Trivial One-Arrow Extensions

• Trivial one-arrow extensions are a construction in bound quiver algebras that add a new arrow with square-zero relations, ensuring no iterated arrow combinatorics.
• The construction precisely selects a submodule V within the radical to enforce admissibility and control the interactions between established paths and the new arrow.
• It is realized as a trivial extension by a bimodule with finite right projective dimension, preserving the finiteness of global and finitistic dimensions.

Trivial one-arrow extensions are a combinatorial construction for bound quiver algebras that adds a single arrow while retaining strong homological control. In the formulation introduced as Construction E, one starts from a bound quiver algebra $\Lambda=kQ_0/I_0$, chooses distinct vertices $i,j$, and a left $\Lambda$-submodule $V\subseteq \Lambda e_i$ satisfying $e_j\Lambda e_i\subseteq V\subseteq \rad_\Lambda \Lambda e_i$. The resulting algebra $\Lambda_{i\to j}^V$ is obtained by adjoining one new arrow $\alpha:i\to j$ and imposing relations $z\alpha=0$ for those $z\in kQ_0e_i$ whose classes lie in $V$. The construction is designed so that the new arrow generates a square-zero ideal, and the finiteness of the little finitistic dimension, big finitistic dimension, and global dimension is preserved (Giatagantzidis, 17 Jul 2025).

1. Definition and ambient framework

The ambient setting is a bound quiver algebra

$i,j$0

where $i,j$1 is a field, $i,j$2 is a finite quiver, and $i,j$3 is an admissible ideal. The defining data for a trivial one-arrow extension are a pair of distinct vertices $i,j$4 of $i,j$5 and a left $i,j$6-submodule

$i,j$7

The algebra is then denoted

$i,j$8

In the concise form stated as Construction E, $i,j$9 is the algebra $\Lambda$0 where $\Lambda$1 is obtained from $\Lambda$2 by adding one new arrow $\Lambda$3, and $\Lambda$4 is generated by

$\Lambda$5

A later equivalent formulation chooses a generating set $\Lambda$6 of $\Lambda$7 as a left $\Lambda$8-module and takes

$\Lambda$9

The paper proves that the resulting algebra does not depend on the chosen generating set (Giatagantzidis, 17 Jul 2025).

The restriction

$V\subseteq \Lambda e_i$0

is structural rather than cosmetic. The inclusion $V\subseteq \Lambda e_i$1 forces all old paths from $V\subseteq \Lambda e_i$2 to $V\subseteq \Lambda e_i$3 to interact trivially with the new arrow, while $V\subseteq \Lambda e_i$4 ensures that the new relations come from paths of positive length, so admissibility is preserved. This is why trivial one-arrow extensions are not arbitrary one-arrow enlargements of quivers, but a controlled subclass of them (Giatagantzidis, 17 Jul 2025).

2. Quiver operation and relation pattern

At quiver level, the construction has three steps. First, one adds a single arrow

$V\subseteq \Lambda e_i$5

to the original quiver. Second, one retains all old relations from $V\subseteq \Lambda e_i$6. Third, one imposes new relations involving the new arrow: $V\subseteq \Lambda e_i$7 If $V\subseteq \Lambda e_i$8 is generated by $V\subseteq \Lambda e_i$9, it suffices to add the relations $e_j\Lambda e_i\subseteq V\subseteq \rad_\Lambda \Lambda e_i$0.

The basic combinatorial consequence is that every old path $e_j\Lambda e_i\subseteq V\subseteq \rad_\Lambda \Lambda e_i$1 from $e_j\Lambda e_i\subseteq V\subseteq \rad_\Lambda \Lambda e_i$2 to $e_j\Lambda e_i\subseteq V\subseteq \rad_\Lambda \Lambda e_i$3 satisfies $e_j\Lambda e_i\subseteq V\subseteq \rad_\Lambda \Lambda e_i$4, because $e_j\Lambda e_i\subseteq V\subseteq \rad_\Lambda \Lambda e_i$5. Hence any path passing through $e_j\Lambda e_i\subseteq V\subseteq \rad_\Lambda \Lambda e_i$6 twice must vanish. The paper states this explicitly: every path passing through $e_j\Lambda e_i\subseteq V\subseteq \rad_\Lambda \Lambda e_i$7 at least twice is zero. Equivalently, the ideal generated by the new arrow is square-zero in the new algebra (Giatagantzidis, 17 Jul 2025).

