Wakamatsu Tilting Conjecture and Huang's Conjecture 4.7: finiteness and symmetry of the injective dimensions of generalized tilting modules over Artin algebras

For Artin algebras $R$ and $S$ and a generalized (Wakamatsu) tilting module $_RU$ with $S=\mathrm{End}(_RU)$, the Wakamatsu Tilting Conjecture (WTC) asserts that every generalized tilting module of finite projective dimension is a classical tilting module; equivalently, for Artin algebras, that finiteness of one of the injective dimensions of $_RU$ and $U_S$ implies finiteness, and hence equality, of the other. Huang's 2007 paper additionally poses Conjecture 4.7: if $_RU$ is (quasi-)Gorenstein, then both injective dimensions are finite (their equality is already proved in that case). Both questions are open as of August 2026: WTC is known for Gorenstein algebras, algebras with ultimately closed projective resolutions, left Artinian local rings, and certain group rings, is implied by the generalized Auslander–Reiten conjecture and by the finitistic dimension conjecture, and implies the Gorenstein Symmetry Conjecture and the Generalized Nakayama Conjecture; but no proof of, and no counterexample to, the general statements is known. A July 2026 claimed proof of the Gorenstein Symmetry Conjecture was withdrawn by its author.

Background

An Artin algebra is a ring that is finitely generated as a module over a commutative Artinian center; its category of finitely generated left modules is denoted mod $R$, and every such module has finite length. The projective (respectively injective) dimension of a module is the minimal length of a projective (respectively injective) resolution; both may be infinite. A module $_RU$ in mod $R$ is called selforthogonal if $\mathrm{Ext}^i_R(U,U)=0$ for all $i\ge 1$; $\mathrm{add}\,_RU$ denotes the modules that are direct summands of finite direct sums of copies of $_RU$, and $\mathrm{gen}^*(_RR)$ denotes the modules admitting an infinite resolution by finitely generated projectives. Following Wakamatsu (1988, 2004), a selforthogonal module $_RU$ in $\mathrm{gen}^*(_RR)$ is a generalized tilting module (also called a Wakamatsu tilting module) if there is an exact sequence $0\to {_RR}\to U_0\to U_1\to\cdots$ with $U_i\in\mathrm{add}\,_RU$ that stays exact after applying $\mathrm{Hom}_R(-,{_RU})$; in that case $_RU_S$ with $S=\mathrm{End}(_RU)$ is a faithfully balanced and selforthogonal bimodule. By contrast, a classical tilting module of finite projective dimension is a selforthogonal $_RT$ in mod $R$ with finite projective dimension for which the corresponding sequence terminates: there is an exact $0\to {_RR}\to T_0\to T_1\to\cdots\to T_m\to 0$ with all $T_i\in\mathrm{add}\,_RT$; cotilting modules are defined dually via injective resolutions. Thus every tilting module is generalized tilting, and the question is whether finite projective dimension forces the converse. Wakamatsu proved that the projective (and injective) dimensions of $_RU$ and $U_S$ coincide whenever both are finite, and Huang (J. Algebra 311, 2007, Theorem 2.7) proved the injective-dimension part over left and right noetherian rings. The conjectural content begins with finiteness of one side: for Artin algebras, the statements (i) every generalized tilting module of finite projective dimension is tilting, (ii) finiteness of one of $l.\mathrm{id}_R(U)$ and $r.\mathrm{id}_S(U)$ (the injective dimensions of $_RU$ and $U_S$) implies finiteness of the other, and (iii) $l.\mathrm{id}_R(U)=r.\mathrm{id}_S(U)$ always, are equivalent; this is the Wakamatsu Tilting Conjecture (WTC). Huang also introduced a finiteness companion: writing $\mathrm{grade}_U X$ for the least $j$ with $\mathrm{Ext}^j_S(X,U)\ne 0$ and $\mathrm{s.grade}$ for the corresponding infimum over all submodules, $_RU$ is an $n$-Gorenstein module if $\mathrm{s.grade}_U\mathrm{Ext}^i_R(M,U)\ge i$ for all $M$ in mod $R$ and $1\le i\le n$ (Gorenstein meaning $n$-Gorenstein for all $n$; by Huang's Theorem 4.4 this condition is left–right symmetric in the bimodule), and $_RU$ is quasi $n$-Gorenstein if the weaker, not left–right symmetric condition $\mathrm{grade}_U\mathrm{Ext}^i_R(M,U)\ge i$ — with $\mathrm{grade}$ in place of $\mathrm{s.grade}$ — holds for all $M$ in mod $R$ and $1\le i\le n$ (quasi Gorenstein meaning quasi $n$-Gorenstein for all $n$). Huang proved $l.\mathrm{id}_R(U)=r.\mathrm{id}_S(U)$ for quasi-Gorenstein generalized tilting modules over two-sided artinian rings, but left their finiteness as his Conjecture 4.7. WTC sits in the classical network of homological conjectures for Artin algebras: it implies the Gorenstein Symmetry Conjecture (GSC: the left and right self-injective dimensions of $R$ are equal, in particular finite on one side iff finite on the other) and the Generalized Nakayama Conjecture (GNC: every indecomposable injective occurs in a minimal injective resolution of $_RR$; Auslander–Reiten 1975), which in turn implies the Nakayama Conjecture (NC); Conjecture 4.7 implies the Auslander Gorenstein Conjecture (AGC: an Artin algebra satisfying the Auslander condition, i.e. with $_RR$ a Gorenstein module, is Iwanaga–Gorenstein), which implies NC; and the finitistic dimension conjecture (FDC: sup of finite projective dimensions of finitely generated modules is finite) implies WTC (Mantese–Reiten 2004). All of GSC, AGC, GNC, NC, and FDC remain open.

Problem Statement

Let $R$ and $S$ be Artin algebras and $_RU$ a generalized tilting module with $S=\mathrm{End}(_RU)$, and write $l.\mathrm{id}_R(U)$ and $r.\mathrm{id}_S(U)$ for the injective dimensions of $_RU$ and $U_S$. Resolve the following two linked but logically independent open questions, both concerning when the two injective dimensions of a generalized tilting bimodule over Artin algebras are finite and equal. (A) Wakamatsu Tilting Conjecture: decide whether the following equivalent statements hold for all Artin algebras: (1) every generalized tilting module with finite projective dimension is a tilting module (equivalently, every generalized tilting module with finite injective dimension is cotilting); (2) if one of $l.\mathrm{id}_R(U)$ and $r.\mathrm{id}_S(U)$ is finite, then so is the other; (3) $l.\mathrm{id}_R(U)=r.\mathrm{id}_S(U)$ always. (B) Huang's Conjecture 4.7: decide whether, whenever $_RU$ is (quasi-)Gorenstein (in the sense of the grade and s.grade inequalities recorded in the Background), $l.\mathrm{id}_R(U)=r.\mathrm{id}_S(U)<\infty$ — the equality is already proved in this case, so the question is whether the two injective dimensions are finite at all. Each of the two questions is settled by a proof in full generality or by an explicit Artin algebra together with a generalized tilting module refuting it; neither question implies the other, so settling one of them does not settle the other, and partial confirmations under additional hypotheses do not resolve either question.

References

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