Chouhy–Solotar Reduction System

• Chouhy–Solotar reduction system is a combinatorial method for constructing minimal and functorial projective bimodule resolutions of associative algebras defined by quivers with relations.
• It employs a structured reduction system and ambiguity analysis to derive explicit differentials that facilitate efficient Hochschild cohomology computations.
• The framework supports diagrammatic representations in Fukaya categories and braiding functor formulations, offering a streamlined alternative to the traditional bar resolution.

The Chouhy–Solotar reduction system is an explicit and combinatorial method for constructing projective bimodule resolutions of associative algebras defined by quivers with relations, particularly where the relations are not necessarily monomial. It yields minimal and functorial resolutions of the diagonal bimodule, enabling rigorous computations in Hochschild cohomology and the categorical paper of natural transformations, as applied to the Fukaya category associated to Coulomb branches of quiver gauge theories and the diagrammatic structure of KLRW categories (Tong, 13 Nov 2025).

1. Quiver, Path Algebra, and Reduction System

Let $Q$ be a quiver with vertices $Q_0$ corresponding to idempotents $e_i$ ($i=0,\dots,N$) and arrows $Q_1$:

• $p_{i+1}$: $e_i \to e_{i+1}$,
• $q_i$: $e_{i+1} \to e_i$,
• $s_i$: $e_i \to e_i$, for $i=0,\dots,N-1$. The path algebra is $A=\mathbb{k}Q/I$, where $I$ encodes the KLRW relations.

A reduction system $R$ is a set of generators of $I$ of the form

$$R = \{ (s, \varphi_s) \mid s\in S\subset Q_2,\ \varphi_s\in\mathbb{k}Q,\ \varphi_s\text{ is irreducible} \},$$

where “irreducible” means no subpath is in $S$. For the KLRW category,

$$S = \{ q_i p_{i+1},\ p_i q_{i-1},\ s_i p_i,\ s_i q_i \},$$

with

$$\varphi_{q_ip_{i+1}} = s_i,\quad \varphi_{p_iq_{i-1}} = s_i,\quad \varphi_{s_ip_i} = p_i s_{i-1},\quad \varphi_{s_iq_i} = q_i s_{i+1}.$$

A preorder "$\preceq$" on monomials is defined by replacing any subpath $s\in S$ by $\varphi_s$. A path is irreducible iff it contains no $s\in S$.

2. Ambiguities and Construction of Projective Bimodules

An $n$-ambiguity is a path $u=u_{n+1}\cdots u_0$ such that each length-2 subpath $u_{i+1}u_i\in S$ and no smaller subpath lies in $S$. Let $S_n$ denote the set of $n$-ambiguities ($S_2=S$).

The sizes are: $$|S_0| = N+1,\quad |S_1| = 3N+1,\quad |S_n| = 4N\text{ for } n\geq 2.$$

Define projective $(A$–$A)$-bimodules by

$$P_n = A \otimes_{\mathbb{k}Q_0} \mathbb{k}[S_n] \otimes_{\mathbb{k}Q_0} A.$$

Thus, $P_0=A\otimes A$, $P_1=A\otimes Q_1\otimes A$, etc. The combinatorial structure of ambiguities forms the backbone of the resolution.

3. Differentials and Exactness

Auxiliary “split” maps on paths $w\in Q$ are defined as

$$\text{split}_n(w) = \sum_{w=u\cdot r\cdot v,\ r\in S_n} \pi(u)\otimes r\otimes\pi(v),$$

with $\text{split}_n^R$, $\text{split}_n^L$ denoting rightmost/leftmost decompositions.

Base cases:

• $\delta_0(x\otimes y)=\pi(xy)$,
• $$\delta_1(1\otimes w\otimes 1)=1\otimes w - w\otimes 1$$.

For $n\geq 2$:

• $n$ even: $$\delta_n(1\otimes w\otimes 1)=\text{split}_{n-1}^L(w) - \text{split}_{n-1}^R(w)$$,
• $n$ odd: $$\delta_n(1\otimes w\otimes 1)=\text{split}_{n-1}(w)$$.

