Equality cases in dominant-dimension comparisons along Frobenius bimodules and Frobenius extensions

For algebras linked by a Frobenius bimodule, Changchang Xi's comparison theorem (Sci. China Math. 64 (2021) 33-44) gives one-sided inequalities between their dominant dimensions, their nu-dominant dimensions, and at ring level their flat-dominant dimensions; equality is known only under additional splitting, freeness, or commutativity hypotheses, and an example due to Morita shows the inequality can be strict. The question left open by Xi's paper is to give a characterization, by necessary and sufficient conditions on the Frobenius bimodule or extension, of when these dominant-dimension and nu-dominant-dimension comparison inequalities are equalities. No general equality criterion is known; the famous conjectures the paper cites as motivation (Nakayama, Auslander-Reiten) remain open in general.

Background

An Artin algebra is an algebra $A$ that is finitely generated as a module over a commutative Artin ring $k$ (a finite-dimensional algebra over a field is the basic example). The dominant dimension $\mathrm{domdim}(M)$ of a finitely generated left $A$-module $M$ is the number of initial terms in its minimal injective resolution $0 \to M \to I^0 \to I^1 \to \cdots$ that are projective (an injective resolution is minimal when each map is an essential monomorphism; $\mathrm{domdim}(A)$ means $\mathrm{domdim}({}_A A)$, and $\mathrm{domdim}(M)=\infty$ if no term fails to be projective). Every self-injective algebra (one whose regular module is injective) has infinite dominant dimension, and the Nakayama conjecture — open since 1958 — asserts the converse for Artin algebras; it is one of the standard open conjectures of the representation theory of Artin algebras, recorded together with the Auslander-Reiten conjecture — which asserts that two algebras whose module categories become equivalent after projective modules are factored out must have the same number of non-isomorphic non-projective simple modules — in the standard reference literature. Dominant dimension is also the invariant behind Morita's notion of a QF-3 ring: for an arbitrary ring $R$, the flat-dominant dimension $\mathrm{fdomdim}(R)$ counts the initial terms of a minimal injective resolution of $R$ that are flat (flat-injective terms), and $R$ is QF-3 when this number is at least 1. A ring extension $B \subseteq A$ is a Frobenius extension when ${}_B A$ is finitely generated projective and $\mathrm{Hom}_B({}_B A, B) \cong {}_A A_B$ as $A$-$B$-bimodules (equivalently, the same conditions hold on the right: $A_B$ is finitely generated projective with the right-handed duality); more generally, a $B$-$A$-bimodule ${}_B M_A$ is a Frobenius bimodule when it is finitely generated projective as a left $B$-module and as a right $A$-module and its two duals $\mathrm{Hom}_B({}_B M, B)$ and $\mathrm{Hom}_{A^{\mathrm{op}}}(M_A, A)$ are isomorphic as $A$-$B$-bimodules (skew group algebras, trivial extensions, Markov extensions from subfactor theory and pairs of stably equivalent algebras all produce such links). Writing $\nu = D\,\mathrm{Hom}_A(-,A)$ for the Nakayama functor ($D$ the Artin duality), a projective is nu-stably projective when $\nu^i P$ is projective for all $i>0$, and the nu-dominant dimension counts initial resolution terms that are nu-stably projective; the endomorphism algebra of a module generating this subcategory is the Frobenius part of $A$, the invariant through which Martinez-Villa approached the Auslander-Reiten conjecture. In this setting Xi proved (Theorem 1.2) that for a Frobenius bimodule ${}_B M_A$ one has $\mathrm{domdim}(A) \le \mathrm{domdim}({}_B M)$ and $\mathrm{domdim}(B) \le \mathrm{domdim}({}_A\mathrm{Hom}_B(M,B))$, hence for a Frobenius extension $B \subseteq A$ that $\mathrm{domdim}(B) \le \mathrm{domdim}(A)$, with the same inequalities for nu-dominant dimensions and, at the level of arbitrary rings, for flat-dominant dimensions. Equality holds when the extension is left-split, or right-split between noetherian rings (the inclusion splits as a one-sided module map; finite-dimensional algebras are noetherian), when it is a free Frobenius extension (one whose ${}_B A$ is free of finite rank as a $B$-module), or when the base ring is commutative; and if $B$ is self-injective then so is $A$, so equality holds automatically at $\mathrm{domdim}=\infty$ on that side. But the inequality can be strict: the paper reproduces an example of Morita (the 7-dimensional subalgebra $B \subset A=M_4(k)$ spanned by $e_{11}+e_{44}$, $e_{22}+e_{33}$, $e_{21}$, $e_{31}$, $e_{41}$, $e_{42}$, $e_{43}$, for which $B \subseteq A$ is a Frobenius extension with $\mathrm{domdim}(B)=1<\infty=\mathrm{domdim}(A)$) and concludes that the inequalities of Theorem 1.2(2) 'cannot be improved as equalities in general'. No condition characterizing equality is given, and the paper's applications (transfer of the Nakayama conjecture along left-split extensions, QF-3 property of universal and quantised enveloping algebras of semisimple Lie algebras, computations for skew group algebras, trivial and Markov extensions) all rest on the sufficient conditions rather than on any general criterion.

