Submodule-closure of higher Ext-images $E_n(R_R)$ over noetherian rings: Auslander's question for $n \ge 3$

For a two-sided noetherian ring $R$, let $E_n(R_R)=\{M\in\mathrm{mod}\,R \mid M \simeq \operatorname{Ext}_R^n(N,R)\ \text{for some}\ N\in\mathrm{mod}\,R^{op}\}$ be the subcategory of finitely generated left $R$-modules arising as $n$-th Ext-images of the regular module. Auslander asked in 1969 whether $E_1(R_R)$ is closed under submodules; Huang (2007) answered this negatively with explicit finite-dimensional algebra counterexamples, proved that an Auslander-Gorenstein ring of self-injective dimension $n$ has $E_n(R_R)$ submodule-closed, and explicitly left open whether $E_n(R_R)$ is submodule-closed in general for any degree $n \ge 3$. The problem is to decide, for each $n \ge 3$ (with $n=3$ the first open instance), whether $E_n(R_R)$ is submodule-closed for every two-sided noetherian ring $R$: either prove a general closure theorem or exhibit an explicit counterexample ring.

Background

For a ring $R$, write $\mathrm{mod}\,R$ for the category of finitely generated left $R$-modules and $\mathrm{mod}\,R^{op}$ for finitely generated right $R$-modules; $R$ is two-sided noetherian if it is both left and right noetherian. For such a ring and a positive integer $n$, Auslander's Ext-image subcategory is $E_n(R_R)=\{M\in\mathrm{mod}\,R \mid M \simeq \operatorname{Ext}_R^n(N,R)\ \text{for some}\ N\in\mathrm{mod}\,R^{op}\}$, where $\operatorname{Ext}_R^n(-,R)$ is the $n$-th derived functor of $\operatorname{Hom}_R(-,R)$ computed on right modules. A full subcategory $\mathcal{X}\subseteq\mathrm{mod}\,R$ is submodule-closed if every non-zero submodule of a module in $\mathcal{X}$ again lies in $\mathcal{X}$. In his 1969 paper Comments on the functor Ext, Auslander proved (Proposition 3.3) that any direct summand of a module in $E_1(R_R)$ again lies in $E_1(R_R)$, and asked whether it is also closed under submodules — equivalently, whether $E_1(R_R)$ is submodule-closed. The question matters because the filtration $\mathrm{mod}\,R \supseteq E_1(R_R) \supseteq E_2(R_R) \supseteq \cdots$ controls the syzygy/torsionfree filtration of modules and is the Ext-theoretic mechanism behind the Auslander condition and injective-dimension symmetry for noetherian rings.

Huang (2007) systematically studied this question inside a tilting-theoretic framework. Recall that a minimal injective resolution $0\to {}_RR\to I^0\to I^1\to\cdots$ is an injective resolution in which each term $I^i$ is the injective envelope of the image of the preceding map (an injective envelope is an essential embedding into an injective module, unique up to isomorphism), and that the flat dimension $\mathrm{fd}(M)$ is the shortest length of a flat resolution of $M$. A two-sided noetherian ring is Iwanaga-Gorenstein if both $R_R$ and $_RR$ have finite injective dimension; it satisfies the Auslander condition if $\mathrm{fd}(I^i)\le i$ for all $i\ge 0$ in the minimal injective resolution of ${}_RR$, and it is Auslander-Gorenstein if both hold. Huang's main criterion (Theorem 3.7 / Corollary 3.8, specialized to the regular bimodule) states: if $R$ is two-sided noetherian with left and right self-injective dimensions equal and at most $n$, then $\mathrm{fd}(I^0\oplus\cdots\oplus I^{n-1})\le n-1$ implies that $E_n(R_R)$ is submodule-closed; the published equivalence pairs this flat-dimension bound with submodule-closure together with the identification $E_n(R_R)=\mathcal{H}_n(_RR)$, where $\mathcal{H}_n(_RR)=\{M\in\mathrm{mod}\,R\mid \operatorname{Ext}_R^i(M,R)=0\ \text{for all}\ i\ne n\}$. Consequently an Auslander-Gorenstein ring — in particular a commutative Iwanaga-Gorenstein ring — of self-injective dimension $n$ has $E_n(R_R)$ submodule-closed (Corollary 3.9); degrees $n$ strictly below the self-injective dimension are not covered by this corollary. In the opposite direction, Huang constructed explicit finite-dimensional algebras (a quiver algebra of shape $2\leftarrow 1\rightarrow 3$ for $n=1$, and a bound quiver algebra with relation $\beta\alpha$ for $n=2$) that are Iwanaga-Gorenstein yet violate the flat-dimension bound, so $E_1(R_R)$ and $E_2(R_R)$ are not submodule-closed in general — a negative answer to Auslander's original question. He then states explicitly that the higher-degree case is unknown: we don't know whether $E_n(R_R)$ (where $n \ge 3$) is submodule-closed or not in general. Thus the boundary between the Auslander-Gorenstein regime and the low-degree counterexamples remains unsettled from degree $3$ onward.

