Projective resolutions of simple modules and Hochschild cohomology for incidence algebras

We give a practical, algorithmic method to calculate minimal projective resolutions of simple modules for a finite dimensional incidence $k$-algebra $Λ$, where $k$ is a field. We apply the method to the calculation of Ext groups between simple $Λ$-modules, Hochschild cohomology groups $\HH^i(Λ, Λ)$, and singular cohomology groups of finite $T_0$ topological spaces with coefficients in $k$.

💡 Research Summary

The paper addresses the problem of computing minimal projective resolutions of simple modules over a finite‑dimensional incidence algebra Λ = kX/I, where X is a finite poset and k a field. Existing methods such as the Bongartz‑Butler algorithm produce projective resolutions but not necessarily minimal ones, and they involve cumbersome ideal calculations. The authors propose a completely elementary, linear‑algebraic algorithm that yields minimal resolutions directly.

The construction begins with the Hasse diagram Q of the poset X, regarded as an acyclic quiver. The incidence algebra Λ is the path algebra kQ modulo the ideal generated by differences of parallel paths. For a fixed vertex x∈X the authors define, for each i≥0, a set C_i of “i‑cycles”. C₀={x}, C₁ consists of all pairs (x,y) with y an immediate successor of x, and the boundary map ∂₁ sends (x,y) to x. For i≥2 the definition is recursive: for each vertex z one looks at the kernel of ∂{i‑1} restricted to elements whose second component is <z, then chooses a complement basis D{i}(z) of the subspace generated by cycles coming from predecessors of z. The new i‑cycles are the pairs (w,z) with w∈D_{i}(z). The maps ∂i simply forget the second component, i.e. ∂i((w,z))=w. Lemma 1 shows that (C·,∂·) is a chain complex; exactness is not guaranteed at this stage, but the subsequent tensoring with Λ produces a genuine projective resolution.

Each kC_i is turned into a right Λ₀‑module (Λ₀ being the semisimple subalgebra generated by the idempotents at each vertex) by letting the idempotent at a vertex act only on cycles whose second component equals that vertex. Tensoring with Λ over Λ₀ yields a right Λ‑module P_i = kC_i⊗{Λ₀}Λ, which is projective because P_i ≅ ⊕{(w,z)∈C_i} P_z. The differential d_i : P_i → P_{i‑1} is defined on generators by d_i((w,z)⊗1)=w⊗q_z, where q_z is the sum of all paths ending at z. Lemma 2 verifies that (P_·,d_·) is a chain complex, and Theorem 3 proves that the augmented complex …→P₂→P₁→P₀→S_x→0 is a minimal projective resolution of the simple module S_x. Minimality follows from the observation that every element of ker d_i lies in the Jacobson radical of P_i.

With the minimal resolution in hand, the authors obtain a very simple description of Ext‑groups between simples. Proposition 4 states that dim_k Ext^i_Λ(S_x,S_b) equals the cardinality |B_i^b|, where B_i^b⊂C_i consists of those i‑cycles whose second component is b. Since the differentials in the Hom complex Hom_Λ(P_·,S_b) are zero (the action of any non‑trivial path on S_b is zero), the homology is just the space of maps, giving the claimed dimension.

The paper then turns to Hochschild cohomology. Using a result of Cibils (Theorem 5), Hochschild cohomology HH^i(Λ,Λ) can be identified with Ext^{i+2}{Λ^*}(S{x^},S_{y^}) for a certain “extended” poset X^* obtained by adding two new vertices x^<X<y^. Applying Proposition 4 to the extended poset yields Theorem 6: for i>0, dim HH^i(Λ,Λ)=|B_{i+2}^{y^}|, while for i=0 the dimension is |B_2^{y^}|+1, which coincides with the number of connected components of the Hasse diagram (the centre of Λ). The proof of the i=0 case involves a careful counting of predecessors and successors in each connected component.

Finally, Remark 4 explains that finite T₀ topological spaces are in bijection with finite posets, and via results of McCord and Gerstenhaber–Schack, the Hochschild cohomology of the incidence algebra equals the singular cohomology of the associated space. Thus the algorithm provides a practical method for computing singular cohomology of finite T₀ spaces with coefficients in k. An implementation in SageMath is mentioned (Remark 5), showing that the i‑cycle algorithm is computationally efficient and readily usable.

In summary, the paper delivers a concrete, linear‑algebraic algorithm that simultaneously produces minimal projective resolutions of simples, computes Ext‑groups, determines all Hochschild cohomology groups of incidence algebras, and, through topological equivalences, yields singular cohomology of finite T₀ spaces. The approach avoids heavy ideal‑theoretic calculations and is amenable to computer implementation, representing a significant practical advance in the homological study of incidence algebras.