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Reflexive equivalence and reflexive-minimal algebras

Primarily AI-generated textHuman understanding: some partsmath.RT — Representation Theorymath.RA — Rings and Algebras

Contributed by Haruhisa Enomoto ↗

Version 1 / Oct 03, 2026 / CC BY 4.0

Abstract

Let AA be a finite-dimensional algebra over a field. A finite-dimensional algebra BB over the same field is called reflexively equivalent to AA if their categories of reflexive modules are equivalent. We prove that there is a basic algebra Amin⁡A_{\min}, unique up to isomorphism, such that BB is reflexively equivalent to AA if and only if B≅End⁡Amin⁡(Amin⁡⊕X)B\cong\operatorname{End}_{A_{\min}}(A_{\min}\oplus X) for some reflexive Amin⁡A_{\min}-module XX. Moreover, we compute Amin⁡A_{\min} explicitly. We extend these results to module-finite algebras over henselian local rings of dimension at most one under a dominant dimension condition at minimal primes. We also introduce reflexive modules over additive categories and prove that taking reflexive modules is idempotent: the reflexive modules over the category of reflexive modules form an equivalent category. As an application, we show that reflexive equivalence classes of algebras with finitely many indecomposable reflexive modules correspond bijectively to Morita equivalence classes of algebras whose reflexive modules are projective.

Provenance statement

The results and proofs were developed by GPT-6-Astra in research directed by the author, and the first drafts of the manuscript were written by GPT-6-Astra. Claude Opus~5.5 revised the manuscript under the author's direction. The author set the structure and much of the wording of the present text. The author is responsible for the mathematical content and the final manuscript.

Tools used

OpenAI
CodexVersion GPT-6-Astra
Anthropic
Claude OpusVersion Opus 5.5

References

  1. M. Asgharzadeh, Reflexivity revisited , preprint, arXiv:1812.00830v7.arXiv
  2. M. Auslander and M. Bridger, Stable module theory , Mem. Amer. Math. Soc. 94 (1969).
  3. T. Avery and T. Leinster, Isbell conjugacy and the reflexive completion , Theory Appl. Categ. 36 (2021), No. 12, 306--347.
  4. R. S. Cunningham, E. A. Rutter, Jr., and D. R. Turnidge, Rings of quotients of endomorphism rings of projective modules , Pacific J. Math. 41 (1972), 647--668.
  5. K. R. Fuller, Double centralizers of injectives and projectives over artinian rings , Illinois J. Math. 14 (1970), 658--664.
  6. N. Hanihara, Reflexive modules and Auslander-type conditions , preprint, arXiv:2412.19625v2.arXiv
  7. M. Hashimoto, Homological aspects of equivariant modules: Matijevic--Roberts and Buchsbaum--Rim , Sūrikaisekikenkyūsho Kōkyūroku 997 (1997), 66--108.
  8. M. Hoshino, On dominant dimension of Noetherian rings , Osaka J. Math. 26 (1989), no. 2, 275--280.
  9. Z. Huang and H. Qin, Homological behavior of Auslander's kk-Gorenstein rings , Algebr. Represent. Theory 15 (2012), 835--853.
  10. J. R. Isbell, Adequate subcategories , Illinois J. Math. 4 (1960), 541--552.
  11. Y. Iwanaga and H. Sato, Minimal injective resolutions of Gorenstein rings , Comm. Algebra 18 (1990), no. 11, 3835--3856.
  12. O. Iyama and I. Reiten, Fomin--Zelevinsky mutation and tilting modules over Calabi--Yau algebras , Amer. J. Math. 130 (2008), 1087--1149.
  13. B. Ma and J. Sauter, On faithfully balanced modules, F-cotilting and F-Auslander algebras , J. Algebra 556 (2020), 1115--1164.
  14. K. Morita, Duality for modules and its applications to the theory of rings with minimum condition , Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 6 (1958), 83--142.
  15. M. Ramras, Betti numbers and reflexive modules , in Ring theory (Proc. Conf., Park City, Utah, 1971) , Academic Press, New York, 1972, 297--308.
  16. C. M. Ringel, The short local algebras of dimension 6 with non-projective reflexive modules , Commun. Math. Stat. 11 (2023), 195--227.
  17. C. M. Ringel and P. Zhang, Gorenstein-projective modules over short local algebras , J. Lond. Math. Soc. (2) 106 (2022), no. 2, 528--589.
  18. H. Tachikawa, Quasi-Frobenius rings and generalizations. QF-3 and QF-1 rings , notes by C. M. Ringel, Lecture Notes in Math. 351 , Springer, Berlin, 1973.

Version history

  1. v1Submitted by Haruhisa EnomotoInitial depositCurrentOct 03, 2026