The differential $\partial_n$ is recursively corrected for exactness: $$\partial_n(1\otimes w \otimes 1) = ( \mathrm{id} - \rho_{n-2} \partial_{n-1} ) \circ \delta_n(1\otimes w\otimes 1),$$ with

$$\rho_{n-1} = \gamma_{n-1} + \sum_{i\geq 1} \gamma_{n-1}( \delta_n \gamma_{n-1} - \partial_n \gamma_{n-1})^i,$$

where $$\gamma_{n-1}(x\otimes w\otimes y)=(-1)^n \text{split}_n(x w) y$$. Theorem 4.1 in [CS] guarantees the resulting complex is exact.

4. Diagrammatics and KLRW Embedding

In the diagrammatic framework compatible with the Fukaya/KLRW embedding, arrows $p_i$, $q_i$ correspond to black strands crossing a fixed red line associated to a puncture. Dots $s_i$ are decorations on stationary strands at red $i$.

The four relations in $S$ map directly to local moves among these strand-dot diagrams. Ambiguities are visualized as oscillations or zig-zag motions of strands around punctures, either as strand oscillations (type I) or those ending in a dot (type II).

5. Projective Resolution of the Diagonal Bimodule

The sequence

$$\cdots \to P_2 \xrightarrow{\partial_2} P_1 \xrightarrow{\partial_1} P_0 \xrightarrow{\partial_0} A \to 0$$

with the above differentials provides a projective resolution of the diagonal bimodule $\Delta(-,-)=\operatorname{Hom}(-,-)$. Each $P_n$ is finitely generated (stabilizing for $n\geq 2$), in contrast to the bar resolution, whose modules grow exponentially.

6. Hochschild Cohomology Computation

The cochain complex for Hochschild cohomology is

$$C^n = \operatorname{Hom}_{A-A}(P_n,\Delta),\qquad d^n(\varphi) = \varphi \circ \partial_{n+1}.$$

Given the explicit structure $$P_n = A \otimes_{\mathbb{k}Q_0} \mathbb{k}[S_n] \otimes_{\mathbb{k}Q_0} A$$, homomorphisms $\varphi$ are determined by their values $\varphi_\ell(s)$ on each $n$-ambiguity $s\in S_n$ and each dot-count $\ell\geq 0$.

Explicit combinatorial formulas for $d^n$ on the coefficient-functions $\varphi_\ell(s)$ are given (see Theorem 7.9 in (Tong, 13 Nov 2025)). For $n\geq 3$, the resolution decomposes into 2-vertex blocks, yielding $HH^n=0$. In low degrees: $$HH^0 (\Delta) \cong \mathbb{k}^{\mathbb{N}},\qquad HH^1 (\Delta) \cong \mathbb{k}^{\mathbb{N}},\qquad HH^2(\Delta) \cong \mathbb{k}^{N-1}.$$ Cocycle representatives are labeled by sequences $\epsilon_\ell$, $\sigma_\ell$, and $\theta_i$.

7. Comparison with Alternative Resolutions and Minimality

The bar resolution $$\operatorname{Bar}_n = A \otimes (A/J)^{\otimes n} \otimes A$$ is universally applicable but large and highly redundant. Monomial-algebra resolutions (Bardzell) are limited to monomial generators.

The Chouhy–Solotar system generalizes Bardzell’s approach to arbitrary quivers with relations, yielding minimal, functorial, and combinatorially efficient resolutions, as $P_n$ stabilizes for $n\geq 2$ and contains no contractible summands.

One can construct a homotopy deformation retract from $\operatorname{Bar}_\bullet$ onto $P_\bullet$, with explicit chain-maps $F$, $G$, $h$, reducing calculations in the bar complex to computations in the smaller Chouhy–Solotar complex.

8. Applications in Fukaya Categories and Braiding Functors

Morita invariance yields $HH^\ast(\Delta) \cong HH^\ast(\mathrm{id})$, $HH^\ast(B_{i^{-}}) \cong HH^\ast(\beta_{i^{-}})$. Low-degree cohomology classes produce explicit $A_\infty$-natural transformations $\mathrm{id}\Rightarrow\mathrm{id}$ and $\mathrm{id}\Rightarrow\beta_{i^{-}}$, whose components $\eta^d$ are given by summations over dot-counts and strand-pictures with coefficients derived from the Hochschild cocycles.

This framework determines the higher $A_\infty$-data encoded in braiding functors and their natural transformations, enabling the categorical formulation of braid cobordism actions within the Fukaya category context, specifically for the Coulomb branch $\mathcal{M}(\bullet,1)$ of the $\mathfrak{sl}_2$ quiver gauge theory (Tong, 13 Nov 2025).