Problem Statement

Let $A$ and $B$ be finite-dimensional algebras over a field $k$ linked by a Frobenius bimodule ${}_B M_A$ (in particular, let $B \subseteq A$ be a Frobenius extension, taking ${}_B M_A = {}_B A_A$). Xi's comparison theorem (Theorem 1.2(2)) gives $\mathrm{domdim}(A) \le \mathrm{domdim}({}_B M), \qquad \mathrm{domdim}(B) \le \mathrm{domdim}({}_A\mathrm{Hom}_B(M,B)),$ and for a Frobenius extension $B \subseteq A$ in particular $\mathrm{domdim}(B) \le \mathrm{domdim}(A)$, with the analogous inequalities for the nu-dominant dimensions of Artin algebras. (The same theorem also gives, for arbitrary rings linked by a Frobenius bimodule, the corresponding inequalities for the flat-dominant dimensions; since $\mathrm{fdomdim}(R) = \mathrm{domdim}(R)$ for an Artin algebra $R$, that ring-level comparison is a parallel question lying outside the setting fixed here.) The objective of this problem is a complete understanding of when these dominant-dimension comparison inequalities are equalities. Specifically, the problem asks to give necessary and sufficient conditions on the structure of the bimodule $M$ and of the two algebras (respectively on the extension $B \subseteq A$) — not a restatement of the values of $\mathrm{domdim}({}_B M)$ and of $\mathrm{domdim}({}_A\mathrm{Hom}_B(M,B))$ — under which these inequalities are equalities; in the Frobenius-extension specialization (the case ${}_B M_A = {}_B A_A$), give the conditions under which $\mathrm{domdim}(B) = \mathrm{domdim}(A)$ holds, equivalently identify the exact mechanism by which $\mathrm{domdim}(A)$ can strictly exceed $\mathrm{domdim}(B)$, as it does in the Morita-type example with $\mathrm{domdim}(B)=1 < \infty = \mathrm{domdim}(A)$; and establish the analogous characterization for the nu-dominant dimension inequalities of Artin algebras. The known sufficient conditions (left-split or right-split extension, free Frobenius extension, Frobenius extension with commutative base ring, and the tensor product $A \otimes_k B$ with $B$ self-injective, for which $\mathrm{domdim}(A \otimes_k B) = \mathrm{domdim}(A)$) and the known strict example are instances covered by such a characterization; the behaviour at infinite dominant dimension is where the question borders on the Nakayama conjecture.

References

1. Changchang Xi, Frobenius bimodules and flat-dominant dimensions, Science China Mathematics 64 (2021) 33-44, DOI 10.1007/s11425-018-9519-2, preprint arXiv:1903.07921, https://doi.org/10.1007/s11425-018-9519-2
2. Changchang Xi, Representation dimensions linked by Frobenius bimodules with applications to group algebras, preprint (2020), arXiv:2007.00139, https://arxiv.org/abs/2007.00139
3. Hongxing Chen and Changchang Xi, Virtually Gorenstein algebras of infinite dominant dimension, Journal of Pure and Applied Algebra 230 (2026), article 108224, DOI 10.1016/j.jpaa.2026.108224, preprint arXiv:2509.04990, https://arxiv.org/abs/2509.04990
4. Hongxing Chen and Changchang Xi, Homological theory of self-orthogonal modules, Transactions of the American Mathematical Society 378 (2025) 7287-7335, DOI 10.1090/tran/9426, https://doi.org/10.1090/tran/9426
5. H. X. Chen, M. Fang and C. C. Xi, Tachikawa's second conjecture, derived recollements, and gendo-symmetric algebras, Compositio Mathematica 160 (2024) 2704-2737, DOI 10.1112/s0010437x24007395, https://doi.org/10.1112/s0010437x24007395
6. Tadasi Nakayama, On algebras with complete homology, Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg 22 (1958) 300-307, DOI 10.1007/BF02941960, https://doi.org/10.1007/BF02941960
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