Simplicial Hochschild cochains as an Amitsur complex

It is shown that the cochain complex of relative Hochschild A-valued cochains of a depth two extension A | B under cup product is isomorphic as a differential graded algebra with the Amitsur complex of the coring S = End {}_BA_B over the centralizer R = A^B with grouplike element 1_S, which itself is isomorphic to the Cartier complex of S with coefficients in the (S,S)-bicomodule R^e. This specializes to finite dimensional algebras, H-separable extensions and Hopf-Galois extensions.

💡 Research Summary

The paper investigates the relationship between relative Hochschild cochains of a depth‑two extension and the Amitsur complex of an associated coring, establishing a precise differential‑graded algebra (DGA) isomorphism. A depth‑two extension A | B is a ring extension for which the tensor square A ⊗_B A decomposes as a finite direct sum of copies of A as a B‑B‑bimodule. This condition guarantees the existence of a centralizer R = A^B and a coring S = End {}_B A_B over R. The coring S carries a natural grouplike element 1_S, which makes the Amitsur complex Ω^·(S,R) well defined.

The authors first construct the relative Hochschild cochain complex C^·(A,B;A) consisting of B‑balanced multilinear maps f : A^{⊗_B n} → A. The cochain differential d is the usual Hochschild coboundary, and the cup product ∪ endows C^· with a graded algebra structure. They verify that (C^·, d, ∪) satisfies the axioms of a DGA.

Next, they recall the Amitsur complex of a coring: Ω^n(S,R) = S^{⊗_R n} with differential built from the comultiplication of S and with multiplication induced by the tensor product over R. The presence of the grouplike element 1_S provides a unit in degree zero.

The central theorem constructs explicit maps φ_n : C^n(A,B;A) → Ω^n(S,R) for each n, using the identification of a Hochschild cochain with an R‑linear map from A^{⊗_B n} to A and then translating it into an element of S^{⊗_R n} via the canonical isomorphism End_B(A) ≅ S. The authors prove that φ = {φ_n} respects both the differential and the cup product, thereby giving a DGA isomorphism \