This square-zero behavior explains the adjective “trivial.” The new arrow is not allowed to create iterated new-arrow combinatorics; it contributes a single nilpotent layer. The same lemma gives a quantitative bound on Loewy length: $e_j\Lambda e_i\subseteq V\subseteq \rad_\Lambda \Lambda e_i$8 The lower bound reflects the survival of old nonzero paths, and the upper bound comes from the fact that sufficiently long paths either contain a long old subpath or pass through $e_j\Lambda e_i\subseteq V\subseteq \rad_\Lambda \Lambda e_i$9 twice (Giatagantzidis, 17 Jul 2025).

The paper’s formal analysis of the construction is summarized in a lemma asserting three properties: independence of the generating set for $\Lambda_{i\to j}^V$0, admissibility of the new presentation as a bound quiver algebra, and the path-vanishing statement above. These are the elementary structural facts on which the later homological theorems rest (Giatagantzidis, 17 Jul 2025).

3. Bimodule-theoretic realization and homological preservation

A central point is that trivial one-arrow extensions are not merely ad hoc quiver operations. They are realized as ordinary trivial extensions by a bimodule. In the proof of Proposition $\Lambda_{i\to j}^V$1, the construction introduces

$\Lambda_{i\to j}^V$2

The ideal generated by the new arrow is then identified with $\Lambda_{i\to j}^V$3, and the algebra $\Lambda_{i\to j}^V$4 is shown to be isomorphic to a trivial extension of $\Lambda_{i\to j}^V$5 by $\Lambda_{i\to j}^V$6 (Giatagantzidis, 17 Jul 2025).

The relevant trivial extension has the usual multiplication. For a multiplicative bimodule $\Lambda_{i\to j}^V$7, the split extension is

$\Lambda_{i\to j}^V$8

with multiplication

$\Lambda_{i\to j}^V$9

In the trivial case $\alpha:i\to j$0, this becomes the ordinary trivial extension. The paper proves that every trivial one-arrow extension is isomorphic to such a trivial extension by a bimodule $\alpha:i\to j$1 of strongly-finite right projective dimension (Giatagantzidis, 17 Jul 2025).

This yields the main preservation theorem, stated in the Introduction as Theorem F and proved later as Proposition $\alpha:i\to j$2: $\alpha:i\to j$3 and the same equivalence holds with $\alpha:i\to j$4 replaced by $\alpha:i\to j$5 or $\alpha:i\to j$6. Thus trivial one-arrow extensions preserve the finiteness of the little finitistic dimension, the big finitistic dimension, and the global dimension (Giatagantzidis, 17 Jul 2025).

The mechanism behind this theorem is the general removable-bimodule framework. The paper proves that if a bimodule has strongly-finite right projective dimension, then it is right perfect and satisfies

$\alpha:i\to j$7

This makes the corresponding trivial multiplicative bimodule removable, so the square-zero ideal $\alpha:i\to j$8 can be analyzed with the general quotient theorem used throughout the paper. In the one-arrow case, the new arrow ideal is therefore not only square-zero at quiver level, but also homologically well behaved (Giatagantzidis, 17 Jul 2025).

4. Relation to arrow removal and generalized split extensions

Trivial one-arrow extensions are introduced as the “opposite direction” to arrow removal. Arrow removal starts from a larger algebra and passes to a quotient by killing an ideal generated by arrows. The trivial one-arrow extension starts from $\alpha:i\to j$9, adds one arrow together with relations, and produces a larger algebra whose new-arrow ideal can be removed again while preserving the finiteness properties under study (Giatagantzidis, 17 Jul 2025).

The paper formulates this using pre-removable and removable sets of arrows, generalized arrow removal algebras, and removable multiplicative bimodules. A set of arrows is pre-removable when the quotient by the ideal it generates canonically splits, and removable when the corresponding ideal also satisfies the homological hypotheses needed for the main quotient theorem. In a trivial one-arrow extension, the newly added arrow $z\alpha=0$0 is designed so that the ideal $z\alpha=0$1 is square-zero and corresponds to a removable trivial multiplicative bimodule (Giatagantzidis, 17 Jul 2025).

