The Algebra Weak Winding Cohomology Of The Coalgebra Structure

Let (A, C,ψ) be a weak entwining structure, where A is an algebra, C a coal-gebra, andψ: C(?)Aâ†'A(?)C a mophism satisfying four relations. In[1], there are examples of weak entwining structure from weak Hopf algebra. In[2], the writer constructs the cochain complex C_ψ(A,M) associated to an entwining structure(A,C)_ψand an A- bimodule M. He studies its relation to the Hochschild complex of A as well as analyse its structure in the case of the canonical entwining structure associated to a C-Galois extension A(B)~C. In paticular showes that if B = k this complex provides a resolution, of M. All these results show that entwining structures admit a rich cohomology theory.This paper wants to show that weak entwining structures also admit a rich cohomology theory. We state the main results as follows.In section 3, we define a chain complex Bar~ψ(A) = ((?) Bar(A),δ) and prove that Bar~ψ(A) is a resolution of both A and C. Furthermore,δis an A-bimodule map. Then we use the resolution to construct the main cochain complex studied in this paper.In section 4, At first, we prove thatthen we prove the main result of this paper: For an weak entwining structure (A,C,ψ), (?) is a projective A-bimodule if and only if (?)= 0, for (?)A-bimodule M. Finally we define the cup product and prove that (?) is an associative algebra and d is a degree 1 derivation in this algebra.