Ringel's Resolution Quiver in Nakayama Algebras

• Ringel's Resolution Quiver is a combinatorial and homological construct that encodes the relationship between projective covers, socles, and syzygies in Nakayama algebras.
• Its cycle structure, periodicity, and weight invariants facilitate precise detection of global dimension, Gorenstein properties, and the magnitude of the Cartan matrix.
• The construct underpins algorithms for verifying homological invariants and unifies combinatorial and cyclic homology criteria in representation theory.

Ringel's resolution quiver is a combinatorial and homological construct that encapsulates the projective, socle, and syzygy structure of finite-dimensional algebras, with a particular focus on Nakayama algebras. For a Nakayama algebra $A$ of order $n$ over a field $k$, the resolution quiver $R(A)$ associates a directed graph to the algebra in which vertices correspond to simple modules and arrows reflect socle composition within projective covers. The cycle structure and numerical invariants of $R(A)$—notably, periodicity and weight—yield decisive criteria for global dimension, Gorenstein property, and magnitude of the Cartan matrix, linking combinatorics with homological properties.

1. Definition and Construction of Ringel’s Resolution Quiver

For a Nakayama algebra $A = kQ_n/I$ with Kupisch series $(\ell_1, \ldots, \ell_n)$, the cyclic quiver $Q_n$ comprises $n$ vertices with arrows $x_i: i \rightarrow i+1 \pmod n$. The ideal $I$ is generated by paths of prescribed length derived from the Kupisch series. The simple $A$-modules $S_i$ correspond to vertices, and $P_i = e_i A$ is the indecomposable projective module of length $\ell_i$.

The resolution quiver $R(A)$ is the directed graph with vertices $1, \ldots, n$, and arrows defined by the map $f(i) \equiv i+\ell_i \pmod n$, indicating the position of the socle of $P_i$. Each arrow $i \to f(i)$ captures the transitions in minimal projective resolutions:

$0 \to \Omega(S_i) \to P_i \to S_i \to 0.$

The socle $\mathrm{soc}\,P_i$ identifies a unique simple $S_{f(i)}$.

2. Cycle Structure, Periodicity, and Weight

$R(A)$ generally decomposes into the disjoint union of oriented cycles, potentially with attached trees (leaves). Fundamental results establish:

• Every component of $R(A)$ contains a unique directed cycle.
• All cycles possess the same length and weight.

If $C$ is a cycle $C = (i_1 \to \cdots \to i_m \to i_1)$, its weight is

$$\mathrm{wt}(C) = \frac{1}{n} \sum_{j=1}^{m} \ell_{i_j} \in \mathbb{Z}_{>0}.$$

The cycle length $m$ and weight are invariant under left retractions and independent of the component, as formalized in (Shen, 2012). In the self-injective case ($\ell_1 = \cdots = \ell_n = c$), there are $\gcd(n, c)$ cycles, each of length $n / \gcd(n, c)$ and weight $c / \gcd(n, c)$.

3. Homological Criteria: Global Dimension and Gorenstein Property

Shen's theorem [She17], as presented in (Hanson et al., 2019), asserts that a cyclic Nakayama algebra $A$ satisfies

$$\mathrm{gl.dim}\,A < \infty \iff R(A) \text{ has exactly one connected component with weight } 1.$$

If $R(A)$ decomposes into multiple cycle-components or has a unique cycle of weight $\geq 2$, the global dimension is infinite. Shen additionally proves that all components of $R(A)$ have the same weight, leading to precise combinatorial detection of homological finiteness.

Gorenstein property is characterized via “black cycles” in $R(A)$: $A$ is Gorenstein iff every vertex on every cycle is black (i.e., $\mathrm{pd}_A S_i \neq 1$). The self-injective dimension is $2 \, \max_i \mathrm{dist}(i)$, where $\mathrm{dist}(i)$ is the minimal distance from $i$ to a cycle vertex (Shen, 18 Nov 2025).

4. Resolution Quiver and Magnitude

Ringel’s resolution quiver is fundamental to the computation of the magnitude of the Cartan matrix $C_A$ of $A$. For Nakayama algebras, magnitude generalizes the Euler characteristic and is defined by the existence of weighting and coweighting vectors $(\alpha, \beta)$ such that $C_A \alpha = \mathbf{1}$ and $\beta C_A = \mathbf{1}$. Every weighting is a rational combination of characteristic vectors of cycles divided by their weight (Shen et al., 2023):

$\operatorname{mag}(C_A) = \frac{p}{w},$

where $p$ is the periodicity (length) of any cycle and $w$ is its weight. These invariants are coprime and independent of cycle.

5. Cyclic-Homology Criterion and its Equivalence

Igusa–Zacharia [IZ92] established a cyclic-homology characterization for finite global dimension. For cyclic Nakayama algebras, the vanishing of relative cyclic homology ($HC_*(J) = 0$ with $J = \mathrm{rad}\,A$) and Euler characteristic $\chi(L(A)) = 1$—where $L(A)$ is the relation simplicial complex—are equivalent to finite global dimension (Hanson et al., 2019). Hanson–Igusa directly proved the equality

$$\#\{\text{components of }R(A)\text{ of weight }1\} = \chi(L(A));$$

thus, the two criteria are equivalent and unify the combinatorial and homological perspectives.

6. Resolution Quiver Criteria for Minimal Auslander-Gorenstein Nakayama Algebras

The structure of $R(A)$ determines whether a Nakayama algebra is minimal Auslander-Gorenstein, relying on the parity of self-injective dimension (Shen, 18 Nov 2025). For dimension $2m-1$, the conditions are:

• Every leaf $i$ satisfies $\mathrm{dist}(i) = m$.
• Every non-cyclic vertex $i$ has at most one predecessor.
• $R(A)$ is connected of weight 1.
• For every cyclic vertex $j$, $\tau^{-1}(j)$ is black.

For even self-injective dimension $2m$, cyclic vertices have at most two predecessors and are all black. These criteria are verified by induction using syzygy-filtered algebras and preserve combinatorial data for $\domdim A \geq 3$.

7. Applications, Worked Examples, and Unified Criteria

The theory provides explicit algorithms for checking homological invariants. For example, the cyclic Nakayama algebra with Kupisch series $(3,2,2,4,3)$ has a connected $R(A)$ with a unique cycle of weight 1, thus finite global dimension (Hanson et al., 2019). In self-injective cases, the number and size of cycles, as well as their weights, afford immediate computation of magnitude (Shen et al., 2023).

Unified criteria equate the Shen/Madsen connectivity and weight-one condition of $R(A)$ with cyclic-homology vanishing, facilitating efficient detection of finite global dimension and rational magnitude.

---

Ringel’s resolution quiver is an essential combinatorial device for Nakayama algebras, acting as an interface between module-theoretic properties, syzygy structure, and homological invariants. Its cycle properties govern the existence of rational magnitude, precisely detect global and self-injective dimension, and unify multiple characterization frameworks within the representation theory of finite-dimensional algebras.