Twisted tensor products of DG algebras

Let A = (A, d) be a differential graded algebra (DGA) over a field k, that is, a Z-graded algebra A = ⊕ q∈Z A q with a k-linear map d : A → A, d = 0, of degree one that satisfies the graded Leibniz rule. Denote by D(A ) the derived category of right A -modules and by perf -A ⊂ D(A ) the triangulated subcategory of perfect modules generated by A , which is equivalent to the subcategory of compact objects D(A ) ⊂ D(A ) [5]. Suppose that A is finite dimensional. We denote by J ⊂ A the (Jacobson) radical of the k-algebra A. The ideal J is graded. Let S be the graded quotient algebra A/J , and let ε : S → A be the canonical homomorphism of algebras. We assume that d ◦ ε = 0 and d(J) ⊆ J , and consider S as a DGA with the trivial differential. In this case there are morphisms ε : S → A and π : A → S of DGAs, and the DGA A will be said to be S-split. Let e ∈ A be an idempotent, and let Pe = eA and Qe = Ae be the right and left projective A-modules. Since d(e) = 0, the A-modules Pe and Qe have the natural structure of DG A -modules. We denote by Pe = (Pe, d) and Qe = (Qe, d) the corresponding right and left DG A -modules. A right (left) DG module Φ will be called semiprojective if there is a filtration 0 = Φ0 ⊂ Φ1 ⊂ · · · = Φ such that every quotient Φi+1/Φi is a direct sum of projective DG-modules Pe (respectively, Qe). The simple right A-modules Se = Pe/eJ with d = 0 become right DG A -modules Se. We consider S as a right DG A -module and denote it by S. For any S-split DGA A , every finite-dimensional DG A -module M has a filtration 0 = Ψ0 ⊂ Ψ1 ⊂ · · · ⊂ Ψk = M such that every quotient Ψi+1/Ψi is isomorphic to some Se. Recall that a DGA A is called smooth if it is perfect as a DG bimodule.