The broader structural statement is that generalized arrow removal algebras are precisely split extensions by removable multiplicative bimodules. Within this landscape, trivial one-arrow extensions form a special concrete subclass: they are split extensions with trivial multiplicative structure, given by an explicit one-arrow quiver construction, and the associated bimodule has strongly-finite right projective dimension. This suggests that the construction is a partial inverse to arrow removal, though not a strict categorical inverse in full generality (Giatagantzidis, 17 Jul 2025).

The same perspective clarifies why the construction preserves finiteness rather than exact homological values. The general quotient theorem compares $z\alpha=0$2 and a quotient $z\alpha=0$3 under structural hypotheses on $z\alpha=0$4, including splitness and either finite projective dimension or square-zero behavior. The one-arrow construction is engineered so that the new ideal falls exactly into this framework (Giatagantzidis, 17 Jul 2025).

5. Position among adjacent arrow-adding constructions

The exact phrase “trivial one-arrow extension” is specific to the 2025 arrow-reduction paper, but it sits within a broader literature on trivial extensions and quiver enlargement. Ordinary trivial extensions of a finite-dimensional algebra $z\alpha=0$5 by $z\alpha=0$6 already have a canonical quiver description: the new quiver has the same vertices as the old one, the old arrows reappear, and additional arrows come from

$z\alpha=0$7

Moreover, any composition of two new arrows vanishes in the trivial extension quiver algebra (Bergh et al., 2015). This is structurally close to the square-zero pattern in trivial one-arrow extensions, but the number and position of new arrows are determined by bimodule-socle data rather than by a chosen pair of vertices and a submodule $z\alpha=0$8.

A more refined quiver-with-relations description of ordinary trivial extensions shows that the new arrows correspond to a basis of $z\alpha=0$9, with relations organized by “elementary cycles” (Fernandez et al., 2022). This suggests a natural single-arrow special case: when the bimodule socle is one-dimensional, the ordinary trivial extension behaves like a single-arrow enlargement, although that paper does not formulate a separate one-arrow theory (Fernandez et al., 2022).

Other arrow-adjoining constructions are broader but differently organized. For almost gentle algebras, the trivial extension is obtained by adding one new arrow for each maximal path, thereby closing each maximal path into a cycle; the paper explicitly notes that this is not a single global one-arrow extension, but a one-per-maximal-path construction (Green et al., 2016). For graded self-injective algebras and their twisted trivial extensions, the quiver is enlarged by adding one returning arrow

$z\in kQ_0e_i$0

for each vertex $z\in kQ_0e_i$1, again not one arrow total (Guo et al., 2010). These comparisons distinguish trivial one-arrow extensions from ordinary trivial extensions, from returning-arrow constructions, and from one-per-maximal-path closures.

6. Examples, variants, and limitations

The paper gives a dedicated example starting from an algebra $z\in kQ_0e_i$2 on a three-vertex quiver with arrows

$z\in kQ_0e_i$3

and loops

$z\in kQ_0e_i$4

with relations

$z\in kQ_0e_i$5

A new arrow

$z\in kQ_0e_i$6

is added, and the new ideal is generated by

$z\in kQ_0e_i$7

The resulting algebra $z\in kQ_0e_i$8 is a trivial one-arrow extension of $z\in kQ_0e_i$9, and the preservation theorem implies

$V$0

because $V$1 had already been established earlier in the paper (Giatagantzidis, 17 Jul 2025).

A variant chooses

$V$2

instead of the smaller submodule $V$3. The resulting trivial one-arrow extension $V$4 is again irreducible in the paper’s technical sense, and $V$5 is again positive and finite (Giatagantzidis, 17 Jul 2025). The paper also notes that when

$V$6

choosing $V$7 recovers the earlier “add one arrow in a zero corner” situation from the literature (Giatagantzidis, 17 Jul 2025).

The limitations of the notion are explicit. A trivial one-arrow extension requires distinct vertices $V$8, a submodule $V$9 satisfying

$i,j$00

and the added-arrow ideal must remain square-zero. The main theorem preserves only the finiteness of $i,j$01, $i,j$02, and $i,j$03, not their exact values. The construction is also distinguished in the paper from general split extensions $i,j$04, from arbitrary multiplicative bimodule extensions, and from triangular matrix algebras. In that sense, trivial one-arrow extensions are a carefully designed and homologically rigid subclass of arrow-adding operations rather than a general theory of adjoining a single arrow (Giatagantzidis, 17 Jul 2025).