Projective resolutions of simple modules and Hochschild cohomology for incidence algebras

1. Problem Statement

The paper addresses the computational challenge of determining minimal projective resolutions for simple modules over finite-dimensional incidence algebras ($\Lambda$) defined over a field $k$.

• Context: Incidence algebras are associated with finite partially ordered sets (posets). While projective resolutions are fundamental for computing homological invariants (like Ext groups and Hochschild cohomology), existing algorithms (e.g., by Bongartz and Butler) are often non-minimal and computationally inefficient because they rely on manipulating ideals within the algebra rather than linear algebra.
• Gap: There was no simple, practical algorithm available to calculate the Hochschild cohomology groups ($HH^i(\Lambda, \Lambda)$) for an arbitrary finite-dimensional incidence algebra. Previous methods were either restricted to specific classes of posets or relied on complex simplicial complex cohomology calculations.

2. Methodology

The authors propose a recursive, algorithmic method based on the construction of "i-cycles" derived directly from the combinatorial structure of the underlying poset $X$.

A. Construction of the Resolution

The core of the method involves defining a sequence of sets $C_i$ (the $i$-cycles) that index the indecomposable summands of the projective modules in the resolution of a simple module $S_x$.

1. Base Cases:
   • $C_0 = \{x\}$.
   • $C_1 = \{(x, y) \mid y \in x^+\}$, where $x^+$ are the immediate successors of $x$.
2. Recursive Step ($i \ge 2$):
   • The authors define a linear map $\partial_i: kC_i \to kC_{i-1}$.
   • They construct subspaces $(\ker \partial_i)_z$ consisting of elements in the kernel where the "target" component is strictly less than $z$.
   • They define $(\ker \partial_i)_{z^-}$ as the sum of kernels for all immediate predecessors of $z$.
   • $C_{i+1}$ is constructed by taking a basis of the complement of $(\ker \partial_i)_{z^-}$ within $(\ker \partial_i)_z$ for each $z \in X$.
3. The Resolution:
   • The minimal projective resolution of $S_x$ is given by the complex:
     $$\cdots \to kC_2 \otimes_{\Lambda_0} \Lambda \xrightarrow{d_2} kC_1 \otimes_{\Lambda_0} \Lambda \xrightarrow{d_1} kC_0 \otimes_{\Lambda_0} \Lambda \to S_x \to 0$$
   • Here, $\Lambda_0$ is the semisimple subalgebra generated by stationary paths. The maps $d_i$ are defined explicitly using the path sums in the algebra.
   • Theorem 3 proves this complex is exact and minimal.

B. Computational Efficiency

The algorithm relies exclusively on basic linear algebra operations:

• Calculating kernels of linear maps.
• Intersections and sums of subspaces.
• Finding bases for complements of subspaces.
  This avoids the heavy ideal-theoretic computations required by previous methods.

3. Key Contributions and Results

A. Minimal Projective Resolutions (Theorem 3)

The paper provides a constructive proof that the sequence defined by the $i$-cycles yields a minimal projective resolution for any simple module $S_x$. This is significant because minimality is required for efficient computation of cohomology groups.

B. Calculation of Ext Groups (Proposition 4)

The authors derive a direct combinatorial formula for the dimension of Ext groups between simple modules $S_x$ and $S_b$:
$$\dim_k \text{Ext}^i_\Lambda(S_x, S_b) = |B^i_b|$$
where $B^i_b$ is the subset of the $i$-cycles indexed by the element $b$. This allows for the immediate calculation of extension groups without performing full homological algebra computations.

C. Hochschild Cohomology (Theorem 6)

By leveraging a result by Cibils (Theorem 5), which relates the Hochschild cohomology of $\Lambda$ to the Ext groups of a related extended algebra $\Lambda^*$ (constructed by adding two points $x^*, y^*$ to the poset), the authors provide a formula for the dimension of Hochschild cohomology groups:
$$\dim HH^i(\Lambda, \Lambda) = \begin{cases} |B^2_{y^*}| + 1 & \text{if } i=0 \\ |B^{i+2}_{y^*}| & \text{if } i > 0 \end{cases}$$
This transforms the problem of computing Hochschild cohomology into a combinatorial counting problem on the $i$-cycles of the extended poset.

D. Topological Application (Remark 4)

The paper establishes a bridge to topology. Since finite $T_0$ topological spaces are in bijection with finite posets, and the Hochschild cohomology of the incidence algebra is isomorphic to the singular cohomology of the space, the algorithm provides a practical method to compute singular cohomology groups of finite $T_0$ spaces.

4. Significance and Performance

• Algorithmic Superiority: The paper demonstrates that the $i$-cycles algorithm is significantly faster than existing implementations in computer algebra systems.
  • Benchmark: In a test with 10 random posets of 30 vertices, the authors' SAGE implementation took an average of 0.034 seconds, whereas Magma's CompactProjectiveResolution took 420.43 seconds.
• Generality: Unlike previous formulas that applied only to specific classes of posets (e.g., those with specific simplicial structures), this method works for arbitrary finite-dimensional incidence algebras.
• Future Work: The authors note that these results provide a theoretical foundation for reducing posets without altering their Hochschild cohomology, a topic for future research.

Conclusion

This paper presents a breakthrough in the computational homological algebra of incidence algebras. By replacing complex ideal-theoretic resolutions with a recursive linear-algebraic construction based on poset combinatorics, the authors provide a highly efficient, minimal, and general algorithm. This not only solves the problem of computing Ext groups and Hochschild cohomology for arbitrary incidence algebras but also offers a powerful tool for calculating the singular cohomology of finite topological spaces.