Problem Statement

For a two-sided noetherian ring $R$ and an integer $n\ge 3$ (with $n=3$ the first open instance), let

$E_n(R_R)=\{M\in\mathrm{mod}\,R \mid M \simeq \operatorname{Ext}_R^n(N,R)\ \text{for some}\ N\in\mathrm{mod}\,R^{op}\}.$

Decide, for each $n \ge 3$, whether $E_n(R_R)$ is submodule-closed (i.e., every non-zero submodule of a module in $E_n(R_R)$ again belongs to $E_n(R_R)$) for every two-sided noetherian ring $R$. The universal claim that $E_n(R_R)$ is submodule-closed for every $n\ge3$ and every two-sided noetherian ring $R$ is refuted by a single explicit counterexample — a two-sided noetherian ring $R$, some $n\ge3$, and a module $M\in E_n(R_R)$ having a submodule outside $E_n(R_R)$ — at that degree, and is established by a proof of this closure for all such degrees and rings. By Huang's published criterion (Theorem 3.7 / Corollary 3.8, specialized to the regular bimodule), for a two-sided noetherian ring $R$ whose left and right self-injective dimensions are equal and at most $n$, the flat-dimension bound $\mathrm{fd}(I^0\oplus\cdots\oplus I^{n-1})\le n-1$ holds if and only if $E_n(R_R)$ is submodule-closed and $E_n(R_R)=\mathcal{H}_n(_RR)$; consequently a submodule-closure counterexample at such a ring and degree must satisfy $\mathrm{fd}(I^0\oplus\cdots\oplus I^{n-1})\ge n$, but this flat-dimension bound alone is only necessary and does not by itself exhibit the required module in $E_n(R_R)$ with a submodule outside it, since the bound may instead fail through $E_n(R_R)\ne\mathcal{H}_n(_RR)$. The problem in its stated generality also covers rings of infinite self-injective dimension, for which no criterion is currently known.

References

1. M. Auslander, Comments on the functor Ext, Topology 8 (1969), DOI 10.1016/0040-9383(69)90006-8, https://doi.org/10.1016/0040-9383(69)90006-8
2. Zhaoyong Huang, Generalized tilting modules with finite injective dimension, Journal of Algebra (2007), DOI 10.1016/j.jalgebra.2006.11.025, https://doi.org/10.1016/j.jalgebra.2006.11.025; preprint arXiv:math/0602572, https://arxiv.org/abs/math/0602572
3. Zhaoyong Huang, Selforthogonal Modules with Finite Injective Dimension III, Algebras and Representation Theory 12 (2009), DOI 10.1007/s10468-009-9157-2, https://doi.org/10.1007/s10468-009-9157-2
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