Strictly Shod Algebras Overview

• Strictly shod algebras are finite-dimensional algebras defined by the property that each indecomposable module has either projective or injective dimension at most one and a global dimension of 3.
• They arise by extending the notion of shod algebras beyond quasi-tilted cases, with constructions involving generalized path algebras and full convex subcategories.
• Their study elucidates the boundary between tame and wild representation theory, as shown by explicit classifications in Dynkin type Dₙ and related combinatorial models.

A strictly shod algebra is a finite-dimensional algebra over a field such that every indecomposable module has either projective dimension at most one or injective dimension at most one, but whose global dimension is precisely three. This notion sits at the intersection of two major classes in the theory of representation-finite and tame algebras: it generalizes tilted (quasi-tilted) algebras but restricts attention to those shod algebras with maximal homological complexity among them. The paper of strictly shod algebras provides insight into the boundaries of tame/wild dichotomies, the structure of module categories, and the finer combinatorics of homological dimensions.

1. Foundational Concepts and Definitions

A finite-dimensional algebra $A$ is termed shod (small homological dimension) if for every indecomposable $A$-module $M$,

$$\operatorname{pd}_A M \leq 1 \quad \text{or} \quad \operatorname{id}_A M \leq 1,$$

where $\operatorname{pd}_A M$ and $\operatorname{id}_A M$ denote the projective and injective dimensions of $M$. When such an algebra $A$ further satisfies $\operatorname{gl.dim} A \le 2$, it is called quasi-tilted.

A strictly shod algebra is a shod algebra that is not quasi-tilted, i.e., it meets the homological criteria for shod, but satisfies

$\operatorname{gl.dim} A = 3.$

This places strictly shod algebras exactly at the homological threshold not attainable by quasi-tilted algebras, distinguishing them in the landscape of finite global dimension.

2. Structural Properties and Constructions

Strictly shod algebras are characterized not only by their homological dimensions, but also by their explicit realizations via constructions that guarantee these properties. In the context of generalized path algebras over acyclic quivers, the following sufficient condition holds (Chust et al., 2022):

• Let $A = k(T, \mathcal{A})$ be a generalized path algebra over an acyclic quiver $T$, where $\mathcal{A} = \{A_i\}_{i \in T_0}$ assigns to each vertex a finite-dimensional algebra $A_i$.
• If all $A_j$ for $j \neq i$ are hereditary and the exceptional vertex algebra $A_i$ is shod (resp. quasi-tilted), then $A$ is shod (resp. quasi-tilted).
• By Theorem 5.1, the global dimension satisfies

$$\mathrm{gl.dim}\,A = \max\{1,\,\mathrm{gl.dim}\,A_i: i \in T_0\},$$

so $\mathrm{gl.dim}\,A = 3$ is possible exactly when some $A_i$ is shod with global dimension $3$, yielding strictly shod behavior.

A core feature is the closure of the class of (strictly) shod algebras under passage to corner algebras and full convex subcategories. Given $A = kQ/I$ and a full convex subquiver $C \subset Q$, form the idempotents $e_C$ (sum over vertices in $C$), $e_C'$ (complement), and consider $A/\langle e_C' \rangle \simeq e_C A e_C$. If $A$ is strictly shod, so is $e_C A e_C$ (Zito, 2020). This localization property allows "zooming in" on substructures while retaining the key homological property.

3. Classification Results and Combinatorial Character

The classification of strictly shod algebras is especially well developed for certain Dynkin types:

• For Dynkin type $\mathbb{A}_n$ and hereditary gentle algebras, there are no strictly shod algebras (Zhang et al., 2022): all silted (i.e., endomorphism algebra of a 2-term silting complex) algebras are either tilted or products of tilted algebras with global dimension at most $2$.
• For Dynkin type $\mathbb{D}_n$, a complete classification identifies strictly shod algebras as those arising from explicit 2-term silting complexes not derived equivalent to tilting complexes (Zhang, 7 Sep 2025). The non-tilting families partitioned via combinatorial and homological conditions yield strictly shod algebras with $\mathrm{gl.dim}=3$.

A striking structural property proved in this setting is the following:

Every strictly shod algebra associated to the standard and mutated Dynkin $\mathbb{D}_n$ quivers is a string algebra (Zhang, 7 Sep 2025). That is, for $A = kQ/I$,

• For all $i \in Q_0$, at most two arrows start or end at $i$,
• For every arrow, at most one extension in either direction does not force a relation, and
• The ideal $I$ is generated by paths of length at least $2$.

