Explicit formulae for the finitistic dominant dimension of Nakayama algebras that are Morita algebras

The finitistic dominant dimension fdomdim(A) of a finite-dimensional algebra A is the supremum of the dominant dimensions of all finitely generated A-modules of finite dominant dimension, where the dominant dimension of a module counts the initial projective terms in its minimal injective coresolution. Marczinzik's study of dominant-dimension bounds for Nakayama algebras (J. Algebra 496, 2018) proves only the general bound fdomdim(A) <= 2n-2 for non-selfinjective Nakayama algebras with n simple modules, and its introduction announces as forthcoming work the determination of explicit formulae for fdomdim on Nakayama algebras that are Morita algebras, together with the claim that for non-selfinjective gendo-symmetric Nakayama algebras the finitistic dominant dimension equals the Gorenstein dimension. No published work carrying out this announcement is known. The open problem is to carry it out: give closed formulae for fdomdim over the entire combinatorially parametrized class of Nakayama algebras that are Morita algebras, and in particular prove or refute the announced gendo-symmetric equality.

Background

Throughout, algebras are finite-dimensional algebras over a field and modules are finitely generated right modules. The dominant dimension of an algebra A is defined via the minimal injective resolution $0\to A\to I_0\to I_1\to\cdots$ of the right regular module: $\mathrm{domdim}(A)$ is $0$ if $I_0$ is not projective, and otherwise one more than the largest $n$ such that $I_0,\dots,I_n$ are all projective. The same convention defines the dominant dimension $\mathrm{domdim}(M)$ of a module $M$ via its minimal injective coresolution, and the finitistic dominant dimension is $\mathrm{fdomdim}(A)=\sup\{\mathrm{domdim}(M)\mid M\in\mathrm{mod}\text{-}A,\ \mathrm{domdim}(M)<\infty\}.$ Dominant dimension originated in work on QF-3 algebras (algebras admitting a faithful module that is both projective and injective): Nakayama conjectured that every finite-dimensional algebra of infinite dominant dimension is selfinjective (an algebra is selfinjective when it is injective as a module over itself), and Yamagata conjectured the stronger statement that the dominant dimensions of non-selfinjective algebras with a fixed number $n$ of simple modules are bounded by a function of $n$; both conjectures remain open in general, as do the Gorenstein symmetry conjecture (left and right injective dimensions of the regular module coincide) and the finitistic dimension conjecture that implies them. A theorem of Mueller states that algebras of dominant dimension at least $2$ are exactly the endomorphism rings $B=\mathrm{End}_A(G)$ of a generator $G$ over an algebra $A\cong eBe$ obtained from an idempotent $e$ with $eB$ a minimal faithful projective-injective module. Following Kerner and Yamagata, $B$ is called a Morita algebra when $A$ can be chosen selfinjective, and a gendo-symmetric algebra when $A$ can be chosen symmetric (a symmetric algebra is one isomorphic to its vector-space dual as a bimodule, the group-algebra-like case). The Gorenstein dimension of $B$ is the injective dimension of the regular module when it is finite; it is left-right symmetric for Nakayama algebras because they have finite finitistic dimension (the supremum of finite projective dimensions of modules). A Nakayama algebra is an algebra all of whose indecomposable modules are uniserial, meaning their submodule lattice is a single chain; connected Nakayama algebras are the algebras whose quiver (the directed graph whose vertices label the simple modules and whose arrows encode generators of the radical) is a directed line or an oriented cycle, with admissible relations, and they are encoded by their Kupisch series $(c_0,\dots,c_{n-1})$, the list of dimensions of the indecomposable projective modules. Nakayama algebras are the canonical computational testing ground for homological conjectures. By a theorem of Yamagata quoted in the source paper, the basic non-selfinjective Nakayama algebras that are Morita algebras are exactly the algebras $B=\mathrm{End}_A(M)$ with $A$ a selfinjective Nakayama algebra with $n$ simple modules and Loewy length $w$ (the Loewy length is the smallest $w\ge 1$ with $J^w=0$, where $J$ is the Jacobson radical of $A$) and $M=\bigoplus_{i=0}^{n-1}e_iA\oplus\bigoplus_{i=1}^{r}e_{x_i}A/e_{x_i}J^{w-1}$ with the points $x_i$ pairwise distinct; the indices $x_i$ are called the special points, so the whole class is parametrized by $n$, $w$ and the special points. Marczinzik (J. Algebra 496, 2018) established the optimal bound $\mathrm{domdim}\le 2n-2$ for non-selfinjective Nakayama algebras with $n$ simple modules, an explicit formula for $\mathrm{domdim}(B)$ in terms of the special points for Nakayama algebras that are Morita algebras, and a graph-theoretic interpretation of the Gorenstein dimension of gendo-symmetric Nakayama algebras as twice the maximal distance between special points, with $\mathrm{injdim}(B)=\mathrm{domdim}(B)$ exactly when the special points are equidistant. For the finitistic dominant dimension, the same paper proves only the bound $\mathrm{fdomdim}(A)\le 2n-2$ for non-selfinjective Nakayama algebras with $n\ge 2$ simple modules, and computes one example, the cyclic Nakayama algebra with Kupisch series $(3s+1,3s+2,3s+2)$, $s\ge 1$, which has dominant dimension and Gorenstein dimension both equal to $2$ but finitistic dominant dimension equal to $4$, showing that the finitistic dominant dimension can strictly exceed the Gorenstein dimension for general Nakayama algebras. The introduction of that paper then announces, as forthcoming work, precisely the determination of the finitistic dominant dimension stated as the open problem; no such publication is known.

Problem Statement

Determine, by explicit formulae in the combinatorial data of the algebra, the finitistic dominant dimension of every non-selfinjective Nakayama algebra that is a Morita algebra. Concretely, by Yamagata's classification these algebras are exactly the endomorphism rings $B=\mathrm{End}_A(M)$ of $M=\bigoplus_{i=0}^{n-1}e_iA\oplus\bigoplus_{i=1}^{r}e_{x_i}A/e_{x_i}J^{w-1}$ over a selfinjective Nakayama algebra $A$ with $n$ simple modules and Loewy length $w$, with pairwise distinct special points $x_i$ and $r\ge 1$; the task is to give a closed formula for $\mathrm{fdomdim}(B)$ in terms of $n$, $w$ and the special points (equivalently, in terms of the Kupisch series of $B$), valid over the whole class. As an integral part of the same task, decide the equality announced in the source: does $\mathrm{fdomdim}(B)=\mathrm{injdim}(B)$, the Gorenstein dimension, hold for every non-selfinjective gendo-symmetric Nakayama algebra $B$ (for which the source paper shows $\mathrm{injdim}(B)$ equals twice the maximal distance between two special points)? Known results constrain any such formula: $\mathrm{fdomdim}(B)\le 2(n+r)-2$ for $B$ with $n+r\ge 2$ simple modules, $\mathrm{fdomdim}(B)\ge\mathrm{domdim}(B)$, and $\mathrm{fdomdim}=4$ for the algebras with Kupisch series $(3s+1,3s+2,3s+2)$, $s\ge 1$.

References

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