In this section we describe the correspondence between the homological conditions of the above type on the morphisms f and f ? . Here is the principal result. Theorem 7. Let f : A – B be a morphism of Koszul algebras and f ? : A? – B ? be the dual morphism of Koszul coalgebras. Then there are natural isomorphisms of the (co)module (co)homology vector spaces Hi,j (A, B) H j-i, j (B ?op , A?op ) compatible with the right action of B and the left coaction of A? on both sides. Proof : According to Proposition 4, the Koszul complex K(A? , A) with the homological grading Ki (A? , A) = A? A is a free graded right A-module resolution of the i trivial A-module k. Therefore, the space Hi,j (A, B) can be computed as the homology space of the complex K(A? , A) A B = A? B at the term A? Bj-i . Analogously, i the Koszul complex K(B ? , B) with the cohomological grading K i (B ? , B) = B ? Bi is a cofree graded left B ? -comodule resolution of the trivial B ? -comodule k. It follows that th
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