Modules in string algebras can be classified combinatorially as string or band modules, facilitating explicit descriptions of indecomposables, and computation of homological invariants.

4. Homological and Categorical Implications

Strictly shod algebras serve as boundaries for the representation-theoretic and homological behavior of module categories:

• The property $\operatorname{pd}_A M \leq 1$ or $\operatorname{id}_A M \leq 1$ for all indecomposable $M$ severely restricts the structure, implying representation-tameness in many cases.
• The property $\operatorname{gl.dim} A = 3$ guarantees that there exist objects of maximal projective or injective dimension $3$, realized via explicit modules.
• These algebras frequently serve as counterexamples or critical cases in tilting theory, derived equivalence, and silting theory due to their maximized homological constraints within the shod class.

A significant categorical phenomenon arises: for strictly shod string algebras coming from specific 2-term silting complexes, the "heart" $\mathcal{C}$ of the associated $t$-structure in $\mathcal{D}^b(\mathrm{mod}\,A)$ may be derived equivalent to $\mathrm{mod}\,\operatorname{End}_A(P)$, but the realization functor $$\mathcal{D}^b(\mathcal{C}) \to \mathcal{D}^b(\mathrm{mod}\,A)$$ need not be a derived equivalence (Zhang, 7 Sep 2025). Explicitly: $$\mathcal{D}^b(\mathcal{C}) \to \mathcal{D}^b(\mathrm{mod}\,A)$$ can fail to be an equivalence even when $\operatorname{End}_A(P)$ and $A$ are derived equivalent, as demonstrated by specific families of 2-term silting complexes.

5. Absence and Presence in the Gentle and String Algebra Settings

In the hereditary gentle setting, strictly shod algebras do not occur (Zhang et al., 2022). This is proved using geometric models:

• Gentle algebras are realized via marked ribbon surfaces, and elementary polygons produced by triangulations limit the global dimension of associated silted algebras to at most $2$.
• The combinatorial geometry (e.g., maximal number of sides in partition polygons) enforces this bound, precluding the existence of strictly shod examples.

By contrast, in certain non-gentle string algebra settings (notably, for $\mathbb{D}_n$ type quivers), the precise configuration of quiver relations allows for the construction of strictly shod string algebras.

Quiver Type	Strictly Shod Algebras Exist?	Structure
Dynkin $\mathbb{A}_n$	No	Tilted or products only
Hereditary gentle	No	Combinatorially excluded
Dynkin $\mathbb{D}_n$	Yes	All are string algebras

6. Methods, Examples, and Further Research Directions

Explicit construction of strictly shod algebras derives from homological and combinatorial data. For instance:

• Take $Q$ the Dynkin quiver of type $\mathbb{D}_n$ and a 2-term silting complex

$$P = \tau^{-1}P(1) \oplus \bigoplus_{i=5}^{n} \tau^{-1}P(i) \oplus I(2) \oplus I(n-1) \oplus I(n)[1].$$

$\operatorname{End}(P)$ is strictly shod (global dimension $3$), string, and exemplifies the key phenomena discussed above (Zhang, 7 Sep 2025). Families arising from mutated quivers (reverse source orientation) are classified into 14 explicit combinatorial types, one of which captures all strictly shod cases.

Potential research directions include:

• Extension of combinatorial and geometric techniques to non-gentle, skew-gentle, or non-hereditary string algebras to determine the occurrence and classification of strictly shod algebras beyond Dynkin diagrams.
• Investigation of the derived and $t$-structure phenomena observed in the string algebra case, particularly the conditions under which the realization functor fails to induce an equivalence.
• Modular paper via the corner algebra method, employing full convex subcategories to localize and detect strictly shod properties within larger or more complicated algebras (Zito, 2020).

7. Summary and Significance

Strictly shod algebras demarcate the sharpest boundary of controlled homological behavior beyond the quasi-tilted class, characterized by the shod module condition and global dimension three. Their classification in Dynkin $\mathbb{D}_n$ and similar settings reveals that they are always string algebras. Their paper elucidates the interplay between quiver combinatorics, silting theory, and homological algebra, and provides critical examples demonstrating subtleties in derived categories, $t$-structures, and functorial equivalence. Their absence from certain classes (notably, hereditary gentle algebras) underscores the combinatorial constraints defining their existence and motivates broader questions about the landscape of tame and wild finite-dimensional algebras.