Building on Wei's work on Miyashita's ordinary tilting pairs, we establish an equivalent characterization for relative tilting pairs, which provides a unified framework that generalizes several important results in tilting theory, including classical tilting pairs, $F$-tilting modules, and Gorenstein tilting pairs.
📌 3 Key Takeaways
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Objective & Hypothesis:Let $\Lambda$ be an artin algebra.
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Key Findings & Data:We extend Miyashita's theory of tilting pairs to the setting of relative tilting pairs with respect to an additive subfunctor $F$ of $\mathrm{Ext}_{\Lambda}^1(-,~-)$.
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Conclusion & Impact:Building on Wei's work on Miyashita's ordinary tilting pairs, we establish an equivalent characterization for relative tilting pairs, which provides a unified framework that generalizes several important results in tilting theory, including classical tilting pairs, $F$-tilting modules, and Gorenstein tilting pairs.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Tilting module · $F$-orthogonal class · $r\textrm{-}F$-tilting pair
Abstract
Let $\Lambda$ be an artin algebra. We extend Miyashita's theory of tilting pairs to the setting of relative tilting pairs with respect to an additive subfunctor $F$ of $\mathrm{Ext}_{\Lambda}^1(-,~-)$. Building on Wei's work on Miyashita's ordinary tilting pairs, we establish an equivalent characterization for relative tilting pairs, which provides a unified framework that generalizes several important results in tilting theory, including classical tilting pairs, $F$-tilting modules, and Gorenstein tilting pairs.
Wang, X., Yao, H. L. (2026). Relative tilting pairs. International Electronic Journal of Algebra, Volume: 40 Issue: 40. https://doi.org/10.24330/ieja.1893161
L. Angeleri Hugel and F. U. Coelho, Infinitely generated tilting modules of finite projective dimension, Forum Math., 13(2) (2001), 239-250.
M. Auslander, M. I. Platzeck and I. Reiten, Coxeter functors without diagrams, Trans. Amer. Math. Soc., 250 (1979), 1-46.
M. Auslander and Ø. Solberg, Relative homology and representation theory I. Relative homology and homologically finite subcategories, Comm. Algebra, 21(9) (1993), 2995-3031.
M. Auslander and Ø. Solberg, Relative homology and representation theory II. Relative cotilting theory, Comm. Algebra, 21(9) (1993), 3033-3079.
M. Auslander and Ø. Solberg, Relative homology and representation theory III. Cotilting modules and Wedderburn correspondence, Comm. Algebra, 21(9) (1993), 3081-3097.
S. Bazzoni, A characterization of $n$-cotilting and $n$-tilting modules, J. Algebra, 273 (2004), 359-372.
I. N. Bernstein, I. M. Gel'fand and V. A. Ponomarev, Coxeter functors, and Gabriel's theorem, Uspehi Mat. Nauk, 28(2) (1973), 19-33.
S. Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gel'fand-Ponomarev reflection functors, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont.) (1979), 103-169.
R. Colpi and J. Trlifaj, Tilting modules and tilting torsion theories, J. Algebra, 178 (1995), 614-634.
L. Dajun and J. Wei, Gorenstein tilting pairs, Comm. Algebra, 48(9) (2020), 3846-3858.
D. Happel, Triangulated Categories in The Representation Theory of Finite-Dimensional Algebras, London Mathematical Society Lecture Note Series, 119, Cambridge University Press, Cambridge, 1988.
D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc., 274 (1982), 399-443.
Y. Hu, H. Yao and X. Fu, Tilting objects in triangulated categories, Comm. Algebra, 48(1) (2020), 410-429.
Y. Miyashita, Tilting modules associated with a series of idempotent ideals, J. Algebra, 238 (2001), 485-501.
J. Wei, A note on relative tilting modules, J. Pure Appl. Algebra, 214(4) (2010), 493-500.
J. Wei and C. Xi, A characterization of the tilting pair, J. Algebra, 317 (2007), 376-391.
L. Yan, W. Li and B. Ouyang, Gorenstein cotilting and tilting modules, Comm. Algebra, 44(2) (2016), 591-603.