-09-01
Accepted: 2023-01-20
Published Online: 2023-03-29
Published in Print: 2023-06-01

Β© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The second cohomology spaces of 𝒦(1) with coefficients in the superspace of weighted densities and deformations of the superspace of symbols on S 1|1
  3. On completely non-Baire unions in category bases
  4. Square-free values of n 2 + n + 1
  5. The stability and convergence analysis for singularly perturbed Sobolev problems with Robin type boundary condition
  6. Sign-changing solutions for parametric Neumann problems with broken symmetry and arbitrary growth
  7. The triharmonic equation on the Heisenberg group
  8. On Busemann–Feller extensions of translation invariant density differentiation bases
  9. Multiplicative bi-skew Jordan triple derivations on prime βˆ—-algebra
  10. Nonmeasurable products of absolutely negligible sets in uncountable solvable groups
  11. The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation
  12. On the stochastic integral representation of Brownian functionals
  13. Generalized Bessel matrix functions
  14. Degenerate time-fractional diffusion equation with initial and initial-boundary conditions
  15. Some new characterizations of EP elements, partial isometries and strongly EP elements in rings with involution
  16. 𝐿𝑝(β‹…) βˆ’ πΏπ‘ž(β‹…) estimates for convolution operators with singular measures supported on surfaces of half the ambient dimension
  17. Tilting pairs and Wakamatsu tilting subcategories over triangular matrix algebras
https://doi.org/10.1515/gmj-2023-2013
Keywords for this article
Triangular matrix algebra; tilting pair; Wakamatsu tiling subcategory

Articles in the same Issue

  1. Frontmatter
  2. The second cohomology spaces of 𝒦(1) with coefficients in the superspace of weighted densities and deformations of the superspace of symbols on S 1|1
  3. On completely non-Baire unions in category bases
  4. Square-free values of n 2 + n + 1
  5. The stability and convergence analysis for singularly perturbed Sobolev problems with Robin type boundary condition
  6. Sign-changing solutions for parametric Neumann problems with broken symmetry and arbitrary growth
  7. The triharmonic equation on the Heisenberg group
  8. On Busemann–Feller extensions of translation invariant density differentiation bases
  9. Multiplicative bi-skew Jordan triple derivations on prime βˆ—-algebra
  10. Nonmeasurable products of absolutely negligible sets in uncountable solvable groups
  11. The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation
  12. On the stochastic integral representation of Brownian functionals
  13. Generalized Bessel matrix functions
  14. Degenerate time-fractional diffusion equation with initial and initial-boundary conditions
  15. Some new characterizations of EP elements, partial isometries and strongly EP elements in rings with involution
  16. 𝐿𝑝(β‹…) βˆ’ πΏπ‘ž(β‹…) estimates for convolution operators with singular measures supported on surfaces of half the ambient dimension
  17. Tilting pairs and Wakamatsu tilting subcategories over triangular matrix algebras
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