Finiteness of injective dimension for (quasi-)Gorenstein generalized tilting modules over Artin algebras (Huang's Conjecture 4.7)

For Artin algebras $R$ and $S$ and a generalized tilting (Wakamatsu tilting) module $_R U$ with $S=\operatorname{End}(_R U)$, Huang proved that the left injective dimension of $U$ over $R$ and the right injective dimension of $U$ over $S$ are equal whenever $U$ is quasi-Gorenstein, and that the two dimensions coincide whenever both are finite. His Conjecture 4.7 predicts more: if $_R U$ is (quasi-)Gorenstein, then $l.\operatorname{id}_R(U)=r.\operatorname{id}_S(U)<\infty$. The equality being known, the open content is precisely finiteness of the injective dimension. The conjecture implies the Auslander–Gorenstein Conjecture of Auslander and Reiten (1994) and hence the 1958 Nakayama conjecture; the conjecture remains open.

Background

An Artin algebra is a ring that is finitely generated as a module over a commutative artinian ring (finite-dimensional algebras over a field are the basic case); its module category is well controlled by finitely generated modules and duality. A guiding theme since Auslander's work is that homological finiteness properties of such algebras — encapsulated in a family of interlocking conjectures going back to Nakayama (1958) — remain unproven in general.

A left $R$-module $U$ is called selforthogonal if $\operatorname{Ext}^i_R(U,U)=0$ for all $i\ge 1$, where $\operatorname{Ext}^i_R$ denotes the derived functors of $\operatorname{Hom}_R$. Following Wakamatsu, a selforthogonal finitely generated module $_R U$ is a generalized tilting module (also called a Wakamatsu tilting module) if there is an exact sequence $0\to {}_R R\to U_0\to U_1\to\cdots$ with each $U_i$ in $\operatorname{add}(U)$, the category of direct summands of finite direct sums of copies of $U$, which remains exact after applying $\operatorname{Hom}_R(-,U)$. Classical tilting modules of finite projective dimension are examples. For such $U$, setting $S=\operatorname{End}(_R U)$ makes $_R U_S$ a faithfully balanced selforthogonal bimodule, and Wakamatsu (1988, "On modules with trivial self-extensions", J. Algebra 114, 302–314) proved that the projective dimensions of $_R U$ and $U_S$ coincide whenever both are finite. Huang (2007) proved the injective-dimensional analogue: the injective dimensions $l.\operatorname{id}_R(U)$ and $r.\operatorname{id}_S(U)$ are equal provided both are finite. What neither result gives is finiteness itself.

The conjecture at issue adds a Gorenstein-type hypothesis. For a finitely generated module $M$, the $U$-grade of $M$ is $\operatorname{grade}_U M=\inf\{i\mid \operatorname{Ext}^i_R(M,U)\neq 0\}$, and its strong $U$-grade is $\operatorname{s.grade}_U M=\inf\{\operatorname{grade}_U B\mid B \text{ a submodule of } M\}$; the strong grade is always at most the plain grade, and may be strictly smaller. As in Theorems 4.4 and 4.5 of the source, the module $_R U$ is an $n$-Gorenstein module if $\operatorname{s.grade}_U \operatorname{Ext}^i_R(M,U)\ge i$ for every finitely generated left $R$-module $M$ and $1\le i\le n$ — equivalently, if $\operatorname{s.grade}_U \operatorname{Ext}^i_S(N,U)\ge i$ for every finitely generated right $S$-module $N$ and $1\le i\le n$ — and a Gorenstein module if it is $n$-Gorenstein for every $n$; for $U={}_R R$ this is exactly the Auslander condition on the ring. The module $_R U$ is quasi-$n$-Gorenstein if $\operatorname{s.grade}_U \operatorname{Ext}^{i+1}_S(N,U)\ge i$ for every finitely generated right $S$-module $N$ and $1\le i\le n$ — equivalently, if $\operatorname{grade}_U \operatorname{Ext}^{i}_R(M,U)\ge i$ for every finitely generated left $R$-module $M$ and $1\le i\le n$ — and quasi-Gorenstein if this holds for every $n$; every Gorenstein module is quasi-Gorenstein, but not conversely. Huang's Proposition 4.6 shows that over two-sided artinian rings, if $_R U$ (or $U_S$) is quasi-Gorenstein, then $l.\operatorname{id}_R(U)=r.\operatorname{id}_S(U)$ — where both sides may still be infinite, finiteness of one forcing finiteness of the other. His Conjecture 4.7 asserts that in this situation both are in fact finite.

The conjecture sits in the classical web of homological conjectures for Artin algebras. Taking $U={}_R R$ (with $S=\operatorname{End}(_R R)$), the Gorenstein hypothesis becomes exactly the Auslander condition on the ring, so Conjecture 4.7 specializes to the Auslander–Gorenstein Conjecture (AGC) of Auslander and Reiten (1994): an Artin algebra satisfying the Auslander condition is Iwanaga–Gorenstein, i.e., has finite self-injective dimension on both sides. Hence Conjecture 4.7 implies AGC, which in turn implies the Nakayama conjecture (NC): if every term of a minimal injective resolution of $R$ (an injective resolution whose terms are injective envelopes of the preceding cokernels) is projective, then $R$ is self-injective. The Generalized Nakayama Conjecture (GNC: every indecomposable injective appears in a minimal injective resolution of $R$) implies AGC. Related but distinct is the Wakamatsu Tilting Conjecture (WTC): every generalized tilting module of finite projective dimension is tilting; equivalently, every generalized tilting module of finite injective dimension is cotilting. WTC implies the Gorenstein Symmetry Conjecture (GSC: the two self-injective dimensions of $R$ are equal) and GNC — but no implication from WTC to Conjecture 4.7 is known. All of NC, GNC, AGC, GSC, and WTC remain open, so any affirmative resolution of Conjecture 4.7 would settle the Nakayama conjecture, which is why its status as open is structural, not merely a gap in the literature.

Problem Statement

Let $R$ and $S$ be Artin algebras and let $_R U$ be a generalized tilting (Wakamatsu tilting) module with $S=\operatorname{End}(_R U)$; that is, $_R U$ is finitely generated and selforthogonal ($\operatorname{Ext}^i_R(U,U)=0$ for all $i\ge 1$) and there is an exact sequence $0\to {}_R R\to U_0\to U_1\to\cdots$ with each $U_i\in\operatorname{add}(U)$ that remains exact after applying $\operatorname{Hom}_R(-,U)$. Suppose that $_R U$ is quasi-Gorenstein, i.e., $\operatorname{s.grade}_U \operatorname{Ext}^{i+1}_S(N,U)\ge i$ for every finitely generated right $S$-module $N$ and every $i\ge 1$, where $\operatorname{s.grade}_U$ denotes the strong $U$-grade (this hypothesis covers the Gorenstein case as well, since every Gorenstein module is quasi-Gorenstein). Must the injective dimensions be finite, i.e., must $l.\operatorname{id}_R(U)=r.\operatorname{id}_S(U)<\infty$? The equality of the two dimensions, allowing the common value $\infty$, is already Proposition 4.6 of the source; the conjectural content is precisely whether the common injective dimension is finite. This is Conjecture 4.7 of Huang (Journal of Algebra 311 (2007) 619–634), whose affirmative resolution would imply the Auslander–Gorenstein Conjecture of Auslander and Reiten and thereby the Nakayama conjecture.

References

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